The Language of Mathematics Education : An Expanded Glossary of Key Terms and Concepts in Mathematics Teaching and Learning [1 ed.] 9789004412651, 9789004409392

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The Language of Mathematics Education

The Language of Education Key Terms and Concepts in Teaching and Learning Series Editor William F. McComas (Parks Family Distinguished Professor of Science Education, University of Arkansas, Fayetteville, AR, USA)

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The titles published in this series are listed at brill.com/tloe

The Language of Mathematics Education An Expanded Glossary of Key Terms and Concepts in Mathematics Teaching and Learning By

Shannon W. Dingman, Laura B. Kent, Kim K. McComas and Cynthia C. Orona

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All chapters in this book have undergone peer review. The Library of Congress Cataloging-in-Publication Data is available online at http://catalog.loc.gov

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CONTENTS

Foreword ............................................................................................................... xi Barbara J. Reys Preface and Introduction ..................................................................................... xii Abstract Thinking ................................................................................................... 1 Action Research...................................................................................................... 2 Active Mathematics Teaching and Learning.......................................................... 3 Additive Reasoning ................................................................................................ 4 Algebraic Reasoning ............................................................................................... 5 Algorithm................................................................................................................ 6 Assessment in Mathematics .................................................................................. 7 Formative Assessment ....................................................................................... 7 Summative Assessment...................................................................................... 7 Progressive Assessment ..................................................................................... 8 Basic (Number) Facts.............................................................................................. 9 Beliefs/Attitudes .................................................................................................. 10 Cognitive Demand ................................................................................................ 11 Cognitive Science.................................................................................................. 12 Cognitively Guided Instruction (CGI) ................................................................... 13 Common Core State Standards for Mathematics (CCSSM) ................................. 14 Computer Algebra Systems (CAS) ........................................................................ 15 Concept Image...................................................................................................... 16 Conceptual Knowledge ........................................................................................ 17 Conjecture ............................................................................................................ 18 Constructivist Theory of Learning ........................................................................ 19 Cooperative Learning ........................................................................................... 21 Council for the Accreditation of Educator Preparation (CAEP) ........................... 22 Counting ............................................................................................................... 23 Covariational Reasoning ...................................................................................... 24 v

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CONTENTS

Curricular Reasoning ............................................................................................ 25 Curriculum ............................................................................................................ 26 Curriculum Alignment .......................................................................................... 27 Curriculum Coherence.......................................................................................... 28 Curriculum Knowledge ......................................................................................... 29 Decentering .......................................................................................................... 30 Deductive Reasoning............................................................................................ 31 Design Research in Education .............................................................................. 32 Didactic ................................................................................................................. 33 Differentiated Instruction .................................................................................... 34 Direct Modeling .................................................................................................... 35 Discourse .............................................................................................................. 36 Discovery Learning ............................................................................................... 37 Dynamic Geometry Software (DGS) .................................................................... 39 Educational Technology ....................................................................................... 40 Epistemology ........................................................................................................ 42 Equity .................................................................................................................... 43 Error Patterns ....................................................................................................... 45 Ethnomathematics ............................................................................................... 46 Fidelity of Implementation .................................................................................. 47 Flipped Classroom ................................................................................................ 48 Functions-Based Approach to Teaching Algebra ................................................. 49 Geometric Reasoning ........................................................................................... 50 High-Stakes Testing .............................................................................................. 51 Inductive Reasoning ............................................................................................. 52 Instructional Strategies and Techniques ............................................................. 53 Direct Instruction/Lecture Method................................................................... 53 Inquiry Based Instruction/Active Learning ....................................................... 53 Three-Act Tasks ................................................................................................ 53 Launch-Explore-Summarize.............................................................................. 54 5 Practices ........................................................................................................ 54 Flipped Classroom Approach ............................................................................ 54

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Learning Trajectory .............................................................................................. 55 Lesson Study ......................................................................................................... 56 Longitudinal Study ............................................................................................... 57 Manipulatives....................................................................................................... 59 Math Anxiety ........................................................................................................ 61 Math Wars ............................................................................................................ 62 Mathematical Identity ......................................................................................... 63 Mathematical Knowledge for Teaching (MKT) .................................................... 65 Mathematical Literacy ......................................................................................... 67 Mathematical Modeling....................................................................................... 68 Mathematics Skills ............................................................................................... 69 Meaningful Learning ............................................................................................ 70 Mental Discipline.................................................................................................. 71 Mental Math ........................................................................................................ 72 Metacognition ...................................................................................................... 73 Misconceptions .................................................................................................... 74 Model-Eliciting Activities (MEA’s)........................................................................ 75 Multiple Embodiment .......................................................................................... 76 Multiplicative Reasoning ..................................................................................... 77 National Assessment of Educational Progress (NAEP) ........................................ 78 NCTM Standards................................................................................................... 79 New Math ............................................................................................................. 80 Non-Anticipatory.................................................................................................. 81 Number Sense/Numeracy.................................................................................... 82 Numerical Estimation........................................................................................... 83 Pedagogical Content Knowledge (PCK) ............................................................... 84 Performance Based Assessments ........................................................................ 85 Prior Knowledge ................................................................................................... 86 Problem Based Learning (PrBL)............................................................................ 87 Problem Solving Heuristics .................................................................................. 88 Problem Structure ................................................................................................ 89 Procedural Knowledge ......................................................................................... 90

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CONTENTS

Productive Struggle .............................................................................................. 91 Professional Development (PD) ........................................................................... 93 Professional Organizations in Mathematics Education....................................... 94 National Council of Teachers of Mathematics (NCTM) .................................... 94 National Council of Supervisors of Mathematics (NCSM) ................................ 94 Association of Mathematics Teacher Educators (AMTE) ................................. 94 Psychology of Mathematics Education (PME).................................................. 94 American Educational Research Association (AERA)........................................ 95 International Commission on Mathematical Instruction (ICMI) ....................... 95 Research Council on Mathematics Learning (RCML) ........................................ 95 Mathematical Association of America (MAA) .................................................. 95 American Mathematical Society (AMS) ........................................................... 95 Program for International Student Assessment (PISA) ....................................... 96 Project Based Learning (PBL) ............................................................................... 97 Proportional Reasoning ....................................................................................... 98 Quantitative Literacy (QL) .................................................................................... 99 Quantitative Reasoning (QR) ............................................................................. 100 Radical Constructivism ....................................................................................... 101 Reification........................................................................................................... 102 Relational Thinking............................................................................................. 103 Representational Fluency .................................................................................. 104 Representations ................................................................................................. 105 Response to Intervention (RtI)........................................................................... 106 Responsive Teaching .......................................................................................... 107 Rigor.................................................................................................................... 108 Rote Learning ..................................................................................................... 109 Scaffolding .......................................................................................................... 110 Sense-Making ..................................................................................................... 111 Situated Learning (Cognition) ............................................................................ 112 Social Constructivism ......................................................................................... 113 Socio-Cultural Learning Theory (SCLT) ............................................................... 114 Sociomathematical Norms ................................................................................. 116

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Spatial Thinking .................................................................................................. 117 Strands of Mathematical Proficiency................................................................. 118 Subitizing ............................................................................................................ 119 Task Analysis ...................................................................................................... 120 Teacher Noticing ................................................................................................ 121 Technological and Pedagogical Content Knowledge (TPACK)........................... 122 Trends in Mathematics and Science Study (TIMMS) ......................................... 124 Van Hiele Levels of Geometric Thinking ............................................................ 125 Zone of Proximal Development (ZPD) ............................................................... 126

FOREWORD

Most of those who study, teach, or do mathematics enjoy it for its logical structure and intrinsic beauty. They consider ideas that are mathematical in nature and often represent these ideas symbolically in order to share and convey them to others. This is also true of those who convey ideas and facts about the nature of teaching and learning mathematics. Representing the acquisition of mathematical knowledge requires not only numbers and symbols but also words. Words matter. Without shared meanings of key words/phrases, we cannot have productive discussions that lead to better teaching and increased learning. This book is a compilation of over 100 important words and phrases that mathematics educators and others use to discuss key features and ideas regarding the current mathematics education system. These include phrases historically used to describe what we teach (e.g., “curriculum” and “standards”) as well as how we teach (e.g., “flipped classroom” and “inquiry-based/discovery learning”). It also includes a delineation of types of knowledge (e.g., “conceptual knowledge” and “procedural knowledge”). These terms give life to fundamental ideas about learning mathematics. For example, unless we understand the difference between conceptual and procedural knowledge, we can’t, as educators, appreciate the need to employ different strategies for helping students acquire this knowledge. All the ideas conveyed in this volume are critical to the improvement of the tools and strategies of teaching and learning mathematics. We need this shared language in order to continue to move forward in the pursuit both of new knowledge about learning mathematics and more effective teaching strategies. The compilation is the result of dedicated effort of a team of scholars at the University of Arkansas. As a field, we owe them our thanks and admiration. While many more words and phrases could be added to this collection, their work is a worthy start and likely to be helpful in many ways. It can help mathematicians and mathematics educators better communicate. It can help teachers better understand the complexity of their job as mathematics educators. It can also convey to lay citizens (non-educators) the meaning of terms used so often within the educational community. Collectively, the volume is an important contribution to the field. I offer my sincere gratitude to the authors for their work in bringing forth this volume. Barbara J. Reys Curators’ Distinguished Professor Emerita University of Missouri – Columbia

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PREFACE AND INTRODUCTION

The language and terminology we use in the field of mathematics education have evolved over time and continue to grow as our knowledge of mathematics teaching and learning grows. Given the diversity of opinions and breadth of experiences, experts in our field often use terms and vocabulary in different ways. However, given the necessity of our scholarship and the dissemination of findings not only across our country but also around the globe, it is vitally important that some basis of shared understanding exists regarding the use and meanings of key terminology. It is with this goal in mind that we share with you our work. In 2016, Dr. William McComas, Parks Family Professor of Science Education at the University of Arkansas, approached the author team regarding the idea and the potential for the book. Dr. McComas had previously edited the first book in this series entitled The Language of Science Education (McComas, 2014), which provided definitions and short synopses for over one hundred common terms used in science education. The author team began by brainstorming a list of terms we often used in our work or words that had been sources of discussion within our work with mathematics education doctoral students. We then sent our list to various groups and researchers working in the field of mathematics education, including the listservs for the Center for the Study of Mathematics Curriculum (CSMC) and the Service Teaching and Research (STaR) program for the Association of Mathematics Teacher Educators (AMTE). Feedback and suggestions from individuals helped expand the list to over 125 terms and provided the basis from which our work began. The format of each page is similar, often beginning with a short, 1-2 sentence definition of the term. We then provide a short summary of some of the work conducted in the area of each term, and then provide some references where the reader might pursue more literature regarding the term. As with the science education book, we endeavored to keep each entry to one page or less. However, with some terms such as assessment, instructional planning and strategies, and constructivist theory of learning, one page could not contain an adequate summary and thus we capped each entry at no more than two pages. These short synopses should not be construed as being exhaustive of the literature surrounding that term but rather give the reader an introduction and background as well as resources where further reading might occur. We hope the final product expands your understanding of these terms. The intended audience for this publication was new initiates into the field of mathematics education: graduate students, early career faculty and those working closely with colleagues or students still emerging in their careers and in their understanding of mathematics education research. However, this should not preclude seasoned faculty in our field who continue to broaden their understanding from holding an interest in this publication. xii

PREFACE AND INTRODUCTION

xiii

We believe that most mathematics educators would agree with the definitions provided here. However, we also believe that, given the vast knowledge that exists in the field, experts may differ or possess further clarification that might continue to enhance the definitions and synopses provided here. To that end, we encourage the readers to reach out to members of the author team with feedback and suggestions regarding entries, including terms that were not included or alternate views regarding the usage of some of these terms. This feedback may serve to inform future revisions and editions of the book. We would also like to thank several individuals who have contributed to and assisted with this project. We would like to thank Dr. William McComas for planting the seed for this project and providing guidance from his experience to help inform our work. We would like to thank Dr. Barbara J. Reys from the University of Missouri for writing the foreword for the book. We would like to extend our thanks to our colleagues Kathy Clark, Doug Grouws, Maryann Huey, Lisa Kasmer, John Kerrigan, James Kratky, Betty Phillips, Robert Reys, Jack Smith, James Tarr, Zalman Usiskin, and Paul Wolfson, who provided feedback and suggestions for terms to include in our book. We would like to thank the team at Brill Sense for their patience and assistance as we completed this book. Finally, we would like to thank our doctoral students Sarah Fredrickson and Steven Homem who researched and authored entries in the book.

CONTRIBUTING AUTHORS The four authors as well as two graduate students who worked along Drs. Kent and Dingman wrote the entries that are included in this book. Multiple members of the writing team have edited each entry, yet the basis of the contribution exists with the original author. As such, the author for each entry is denoted with his/her initials at the conclusion of the short synopses of the term. The following initials indicate the author for the entry: SWD SRF SJH

Shannon Dingman Sarah Fredrickson Steven Homem

LBK KKM CCO

Laura Kent Kim McComas Cynthia Orona

SUGGESTED REFERENCE FORMAT Readers may wish to cite specific entries and give credit to the appropriate author. Here is an example of how the term Action Research might be cited: Kent, L. B. (2019). Action research. In S. W. Dingman, L. B. Kent, K. K. McComas, & C. C. Orona (Eds.), The language of mathematics education (p. 2). Leiden/ Boston: Brill Sense.

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Abstract Thinking involves thinking beyond physical representations or instances of a mathematical concept, in a process of extracting the underlying essence of an idea. Thinking becomes more abstract as it is removed from the specific contexts that initiate it, becoming a level of its own intangible nature. It is often contrasted with Concrete Thinking which involves thinking within physical representations. The term abstract gets at the very nature of mathematics while the other, concrete, is often referred to as the ‘other side of a bridge’ that connects physical representations to abstract thinking, if adequate experiences allow. “As a science of abstract objects, mathematics relies on logic rather than observation as its standard of truth, yet employs observation, simulation, and even experimentation as a means of discovering truth” (NRC, 1989, p. 31). Such observation, simulation, and experimentation are common actions of students exploring mathematics with concrete materials and representations. When patterns are detected from observing concrete representations and generalizations are made, students are engaged in a process of abstracting. As a concept is an abstraction of ideas or things that relate, abstract thinking is at the heart of conceptual understanding. Clements (1999) describes “integrated-concrete thinking” as a type of interconnected knowledge in which “physical objects, actions performed on them, and abstractions are all interrelated in a strong mental structure” (p. 48). Counting is an example of such an interaction in a child’s cognitive development: The idea that ‘one more’ can always be applied to a number … that, therefore, the counting numbers must go on forever – is an abstract idea. It is this movement from the individual instance to the generalization that we are striving for …, reasoning about a whole class of mathematical objects …. It is when children ‘lift off’ from the specific instance to consider the general case that they are really doing mathematical reasoning. (Russell, 1999, p. 3) The teacher is tasked with providing experiences that support this ‘lift off’ for the essence of the concept to be abstracted. Dienes’ principles of learning (Lesh et al., 1987) inform these experiences: 1) Concrete materials do not inherently hold the mathematical idea – students construct the ideas from the relationships they impose on the materials, and it is within this mental activity that abstracting emerges; 2) Exploring a variety of models and contexts provides opportunities for structural similarities to be abstracted. (KKM) Clements, D. H. (1999). ‘Concrete’ manipulatives, concrete ideas. Contemporary Issues in Early Childhood, 1(1), 45-60. Lesh, R., Post, T., & Behr, M. (1987). Dienes revisited: Multiple embodiments in computer environments. In I. Wirsup & R. Streit (Eds.), Development in school mathematics education around the world (pp. 647-680). Reston, VA: National Council of Teachers of Mathematics. National Research Council (NRC). 1989. Everybody counts: A report to the nation on the future of mathematics education. Washington, DC: The National Academies Press. Russell, S. J. (1999). Mathematical reasoning in the elementary grades. In L. Stiff (Ed.), Developing mathematical reasoning in grades K-12 (pp. 1-12). Reston, VA: NCTM.

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Action Research was used in the early to mid-1900s to describe an approach to studying various group dynamics for the purpose of bringing about changes within those groups. Action research has evolved from this early definition to include the systematic inquiry into a problem or situation within a given field or practice (Lewis, 1944). In education, action research is often associated with studying some aspect of schooling for the purpose of improving teaching and learning (Stringer, 2008). Action research provides a structure for teachers to reflect and make changes in some aspect of their practice, such as curriculum and classroom culture. Efron and Ravin (2013) identified five characteristics of action research: constructivist, situational, practical, systematic, and cyclical (p. 7). These characteristics are considered unique to action research in the sense that new knowledge is gained from the intentional study of a problem within the context of the researcher’s work or practice and new research is based on the results of the prior findings. Teachers studying aspects of their own practice are able to continuously refine their research as they test out new iterations/variations of curricular and/or instructional interventions with their students. Action research differs from other types of research in terms of goal orientation and expectations. Research associated with degree programs such as dissertations and theses are meant to fulfill graduation requirements and must meet rigorous criteria in terms of design and methodologies. Action research is meant to be pragmatic with the goal of making improvements to one’s practice. Reports of results of action research within K-12 mathematics classrooms is reported in journals such as Teaching Children Mathematics, Mathematics Teaching in the Middle School, and Mathematics Teacher, published by the National Council of Teachers of Mathematics. (LBK) Efron, S. E., & Ravin, R. (2013). Action research in education: A practical guide. New York, NY: Guilford Press. Lewin, K. (1944). The dynamics of group action. Educational Leadership, 1(4), 195-200. Stringer, E. T. (2008). Action research in education. Upper Saddle River, NJ: Pearson Prentice Hall.

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Active Mathematics Teaching and Learning is a pedagogical approach that places the student in the role of dynamically engaging with the content in order to learn. This contrasts with traditional instructional methods, where students passively watch as the teacher shows procedures or reviews mathematical concepts. Elements of active mathematics teaching and learning (also called “Active Learning”) have long permeated mathematics education. However, as the constructivist theory of learning became more established and understood throughout the latter part of the 20th century, these pedagogical methods became more prominent in mathematics education. Spurred by reform efforts such as the NCTM’s Standards documents, educators worked to create learning environments where students were asked to investigate and explore, reason through challenging problems, justify their work and discuss their thinking with their peers in order to develop and utilize higher-order thinking skills (CBMS, 2016). In these settings, students are actively constructing their knowledge and understanding of the mathematics under study in much the same way mathematicians engage with and discover new mathematical theory. This approach to teaching and learning stands in contrast to passive learning (Boaler, 2015), where students are expected to watch and listen while a teacher demonstrates a procedure for solving problems. The students then practice the procedure with a different set of problems in order to master the skill. With passive learning, the teacher uses a “teaching by telling” model of instruction (Freeman et al., 2014) and students are often expected to master facts and methods, in contrast to active learning where students solve problems through inquiry, model-development, and questioning (Boaler, 2015). Research on active teaching and learning has illustrated positive results related to student achievement. Freeman et al. (2014) compiled 225 studies that examined active versus traditional learning approaches in undergraduate STEM courses. The combined results suggest that active learning approaches produce learning gains that increase exam scores and raise average grades by a half letter, and that failure rates with traditional approaches are 55% higher than those with active learning approaches. The CBMS (2016) reports that active learning methods “have been shown to strengthen student learning and achievement in mathematics, to foster students’ confidence in their ability to do mathematics, and to increase the diversity of the mathematical community” (p. 1). These findings have propelled calls for greater use of active teaching and learning approaches in mathematics teaching and learning (CBMS, 2016; Boaler, 2015). (SWD) Boaler, J. (2015). What’s math got to do with it? New York, NY: Penguin Books. Conference Board of the Mathematical Sciences (CBMS) (2016). Active learning in postsecondary mathematics education. Retrieved from http://www.cbmsweb.org/Statements/Active_Learning_Statement.pdf. Freeman, S., Eddy, S. L., McDonough, M., Smith, M. K., Okoroafor, N., Jordt, H., & Wenderoth, M. P. (2014). Active learning increases student performance in science, engineering, and mathematics. Proceedings of the National Academy of Sciences, 111(23). Retrieved from http://www.pnas.org/content/111/23/8410.

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Additive Reasoning is the absolute comparison of two or more quantities (numbers) through determining which is larger (smaller) and/or calculating the difference. Additive reasoning is described in a variety of mathematics content areas including early learning of number concepts, proportional reasoning, fractions, and data analysis (Cobb, 1999; Dooren et al., 2010; Lamon, 1993; Smith & Thompson, 2017; Streefland, 1991, Tourniaire & Pulos, 1985). For example, Smith and Thompson (2017) advocate for early experiences in all types of quantitative reasoning as crucial for developing algebraic reasoning skills. Dooren et al. (2010) report that upper elementary students are in a transitional stage in terms of their additive reasoning and that they were often found to use additive reasoning in proportional situations and proportional reasoning in situations that called for the use of additive reasoning. Much of the discussion of the use or misuse of additive reasoning is associated with learning fractions and proportions. Streefland (1991) found that students’ understanding of fractions was confounded by their additive strategies for whole numbers. For example, students know 1 + 2 = 3, so they may incorrectly assume ଵ ଵ ଶ that + = . In her study of sixth grade students’ intuitive strategies for solving ଶ ଷ ହ proportion problems, Lamon (1993) found that students applied additive reasoning strategies to stretcher/shrinker problems. Langrall & Swafford (2000) also reported incorrect additive reasoning strategies for solving other types of proportion problems. Pattern building strategies have an additive component and are considered a bridge between additive and multiplicative strategies for problem solving (Steinthorsdottir, 2005). (LBK) Cobb, P. (1999). Individual and collective mathematical development: The case of statistical data analysis. Mathematical Thinking and Learning, 1(1), 5-43. Dooren, W. V., Bock, D. D., & Verschaffel, L. (2010). From addition to multiplication … and back: The development of students’ additive and multiplicative reasoning skills. Cognition and Instruction, 28(3), 360-381. Langrall, C. W., & Swafford, J. (2000). Three balloons for two dollars: Developing proportional reasoning. Mathematics Teaching in the Middle School, 6(4), 254. Smith III, J. P. J., & Thompson, P. W. (2017). Quantitative reasoning and the development of algebraic reasoning. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 117-154). New York, NY: Routledge. Steinthorsdottir, O. B. (2005). Girls journey towards proportional reasoning. International Group for the Psychology of Mathematics Education, 4, 225-232. Streefland, L. (1991). Fractions in realistic mathematics education: A paradigm of developmental research. Dordrecht, The Netherlands: Springer Science + Business Media. Tourniaire, F., & Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational Studies in Mathematics, 16(2), 181-204.

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Algebraic Reasoning has been described as a process in which students generalize mathematics concepts and procedures from a set of particular instances. Algebraic reasoning exists across the mathematics curriculum, beginning in elementary grades and growing in sophistication into the middle and secondary grades (Blanton & Kaput, 2005). In secondary algebra courses, generalizing procedures that work for all real numbers is a majority of the curriculum. One of the most famous cases of generalizing arithmetic came from a challenge given to Carl Friedrich Gauss as a young child. The story is that his elementary teacher had exhausted number activities for Gauss and decided to try to occupy him by asking him to add up the numbers from 1 to 100. He quickly discovered a pattern of combining numbers that added to 101, (i.e., 100 + 1, 99 + 2, 98 + 3, …) and realized that there were 50 of these sums and returned to his teacher with the result of 50 x 101 or 5050. This pattern is generalized as the formula to find the sum S of the numbers from 1 to n: S = [n(n+1)]/2 (where n is a natural number) (Burton, 1985). Historically, algebraic reasoning was associated with formal algebra courses in high school. However, mathematics educators have described and advocated for attention to algebraic reasoning in elementary and middle grades (Carpenter et al., 2003; Kaput, 2017). Carpenter et al. (2003) describe children’s intuitive use of properties such as the distributive property and associative properties as they engage in more efficient computation strategies for whole number operations. Examples of these strategies could be using the distributive property of multiplication over addition for 8 x 15 by decomposing 15 into 10 and 5, then multiplying 8 x 10 and 8 x 5 and finally adding those products together (80 + 40 = 120). Students who intuitively use the associative property of multiplication might continuously double 15 to get the answer (2 x (2 x (2 x 15)) = 120). Algebraic reasoning has been associated with quantitative reasoning (QR) and functions. Smith and Thompson (2017) connect algebraic reasoning to QR with traditional algebra tasks involving distance, rate, and time. Carraher, Schliemann, & Brizuela (2000) advocate for approaches to operations as functions to enhance student understanding of later function concepts. Algebraic reasoning extends to courses beyond secondary algebra such as precalculus and calculus. (LBK) Blanton, M., & Kaput, J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412-446. Burton, D. M. (1985). The history of mathematics: An introduction. Dubuque, IA: Allyn & Bacon. Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann. Carraher, D., Schliemann, A. D., & Brizuela, B. M. (2000, October). Early algebra, early arithmetic: Treating operations as functions. Paper presented at the Twenty–second annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Tucson, Arizona. Kaput, J. J. (2017). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 27-40). New York, NY: Routledge. Smith III, J. P. J., & Thompson, P. W. (2017). Quantitative reasoning and the development of algebraic reasoning. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 117-154). New York, NY: Routledge.

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Algorithm is defined as “a precise, systematic method for solving a class of problems” (Maurer, 1998, p. 21). A class of problems includes procedures involving numbers, variables, etc. Algorithms have evolved historically as mathematicians have discovered more efficient methods and notations for performing a wide variety of computations. The word algorithm is derived from the Latin word algorism after the Iraqi mathematician and astronomer Mûsa âl-Khowarizmi (Burton, 2011). Around the ninth century it was thought to have been used to describe the “art of computing with Hindu-Arabic numerals” (p. 174). The use of Hindu-Arabic numerals provided a notation for mathematicians to use to record each step of the process (Barnett, 1998). As mathematicians discovered new methods for solving problems and representing solution strategies using new notations, the term algorithm became associated more for generalized procedures for number computations. These algorithms for computing with numbers are described or standardized with variables and represent fundamental properties associated with formalized algebra. For example, 382 x 43 could be solved by the following six multiplications: (300 + 80 + 2) x (40 + 3) = n 300 x 40 + 300 x 3 + 80 x 40 + 80 x 3 + 2 x 40 + 2 x 3 = n This process is generalized as the distributive property of multiplication over addition for all real numbers. In recent decades, some mathematics educators have cautioned against the over use of algorithms. Kamii (1998) cautioned that rote learning of standard algorithms can be harmful in that students may abandon their own productive thinking and mathematical strategies to perform an algorithm that they do not understand. Student-invented algorithms have components of generalized methods but often involve novel representations and different uses of mathematical notations compared to standardized recordings of algorithms (Kent, 2017). (LBK) Barnett, J. H. (1998). A brief history of algorithms in mathematics. In L. J. Morrow & M. J. Kenney (Eds.), Teaching and learning of algorithms in school mathematics (pp. 69-77). Reston, VA: National Council of Teachers of Mathematics. Burton, D. (2011). The history of mathematics: An introduction. New York, NY: McGraw-Hill. Kamii, C. (1998). The harmful effects of algorithms in grades 1-4. In L. J. Morrow & M. J. Kenney (Eds.), Teaching and learning of algorithms in school mathematics (pp. 130-140). Reston, VA: National Council of Teachers of Mathematics. Kent, L. B. (2017). Reinventing the wheel: Mathematics comes full circle. Arkansas Council of Teachers of Mathematics, 4(1), 5-8. Maurer, S. B. (1998). What is an algorithm? What is an answer? In L. J. Morrow & M. J. Kenney (Eds.), Teaching and learning of algorithms in school mathematics (pp. 21-31). Reston, VA: National Council of Teachers of Mathematics.

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Assessment in Mathematics involves the measurement and evaluation of student learning through formal means (e.g., quizzes, exams) as well as informal procedures (portfolios, student interviews, etc.). Organized by purpose, there are three overarching categories of assessments in mathematics: formative assessment, summative assessment, and progressive assessment. In the realm of mathematics education, all assessment involves being able to make decisions about what are the appropriate pieces of evidence that dictate a thorough understanding of a mathematical concept, how to collect that evidence from the students learning the mathematical concept, and how to interpret and communicate appropriately the results of the assessment. These decisions are all driven by the purpose for which one would be assessing the students during the learning process. This concept of purpose provides the outline for the following types of assessments in mathematics: Formative Assessment: This is the assessment for learning. Assessments that are of the formative type are typically used to help the learning process while it is occurring. Its purpose is to help the student and teacher understand how the students are understanding the material, how to instructionally sequence future lessons, and what needs to be improved upon for both the teacher and the student at the present time. Perhaps the most important characteristic of a formative assessment is that it can be a catalyst for change when productive change can still occur, hence making them extremely beneficial for student achievement and motivation. As discussed by Harlen (2005), an important question to ask about formative assessments is: “Can this rich but sometimes inconsistent information be used for summative assessment purposes as well as for formative assessment, for which it is so well suited?” (p. 218). Harlen and James (1997) propose that both purposes can be served providing that a distinction is made between the evidence and the interpretation of the evidence, meaning there needs to be a clear distinction about what the evidence is and how the evidence can be used to enhance the learning process of the student. Examples of formative assessments include setting student goals and standards, observations, peer review, peer assessment, and student journals. Summative Assessment: This is the assessment of learning. Assessments that are of the summative type are used from the professional judgement of the teacher to draw inferences about student understanding. Summative assessments are typically guided by a set of curriculum standards and are only indicative of a student’s understanding at a particular time, in contrast to formative assessment. It is probably best to think of summative assessments in two groups: (1) assessments that are used in the classroom/school and (2) assessments that are used outside of the classroom/school. Inside of the classroom/school, summative assessments are used to record students’ progress, help inform decisions about student placement in future math courses, and to inform parents about where their child stands in the subject being taught to them. Outside of the

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classroom/school, summative assessments are used to assess whether schools are meeting district benchmarks, to decide new curricula for subject areas of concern, and to assess teachers. Because of the purpose and use of summative assessments, the results are less likely to help students who are currently in the learning process. Any changes that could be or are made by the teacher instructionally will only happen after the changes were needed by the student. Some examples and uses of summative assessments include state mandated testing, end-of-chapter tests, district benchmarks, and certification examinations. Progressive Assessment: Progressive assessments blur the lines between formative and summative assessments. Ideally, this type of assessment is one that serves the student and teacher formatively while also being capable for use as a summative assessment. Progressive assessments are usually created and formed using stated learning goals and expectations that need to be made clear to the students during the learning process. Maxwell (2004) describes the approach to assessment used in the Senior Certificate in Queensland (Australia) as progressive assessment, in which evidence of the students learning is collected over time in a portfolio. He states All progressive assessment necessarily involves feedback to the student about the quality of their work (sic) performance. This can be expressed in terms of the students' progress towards desired learning outcomes and suggested steps for further development and improvement … For this approach to work, it is necessary to express the learning expectations in terms of common dimensions of learning (criteria). Then there can be discussion about whether the student is on-target with respect to the learning expectations and what needs to be done to improve performance on future assessment where the same dimensions appear. (pp. 2-3) Examples of progressive assessments include specifications (or specs) grading and performance-based assessment. (SJH) Harlen, W. (2005). Teachers' summative practices and assessment for learning - tensions and synergies. The Curriculum Journal, 16(2), 207-233. Harlen, W., & James, M. (1997). Assessment and learning: Differences and relationships between formative and summative assessment. Assessment in Education: Principles, Policies and Practice, 4(3), 365-379. Maxwell, G. S. (2004). Progressive assessment for learning and certification: Some lessons from school-based assessment in Queensland. Paper presented at Third Conference of the Association of Commonwealth Examination and Assessment Boards, Redefining the Roles of Educational Assessment, Nadi, Fiji.

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Basic (Number) Facts refer to knowledge of the four arithmetic operations (+, -, x, ÷) with single digit numbers (0-9). Learning basic facts has been a subject of much debate over the last century. Early learning theorists focused on training students to memorize their number facts using methods of repetition (Thorndike, 1922). These approaches were pervasive in the early 1900s during the industrial age when speed and repetition were considered important for later job performance. Baroody (1985) notes that other strategies based on counting or number relationships were characterized as limiting factors to learning number facts. While memorizing basic facts dominated instruction, other ideas emerged as cognitive psychologists studied learning from a process perspective, including discovery learning (Brownell, 1935). Counting strategies and use of familiar combinations were explored as intermediary steps to basic fact memorization (Baroody, 1985). Constructivist learning theories focusing on how students interpret new information based on current knowledge were providing new lenses to consider instructional approaches to teaching young learners (Kamii et al., 1993). These researchers proposed that children were capable of inventing their own ways to learn number facts. Professional development (PD) programs such as Cognitively Guided Instruction (CGI) utilized both the information about the progression of learning number facts and the problem type taxonomies for whole number operations to enhance elementary teachers’ approaches in using students’ thinking to guide their instructional practice of these concepts (Carpenter et al., 1989). One of the major impacts of CGI has been the role of direct modeling as a fundamental component of children’s learning of number facts (Carpenter et al., 1999). The CGI Kindergarten study provided evidence that children were capable of solving addition and subtraction problems through direct modeling as well as multiplication and division problems (Carpenter et al., 1993). CGI PD programs outlined a general trajectory of learning single digit number facts from direct modeling to counting strategies to derived facts (Carpenter et al., 1999). (LBK) Baroody, A. (1985). Mastery of basic number combinations: Internalization of relationships or facts? Journal for Research in Mathematics Education, 16(2), 83-98. Brownell, W. A. (1935). Psychological considerations in the learning and teaching of arithmetic. In W. D. Reeve (Ed.), Tenth yearbook: The teaching of arithmetic (pp. 1-31). Washington, DC: NCTM. Carpenter, T. P., Ansell, E., Franke, M. L., Fennema, E., & Weisbeck, L. (1993). Models of problem solving: A study of kindergarten children's problem-solving processes. Journal for Research in Mathematics Education, 24(5), 428-441. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann. Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C. P., & Loef, M. (1989). Using knowledge of children’s mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal, 26(4), 499-531. Kamii, C., Lewis, B., & Livingston, S. (1993). Primary arithmetic: Children inventing their own procedures. The Arithmetic Teacher, 41(4), 200-203. Thorndike, E. L. (1922). The psychology of arithmetic. New York, NY: The Macmillan Company.

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Beliefs/Attitudes, often referred to as Affect in education research, concerns the principles, interests and viewpoints one holds when encountering various phenomena and which influence how one approaches these phenomena in educational settings. The study of beliefs/attitudes has an extensive history in mathematics education. Beliefs/attitudes are often associated with the description of affect issues in mathematics teaching and learning (Gal & Ginsburg, 1994; Hart, 1989; McLeod, 1989). In their critique of instruments used to assess students’ beliefs/attitudes about learning statistics, Gal and Ginsburg (1994) recommend open response items in addition to Likert scale items to gather more elaborate information about students’ thoughts and feelings about the content. In one of the seminal studies of student achievement and attitudes about mathematics, Fennema and Sherman (1978) found significant gender differences in attitudes towards mathematics even though significant differences were not found between males and females on achievement measures. Mathematics teachers’ beliefs/attitudes have also been studied from a variety of perspectives (Ernest, 1989; Wilkins, 2008). Ernest (1989) described the influence of teachers’ beliefs on their instructional practices. Wilkins (2008) studied the relationship between beliefs, knowledge, attitudes and practices of 481 inservice elementary teachers and found that that beliefs and attitudes were positively related to inquiry practices while knowledge was found to be negatively associated with inquiry practices. (LBK) Ernest, P. (1989). The knowledge, beliefs and attitudes of the mathematics teacher: A model. Journal of Education for Teaching, 15(1), 13-33. Fennema, E., & Sherman, J. (1978). Sex-related differences in mathematics achievement and related factors: A further study. Journal for Research in Mathematics Education, 9(3), 189203. Gal, I., & Ginsburg, L. (1994). The role of beliefs and attitudes in learning statistics: Towards an assessment framework. Journal of Statistics Education, 2(2). Retrieved from http://www.amstat.org/publications/jse/v2n2/gal.html Hart, L. E. (1989). Describing the affective domain: Saying what we mean. In D. B. McLeod & V. M. Adams (Eds.), Affect and mathematical problem solving (pp. 37-45). New York, NY: Springer. McLeod, D. B. (1989). Beliefs, attitudes, and emotions: New views of affect in mathematics education. In D. B. McLeod & V. M. Adams (Eds.), Affect and mathematical problem solving (pp. 245-258). New York, NY: Springer. Wilkins, J. L. (2008). The relationship among elementary teachers’ content knowledge, attitudes, beliefs, and practices. Journal of Mathematics Teacher Education, 11(2), 139-164.

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Cognitive Demand is the level of mathematical thinking and reasoning required to work mathematical problems. Cognitive demand ranges from lower-level tasks that require a memorized fact or a simplistic procedural solution to higher-level tasks that necessitate generalizing patterns or non-algorithmic thinking. Prior to the late 1970s, much research on the improvement of mathematics teaching centered on “process-product” studies that examined the effects of various classroom features (e.g., class size, amount of time spent lecturing, teacher behaviors) on student learning. In the late 1970s and 1980s, researchers began to focus more on the kinds of instructional tasks given to students to promote student learning. Doyle (1983) explored the role of problems in student learning, noting “tasks influence learners by directing their attention to particular aspects of content and by specifying ways of processing information” (p. 161). He noted that the selection of tasks inevitably defines what students do and learn, and that a student will acquire information and practice operations and skills based upon what is required to solve given tasks. He outlined four types of academic tasks based on the cognitive operations required: memory tasks, where a student reproduces content from memory; procedural or routine tasks, where a student uses a standardized formula to produce an answer; comprehension or understanding tasks, where students may be required to decide among several procedures in solving a problem; and opinion tasks, where students communicate their thinking on the problem (pp. 162-163). Further work on the impact of the cognitive demand of tasks on student learning has been conducted by the QUASAR project (Quantitative Understanding: Amplifying Student Achievement and Reasoning), a research project examining mathematics education reform efforts in urban U.S. middle schools. QUASAR researchers created a Task Analysis Guide highlighting the features of problems that required various levels of cognitive demand (Smith & Stein, 1998). The guide was divided into low-level (memorization and procedures without connections to meaning) and high-level cognitive demand tasks (procedures with connections to meaning and doing mathematics). Underscoring the critical importance of tasks in student learning, the researchers found that the greatest gains for students in the study were directly related to the use of tasks that were designed and implemented in ways that engaged students at high levels of cognitive demand (Smith & Stein, 1998). QUASAR researchers also identified instructional factors that impacted and supported the implementation of high-cognitive level tasks, including appropriate scaffolding, sufficient time provided, and the use of tasks that use students’ prior knowledge (Henningsen & Stein, 1997). (SWD) Doyle, W. (1983). Academic work. Review of Educational Research, 53(2), 159-199. Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroombased factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524-549. Smith, M. S., & Stein, M. K. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3(5), 344-350.

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Cognitive Science is an interdisciplinary study of the mind and its processes that draws from psychology, linguistics, computer science, neuroscience, anthropology, and philosophy. As technological advances ushered in the disciplines of neuroscience and artificial intelligence in the 1950s, and as the interest in behaviorism began to decline, opportunities for cross-disciplinary studies of how the mind works and how it translates to behaviors became evident. The flourishing interest became known as the cognitive revolution (Miller, 2003). As an adjective form of cognition, cognitive refers to the mental actions used in reasoning, learning, understanding, remembering, abstracting, and mental representations – all important for mathematical thinking. Turner (2010) identified six cognitive processes critical for the activation of mathematics learning: communication, mathematizing, representation, reasoning and argument, devising strategies, and using symbolic, formal, and technical language and operations. Much research in cognitive science has focused on or has been relevant to the learning of mathematics and has supported the principles suggested by the National Council of Teachers of Mathematics (Siegler, 2003). Siegler describes eight of the areas supported by research: 1. Mathematical understanding before children enter school – children have prior knowledge and experience with mathematics. 2. Pitfalls in mathematics learning – Systematic misconceptions can result when abstract concepts and procedures are not connected. 3. Cognitive variability and strategy choice – children use a variety of strategies and may choose less sophisticated strategies even after learning more advanced ones. 4. Individual differences – differences in working memory capacity, processing, stylistic preferences, and prior knowledge need to be addressed for students to progress. 5. Discovery and insight – children can generate new ideas and strategies on their own without formal discovery situations set up for them. 6. Relations between conceptual and procedural knowledge – success in procedural skills is positively correlated with conceptual understanding. 7. Cooperative learning – with careful structuring, students benefit from the discourse involved in shared problem solving. 8. Promoting analytic thinking and transfer – analytic reasoning and purposeful engagement are related, and promote transfer. (KKM) Miller, G. A. (2003). The cognitive revolution: A historical perspective. TRENDS in Cognitive Sciences, 7(3), 141-144. Siegler, R. S. (2003). Implications of cognitive science research for mathematics education. In J. Kilpatrick, W. B. Martin, & D. E. Schifter (Eds.), A research companion to Principles and Standards for School Mathematics (pp. 219-233). Reston, VA: NCTM. Turner, R. (2011). Identifying cognitive processes important to mathematics learning but often overlooked. The Australian Mathematics Teacher, 67(2), 22–26.

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Cognitively Guided Instruction (CGI) is a mathematics professional development program for elementary teachers, the goal of which is to enhance teachers’ understanding of how young children solve problems and reason algebraically. CGI is based on a robust body of knowledge of how children think about number concepts and progress in their problem solving strategies. “The thesis of CGI is that children enter school with a great deal of informal or intuitive knowledge of mathematics that can serve as the basis for developing understanding of the mathematics of the primary school curriculum” (Carpenter et al., 1999, p. 4). Initially, taxonomies of problem types were identified and organized in a manner that reflected students’ strategies for whole number addition and subtraction situations (Carpenter & Moser, 1984) and similarly for whole number multiplication and division problem types. Participants in CGI engage in activities that provide them with the opportunity to learn about the distinctions in problem types and distinctions between problem solving strategies for the different problem types. These activities are based on studies that show that teachers who participated in CGI knew more about their students’ problem solving strategies and were more likely to encourage them to solve problems without direct instruction on possible strategies (Fennema et al., 1996). One early study on CGI showed that students from classes in which the teacher participated in CGI outperformed students from control classes in problem solving tasks (Carpenter et al., 1989). The CGI kindergarten study shed new light on the capacity of children to solve addition, subtraction, multiplication and division problems by directly modeling the quantities and structure of the situation (Carpenter, et al., 1993). CGI principles have expanded to include both pre-K and upper elementary content (Carpenter et al., 2016; Empson & Levi, 2011). (LBK) Carpenter, T. P., Ansell, E., Franke, M. L., Fennema, E., & Weisbeck, L. (1993). Models of problem solving: A study of kindergarten children's problem-solving processes. Journal for Research in Mathematics Education, 24(5), 428-441. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann. Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C. P., & Loef, M. (1989). Using knowledge of children’s mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal, 26(4), 499-531. Carpenter, T. P., Franke, M. L., Johnson, N. C., Turrou, A. C., & Wager, A. (2016). Young children’s mathematics: Cognitively guided instruction in early education. Portsmouth, NH: Heinemann. Carpenter, T. P., & Moser, J. M. (1984). The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education, 15(3), 179-202. Empson, S. B., & Levi, L. (2011). Extending children’s mathematics: Fractions and decimals. Portsmouth, NH: Heinemann. Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use children's thinking in mathematics instruction. Journal for Research in Mathematics Education, 27(4), 403-434.

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Common Core State Standards for Mathematics (CCSSM) refer to the mathematical standards for grades K-12 created by the National Governors Association (NGA) and the Council of Chief State School Offices (CCSSO) as a result of the Common Core State Standards Initiatives (CCSSI). The CCSSI began in the 1990s when states began working together in the “Standards and Accountability Movement” (Achieve, 2011). In 1996, governors and corporate leaders created Achieve, a nonprofit education reform organization with three primary goals: increase state academic standards and graduation requirements; improve assessments; and strengthen accountability. The formal writing of CCSSM began in 2009 as the NGA convened a group of people tasked with writing standards that are clear and indicative of what students should be able to know and perform prior to the end of each grade level. A major component of the CCSSI was to formally assess the standards. These assessments were to go beyond measuring low level skills in order to assess CCSSM. Two consortiums (the Partnership for Assessment of Readiness for College and Careers (PARCC) and Smarter Balanced) were formed to create the assessments aligned to CCSSM, with state education agencies using the consortium of their choice to assess CCSSM. PARCC uses a computer-based, end-of-year test (PARCC, 2018). The results allow parents and teachers to know current student performance as well as what students need to know in order to be ready for college programs or the workforce. Smarter Balanced uses a systems approach to assessment in which there are three components: a digital library, interim assessments, and summative assessments (Smarter Balanced, 2018). The interim assessments are designed to be used throughout the year and allow for progress monitoring of students. In the initial development of CCSSM, two categories of standards were written: college- and career-readiness (CCR) standards and K-12 standards. The final version of CCSSM consisted of the K-12 standards with the CCR standards integrated at the individual grade levels. CCSSM provides details for what K-12 students should know prior to graduation from high school, in preparation for entering either a two- or four-year college program, or the workforce. CCSSM was initially adopted by 45 states, the District of Columbia and several other U.S. territories, but for various reasons, states have opted out of or repealed various aspects of the initiative, including participation in one of the assessment consortiums. Many states are replacing CCSSM with adapted versions of the standards. In terms of assessment, some states are working collectively to create a common assessment, while other states are working independently to do so or in conjunction with one of the consortiums. (CCO) Achieve. (2011). Closing the expectations gap: Sixth annual 50-state progress report on the alignment of high school policies with the demands of college and careers. Washington, DC: Author. Retrieved from https://www.achieve.org/files/AchieveClosingthe Expectations Gap 2011.pdf PARCC. (2018). Retrieved from http://www.parcconline.org/about/the-parcc-tests. Smarter Balanced. (2018). Retrieved from http://www.smarterbalanced.org/about/history/.

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Computer Algebra Systems (CAS) refer to numerical or symbolic mathematical software programs (Artigue, 2002). There are many examples of CAS programs including Mathematica, TI-Nspire, and SINGULAR, to name a few. The main purpose of these programs is to simplify cumbersome algorithms but they also may enhance learning of concepts. Some studies have shown that students who use CAS as part of their mathematics curriculum outperform students who were exposed to traditional methods only. For example, Palmiter (1991) examined student performance in an integral calculus course with and without CAS. In her study, students who used the MACSYMA CAS system outperformed the students taught through traditional methods on both conceptual and computational tasks. Kramarski and Hirsch (2003) found that students who used the CAS system DERIVE and engaged in selfregulated learning strategies outperformed CAS only students on algebraic tasks. Heid and Edwards (2001) offer four possible roles of CAS systems within learning mathematics: generating symbolic results, generating symbolic procedures, looking for patterns from the generation of multiple examples, and generating results from abstract problem solving These roles decrease the amount of time that students would need to calculate answers and increase the amount of time they could spend making conjectures and focusing on big ideas of mathematics. They also advocated for studies of students’ use of CAS systems that focused on the details of their thinking and understanding in addition to experimental or quasi-experimental studies. Thompson et al. (2013) demonstrate that conceptual learning of calculus topics such as limits could be significantly enhanced by the use of CAS technologies. As technology continues to evolve at an ever-increasing rate, the role of CAS technology and will continue to impact teaching and learning of mathematics. (LBK) Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7(3), 245-274. Heid, M. K., & Edwards, M. T. (2001). Computer algebra systems: Revolution or retrofit for today's mathematics classrooms? Theory into Practice, 40(2), 128-136. Kramarski, B., & Hirsch, C. (2003). Using computer algebra systems in mathematical classrooms. Journal of Computer Assisted Learning, 19(1), 35-45. Palmiter, J. (1991). Effects of computer algebra systems on concept and skill acquisition in calculus. Journal for Research in Mathematics Education, 22(2), 151-156. Thompson, P. W., Byerley, C., & Hatfield, N. (2013). A conceptual approach to calculus made possible by technology. Computers in the Schools, 30(1-2), 124-147.

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Concept Image is defined as “a total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes” (Tall & Vinner, 1981, p. 152). The term concept image was created to describe ideas and thinking students may have about a mathematics topic prior to formal introduction (Tall, 1992). Tall & Vinner (1981) define concept definition as the words and terminology used to describe mathematical topics and concepts. Vinner & Hershkovitz (1980) describe concept definitions as external sources that can be influential, both positively and negatively, in terms of students’ development of concept images for mathematical topics. Examples of concept images are possible with all levels of mathematical content. However, the theoretical framework of concept image and concept definition is most frequently associated with preservice teachers’ knowledge and secondary level mathematics content areas (Da Ponte & Chapman, 2006; Even, 1993). Some of the discussion relates to mathematics concepts such as functions and limits. Even (1993) describes the inconsistency between preservice teachers’ concept image of “functions as a graph” and the modern definition of a function as an arbitrary relationship that requires the univalence component. For example, some preservice teachers in her study would characterize a circle as a function because “circles are smooth graphs” but would not characterize discontinuous graphs as functions because they were not smooth curves even though each input variable contained a unique output variable. Tall (1992) also discussed the conflicts between students’ concept images of limits and concept definitions based on colloquial uses of the terms limit. Students may have a mental image of a limit as not actually reaching the value while the limits of some sequences actually equal the limit as in the example of the limit of the sequence of 0.9, 0.99, 0.999, 0.9999, 0.99999, …, eventually being equal to one (p. 502). (LBK) Da Ponte, J. P., & Chapman, O. (2006). Mathematics teachers’ knowledge and practices. In A. Gutierrez, & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 461-494). Rotterdam, The Netherlands: Sense Publishers. Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept. Journal for Research in Mathematics Education, 24(2), 94-116. Tall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity, and proof. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (1-26). Reston, VA: National Council of Teachers of Mathematics. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151-169. Vinner, S., & Hershkowitz, R. (1980, August). Concept images and common cognitive paths in the development of some simple geometrical concepts. In Proceedings of the fourth international conference for the psychology of mathematics education, Berkeley, CA, (pp. 177-184).

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Conceptual Knowledge goes beyond knowledge of facts and individual pieces of information to include knowledge of relationships and networks of connected ideas. Conceptual knowledge … can be thought of as a connected web of knowledge, a network in which the linking relationships are as prominent as the discrete pieces of information. Relationships pervade the individual facts and propositions so that all pieces of information are linked to some network. (Hiebert & Lefevre, 1986, pp. 3-4) Considering the term concept as an abstraction that comes from generalizing related ideas or things, the focus on relationships is inherent in conceptual understanding. As the first strand of mathematical proficiency, “conceptual understanding refers to an integrated and functional grasp of mathematical ideas” (NRC, 2001, p. 118). When knowledge is organized in a network of relationships, information can be more readily retrieved and flexibly applied by students, as when they invent their own strategies (Carpenter, 1986). “The development to more advanced levels of problem solving is characterized by an increase in flexibility … made possible by an increasingly rich conceptual base, more efficient procedures, and the maintenance of links between them” (Carpenter, 1986, p. 115). Children transition from utilizing relationships within problems to generalizing relationships across problem types (Carpenter, 1986) moving from what Hiebert calls the primary level to the reflective level of relationships (Hiebert & Lefevre, 1986). Constructivist theory explains how conceptual knowledge develops. Insight is gained when students find a way to connect ideas that they did not previously see as related, either constructing relationships between new pieces of information, or new information with prior knowledge. For example, students who discover that decimal place value is connected to understanding fractions with denominators of powers of 10, have broadened their conceptual understanding of rational numbers as well as the base 10 number system. New knowledge is assimilated into an existing network and cognitive reorganization occurs as previously independent networks are related (Hiebert & Lefevre, 1986). For over a century, mathematics education researchers have studied conceptual and procedural knowledge as two major types of knowledge involved in mathematical learning. The current consensus is that both types of knowledge interrelate and support each other (Hiebert & Lefevre, 1986). (KKM) Carpenter, T. P. (1986). Conceptual knowledge as a foundation for procedural knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 113132). Hillsdale, NJ: Erlbaum. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1-27). Hillsdale, NJ: Erlbaum. National Research Council (NRC). (2001. Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, & B. Findell (Eds.). Washington, DC: National Academy Press.

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Conjecture is a statement of generalization based on empirical evidence, such as observing a pattern based on examples. Although a conjecture may be true, it is not accepted as truth without formal proof. “Conjecturing and demonstrating the logical validity of conjectures are the essence of the creative act of doing mathematics” (NCTM, 1989, p. 81). Making conjectures plays a pivotal role in students’ mathematical thinking as it is a result of an inductive reasoning process that can prompt the asking of “why?” or “will this always be true?” which can motivate the search for validity through a deductive reasoning process. The term conjecture was popularized in Serra’s (2008) Discovering Geometry text, first published in 1989. Serra’s curriculum emphasized an inductive approach for building geometric reasoning skills as a way to scaffold the development of students’ deductive reasoning for more formal geometry proofs. In Discovering Geometry, students experience a series of exploratory activities, often using such tools as paper folding or dynamic geometry software to examine multiple examples that lead to making conjectures of geometric concepts that in more traditional textbooks are already laid out as theorems and postulates. For instance, one investigation involves measuring angles leading to a conjecture that a linear pair of angles has a sum of 180°. In a subsequent exploration, students fold paper to conjecture that vertical angles are congruent. They are then challenged to give a deductive argument for why the Vertical Angles Conjecture is true based on the Linear Pair Conjecture (Serra, 2008). Boats et al. (2003) point out that as students examine examples that may lead to a conjecture, they should understand the power of a counterexample as being sufficient to disprove a conjecture. Learning that a statement is true only if it is always true helps students understand the limitations of a conjecture based on a small number of examples. Cantlon (1998) describes how a focus on making conjectures has empowered elementary students as mathematical thinkers. As students construct their own knowledge in the process of making a conjecture, they feel ownership of their own ideas and are willing to discuss, defend, and further investigate them. The validation of the conjecture is done collaboratively as it is subjected to class discussion, and students decide if the conjecture is worthy of being written on the class ‘conjecture wall.’ Students learn, however, that their informal validation process is tentative as new knowledge may cause them to revise or reject a conjecture, as when an initial conjecture of ‘larger denominators imply smaller fractions’ is revised to acknowledge that it is true only for unit fractions. (KKM) Boats, J. J., Dwyer, N. K., Laing, S., & Fratella, M. P. (2003). Geometric conjectures: The importance of counterexamples. Mathematics Teaching in the Middle School, 9(4), 210 – 215. Cantlon, D. (1998). Kids + conjecture = mathematics power. Teaching Children Mathematics, 5(2), 108-122. National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. Serra, M. (2008). Discovering geometry: An investigative approach. Emeryville, CA: Key Curriculum Press.

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Constructivist Theory of Learning explains that individuals construct their own knowledge and build understanding as they connect their prior knowledge with experiences, new information, and interactions with others. The theory posits that knowledge acquisition is an active process with the learner at the center of the action, interpreting, reflecting, and organizing new information, rather than passively absorbing external information. With the growth of cognitive psychology in the 1960s and 1970s, and the shift from behaviorist views of learning, constructivism emerged with new ideas of how people learn, calling into question the common practice of direct instruction as the primary instructional strategy. NCTM embraced these new ideas as foundational for learning mathematics. The consequent publication of Curriculum and Evaluation Standards (NCTM, 1989) sparked a reform movement in mathematics education based on constructivist principles and suggested a variety of instructional strategies to help children build mathematical understanding. Clements and Battista (1990) describe how a classroom based on constructivist principles compares to one with a direct instruction focus: In constructivist instruction, students are encouraged to use their own methods for solving problems. They are not asked to adopt someone else's thinking but encouraged to refine their own. Although the teacher presents tasks that promote the invention or adoption of more sophisticated techniques, all methods are valued and supported. Through interaction with mathematical tasks and other students, the student's own intuitive mathematical thinking gradually becomes more abstract and powerful. (p. 35) Ndluvo (2013) articulates five basic tenets of constructivism. These tenets are: 1. 2.

3.

4.

Students construct their own knowledge Independence of uniqueness of constructions No matter how they are taught or communicated with, students will always form their own understandings and do so even idiosyncratically. This partly explains why two learners in the same classroom under the same instructor at the same time do not necessarily attain the same quality of understandings because they bring to the learning context different prior understandings upon which to construct their new mathematical knowledge (p. 4). Recognition of prior understandings “… prior understandings and predispositions invariably become the prism by which new knowledge is viewed, interpreted and assimilated” (p. 5). The process of knowledge transformation As a result of years of studying child development, Piaget theorized that children construct their own knowledge based on their experiences. New information is either assimilated into existing schema (organized blocks of knowledge), or accommodated, which may require conceptual change. The disequilibrium, or cognitive dissonance, that is experienced when information

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does not fit into existing schema motivates the learner to make changes to make it fit (accommodation). Piaget believed that the process of accommodation is where new learning occurs. “Learning is therefore distilled into a human activity driven and propelled by self-reliant, self-reflexive cognitive actions of equilibration and re-equilibration which cause movement from one level of understanding to a new level of sense making” (p. 6). The social dimension of construction and reconstruction The process of sharing implies explaining and justifying one’s mathematical understanding or problem solving procedure to others. In turn, other members of the group have an obligation to subject the explanation to scrutiny, critical reflection, before reaching consensus. This implies group construction, and reconstruction which inherently embodies group reflective thinking. In other words, the attainment of a new level of conceptualization is a product of collaborative constitution and reconstitution, co-responsibility and co-ownership. That is, dialogue and the negotiation of meaning provide the basis for the individuals to develop, test and refine their ideas (p. 6).

In a classroom based on these principles, the teacher has the responsibility to create opportunities for students to construct mathematical knowledge. Providing student-centered exploratory experiences, multiple embodiments of concepts, relevant problems, and ample time for sense-making, both individually and shared, allows for multiple pathways to making connections and building understanding. In a study in which teachers were trained to use such strategies to allow more autonomy for student thinking, teachers felt they and their students became co-creators of the mathematics learning environment. They used their authority “to guide and sustain mathematical communication in both whole class and small group settings by listening, offering suggestions, and clarifying children’s meanings” (Cobb et al., 1990, p. 137). Consequently, the teachers reconceptualized their role as facilitators and their view of what mathematical activity is. (KKM) Clements, D. H., & Battista, M. T. (1990). Constructivist learning and teaching. Arithmetic Teacher, 38(1), 34-35. Cobb, P., Wood, R., & Yackel, E. (1990). Classrooms as learning environments for teachers and researchers [Monograph]. In R. B. Davis, C. A. Maher, & N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics. Journal for Research in Mathematics Education (pp. 125-146). Reston, VA: National Council of Teachers of Mathematics. National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. Ndlovu, M. (2013). Revisiting the efficacy of constructivism in mathematics education. Philosophy of Mathematics Education Journal, 27, 1-13.

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Cooperative Learning is an instructional strategy in which students work together in small groups on a common goal, committed to helping one another reach that goal and maximize learning for each group member. The term cooperative learning is more purposeful than ‘group work’ as it demands certain elements to be in place to ensure interaction among students, minimize distracted and unproductive behavior, and maximize learning. Kagan (2003) created a variety of structures (prescribed situations that elicit cooperation) that promote equal and frequent participation in a cooperative group, such as “Numbered Heads Together”. Johnson et al. (2014) describe five elements that must be present for group work to be elevated to cooperative learning: 1. Positive interdependence – The group effort is motivated by the attitude that an individual’s success is tied to how well other individuals do in the group, and that each has a role in ensuring that everyone understands the task and is contributing to the common goal. 2. Individual accountability – Each student is held accountable for mastering the learning objectives, as well as for fulfilling their role in the group. 3. Promotive interaction – Students are active in discourse, brainstorming solutions, critiquing others’ ideas, asking for or providing justification, encouraging and helping as needed to promote one another’s success. 4. Appropriate use of social skills – For successful collaboration, students must develop their skills in communication, trust-building, conflict management, decision-making, and leadership. Teachers should explicitly teach these skills if necessary. 5. Group processing – Group members reflect upon their team efforts, identifying processes that are productive and those that are not. Vygotsky’s sociocultural theory describes learning as a social process in which understanding develops from interaction with others that may not have occurred alone (Eggen & Kauchak, 2010). Several decades of evidence support the benefits of cooperative learning over learning individually, including greater achievement, retention, transfer of learning, higher level reasoning and more creative problem solving. Students in cooperative groups are more willing to take on difficult tasks and persevere through solving them, are more intrinsically motivated, and are more engaged in their learning (Johnson et al., 2014). (KKM) Eggen, P., & Kauchak, D. (2010). Educational psychology: Windows on classrooms. Upper Saddle River, NJ: Pearson. Johnson, D. W., Johnson, R. T., & Smith, K. A. (2014). Cooperative learning: Improving university instruction by basing practice on validated theory. Journal on Excellence in College Teaching, 25(3&4), 85-118. Kagan, S. (2003). A brief history of Kagan structures. Kagan Online Magazine. San Clemente, CA: Kagan Publishing. Retrieved from https://www.kaganonline.com/free_articles/dr_ spencer_kagan/256/A-Brief-History-of-Kagan-Structures

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Council for the Accreditation of Educator Preparation (CAEP) [formerly known as NCATE] is the primary accrediting organization for teacher preparation programs in the United States. Traditional licensure programs within universities across the country are responsible for maintaining quality control of their certification programs through the review and recommendation process of CAEP. The goal according to the caepnet.org website is “creating standards to ensure educator preparation providers impart future teachers with the knowledge and skills to support the development of all students”. As of spring 2018, over 800 educator preparation providers participate in the CAEP Accreditation system, including many previously accredited through former standards. There are five standards related to CAEP accreditation. x x x x x

Standard 1: Content and Pedagogical Knowledge Standard 2: Clinical Partnerships and Practice Standard 3: Candidate Quality, Recruitment, and Selectivity Standard 4: Program Impact Standard 5: Provider Quality, Continuous Improvement, and Capacity

These are general standards that involve all subjects across K-12. Peer-reviewed teacher preparation programs are responsible for meeting these general standards. However, there are related Specialized Professional Association (SPA) standards within specific grade bands or content areas. Many of the components of NCATE and CAEP are similar. One difference is in the expectations for technology integration in secondary mathematics. For NCATE, the technology requirements were specific to a particular device or software. For example, “graphing calculator” was specifically required for NCATE. However, as a result of all of the changes in technology, the CAEP requirements for technology use are not as specific. Teacher candidates are still required to integrate technology related to specific lessons but do not have to include a particular type. “They use instructional tools such as manipulatives, digital tools, and virtual resources to enhance learning while recognizing the possible limitations of such tools” (NCTM SPA Standard 4: Learning Environment; https://www.nctm.org/Standards-and-Positions/CAEP-Standards/). CAEP accreditation covers both initial and advanced certification programs. Teacher preparation programs involved in accreditation through CAEP are responsible for documentation related to their related SPA standards. Systematic studies of the role of CAEP standards are not prevalent. However, CAEP has been critiqued as possibly diminishing aspects of a variety of teacher preparation programs (Schwarz, 2016). (LBK) Schwarz, G. (2016). CAEP and the decline of curriculum and teaching in an age of technology. Curriculum and Teaching Dialogue, 18(1 & 2).

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Counting is typically associated with the process of finding the number of elements in a finite set. In a strictly mathematical sense, the counting numbers are also called natural numbers from 1 to N. In mathematics education, counting is used in a variety of ways and topics. For example, counting within Cognitively Guided Professional (CGI) Development is described in terms of a strategy in which a child can mentally hold one of the numbers from an addition or subtraction problem and count up or down from the number. Counting is also used to distinguish between direct modeling strategies in which all quantities are represented by the child (Carpenter et al., 1999, 2016). For example, to add 4 + 3, a direct modeler would represent and count to four by ones, then count to three by ones, then combine the two sets and count to seven by ones. However, a child who uses a counting strategy would mentally hold either the three or the four and count on to seven, (i.e., “five, six, seven”) to get the answer. Skip counting is used to describe repeated addition of an equal group as a strategy for multiplication of whole numbers or for multiple groups problems (Empson & Levi, 2011). Counting is also an important term in combinatorics and discrete mathematics and references counting the number of combinations or permutations possible of an event (Cameron, 1994). Moreover, “countably infinite” is a term in advanced level analysis courses and topology. (Freedman, 1965; Higman et al., 1949). (LBK) Cameron, P. J. (1994). Combinatorics: Topics, techniques, algorithms. Cambridge, England: Cambridge University Press. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann. Carpenter, T. P., Franke, M. L., Johnson, N. C., Turrou, A. C., & Wager, A. (2016). Young children’s mathematics: Cognitively guided instruction in early education. Portsmouth, NH: Heinemann. Empson, S. B., & Levi, L. (2011). Extending children’s mathematics: Fractions and decimals. Portsmouth, NH: Heinemann. Freedman, D. A. (1965). On the asymptotic behavior of Bayes estimates in the discrete case II. The Annals of Mathematical Statistics, 36(2), 454-456. Higman, G., Neumann, B. H., & Neuman, H. (1949). Embedding theorems for groups. Journal of the London Mathematical Society, 1(4), 247-254.

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Covariational Reasoning is the mental activity used to coordinate two varying quantities while considering how they change in relation to each other. Madison et al. (2015) write that “quantitative and covariational reasoning are two foundational ways of thinking that students engage in when constructing, interpreting, and using functions meaningfully” (p. 58). To better understand the development of this critical ability, Carlson et al. (2002) created a framework to identify five levels of covariational reasoning. They describe distinct “mental actions” characterizing each level (pp. 357-358) and suggest questions to elicit student reasoning about dynamic function situations. These levels are: 1. Coordination Level: Coordinating the value of one variable with changes in the other. Which quantities are constant, which vary, and which vary together? 2. Direction Level: Coordinating the direction of change in one variable with changes in the other variable. Does a function increase or decrease as the independent variable increases or decreases? 3. Quantitative Coordination Level: Coordinating the amount of change of one variable with changes in the other variable. Considering equal intervals of change in input, what amount of change happens to the output? 4. Average Rate Level: Coordinating the average rate of change of the function with uniform increments of change in the input variable. How can the amounts of change in output per interval be described as a rate of change? 5. Instantaneous Rate Level: Coordinating the instantaneous rate of change of the function with continuous changes in the independent variable for the entire domain of the function. What do inflection points tell you about the rate of change throughout the domain? In a study of calculus students, Carlson et al. (2002) used the framework to analyze students’ covariational reasoning. They found only 25% of the students could adequately describe how the height of the water in an irregularly shaped bottle changed in relation to the increase in volume of water, beyond the Level 2 description of height increasing as volume increases. Most students struggled with attempts to think in terms of increments and rate of change. Increased attention to each level of the framework was recommended to support success at the higher levels. (KKM) Carlson, M. P., Jacobs, S., Coe, E., Larsen, S. & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352–378. Madison, B. L., Carlson, M., Oehrtman, M., & Tallman, M. (2015). Strengthening students’ quantitative and covariational reasoning. Mathematics Teacher, 109(1), 54-61.

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Curricular Reasoning concerns the thinking and decision-making teachers perform as they utilize curriculum materials (e.g., textbooks, standards, supplemental materials) to create and enact mathematical learning opportunities for their students. As teachers work to implement mathematics instruction in their classrooms, there are a number of curricular resources that inform and guide their decision-making processes. These resources include state standards, district curriculum guides, the district-adopted mathematics textbook, and supplemental materials and activities at the teacher’s disposal. These various resources can pull a teacher in a multitude of directions, necessitating the teacher to make a number of decisions regarding a variety of instructional issues. These issues include on which problems or tasks to focus, which questions to pose, how to sequence and facilitate student discussion of the mathematical ideas, and how to connect and assess various student lines of thinking (Breyfogle et al., 2010). Each of these decisions guides how the teacher enacts instruction and ultimately holds considerable influence regarding what students have the opportunity to learn (Stein et al., 2007). As teachers make these decisions, they utilize their “curricular reasoning” to inform their instructional decisions. Breyfogle et al. (2010) define curricular reasoning as “the thinking processes that teachers engage in as they work with curriculum materials to plan, implement, and reflect on instruction” (p. 308). These thinking processes are based in one’s curriculum knowledge and are an integral part of the pedagogical plan the teacher creates and implements in the classroom. These thinking processes also underlie the goals for the lesson and can assist a teacher in deciding what lessons to teach, when to use the mathematics textbook or when to supplement with outside activities and materials, and how to gauge the overall effectiveness of the materials in teaching to the adopted standards (McDuffie & Mather, 2009). Given that there exist important discrepancies across textbooks with regards to matters such as the development of mathematical content as well as the instructional approaches supported, the ability for teachers to reason and make sense about the various resources available to them is critically important in ultimately determining what and how students learn in mathematics. (SWD) Breyfogle, M. L., McDuffie, A. R., & Wohlhuter, K. A. (2010). Developing curricular reasoning for grades Pre-K-12 mathematics instruction. In B. J. Reys, R. E. Reys, & R. Rubenstein (Eds.), Mathematics curriculum: Issues, trends, and future directions – 72nd NCTM Yearbook (pp. 307-320). Reston, VA: National Council of Teachers of Mathematics. McDuffie, A. R., & Mather, M. (2009). Middle school mathematics teachers’ use of curricular reasoning in a collaborative professional development project. In J. T. Remillard, B. A. Herbal-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 302-320). New York, NY: Routledge. Stein, M. K., Remillard, J. T., & Smith, M. S. (2007). How curriculum influences student learning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 319-370). Charlotte, NC: Information Age Publishing.

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Curriculum refers to the mathematical topics and skills that comprise a specific course of study, either in a single class (such as an Algebra I curriculum) or across a range of courses or a grade band (such as the middle school curriculum). Curriculum in its purest definition encompasses a broad expanse that illustrates the “what” of mathematics teaching and learning (Stein et al., 2007). In the guiding principles of their research agenda, researchers for the Center for the Study of Mathematics Curriculum (CSMC) state, “Mathematics curriculum: what is important, what is expected, how it is organized and sequenced, how it is taught, and what students learn is the core around which mathematics education revolves”. This statement captures the multi-faceted nature of this term, as curriculum can refer to textbooks, standards, assessments, content goals, instructional methods, and recommendations from national organizations (e.g., NCTM). Curriculum also focuses on both the knowledge students should gain from studying a particular subject (e.g., linear functions in Algebra I) as well as the skills that should be developed (e.g., proof and justification in a Geometry course). Researchers have outlined a “curricular chain” (Venezky, 1992; Glatthorn, 1999) that depicts the different forms in which curriculum exists: x x x

x

x

The needed or recommended curriculum refers to the suggestions and recommendations experts propose should be taught to students. The intended or desired curriculum refers most commonly to standards documents that outline the content and skills to be taught to students. The written or prescribed curriculum concerns the curriculum as it appears in textbooks, educational software, or in other supplementary instructional materials that provide a day-to-day plan for teachers. The enacted or implemented curriculum refers to the instructional activities, lectures, and homework assignments used by teachers to instruct students and help them learn the mathematical content and skills under study. The assessed or tested curriculum is the content and skills that students are ultimately tested upon, whether it is through teacher-designed exams, statemandated assessments, or national examinations.

Curriculum is also organized into the horizontal curriculum, which details how topics and lessons in a single course build upon one another, and the vertical curriculum, which refers to how mathematical content is developed across multiple classes over a student’s mathematical preparation. (SWD) Glatthorn, A. A. (1999). Curriculum alignment revisited. Journal of Curriculum and Supervision, 15(1), 26-34. Stein, M. K., Remillard, J., & Smith, M. S. (2007). How curriculum influences student learning. In F. K. Lester (Ed.), Second handbook of research of mathematics teaching and learning (pp. 319-370). Charlotte, NC: Information Age Publishing. Venezky, R. L. (1992). Textbooks in school and society. In P.W. Jackson (Ed.), Handbook of research on curriculum (pp. 436-460). New York: Macmillan Publishing Company.

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Curriculum Alignment is the extent to which pieces of the curricular landscape match up to provide a coherent picture of what is to be taught and learned. With the elevation of status and importance of curriculum standards and statemandated assessments as accountability measures during the 1980s, greater attention has been paid to ensuring that the standards and assessments portray the same focus regarding what students should know and be able to perform. In addition, to carry out the day-to-day instructional planning to meet the goals specified in standards and to prepare students for assessments, teachers generally use textbooks or supplementary materials to structure learning experiences for students. Researchers and policymakers use the term curriculum alignment to describe the degree to which these three forms of curricula – the intended curriculum (standards), the written curriculum (textbooks), and the assessed curriculum (state-mandated examinations) – are linked and provide a common picture regarding the important goals of teaching and learning (Anderson, 2002). The underlying hypothesis regarding the importance of curriculum alignment is that consistent themes outlined in standards, textbooks, and assessments will influence what teachers teach, which will then influence teachers’ instructional practice and ultimately guide what students learn (Porter, 2002). To measure curriculum alignment, researchers have developed tools that allow for the comparison of different forms of curricula. Porter (2002) describes the development of alignment measures and indices in order to document and quantify the extent to which various curricula are aligned. One such tool is the Survey of Enacted Curriculum (SEC), which uses uniform categories of mathematical content along with levels of cognitive demand in a two-by-two table in order to describe the curriculum. After all coding is complete, an alignment index is calculated to denote the strength of alignment between any two forms of curriculum. Webb (2007) developed an alignment tool specifically to document the alignment between standards and assessments. The tool focuses on four alignment criteria: categorical concurrence (both curricula focus on the same categories of content), depth of knowledge consistency (the level of cognitive demand represented in both curricula), range of knowledge correspondence (the span of knowledge expected and needed in both curricula), and balance of representation (the degree of emphasis given to individual standards on an assessment). For standards and assessments to be aligned, all four criteria must be met. These curriculum alignment tools along with others are used by researchers and other stakeholders to study the extent to which various curricula portray similar messages regarding mathematical content and skills. (SWD) Anderson, L. W. (2002). Curricular alignment: A re-examination. Theory into Practice, 41(4), 255260. Porter, A. C. (2002). Measuring the content of instruction: Uses in research and practice. Educational Researcher, 31(7), 3-14. Webb, N. L. (2007). Issues related to judging the alignment of curriculum standards and assessments. Applied Measurement in Education, 20(1), 7-25.

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Curriculum Coherence refers to the extent to which the topics and skills that students are to learn build upon one another as the student progresses both within a specific grade level as well as across multiple grade levels. In their analysis of the findings from the Third International Mathematics and Science Study (TIMSS), Schmidt et al. (2002) argue that many of the topperforming countries benefit from a mathematics curriculum that is similar, coherent, and taught to all mathematics students. They identified topics taught in at least 2/3 of the top-achieving TIMSS countries (called A+ countries) and found those topics progressed from the basic building blocks in the early grade levels (such as the meaning and use of basic arithmetic operations with whole numbers) to more sophisticated concepts. To this point, Schmidt et al. (2002) highlight the important nature of curriculum coherence, which in terms of standards or textbooks is defined as “articulated over time as a sequence of topics and performances that are logical and reflect, where appropriate, the sequential or hierarchical nature of the disciplinary content from which the subject matter derives” (p. 18). This coherent curriculum stands in stark contrast to what they found in the United States, where standards documents read as checklists of unrelated topics that often repeated themselves over grade levels, which contributed to a mathematics curriculum characterized as “a mile wide and an inch deep” (Schmidt et al., 1997, p. 122). The incoherence of U.S. state standards documents was commonly attributed to the decentralized organization of education in the U.S., where states were given latitude to create their own standards for mathematics and there was little coordination among states regarding the creation of these documents. Given the lack of consensus across states with regard to when certain mathematical topics were to be taught (Reys, 2006), NCTM worked to provide guidance by publishing Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence (NCTM, 2006). This document outlines specific mathematical content that was to be the focus of learning at each grade level and was to be used by states in revising standards. The effort to build a more coherent mathematics curriculum was further advanced with the 2010 release of the Common Core State Standards for Mathematics (CCSSM). Schmidt and Houang (2012) studied CCSSM using the same TIMSS techniques and found CCSSM to be coherent, focused, and consistent with the curriculum found in A+ TIMSS countries. (SWD) National Council of Teachers of Mathematics (NCTM). (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: Author. Reys, B. J. (2006). The intended mathematics curriculum as represented in state-level curriculum standards: Consensus or confusion? Charlotte, NC: Information Age Publishers. Schmidt, W., Houang, R., & Cogan, L. (2002). A coherent curriculum: The case of mathematics. American Educator, 26(2), 10-26. Schmidt, W. H., & Houang, R. T. (2012). Curricular coherence and the Common Core State Standards for Mathematics. Educational Researcher, 41(8), 294-308. Schmidt, W. H., McKnight, C. C., & Raizen, S. A. (1997). A splintered vision: An investigation of U.S. science and mathematics education. Boston, MA: Kluwer Academic Publishers.

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Curriculum Knowledge is the knowledge teachers possess about the content that comprises the mathematics curriculum, namely how topics and skills are connected at a given grade level, how they build across grade levels, and an understanding of the resources available that can support teaching and learning. Research on the knowledge teachers need in order to carry out the work of their profession has highlighted the multi-faceted knowledge base required to be successful (Shulman, 1986; Grossman, 1990). Beyond knowledge of mathematics (content knowledge) and of how to teach it (pedagogical knowledge), researchers have further refined the categories of knowledge teachers should possess). Shulman (1986) discusses curriculum knowledge as one of these specialized forms of teacher knowledge, defining this knowledge as: the full range of programs designed for the teaching of particular subjects and topics at a given level, the variety of instructional materials available in relation to those programs, and the set of characteristics that serve as both the indications and contraindications for the use of particular curriculum or program materials in particular circumstances. (p. 10) Shulman (1986) and Grossman (1990) further delineate curriculum knowledge into horizontal or lateral curriculum knowledge, referring to how the topics and skills at a given grade level should be sequenced and addressed, and vertical curriculum knowledge, which is an understanding of how the “topics and issues that have been and will be taught in the same subject area during the preceding and later years in school” connect together (Shulman, 1986, p. 10). Ball et al. (2008) also detail curriculum knowledge in their model of teacher MKT under the category of PCK, referring to this knowledge as Knowledge of Content and Curriculum. Teachers’ curriculum knowledge is a vital component of their practice, informing the decisions regarding how topics may be approached. For example, an elementary teacher working with students to add and subtract fractions with unlike denominators would use curriculum knowledge in knowing the topics students would have mastered prior to this topic (e.g., understanding of fractions as part of a whole, equivalence of fractions, adding and subtracting fractions with like denominators) as well as the topics that students will later learn (e.g., percents, ratios, multiplication and division of fractions). In addition, a teacher would also need to know what resources – textbooks, supplementary resources, manipulatives – that could be brought to the assistance of students in learning the topic. All of these aspects comprise the teacher’s curriculum knowledge. (SWD) Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407. Grossman, P. (1990). The making of a teacher: Teacher knowledge and teacher education. New York, NY: Teachers College Press. Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.

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Decentering is to the ability to retreat from one’s own thoughts and viewpoint and to consider phenomena from multiple perspectives, including that of other people. One of the pioneers in the study of human cognitive development and, subsequently, in the constructivist theory of learning, is the Swiss psychologist Jean Piaget (1896-1980), who devoted much of his life to the study of childhood education and development. One of his major contributions to the knowledge of how children mature is his articulation of the four stages of child development. These four stages are sensorimotor (birth to 2), preoperational (ages 2-7), concrete operational (ages 7-11), and formal operational (ages 11-16 and up). Each stage illustrates varying characteristics of children as they grow and move from stage to stage. One feature of the transition from the preoperational stage to the concrete operational stage is the child’s movement away from egocentric language behavior, where a child can focus only on their thoughts, feelings, and viewpoints while struggling to see and understand those aspects from another child’s or adult’s perspective (Piaget, 1959). In addition, children with egocentric behaviors are often unable to spatially visualize beyond what is seen or attend to multiple attributes of an object (Piaget & Inhelder, 1963). As they move into more logical thinking in the concrete operational phase, they are able to decenter not only their visual perceptions but also the way they interact with others (Teuscher et al., 2016). The action of decentering or decentration allows one to understand other’s perspectives as well as to spatially process multiple attributes of an object. As an example, Piaget and Inhelder (1963) describe how a 4 ½ year old boy struggled to understand how a mountain could look different when standing at different vantage points, as the boy felt that the mountain had undergone a change rather than noticing the impact of the change in his perspective. Piaget and Inhelder found that as children age and progress in development, they are able to decenter and understand how the mountain could look different based upon changing viewpoints (pp. 216-217). A child who can decenter also can view and attend to various features of the mountain, such as its height, shape, size, and color, when describing the object to others. The notion of decentering is prevalent not only in learning mathematics but also in the practice of teaching. Teuscher et al. (2016) use the construct of decentering to examine how teachers interpret and use student thinking to make decisions during instruction. As they decenter while teaching, teachers attempt to take on the thinking and perspective of the student in order to be able to pose questions that can assist all students in making connections and to eventually develop an understanding of the concept under investigation. (SWD) Piaget, J. (1959). The language and thought of the child (3rd ed.). London, England: Routledge & Kegan Paul LTD. Piaget, J., & Inhelder, B. (1963). The child’s conception of space. London, England: Routledge & Kegan Paul. Teuscher, D., Moore, K. C., & Carlson, M. P. (2016). Decentering: A construct to analyze and explain teacher actions as they relate to student thinking. Journal of Mathematics Teacher Education, 19(5), 433-456.

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Deductive Reasoning is a process in which each statement follows necessarily from previous statements that are known or assumed to be true, leading to a logically certain conclusion. The importance of deductive reasoning in mathematics has its roots in Euclid’s Elements. Euclid organized the study of geometry into an axiomatic system of undefined terms, definitions, and postulates (self-evident ideas accepted without proof) that are used to logically deduce and prove theorems. Once proven, a theorem can then be used to prove other theorems. Thus, deductive reasoning involves using what is already known to be true to formally prove other ideas. The process is considered rigorous with a conclusion that is accepted as truth, unlike a conclusion that is inductively reached based on empirical evidence, and is not necessarily true. Szydlik et al. (2016) describe deductive proof as an aspect of mathematical culture that gets at “the heart of geometric thinking” (p. 508). “Students should see the power of deductive proofs in establishing results. They should be able to produce logical arguments and present formal proofs that effectively explain their reasoning …” (NCTM, 2000, p. 345). Deduction is the second-highest level of the Van Hiele levels of geometric thinking. Students who reach this level understand the structure of the Euclidean axiomatic system to construct a formal deductive proof and appreciate the significance of proof. However, research has shown that many students who enter a high school geometry course are not at this level, and their success with deductive proof is correlated with their Van Hiele level (Senk, 1989). DeVilliers (1999) suggests that motivational issues may explain students’ lack of success with deductive proofs. He maintains that exploring the functions of proof beyond verification can lead to understanding what kinds of meaningful activities could motivate students to engage in and value deductive reasoning. In addition to verification, other functions of proof include explanation (providing insight into why it is true), systematization (the organized, deductive system), discovery, communication of mathematical knowledge, and intellectual challenge (the fulfillment derived from constructing a proof). Suggesting the use of dynamic geometry software as a meaningful activity to inductively discover geometric relationships and broaden the approaches to proof, de Villiers found that students were more motivated to find an explanation for their conjectures and engage in deductive reasoning rather than to simply verify them. (KKM) De Villiers, M. (1999). Rethinking proof with the Geometer’s Sketchpad. Emeryville, CA: Key Curriculum Press. National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: Author. Senk, S. L., (1989). Van Hiele levels and achievement in writing geometry proofs. Journal for Research in Mathematics Education, 20(3), 309-321. Szydlik, J. E., Parrott, A., & Belnap, J. K. (2016). Conversations to transform geometry class. Mathematics Teacher, 109(7), 507-513.

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Design Research in Education is defined in terms of both developmental studies and validation studies with developmental studies used to describe complex problems and validation studies used to validate theories (Plomp, 2013). Design research studies have gained popularity over the last several decades as a lens through which the complexities of mathematics teaching and learning can be captured and characterized. A variety of purposes for design research have been described including collaborative aspects of conducting research in real time with teachers and students. Cobb et al. (2003) summarize design research as “… extended (iterative), interventionist (innovative and design-based), and theoryoriented enterprises whose ‘theories’ do real work in practical educational context” (p. 13). Design research studies are typically qualitative in nature and are used to explore student learning trajectories in specific content areas. It has also been used to describe professional development programs for teachers (Swan, 2007) and as an approach to research on technology use in educational settings (Reeves et al., 2005). Educational researchers have refined the idea of design research and offered systematic approaches to conducting this type of research. For example, Lamburg and Middleton (2009) describe a Compleat model as “a cycle of inquiry practices involved in conceptualizing and conducting design research from inception of an idea, to creation of products and artifacts, to testing and upscaling innovations for a broader audience” (p. 233). The Compleat model involves seven phases including: grounded models, development of an artifact, feasibility study, prototyping and trialing, field study, definitive test, and dissemination and impact (p. 234). While not all design research studies involve implementation of all of these phases, these seven phases provide structured guidance for this type of educational research in mathematics and other content areas. (LBK) Cobb, P., Confrey, J., DiSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9-13. Lamberg, T. D., & Middleton, J. A. (2009). Design research perspectives on transitioning from individual microgenetic interviews to a whole-class teaching experiment. Educational Researcher, 38(4), 233-245. Plomp, T. (2013). Educational design research: An introduction. In, T. Plomp & N. Nieveen (eds.), Educational design research, SLO: Netherlands Institute for Curriculum Development, 1050. Reeves, T. C., Herrington, J., & Oliver, R. (2005). Design research: A socially responsible approach to instructional technology research in higher education. Journal of Computing in Higher Education, 16(2), 96. Swan, M. (2007). The impact of task-based professional development on teachers’ practices and beliefs: A design research study. Journal of Mathematics Teacher Education, 10(4-6), 217237.

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Didactic is a general education term associated with teaching and informing that follows a consistent approach or educational style to engage the student's mind. The didactic method of instruction is often contrasted with dialectics and the Socratic method; the term can also be used to refer to a specific didactic method, such as constructivist didactics. Didactic has specific applications to mathematics teaching and learning and is frequently used to frame models of mathematics instruction. Historically it has been used to describe instructional approaches involving the teacher telling students what and how to do mathematics using prescribed steps (Stoddart et al., 1993). However, other uses of the term didactic in mathematics education portray alternative teaching strategies. For example, didactical phenomenology is an instructional approach advocated by Hans Freudenthal in which mathematics is taught by engaging students in phenomena that allow them to make sense of the ideas prior to formalization (Van den HeuvelPanhuizen & Drijvers, 2014). These initial ideas were further characterized as a dual process of horizontal and vertical mathematization from realistic phenomena to formal mathematical concepts and notations (Streefland, 1991). Tichá and Hošpesová (2013) use the phrase didactic competence to describe mathematical practices associated with problem posing. Their research examined mathematical ideas associated with preservice teachers’ constructed problems. Their study showed that elementary preservice teachers’ constructed fraction problems highlighted limitations in their own conceptions of fraction topics and posited that this could limit their teaching of fraction concepts. Radford (2008) describes The Theory of Didactic Situations as the relationship between learning in a social setting and ways in which individuals logically process mathematical ideas. This theory is associated with studies of mathematics teaching and learning. It emphasizes contextual factors such as the types of problems and how they are posed and explored within the classroom setting. Schoenfeld (2012) further delineated and specified these ideas in relation to the mathematics, the teacher, and the learner, in what he characterized as the didactic triangle. He used this term as a framework to describe a variety of productive studies of mathematics teaching and learning (p. 587). (LBK) Radford, L. (2008). Connecting theories in mathematics education: Challenges and possibilities. ZDM, 40(2), 317-327. Schoenfeld, A. H. (2012). Problematizing the didactic triangle. ZDM, 44(5), 587-599. Stoddart, T., Connell, M., Stofflett, R., & Peck, D. (1993). Reconstructing elementary teacher candidates' understanding of mathematics and science content. Teaching and Teacher Education, 9(3), 229-241. Streeefland, L., (1991). Fractions in realistic mathematics education: A paradigm of developmental research. Dordrecht, The Netherlands: Kluwer Academic Publishers. Tichá, M., & Hošpesová, A. (2013). Developing teachers’ subject didactic competence through problem posing. Educational Studies in Mathematics, 83(1), 133-143. Van den Heuvel-Panhuizen, M., & Drijvers, P. (2014). Realistic mathematics education. Encyclopedia of Mathematics Education, 521-525.

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Differentiated Instruction is an instructional design model used by teachers to adjust their teaching to the academic needs of a varied group of learners in order to maximize the knowledge capacity of every student. The differentiated instruction model arose from the needs of an increasingly diverse learner profile (Subban, 2006). It is based on the premise that students learn better when teachers accommodate for differences in readiness (Pham, 2011), interest, and learning profile (Tomlinson & Imbeau, 2010). All students have the capability to learn given an appropriate combination of teacher controlled variables (Morgan, 2014). Each student is valued for his or her unique strengths, while being offered opportunity to learn (Tomlinson, 2014; Patterson et al., 2009). Differentiated instruction places an emphasis on four classroom elements: instruction, curriculum, assessment, and learning environment. These elements are viewed as interdependent processes and procedures. The teacher is the specialist in the classroom, an individual who has been proficiently trained to guide and lead his or her students, using appropriate techniques, assisting each learner to reach his or her potential within the learning environment (Tomlinson, 2004). Teachers are legally and ethically bound to be the expert leading the student to full contextual development (Lawrence-Brown, 2004). At its core, differentiated instruction is how a teacher studies and responds to learner variance in those elements (Subban, 2006). It is assumed there is a high level of reciprocity between the teacher and student. The teacher can adjust any of the aspects of the elements to effectively serve the learner. The teacher makes judgements on the potential and the approach of learning for students (Tomlinson, 2005). Based on teacher conclusions, personalized instruction for students is then implemented. (SRF) Lawrence-Brown, D. (2004). Differentiated instruction: Inclusive strategies for standards-based learning that benefit the whole class. American Secondary Education , 32(3), 34–62. Morgan, H. (2014). Maximizing student success with differentiated learning. The Clearing House, 87(1), 34-38. Patterson, J. L., Connolly, M. C., & Ritter, S. A. (2009). Restructuring the inclusion classroom to facilitate differentiated instruction. Middle School Journal, 41(1), 46-52. Retrieved from https://eric.ed.gov/?q=inclusion&pr=on&ff1=audteachers&id=ej854575 Pham, H. L. (2012). Differentiated instruction and the need to integrate teaching and practice. Journal of College Teaching and Learning, 9(1), 13-20. Retrieved from https://eric.ed.gov/?q=differentiated+instruction&id=ej979186 Subban, P. (2006). Differentiated instruction: A research basis. International Education Journal, 7(7), 935-947. Retrieved from https://eric.ed.gov/?id=ej854351 Tomlinson, C. A. (2004). The Möbius effect addressing learner variance in schools. Journal of Learning Disabilities , 37(6), 516-524. Tomlinson, C. A. (2005). This Issue: Differentiated instruction. Theory into Practice , 44(3), 183184. Tomlinson, C. A. (2014). The differentiated classroom: Responding to the needs of all learners (2nd ed.). Alexandria, VA: Association for Supervision and Curriculum Development. Tomlinson, C. A., & Imbeau, M. B. (2010). Leading and managing a differentiated classroom. Alexandria, Virginia: Association for Supervision and Curriculum Development.

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Direct Modeling is a process students use to solve problems by following the structure of the problem and sequence of numbers in the problem (Carpenter et al., 1999, 2015), typically by drawing a picture or model and then simulating the mathematical operation needed to solve the problem. The term became prevalent as research demonstrated the capacity of young children to use direct modeling to solve a variety of word problems (Carpenter et al., 1999, 2015; NGA, 2010). It is the basis for the problem type taxonomies that distinguish word problems based on the unknown quantity. Direct modeling however, describes a level of problem solving in which all of the quantities in the problem are represented. For example, a word problem such as “There are four baskets with five apples in each basket, how many apples altogether” could be directly modeled by representing the four baskets in some way such as circles, representing each of the five apples in each basket, and then counting up all of the apples by ones to get the total number of apples in the four baskets. The term, direct modeling, is distinguished from another term in education, “modeling”. The term, modeling, is often used to describe a teacher showing or modeling an idea for a lesson and then the students mimicking the same process on similar problems. It also contrasts with the term “mathematical model” which refers to a representation of a mathematical idea or concept. Mathematical model has been used to describe a representation of a real-world phenomenon (see mathematical modeling description). (LBK) Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., Empson, S. B. (2015). Children’s mathematics: Cognitively guided instruction (2nd ed.). Portsmouth, NH: Heinemann. National Governors Association Center for Best Practices (NGA), Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Author. Retrieved from http://www.corestandards.org/

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Discourse is an expression of thought, verbal or written, and is an area of analysis in a variety of educational contexts. Gee (1996) defined “discourses” as a socially accepted connection among an individual’s ways of participating in a social setting. Moschkovich (2007) referred to “discourse practices” to describe interactions involving language, symbols, and other artifacts central to an individual’s participation in a mathematics community. An aspect common to the role of discourse in mathematics teaching and learning is the idea of “taken-as-shared” understandings of terms, symbols, and other tools that provide opportunities to advance understanding. Watson and Mason (2007) used this idea to describe the purpose and implementation of mathematical tasks. Others refer to this idea in terms of students’ learning and development (McClain & Cobb, 1998). For example, there is a shared understanding of the equal sign symbol that students and teachers alike discuss within the context of understanding that “=” represents quantitative equivalence. Discourse practices have also been studied with respect to bilingual students in mathematics classes (Moschkovich, 2002; Setati, 2005). Setati (2005) described a variety of discourses including conceptual, procedural, and contextual in terms of students’ mathematics learning outside of their target language in South Africa. Moschkovich (2002) explored bilingual students’ mathematics learning within the framework of socio-cultural and situated perspectives. Discourse has also been described as mathematical discussions. Several books with practical implications for mathematics classrooms have enhanced teachers’ efforts to cultivate mathematical communication (Kazemi & Hintz 2014; Smith & Stein, 2011). (LBK) Gee, J. (1996). Social linguistics and literacies: ideology in discourses. London, UK: Taylor and Francis. Kazemi, E., & Hintz, A. (2014). Intentional talk: How to structure and lead productive mathematical discussions. Portland, ME: Stenhouse Publishers. McClain, K., & Cobb, P. (1998). The role of imagery and discourse in supporting students’ mathematical development. In M. Lampert and M. Blunk (Eds.), Talking mathematics in school: Studies of teaching and learning (pp. 56-81). Cambridge, England: Cambridge University Press. Moschkovich, J. (2002). A situated and sociocultural perspective on bilingual mathematics learners. Mathematical Thinking and Learning, 4(2-3), 189-212. Moschkovich, J. (2007). Examining mathematical discourse practices. For the Learning of Mathematics, 27(1), 24-30. Setati, M. (2005). Teaching mathematics in a primary multilingual classroom. Journal for Research in Mathematics Education, 36(5), 447-466. Smith, M. S., & Stein, M. K. (2011). 5 practices for orchestrating productive mathematical discussions. Reston, VA: National Council of Teacher of Mathematics. Watson, A., & Mason, J. (2007). Taken-as-shared: A review of common assumptions about mathematical tasks in teacher education. Journal of Mathematics Teacher Education, 10(46), 205-215.

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Discovery Learning is an Inquiry Based Approach to learning mathematics in which students investigate mathematical relationships and make generalizations based on their exploration. In this student-centered approach, the teacher serves as a facilitator, in contrast to direct instruction in which a teacher presents information without prior student exploration or autonomy. In a guided discovery experience, teachers have specific goals in mind for student investigations and provide supporting activities to lead to desired conclusions. In a more open inquiry, students are stimulated to pose their own questions and problems with a result that may not be apparent at the onset, then consequently investigate and seek resolution. Based on constructivist theory, discovered concepts hold deeper meaning to the individual as students make connections in their own terms, are better-retained, and more easilyretrieved when problem solving (Bruner, 1961). Students engaged in the inquiry process are more apt to take risks, attempt more complex problems, and gain confidence in their mathematical ability (Borasi, 1992). A meta-analysis reports that guided discovery leads to greater learning over other instructional methods, including that of unassisted discovery which can cause frustration and cognitive overload (Alfieri et al., 2011). Bruner suggested that rather than define discovery as a product discovered, the focus should be on the process and how a discovery-minded attitude develops as students are actively involved in mathematical reasoning. Furthermore, “discovery, with the understanding and mastery it implies, becomes its own reward, a reward that is intrinsic to the activity of working” (Bruner, 1960, p. 611). The guided discovery process for learning mathematics was exemplified in an innovative series of textbooks, beginning with Discovering Geometry: an Investigative Approach (Serra, 1990). Maintaining that many high school geometry students are not cognitively ready for the rigor of deductive proofs, Serra designed a geometry curriculum with an emphasis on inductive reasoning. In a series of investigations, students make sense of definitions, explore concepts, look for patterns, discover relationships, make conjectures, and justify their generalizations. As their reasoning grows in sophistication, they are more prepared and motivated to prove their generalizations deductively. In mathematics education, a distinction between discovery learning and inquiry has not been clearly defined and some have used the terms interchangeably. The term ‘inquiry’ has been used in a broader sense referring to student-centered approaches in which students are actively engaged in problem solving and investigation. Discovery learning tends to imply that something is discovered, such as when an exploration of triangles results in a generalization that the sum of the angles is 180 degrees. Borasi, however, makes a distinction based on case studies of mathematical inquiries in which the goal was not for students to discover “preestablished truths” under the guidance of an instructor, rather to “engage the students in genuine attempts to make sense of mathematics phenomena” through the processes of inquiry (1992, p. 180). An inquiry approach has been foundational to the learning and doing of science as a pursuit for studying the natural world.

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Degrees of inquiry have been well-defined in science education from open inquiry, in which a student poses the question and investigates it, to a teacher-guided or more structured inquiry (Martin-Hansen, 2002). Instructional models commonly used in science education such as the learning cycle and the 5E (Engage, Explore, Explain, Elaborate, Evaluate) (Bybee et al., 2006) can be applied to mathematics learning. However, mathematics is different from science: “As a science of abstract objects, mathematics relies on logic rather than observation as its standard of truth, yet employs observation, simulation, and even experimentation as a means of discovering truth” (NRC, 1989, p. 31). Thus, it would be reasonable to expect that inquiry would look different in math learning than in science learning. Borasi (1992) maintains that how we think of a discipline must dictate how we teach it. With mathematics viewed as more of a process than a set of rules to be followed, an inquiry approach with opportunities for discovery, critical thinking, and problem solving is a good fit. Borasi (1992) suggests these strategies for initiating and supporting students’ mathematical inquiry: x x x x x x x x x

Exploiting the complexity of real-life problematic situations Focusing on nontraditional mathematical topics where uncertainty and limitations are most evident Uncovering humanistic elements within the traditional mathematics curriculum Using errors as ‘springboards for inquiry’ Exploiting the surprises elicited by working in a new domain Creating ambiguity and conflict by proposing alternatives to the status quo Generative reading activities as a means of sustaining inquiry Providing occasions for reflecting on the significance of one’s inquiry Promoting exchanges among students (pp. 190-201) (KKM)

Alfieri, L., Brooks, P. J., Aldrich, N. J., & Tenenbaum, H. R. (2011). Does discovery learning enhance learning? Journal of Educational Psychology (103)1, 1-18. Borasi, R. (1992). Learning mathematics through inquiry. Portsmouth, NH: Heinemann. Bruner, J. (1960). On learning mathematics. Mathematics Teacher, 53, 610-619. Bruner, J. S. (1961). The act of discovery. Harvard Educational Review (31)1, 21–32. Bybee, R. W., Taylor, J. A., Gardner, A., Van Scotter, P., Powell, J. C., Westbrook, A., & Landes, N. (2006). The BSCS 5E Instructional Model: Origins and effectiveness. Colorado Springs, CO: Biological Science Curriculum Study. Martin-Hansen, L. (2002). Defining inquiry. Science Teacher (69)2, 34-37. Serra, M. (1990). Discovery geometry: An investigative approach. Emeryville, CA: Key Curriculum Press. National Research Council (NRC). (1989). Everybody counts: A report to the nation on the future of mathematics education. Washington, DC: National Academy Press.

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Dynamic Geometry Software (DGS) provides an electronic version of the classic compass and straightedge to perform accurate construction of geometric figures that can be easily and continuously manipulated allowing for exploration and discovery of geometric relationships. By the late 1980s, as computers were entering schools for student use, two dynamic geometry software products were at the forefront of math visualization software: Cabri Geometry and The Geometer’s Sketchpad. The term dynamic geometry® was coined and trademarked by Steve Rasmussen and Sketchpad’s inventor, Nicholas Jackiw, to describe the non-static and interactive nature of the software. The developer of Cabri Geometry, Jean-Marie Laborde, coined a similar term, drag mode geometry, to highlight how dragging key parts of a figure, thereby making it dynamic, reveals an infinite number of related cases, allowing for exploration, prediction, and generalization. An open source DGS, GeoGebra, created by Markus Hohenwarter, later became available as an online option, along with others. The terms dynamic geometry and dragging have evolved into common use referring to any products with these capabilities. These DGS programs have algebra, trigonometry, and calculus applications in addition to geometry, and the term dynamic geometry software is sometimes extended analogically to dynamic mathematics software. The use of DGS promotes geometric reasoning (Battista, 2002) as the following example describes. Using construction tools, a rectangle can be formed by constructing a quadrilateral with four right angles. Students learn the distinction between deliberately constructing perpendicular segments and drawing segments that appear to meet at right angles – the right angles of the construction will not collapse. They can manipulate the rectangle by dragging it by a vertex or side to change its size, orientation, and shape, thus creating many examples of a rectangle leading to a generalization that opposite sides remain parallel and congruent. Adding diagonals to the rectangle, students can examine multiple cases to discover which properties of the diagonals appear to be preserved despite the changes. They may conjecture that the diagonals of a rectangle are congruent. If the dragging creates a rectangle that is also a square, a student may notice that the diagonals are also perpendicular. Yet when the figure no longer has congruent sides, the diagonals are no longer perpendicular, thus the generalization of perpendicular diagonals cannot be made for a rectangle. As seen in this example, DGS lends itself to inductive reasoning through examining many examples and counterexamples. Students tend to recognize that verification of many examples is not sufficient for a proof, and once they are convinced that their conjecture appears to be true based on their exploration, they have the motivation to justify with the formal explanation of a deductive proof (de Villiers, 2003). (KKM) Battista, M. T. (2002). Shape makers: A computer environment that engenders students’ construction of geometric ideas and reasoning. Computers in Schools, 17(1/2), 105–120. deVilliers, M. D. (2003). Rethinking proof with Geometer’s Sketchpad. Emeryville, CA: Key Curriculum Press.

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Educational Technology (also referred to as instructional technology) is a term that encompasses the broad spectrum of tools (predominantly electronic) that promote and assist the teaching and learning of mathematics. Across the history of mathematics education, a wide array of tools has been used to aid students in learning mathematics. From the earliest use of the pencil and paper or the chalkboard to the development of the abacus and the slide rule, teachers have incorporated these devices into their instruction in support of student learning. With the increased availability of computer technology in the latter part of the 20th century, the resources available to teachers and students greatly expanded, as the field of mathematics education hoped that the promise of this new technology could transform mathematics teaching and learning. In fact, NCTM’s Agenda for Action (1980) encouraged schools to “take full advantage of the power of calculators and computers at all grade levels” (p. 8), imploring greater access to these devices as well as urging the consideration of how the mathematics curriculum could be changed to more greatly incorporate these new forms of educational technology. The integration of these technologies into the mathematics classroom exploded in the 1990s with the advent of the worldwide web and continued into the 21st century, as schools invested heavily into desktop and laptop computers, tablets, and faster internet access to support student learning (Cheung & Slavin, 2013). Educational software, digital games, and computer assisted instruction now permeate the mathematics education landscape, altering not only what mathematics is learned but also how the mathematics is learned. Beyond the tools available to the student and teacher, the term “educational technology” also implicitly points to how these tools are integrated into instruction. Aziz (2010) defines educational technology as “the considered implementation of appropriate tools, techniques, or processes that facilitate the application of senses, memory, and cognition to enhance teaching practices and improve learning outcomes” (p. 1). To this point, considerable research has been undertaken to understand the effect of educational technology on mathematics teaching and learning, including its successful implementation into classroom practice, its support and promotion of greater student understanding, and its impact on student learning and achievement (Zbiek et al., 2007; Heid & Blume, 2008). Although educational technology has greatly transformed learning of all school subjects, specific forms of technology have directly impacted mathematics teaching and learning. The following categories highlight a variety of educational technology that are used to support mathematical learning. Though not exhaustive nor mutually exclusive, these categories underscore the diversity of tools that fall under the umbrella of educational technology. Dynamic Geometry Software: These technologies allow the user to manipulate geometric figures and provide for student exploration and discovery of geometric

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concepts. Examples include Geometer’s Sketchpad, Cabri Geometry, GeoGebra, Geometric Supposer, and Logo-based software. Statistical Software: These software packages provide students the setting to plot and analyze data, calculate statistical measures, interpret results, and report findings stemming from their data analysis. Examples include Tinkerplots, Fathom, SAS (Statistical Analysis Software), and SPSS. Calculators and Graphical Software: These devices and programs allow students to perform a variety of activities, ranging from basic arithmetic calculations to the visual display and analysis of graphical representations. Examples include Scientific Calculators, Graphing Calculators (such as the Casio, HP, and Texas Instruments series), Desmos, and GeoGebra. Computer Algebra Systems (CAS): These technologies permit students to complete symbolic operations of algebraic expressions and equations, thereby making solving algebraic equations quicker and more efficient. Examples include Maple, Mathematica, MATLAB, Wolfram Alpha, and the TI-Nspire series calculators. Computer Assisted Instruction and Software: These software programs center on the use of computers to serve as the instructional setting in which students engage with mathematical concepts. These settings may take the form of interactive games, virtual manipulatives, tutorials, or continued practice with mathematical content through the use of computers or tablets. Examples include Mathlets, Spreadsheets, and Tablet Applications. (SWD) Aziz, H. (2010). The 5 keys to educational technology. THE Journal. Retrieved from https://thejournal.com/articles/2010/09/16/the-5-keys-to-educational-technology.aspx Chueng, A. C., & Slavin, R. E. (2013). The effectiveness of educational technology applications for enhancing mathematics achievement in K-12 classroom: A meta-analysis. Educational Research Review, 9, 88-113. Heid, M. K., & Blume, G. W. (Eds.). (2008). Research on technology and the teaching and learning of mathematics: Volume 1. Research syntheses. Charlotte, NC: Information Age Publishing. National Council of Teachers of Mathematics (NCTM). (1980). An agenda for action: Recommendations for school mathematics of the 1980s. Reston, VA: Author. Zbiek, R. M., Heid, M. K., Blume, G. W., & Dick, T. P. (2007). Research on technology in mathematics education: A perspective of constructs. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning, Vol 2 (pp. 1169-1207). Charlotte, NC: Information Age Publishing.

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Epistemology is the study of the origins and nature of knowledge. .

The term epistemology is used across the curriculum in a variety of subject matter areas to describe learning, knowledge, and understanding. With respect to mathematics education, it has been used to describe changes in views about how students learn and make connections to new content (Kilpatrick, 1987; Ernest, 2003; Lerman, 1989, 1996). Lerman (1989) used the term genetic epistemology to characterize teaching and learning mathematics within constructivism and radical constructivism paradigms (p. 213). Cobb (1994) and others further describe extensions of epistemological ideas to social constructivist theories of learning. Epistemological ideas are mostly discussed in terms of theoretical views of teaching and learning. However, these ideas have also been used as the conceptual basis for empirical studies (Gill et al., 2004; Ravindran et al., 2005). For example, Ravindran et al. (2005) described “epistemological beliefs” in describing preservice teachers’ belief systems that influence their instructional strategies. Their survey results indicate the influence of belief systems in determining preservice teachers’ distinguishing between deep and shallow learning experiences of students. (LBK) Cobb, P. (1994). Where is the mind? Constructivist and sociocultural perspectives on mathematical development. Educational Researcher, 23(7), 13-20. Ernest, P. (2003). Constructing mathematical knowledge: Epistemology and mathematics education. London, England: Routledge. Gill, M. G., Ashton, P. T., & Algina, J. (2004). Changing preservice teachers’ epistemological beliefs about teaching and learning in mathematics: An intervention study. Contemporary Educational Psychology, 29(2), 164-185. Kilpatrick, J. (1987). What constructivism might be in mathematics education. In Proceedings of the eleventh conference of the international group for the psychology of mathematics education (PME 11). Volume 1 (pp. 3-27). Montreal, Canada: University of Montreal. Lerman, S. (1989). Constructivism, mathematics and mathematics education. Educational Studies in Mathematics, 20(2), 211-223. Lerman, S. (1996). Intersubjectivity in mathematics learning: A challenge to the radical constructivist paradigm? Journal for Research in Mathematics Education, 27(2), 133-150. Ravindran, B., Greene, B. A., & Debacker, T. K. (2005). Predicting preservice teachers' cognitive engagement with goals and epistemological beliefs. The Journal of Educational Research, 98(4), 222-233.

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Equity in mathematics education refers to providing all students access to the opportunities, resources, and support necessary to learn mathematics. “Access and Equity” is a guiding principle of mathematics education highlighted in NCTM’s Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014). The authors of this publication state that “an excellent mathematics program requires that all students have access to a high-quality mathematics curriculum, effective teaching and learning, high expectations, and the support and resources needed to maximize their learning potential” (p. 5). Given the diverse needs of the student population, attention to equity is central to meeting this vision for mathematics education. In fact, the authors underscore the role of the educators in ensuring equity for all students, pointing out that “the question is not whether all students can succeed in mathematics but whether the adults organizing mathematics learning opportunities can alter traditional beliefs and practices to promote success for all” (p. 61). The authors suggest the following as “productive beliefs”: x

x x

x x x

x

Math ability is a function of opportunity, experience, and effort and all students are capable of participating and achieving in mathematics, and deserve the support to achieve. Equity is not synonymous with equality. Students should receive the differentiated support they need to be successful in mathematics. Although children from poverty, racial minorities, females, and English language learners have commonly experienced equity issues, any student or group could be in an inequitable situation if the teacher is not providing a rigorous and meaningful mathematics curriculum. English language learners can learn math at grade level or beyond. Effective mathematics instruction is not separate from culture – it can leverage it. All students, not just those of high-ability, should engage in challenging tasks and open-ended problem solving to experience higher order thinking and raise mathematics achievement. Heterogeneous classes allow greater opportunities for low-achievers to engage in meaningful mathematics compared to isolation in a low-tracked class with lower expectations. (p. 63)

Gutierrez (2007) describes the importance of looking more broadly at equity to understand contexts in which students experience mathematics that can influence learning, recognizing that no group of students is homogeneous and many factors influence how a student interacts in a classroom setting. This necessitates considering four dimensions of equity: access, achievement, identity, and power. Access to opportunities for learning that leads to achievement has been a primary focus of researchers, highlighting variables such as low income correlating with low achievement and access. However, a focus on variables that teachers and students have little control over may not be sufficient to leverage change;

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examining issues of identity and power within context of individual, class, and community experiences may hold more potential. Teachers can provide an environment that nurtures development of students’ identities as learners of mathematics who relate to the world in multiple ways. The final dimension of equity is power to enact individual and social transformation, from student voice in the classroom to bringing new meaning to mathematics as a humanizing force that is stronger with a diverse body of participants. A math teacher community that shares a vision of empowering students is a promising model for advancing equity in all dimensions. (KKM) Gutierrez, R. (2007). Context matters: Equity, success, and the future of mathematics education. In T. Lamberg & L. R. Wiest (Eds.), Proceedings of the 29th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1-18). Reno, NV: University of Nevada. National Council of Teachers of Mathematics (NCTM). (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.

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Error Patterns are systematic mistakes that have some misunderstanding or misconception behind the erroneous procedure or conclusion. Error patterns are distinguished from careless mistakes in that unless the misconception is cleared up, a student would continue to make the same mistake using the same incorrect procedure or reasoning under the same conditions, thus a pattern of errors. The authors of the National Research Council’s Adding It Up (NRC, 2001) allude to this idea when they state that “when students fail to grasp the concepts that underlie procedures or cannot connect the concepts to the procedures, they frequently generate flawed procedures that result in systematic patterns of errors” (p. 196). Below are examples of common mathematical errors that illustrate the systemic mistakes that characterize error patterns:

In the example above, prior knowledge of whole number addition is incorrectly applied to a new situation (fraction addition), indicating that the student has not conceptualized the meaning of the denominator as representing the number of parts of a whole that defines the type or relative size of fraction in relation to the whole. The example below shows how a student could overgeneralize knowledge of place value to apply it to a non-base 10 situation:

Diagnosing the underlying cause for error patterns is an important step in furthering student understanding of mathematics (Ashlock, 2006). Allowing students to explain their reasoning can allow both the teacher and student to detect the faulty reasoning. Graphic organizers, concept maps, and flowcharts are tools students can use to clarify their own thinking. By understanding the common misconceptions behind error patterns, teachers can attend to the development of students’ procedural knowledge by embedding it in solid conceptual knowledge. Borasi (1994) found that the doubt or cognitive dissonance created as a student discovers an error can serve as a springboard for mathematical exploration in pursuit of clarification. Lim (2014) proposed using “error-eliciting problems” so that students have the “opportunity to tackle certain errors head-on, discuss the mathematics underlying those errors, learn from mistakes, and deepen mathematical understanding” (p. 111). (KKM) Ashlock, R. B. (1998). Error patterns in computation. Upper Saddle River, NJ: Prentice-Hall. Borasi, R. (1994). Capitalizing on errors as ‘springboards for inquiry’: A teaching experiment. Journal for Research in Mathematics Education, 25(2), 166-208. Lim, K. H. (2014). Error-eliciting problems: Fostering understanding and thinking. Mathematics Teaching in the Middle School, 20(2), 106-114. National Research Council (NRC). (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Washington, DC: National Academy Press.

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Ethnomathematics describes the relationship between culture and mathematics (d’Ambrosio, 2001). There are levels of ethnomathematics studies and discussions. At one level, this term has been used in descriptions of specific cultures and mathematics teaching and learning (Gerdes, 1988, 1999; Orey, 2000). Gerdes (1988, 1999) explored the intersection between culture and learning mathematics in a variety of African countries. For example, he explored cultural differences in the development of Euclidean geometry in Mozambique and traditional approaches in other countries (1988). Orey (2000) described differences in emphasis in topics of geometry between Native American groups and eurocentric America. Beyond teaching and learning specific mathematics topics, ethnomathematics also encompasses political and critical perspectives (d’Ambrosio, 2007; Francois, 2016; Knijnik, 1993, 1999). In general, advocates for ethnomathematics approaches counter historical perspectives that mathematics is a subject devoid of context and embrace mathematics and learners as inextricably linked by culture and diversity (d’Ambrosio, 2007). Knijnik (1999) applied the social and political aspects of ethnomathematics by investigating teaching and learning mathematics within the context of farming and land ownership in Brazil. In their critique of various perspectives of ethnomathematics, Vithal and Skovsmose (1997) describe four strands: traditional history of mathematics, cultural aspects of specific mathematics topics such as the Arab influence of the subject of algebra, mathematics learning in everyday settings, and the overlay of authentic mathematics content and mathematics education. They assert that the fourth strand may currently be viewed in a limiting factor and should be built on a more critical view of current mathematics education practices. (LBK) d'Ambrosio, U. (2001). What is ethnomathematics, and how can it help children in schools? Teaching children mathematics, 7(6), 308. d’Ambrosio, U. (2007). Peace, social justice and ethnomathematics [Monograph]. The Montana Mathematics Enthusiast, 1, 125-134. François, K. (2016). Ethnomathematics as a human right. Critical mathematics education: Theory, praxis and reality, 187-198. Gerdes, P. (1988). On culture, geometrical thinking and mathematics education. Educational studies in mathematics, 19(2), 137-162. Gerdes, P. (1999). Geometry from Africa: Mathematical and educational explorations. Washington, DC: Mathematical Association of America. Knijnik, G. (1993). An ethnomathematical approach in mathematical education: A matter of political power. For the learning of mathematics, 13(2), 23-25. Knijnik, G. (1999). Ethnomathematics and the Brazilian landless people education. ZDM, 31(3), 9699. Orey, D. C. (2000). The ethnomathematics of the Sioux tipi and cone. In H. Selin (Ed.), Mathematics across cultures (pp. 239-252). Dordrecht, The Netherlands: Springer. Vithal, R., & Skovsmose, O. (1997). The end of innocence: A critique of ‘ethnomathematics’. Educational Studies in Mathematics, 34(2), 131-157.

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Fidelity of Implementation refers to the degree to which an educational intervention (e.g., professional development, curriculum, instructional practice) is used in comparison to how the intervention was designed to be implemented. As state and federal policymakers have considered ways to improve mathematics education in the United States, one important variable that researchers have examined concerns the extent to which the enactment of a given policy has aligned or matched with the implementation of this policy under ideal conditions (O’Donnell, 2008). This variable, commonly referred to as “fidelity of implementation”, has come to represent a key factor when determining the overall effectiveness in the success of the intervention. O’Donnell (2008) states that the idea of implementation fidelity is rather new to K-12 curriculum intervention research, dating back to the 1970s, as researchers began to consider the impact of local factors and influences on how teachers interpreted and then incorporated ideas or materials into their instructional practice. Also called program integrity, the notion of fidelity of implementation has become important in several areas of mathematics education, most notably in curriculum interventions. The National Research Council (2004), in its recommendations regarding the examination of curricular effectiveness, identified fidelity of implementation as one of seven critical decision points that researchers should consider that would influence the rigor and efficacy of curricular effectiveness studies. The notion of implementation fidelity has been documented in different ways, including studies of curricular coverage, of alignment between textbooks and classroom lessons, and of teachers’ instructional practices (Brown et al., 2009). Jacobs et al. (2006) examined TIMSS data to determine the extent to which U.S. teachers incorporated elements of classroom instruction outlined in the NCTM’s Principles and Standards for School Mathematics (PSSM). The researchers found that U.S. teachers for the most part did not implement these practices with high degrees of fidelity in comparison to how they were described in PSSM but more closely reflect traditional techniques long pervasive in K-12 mathematics education. (SWD) Brown, S. A., Pitvorec, K., Ditto, C., & Kelso, C. R. (2009). Reconceiving fidelity of implementation: An investigation of elementary whole-number lessons. Journal for Research in Mathematics Education, 40(4), 363-395. Jacobs, J. K., Heibert, J., Givvin, K. B., Hollingsworth, H., Garnier, H., & Wearne, D. (2006). Does eighth-grade mathematics teaching in the United States align with the NCTM standards? Results from the TIMSS 1995 and 1999 video studies. Journal for Research in Mathematics Education, 37(1), 5-32. National Research Council. (2004). On evaluating curricular effectiveness: Judging the quality of K12 mathematics evaluations (J. Confrey & V. Stohl, Eds.). Washington, DC: National Academies Press. O’Donnell, C. L. (2008). Defining, conceptualizing, and measuring fidelity of implementation and its relationship to outcomes in K-12 curriculum intervention research. Review of Educational Research, 78(1), 33-84.

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Flipped Classroom is an instructional model in which the teacher’s delivery of information that traditionally occurred during class occurs outside of class time, often by viewing video, in order to have more time for student interaction, discourse, and sense-making during class time. The availability of video technology as well as the presence of the internet by the late 1990s presented new options for rethinking the homework experience. Considering the traditional classroom with a teacher lecture during class followed by homework, the most basic change within a flipped classroom entails having students view a video presentation of the topic as their homework, then doing what used to be assigned as homework practice during class time. This approach frees up the teacher to assist students on problem solving in class, avoiding situations where students get stuck on problems at home without access to help. Early proponents of the flipped classroom, high school chemistry teachers Bergmann and Sams (2012) originally began video recording and posting their lessons on a website to assist absentees, then discovered the rearrangement of the direct instruction component to be an opportunity to increase teacher-student and peer interaction, active learning, and reasoning during class time. They found that students valued having the teacher as a resource as they worked on investigations or assignments, and saw student motivation, engagement, and achievement increase. Student autonomy and responsibility for their own learning increased as students could view videos and make sense of the information at their own pace, pausing and replaying, looking up supplementary information as needed, or playing at double speed if they were already familiar with the content, an option that was not possible in a whole class presentation. Moore et al. (2014) found similar results, noting that homework completion increased by an average of 13% and that class lecture was replaced with student engagement in rich problem solving and applying concepts from the videos. Others warn that “having students watch an online lecture does no more to promote the mathematical practices than watching a live lecture …. The key issue is whether students in the flipped classroom are engaged in active learning, solving problems that promote reasoning and build understanding” (NCTM, 2014, p. 80). Strayer et al. (2016) describe a flipped classroom in which the homework video launches the next day’s mathematical task. Prerequisite information is recapped, student exploration is prompted and initial thoughts are solicited to prepare students for the in-class investigation and provide formative assessment for the teacher before the class begins. (KKM) Bergmann, J., & Sams, A. (2012). Flip your classroom: Reach every student in every class every day. Washington, D.C.: International Society for Technology in Education. Moore, A. J., Gillet, M. R., & Steele, M. D. (2014). Fostering student engagement with the flip. Mathematics Teacher, (107)6, 420-425. National Council of Teachers of Mathematics (NCTM). (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. Strayer, J. F., Hart, J. B., & Bleiler-Baxter, S. K. (2016). Kick-starting discussions with the flipped classroom. Mathematics Teacher, 109(9), 662-668.

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Functions-Based Approach to Teaching Algebra uses the function concept (a relationship between quantities where the output variables are uniquely dependent upon the input variables) as the unifying idea around which all algebraic concepts are taught. Traditional approaches to teaching algebra have stressed the importance of symbolic manipulation and procedural skills in solving equations in problems devoid of context, without the use of technology and with little to no emphasis on conceptual understanding (Chazan, 1993). However, given the prevalent lack of student understanding of algebraic techniques, reformers appealed for changes in the teaching of algebra. Fey and Good (1985) called for school mathematics to reflect the technology-rich settings in which mathematics is used in society, urging that reforms put “the function concept at the heart of the curriculum, preparing students for the dynamic, global quantitative thinking that typifies most models they encounter in future mathematics and its applications” (pp. 49-50). Kieran et al. (1996) describe features of a functions-based approach to algebra, noting that it is not solely a study of functions. This approach utilizes letters as variables rather than just as unknowns for which to solve. It focuses on the covariational relationship between x-values (or independent variables) and the corresponding y-values (or dependent variables), noting how changes in x produce a change in y. This approach uses multiple function representations, namely the graphical, symbolic, and tabular or numeric depictions of the function. Chazan (1993) adds that a function-based approach moves beyond seeing equations as the central tenet of algebra and places a stronger emphasis on modeling with functions. This approach to teaching algebra was incorporated into the NCTM’s Standards and the curriculum reform of the 1980s and 1990s. Research on the effectiveness of these curricula revealed that students learning algebra from a functions-based curriculum possessed stronger conceptual understanding and more successfully solved algebraic problems in context and with technology than their peers who learned in a traditional manner. Students of the traditional approach were stronger with paper-and-pencil symbolic manipulation in problems without context and without technology (Schoen & Hirsch, 2003). Given the importance of problem solving and modeling within standards such as CCSSM, curricula using a functions-based approach to algebra have become more prevalent. (SWD) Chazan, D. (1993). F(x)=G(x)? An approach to modeling with algebra. For the Learning of Mathematics, 13(3), 22-26. Fey, J. T., & Good, R. A. (1985). Rethinking the sequence and priorities of high school mathematics curricula. In C. R. Hirsch & M. J. Zweng (Eds.), The secondary school mathematics curriculum (pp. 43-52). Reston, VA: NCTM. Kieran, C., Boileau, A., & Garancon, M. (1996). Introducing algebra by means of a technologysupported, functional approach. In N. Bednarz et al. (Eds.), Approaches to algebra (pp. 257293). Dordrecht, The Netherlands: Kluwer Academic Publishers. Schoen, H.L., & Hirsch, C. R. (2003). The Core-Plus Mathematics Project: Perspectives and student achievement. In S. L. Senk & D. R. Thompson (Eds.), Standards-based school mathematics curricula: What are they? What do students learn? (pp. 311-343). Mahwah, NJ: Erlbaum.

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Geometric Reasoning is the ability to reason through relationships, generalize geometric ideas, investigate invariants, and balance exploration and reflection. In Fostering Geometric Thinking: A Guide for Teachers, Grades 5-10, Driscoll (2007) argues that U.S. students have had too little exposure to geometric ideas and habits of mind and consequently compare poorly to other countries in international assessments of geometry. He explains that the four habits of mind that encompass geometric reasoning and thinking are as follows: 1. Reasoning with relationships – Being able to reason with relationships is being able to actively look for relationships within figures and between geometric figures and thinking about how these relationships may or may not help in the given situation or problem. 2. Generalize geometric ideas – Generalizing geometric ideas is exactly that – the ability to rationalize why a phenomenon does or does not happen every time. 3. Investigating invariants – This is the ability to figure out what aspects of a certain situation will change or what will stay the same given slightly different scenarios. 4. Balancing exploration and reflection – This is the ability to approach a problem and explore it as much as possible, pause and gather the information about what is known, and use this information to progress further into the solution or regress and start over because a dead end was reached. Driscoll explains that if teachers can guide students in fostering these habits of mind, students will increase their capacity to think and reason geometrically. Pierre van Hiele and Dina van Hiele-Geldof proposed a theory that explains the types of experiences students need to have in order to develop their geometric reasoning capabilities. This theory is called the Van Hiele Levels of Geometric Thinking (Clements & Battista, 1992) and provide a sequential, non-age dependent progression of levels that allow teachers to understand what teachers or more knowledgeable others need to cultivate in the students to promote successful growth in their geometric reasoning capabilities. (SJH) Clements, D. H., & Battista M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 420-464). New York, NY: Macmillan. Driscoll, M. (2007). Fostering geometric thinking: A guide for teachers, grades 5-10. Portsmouth, NH: Heinemann.

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High-Stakes Testing refers to the use of results on examinations or standardized assessments to make either rewarding or punitive decisions toward various populations such as students, teachers, or schools. High-stakes testing has a long history in U.S. education and in society. In the early 1900s, standardized test scores were used in a variety of ways, such as to determine whether immigrants could enter into the United States or were ultimately rejected, to discover exceptional students or those that would be placed in vocational tracks, or to determine entrance or rejection into the military (Amrein & Berliner, 2002). More recently the use of high-stakes testing has focused on educational reform, namely measuring student progress toward academic standards and holding teachers and schools accountable for student learning. Due to poor performances by U.S. students on international assessments such as SIMS and TIMSS as well as the bleak picture of U.S. education painted by A Nation at Risk, many states moved to enhance accountability policies. In the 1980s alone, 40 states developed or enhanced their accountability measures, and by 1992, 46 states were using mandated testing to reform public school education (Bauer, 2000). In many of these states, serious consequences are associated with results of these exams in order to hold schools, teachers, and students responsible for meeting the raised expectations (Amrein & Berliner, 2002). The results from these high-stakes tests can be used to determine the promotion of students or if students can graduate, and to offer rewards or sanctions to schools and teachers based on how students performed (Carpenter, 2001). Proponents of high-stakes testing state that when used properly, mandated tests that are aligned with rigorous state standards can serve as a vehicle to push schools toward higher levels of achievement. Proponents also suggest that tying rewards and punishments to the test results will produce greater effort from all stakeholders to perform at their highest level, and point to other high achieving, industrialized countries such as England, Germany, Singapore, and Japan that have long histories of using high-stakes tests to measure student understanding. Opponents of high-stakes testing criticize the notion that a single assessment can measure everything a student knows and thus should not be used as a reward or punishment. Opponents also stress that teachers are unfairly judged on the sole basis of results on an exam for which students do not possess strong incentives to perform well, and that high-stakes tests typically punish those schools that serve the most vulnerable populations and need the most assistance. These collective arguments point to the contentious nature of high-stakes testing and its potential advantages and disadvantages as an education reform tool. (SWD) Amrein, A. L., & Berliner, D. C. (2002). High-stakes testing, uncertainty, and student learning. Education Policy Analysis Archives, 10(18). Retrieved from http://epaa.asu.edu/epaa/v10n18/ Bauer, S. (2000). Should achievement tests be used to judge school quality? Education Policy Analysis Archives, 8(46). Retrieved from https://epaa.asu.edu/ojs/article/view/437 Carpenter, S. (2001). The high stakes of educational testing. Monitor on Psychology, 32(5). Retrieved from http://www.apa.org/monitor/may01/edtesting.aspx

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Inductive Reasoning is a thought process that involves examining data or multiple examples to discern a trend, pattern, or regularity that leads to making a conjecture or tentative generalization, based on the empirical evidence. “Mathematics involves examining patterns and noting regularities, making conjectures about possible generalizations, and evaluating the conjectures” (NCTM, 2000, p. 262). Inductive reasoning skills are confirmed as part of the math practice of constructing viable arguments in this statement: “mathematically proficient students … reason inductively about data, making plausible arguments that take into account the context from which the data arose” (NGA, 2010). An example of inductive reasoning is when a student measures and sums the angles of a variety of triangles, then makes a conjecture that the sum of the angles of any triangle is 180 degrees. Although this argument by example, based on empirical evidence, may be persuasive and is a worthy endeavor of mathematical thinking, the conclusion can only be tentative as it is not a formal proof that generalizes for all cases. Yopp (2010) suggests that teachers can leverage inductive reasoning tasks to pave the way for the more rigorous process of deductive proofs, by looking for key insights in students’ inductive arguments. He gives the example of an inductive exploration of divisibility by 3. One argument resulted from testing several numbers by summing their digits leading to a tentative conclusion that the sums that were divisible by 3 came from numbers that were divisible by 3. A second argument provided examples of numbers decomposed into place value components: 471 = 4(99+1) + 7(9+1) + 1(1), then rearranged to have a sum of a multiple of 3 plus the sum of the 3 digits. Both of these are empirical arguments, but the latter one provides insight into why the divisibility works. “The strategy of identifying inductive arguments that prompt formal arguments is an important skill” for teachers to have (Yopp, 2010, p. 287). Recognizing the potential for false generalizations, understanding that inductive reasoning is not in itself a formal proof, and having a knowledgeable teacher who can elicit and support this kind of reasoning are important in addressing the limitations of inductive reasoning (Murawska & Zollman, 2015). It should also be noted that inductive reasoning should not be confused with mathematical induction, a formal proof process which involves recursive reasoning to prove for the nth and the n+1 case, thus for all cases. (KKM) Murawska, J. M., & Zollman, A. (2015). Taking it to the next level: Students using inductive reasoning. Mathematics Teaching in the Middle School, 20(7), 416-422. National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: Author. National Governors Association Center for Best Practices (NGA), Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Author. Retrieved from http://www.corestandards.org/ Yopp, D. A. (2010). From inductive reasoning to proof. Mathematics Teaching in the Middle School, 15(5), 286-291.

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Instructional Strategies and Techniques entail the various pedagogical methods and frameworks a teacher might employ to structure learning experiences for students. These techniques differ with respect to the amount of teacher and student involvement and the role the teacher plays in overall student learning. One of a mathematics teacher’s most critical responsibilities is planning and enacting instruction that supports student learning. Teachers often enter the profession with ideas regarding how mathematics should be taught and learned. This phenomenon, termed the “apprenticeship of observation”, is a result of a teacher’s experiences as a student observing their own teachers in action (Lortie, 1975). With greater experience as well as additional learning and professional development, teachers broaden their repertoire with additional instructional strategies they can use in their classroom. Some of these techniques are researchbased strategies stemming from studies of how students best learn mathematics. Some hold their roots in approaches used in other subjects, namely science. The following are popular instructional planning strategies and techniques used by teachers to facilitate classroom discussion and to support student learning: Direct Instruction/Lecture Method: This method is one of the most prevalent instructional methods used in mathematics teaching, and involves the teacher working problems and telling students about the mathematical topics under study while students watch, listen and take notes for future study. This method has a long history in mathematics teaching and is effective for imparting sizeable amounts of information quickly and efficiently to a large audience, yet may not be as effective for student learning and achievement (Freeman et al., 2014). Inquiry Based Instruction/Active Learning: This approach to teaching places students at the center of instruction as they actively engage with the content. A teacher may pose a problem and allow students to investigate various solutions or pose their own problems stemming from their investigations. The teacher works more as a facilitator of the lesson rather than as an authority figure. Three-Act Tasks: Introduced by Dan Meyer in 2010, Three-Act Tasks is an approach to engage students with mathematical modeling and problem solving through a story-telling approach. The Three-Acts follow a similar script: Act 1: Introduce the conflict. This may involve watching a video that draws the student into the contextual situation and introduces the problem on which they will be working. Act 2: The student overcomes obstacles, looks for resources, and develops new tools that can be used to model the situation and work to resolve the conflict. Act 3: Resolve the conflict and set up a sequel/extension. Students discuss their solutions and come to a consensus regarding their investigation, or discuss further conflicts they are investigating as a result of the original problem (Meyer, 2011).

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Launch-Explore-Summarize: Similar to Three-Act Tasks, this pedagogical approach is the instructional model promoted by the Connected Mathematics Project (CMP) middle grades curriculum. The lesson is comprised of three sections: Launch: This is the beginning of the lesson, where the teacher introduces the problem and makes sure students understand the context of the exercise. Teachers may also connect prior topics or provide background at this stage. Explore: This part of the lesson allows students to work either individually or as a team to explore and attempt to solve the problem. A teacher will work to assess and monitor student progress toward the goal of solving the problem as well as plan ahead to the Summarize part of the lesson. Summarize: The teacher facilitates a class discussion regarding the findings of the students stemming from their explorations. The teacher works to guide discussion toward consensus regarding solutions as well as to connections not only among solution strategies but also to broader mathematical content. 5 Practices: The 5 Practices are a pedagogical framework used to organize instruction in order to facilitate mathematical discussions (Smith & Stein, 2011). This framework outlines “a set of instructional practices that will help teachers achieve high-cognitive demand learning objectives by using student work as the launching point for discussions” (p. vii). The 5 Practices are: 1. 2. 3. 4. 5.

anticipating likely student responses to challenging mathematical tasks; monitoring students’ actual responses to the tasks; selecting particular students to present their work during class discussion; sequencing the student responses that will be displayed in a specific order; connecting different students’ responses and connecting the responses to key mathematical ideas (p. 8).

Flipped Classroom Approach: In a traditional classroom, teachers would present new material at school with students practicing with out-of-class homework. In a flipped classroom approach, the students watch a video of the teacher’s lesson outside of class, and then do their homework during the regular class period, thereby flipping the traditional organizational structure. This instructional model allows the teacher to be present should students have difficulty solving problems and increases the amount of assistance provided to students at the moments where it is most needed. (SWD) Freeman, S., Eddy, S. L., McDonough, M., Smith, M. K., Okoroafor, N., Jordt, H., & Wenderoth, M. P. (2014). Active learning increases student performance in science, engineering, and mathematics. Proceedings of the National Academy of Sciences, 111(23). Retrieved from http://www.pnas.org/content/111/23/8410. Lortie, D. (1975). Schoolteacher: A sociological study. London, England: University of Chicago Press. Meyer, D. (2011). The three acts of a mathematical story. Retrieved from http://blog.mrmeyer.com/2011/the-three-acts-of-a-mathematical-story/ Smith, M. S., & Stein, M. K. (2011). 5 practices for orchestrating productive mathematical discussions. Reston, VA: National Council of Teachers of Mathematics.

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Learning Trajectory is used to describe the reflexive relationship between the teacher's design of activities and the teacher’s consideration of the thinking that students might engage in as they participate in those activities. Though the type of thinking required to successfully create a Learning Trajectory has long been considered, directly or indirectly, by many prominent educational figureheads such as Dewey, Vygotsky, and von Glasersfeld, it was Martin Simon’s work (1995) that motivated significant research into the areas of hypothetical learning trajectories (HLT), learning trajectories (LT), and learning progressions (LP). The deeper understanding of HLT/LT/LP became so prominent and important to the pedagogy of teaching that NCTM intertwined the motivation of HLTs into their principles of effective mathematical teaching practices (NCTM, 1991; 2014). Initially, Simon (1995) proposed that a HLT/LT/LP was composed of three components: (1) the learning goal which defines the direction; (2) the activities involved in the learning process; and (3) the hypothetical learning process, which is a prediction of how the students’ thinking and understanding will evolve in the context of the learning activities. Clements and Sarama (2004) emphasize a similar model: (1) the learning goal; (2) developmental progressions of thinking and learning; and (3) the sequence of instructional tasks. However, they proposed that research focusing on the inextricable connection between the psychological developmental progressions and the instructional sequences would produce more valuable results, leading them to conceptualize HLT/LT/LPs as defined above. Currently, it is easy to see how the nature of HLTs/LTs/LPs has become an essential piece of pedagogical content knowledge just by considering current mathematics education reform efforts and their idea of effective teaching. For example, in the NCTM’s Professional Standards for Teaching Mathematics (1991), the first standard describes the teacher’s role in providing meaningful and correctly sequenced tasks based on sound and significant mathematics, on the knowledge of how students learn mathematics, and on the knowledge of the students’ understandings, interests, and experiences (NCTM, 1991). Also, as mentioned above, the idea of HLTs/LTs/LPs can be seen in the NCTM’s Principles to Actions: Ensuring Mathematical Success for All (2014). A few of the principles for effective mathematics teaching motivated by these ideas include: (a) implementing tasks that promote reasoning and problem solving; (b) facilitating meaningful mathematical discourse; and (c) posing purposeful questions (NCTM, 2014), all of which are characteristics of well-developed HLTs/LTs/LPs. (SJH) Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6(2), 81-89. National Council of Teachers of Mathematics (NCTM). (1991). Professional standards for teaching mathematics: Executive summary. Reston, VA: Author. National Council of Teachers of Mathematics (NCTM). (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114-145.

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Lesson Study is a type of professional development in which a structured process through which teachers develop lessons, observe the implementation in one teacher’s classroom, and revise the lesson based on evaluating the success of enactment with the goal to enhance student learning. (Loucks-Horsely et al., 2003). Many of the ideas involved in Lesson Study professional development originated from analyzing mathematics lessons in Japanese classrooms as part of the Third International Mathematics and Science Study (TIMSS). One of the major findings of the TIMSS highlighted the differences in approach to content instruction between the two countries. In these comparison countries, fewer concepts were addressed and in more depth than in the US (Hiebert et al., 2005). These comparisons of individual lessons kindled the adaption of Lesson Study as part of formal professional development for teachers. Lesson Study professional development provides the opportunity for teachers to observe lesson enactment by peer teachers with their actual students. Key elements of Lesson Study professional development include teacher collaboration, research connections, critical feedback on the effectiveness of the lesson, and opportunities to revise and re-enact the lesson (Loucks-Horsely et al., 2003). Variations of Lesson Study professional development have evolved as research on teaching and learning has progressed. Students taught by an expert mathematics teacher within a laboratory setting while other teachers observe and participate in aspects of the lesson provide opportunities to enhance teachers’ knowledge of questioning and scaffolding students’ ideas about the mathematics (Naik & Ball, 2014). Another variation, Classroom Embedded (CE) Professional Development, is a type of Lesson Study structure built around research on students’ thinking in specific mathematics content areas (Nielsen et al., 2016). (LBK) Hiebert, J., Stigler, J. W., Jacobs, J. K., Givvin, K. B., Garnier, H., Smith, M., … Gallimore, R. (2005). Mathematics teaching in the United States today (and tomorrow): Results from the TIMSS 1999 video study. Educational Evaluation and Policy Analysis, 27(2), 111-132. Loucks-Horsley, S., Love, N., Stiles, K. E., Mundry, S., & Hewson, P. W. (2003). Designing professional development for teachers of science and mathematics. Thousand Oaks, CA: Corwin Press. Naik, S. S., & Ball, D. L. (2014). Professional development in a laboratory setting examining evolution in teachers’ questioning and participation. Journal of Mathematics Education, 7(2), 40-54. Nielsen, L., Steinthorsdottir, O. B., & Kent, L. B. (2016). Responding to student thinking: Enhancing mathematics instruction through classroom based professional development. Middle School Journal, 47(3), 17-24.

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Longitudinal Study is a term in mathematics education referring to a specific form of research where data is collected on a certain population (e.g., teachers, students, schools) over an extended period of time. The key feature of longitudinal studies is the repeated observation and data collections of the same research subjects or on a similar group of students under study over time. Longitudinal studies in mathematics education are designed to study a specific phenomenon over a period of time. One critical characteristic of longitudinal studies is the relative consistency of the population under study in order to document potential changes that occur over the duration of the study. The length of longitudinal studies can vary – some may last only a few months, while others can last many years. Longitudinal research studies are inherently difficult to conduct, as sampling issues, retention and dropout problems, and the timeconsuming nature of such work produce challenges beyond the scope of other research designs. Bauer (2004) suggests that longitudinal studies should be conducted when one of the following criteria are met: x x x x x

The stated objects require measurement of individual change over time; A causal relationship can be postulated between an earlier and subsequent event; The measurement of traits, characteristics, or events proposed are meaningful and of reasonable validity; Results of the study will permit generalization; The analytic technique proposed will permit the exploration of the data gathered at intermediate stages as well as at end of inquiry. (p. 80)

Despite the challenges presented in conducting longitudinal studies, the impact of such work is vitally important on a number of issues in mathematics education. For example, Boaler (2002) studied the impact of different instructional methods on students’ opportunity to learn at two schools in England. Over a three-year span, Boaler collected classroom observation data, interview data with students and teachers, as well as student achievement data at a school that utilized traditional lecture-based instruction and at a school that heavily emphasized project based work. Boaler found that the different instructional approaches led to the students at the two schools developing different kinds of mathematical knowledge, and that the students from the school that used project based work possessed a mathematical understanding that they were able to use on both procedural assessment questions as well as in applied and conceptual tasks. Students from this school performed as well or better on mandated assessments than their peers that learned in a lecture-based format, and possessed greater confidence and enjoyment with mathematics as well. Fennema et al. (1996) used a longitudinal design to explore primary teachers’ beliefs and instruction over a four-year period while participating in a Cognitively Guided Instruction (CGI) teacher professional development program. The researchers collected baseline data in year 1 and then immersed teachers in the

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CGI program over the following three years. Throughout the study, the researchers collected observational and interview data as well as student written work from project-designed tests. The study indicated that the teachers had dramatic shifts in beliefs regarding how mathematics should be taught which impacted their instruction. These changes in instruction corresponded to changes in student achievement, as students’ abilities to problem solve and to understand mathematical concepts increased over time. Other longitudinal studies in mathematics education have examined large-scale assessment data to examine trends in student learning. An early predecessor to modern-day studies was the National Longitudinal Study of Mathematical Abilities (NLSMA), conducted by the School Mathematics Study Group. Conducted for the first time in 1962, this research spanned five years in order to study the evolution and potential revision of mathematics curriculum as well as to examine the nature of student mathematical abilities (Cahen, 1965). Today, several large-scale assessments, including the National Assessment of Educational Progress (NAEP), the Program for International Student Assessment (PISA), and the Trends in International Mathematics and Science Study (TIMSS), provide data for researchers to analyze changes in student achievement over time. Although the student population changes between administrations of the exams, researchers use representative samples to chart how similar groups of students perform over time. (SWD) Bauer, K. W. (2004). Conducting longitudinal studies. New Directions for Institutional Research, 121, 75-90. Boaler, J. (2002). Experiencing school mathematics: Traditional and reform approaches to teaching and their impact on student learning. New York, NY: Routledge. Cahen, L. S. (1965). An interim report on the national longitudinal study of mathematical abilities. The Mathematics Teacher, 58, 522-526. Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction. Journal for Research in Mathematics Education, 27(4), 403-434.

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Manipulatives refer to a variety of objects that learners can utilize to help provide concrete representations of various mathematical concepts. At the very basic level, manipulatives are typically objects chosen with a purpose when teaching a given concept. For example, when teaching counting, there are numerous objects that can be utilized, such as beans, counters, tiles, place value unit cubes, etc. On the other hand, when working with factoring polynomials, Algebra tiles may be a much more effective tool to help students grasp the concept. Concrete manipulatives typically found in classrooms are counters – colored or translucent, Cuisenaire rods, place value or base ten blocks, fraction tiles, money (bills and coins), number lines, spinners, dice, etc. With computerized enhancements, these concrete manipulatives have morphed into virtual manipulatives with a powerful effect on teaching and learning. Instead of having number lines limited to whole numbers, the possibilities are endless and are able to accommodate a variety of types of numbers with minimal changes, allowing for ease of use in instruction for teachers. With the wide-spread availability of the Internet and technological advances, there are more options available for teaching in terms of manipulatives, these are called virtual manipulatives. Virtual manipulatives can be categorized as either static or dynamic visual representations of traditional concrete manipulatives (Moyer et al., 2002). Static visual representations are merely computer generated images of the traditional concrete manipulatives, but cannot be utilized in manner similar to their concrete counterparts. Virtual manipulatives that are categorized as dynamic visual representations are a mix of their static visual representations, along with the ability to be manipulated like their concrete counterparts. Clements and McMillen (1996) argued that virtual manipulatives can be as concrete as tactile manipulatives. Examples with brief descriptions: x

x

x

x

Base Ten Blocks: These are proportional blocks that contain individual units and pre-grouped units, such as ten sticks and 100 flats. The pre-grouped units are not able to be taken apart, thus remaining one whole. Demonstration Clocks: These typically consist of a larger clock for teacher use and then a class set of smaller clocks which the student can manipulate times on. Geoboards: These are boards that contain pegs that are set equidistant from each other. Rubber bands are typically placed around the pegs to create shapes or manipulated shapes already constructed on the board. Pattern Blocks: These are typically wooden or plastic blocks that come in the following shapes: equilateral triangle, rhombus, trapezoid, hexagon, small rhombus, and square. They can be used for teaching a variety of topics including shape composition and decomposition and fractional parts.

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Cuisenaire Rods: These are a set of colored number rods used in an experiment in 1931 to help primary grades students understand arithmetic. The rods were popularized around the world in the 1950s. Algebra Tiles: These are a set of tiles that include individual units, rectangles, and large squares. The individual unit represents the number one, the rectangle represents the variable x and the large square represents x2. Fraction Tiles: These are a set of tiles that illustrate one and several fractional variations of a whole. The tiles are color coded, so that all the unit fractions of one denominator can be grouped together, illustrating a whole. Ten Frames: This is a five column table with two rows. It is typically used to help people “see” ten. This visual aid is used to help students get a strong sense of “ten”. Once students have a strong sense of “ten”, place-value understanding and mental math calculations become easier. Interlocking Cubes: These cubes are designed so that one side easily fits into the next cube. There are variations of these cubes in which multiples sides can be connected, thus providing more options for users. (CCO)

Clements, D. H., and McMillen, S. (1996). Rethinking “concrete” manipulatives. Teaching Children Mathematics, 2(5), 270-279. Moyer, P. S., Bolyard, J. J., and Spikell, M. A. (2002). What are virtual manipulatives? Teaching Children Mathematics, 8(6), 372-377.

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Math Anxiety has been defined as feelings of tension that impede an individual’s manipulation of numbers and the solving of mathematical problems in many different situations (Tobias, 1993). Researchers have examined both cognitive and affective factors. For example, Wigfield and Meece (1988) assessed 564 secondary students using the Mathematics Anxiety Rating Scale (MARS) and the Mathematics Anxiety Questionnaire (MAQ). They determined two components that influenced students’ math anxiety levels. One component was related to negative reactions to math, involving nerves, fear, and discomfort. The other component was related to concerns about doing well in mathematics. In terms of cognitive factors, math anxiety has been linked to the amount of working memory needed to solve a mathematics problem. Ashcraft (2002) argued that reproducing the steps of a standard algorithm such as “borrowing and carrying” were more likely to increase anxiety for someone who is already anxious about math than recall of single digit number facts (p. 184). However, this perspective does not include alternative strategies to the standard algorithm in terms of number computation strategies. There has also been discussion of the role that teachers and others might play in increasing students’ math anxiety. Much of the discussions and studies were conducted with preservice elementary teachers. For example, Jackson and Leffingwell (1999) surveyed 157 elementary preservice teachers and found the major factors influencing their potential for math anxiety were math instructors’ negative behaviors while they were teaching. Philipp (2007) posited that more reform oriented and sense-making approaches to teaching mathematics could lessen the mathematics anxiety of elementary preservice teachers. (LBK) Ashcraft, M. H. (2002). Math anxiety: Personal, educational, and cognitive consequences. Current Directions in Psychological Science, 11(5), 181-185. Jackson, C., & Leffingwell, R. (1999). The role of instructors in creating math anxiety in students from kindergarten through college. The Mathematics Teacher, 92(7), 583-586. Maloney, E. A., & Beilock, S. L. (2012). Math anxiety: Who has it, why it develops, and how to guard against it. Trends in Cognitive Sciences, 16(8), 404-406. Philipp, R. A. (2007). Mathematics teachers’ beliefs and affect. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 257-315). Charlotte, NC: Information Age Publishing. Tobias, S. (1993). Overcoming math anxiety. New York, NY: W.W. Norton & Company. Wigfield, A., & Meece, J. L. (1988). Math anxiety in elementary and secondary school students. Journal of Educational Psychology, 80(2), 210.

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Math Wars refers to the ongoing debate between teacher-centered traditional methods of teaching math as a collection of procedures and algorithms and student-centered methods emphasizing conceptual understanding, discovery of relationships, cooperative learning, and multiple strategies for solving problems. Although the pendulum has swung several times in the history of U.S. education from a traditional, back-to-basics curriculum to reform-oriented, the term math wars was coined in the 1990s following reaction in California to the 1989 publication and dissemination of the NCTM Curriculum and Evaluation Standards. What became known as the ‘NCTM Standards’ was, in itself, a reaction to national data of low problem solving scores from the previous decade of back-to-basics curriculum. The Standards document suggested what should be done to improve students’ problem solving skills and mathematical understanding. It was based on a constructivist view that mathematics learning is an active process, with students at the center of the thinking, thus much more than memorization and procedures. With the growing body of research on mathematical thinking, California was at the forefront of the reform movement in mathematics embracing reform-minded ideas in their 1992 mathematics framework. Publishers followed with curriculum materials that looked different from traditional textbooks, teachers adopted different practices, and parents began to complain. Some solicited help from legislators and mathematicians and formed anti-reform organizations such as Mathematically Correct. Criticism increased when the state assessment included a question that gave more points for a wrong answer with a coherent explanation than a right answer with inadequate explanation, and the term ‘fuzzy math’ arose. When a draft of the California state standards, based on the NCTM standards, was proposed to the state board in 1997, it had been vetted through public review and was research-based. The conservative majority of the state board rejected the proposal, rewrote the elementary standards to conform to the conservative agenda and hired mathematicians to rewrite the secondary standards. The math education community was shocked by the negation of their expertise and disregard for research, in favor of a politically-motivated agenda. This action began to play out in other parts of the country, as parents began to demand a return to emphasis on procedures and direct instruction (Schoenfeld, 2004). Another wave of math wars, with new political motivations, cycled through with criticism of the 2010 publication of the Common Core Math Standards. “To move mathematics teaching and learning forward, we have to resist the urge to be pushed to extremes. We have to do our part to break the historic cycle of pendulum swings …. When we stay the course and let students engage, learn, and develop their understanding, skills, and abilities to use mathematics, our students will be the beneficiaries” (Larson, 2017). (KKM) Larson, M. (2017). The elusive search for balance. Retrieved from https://www.nctm.org/Newsand-Calendar/Messages-from-the-President/Archive/Matt-Larson/Math-Education-IsSTEM-Education!/ Schoenfeld, A. H. (2004). The math wars. Educational Policy, 18(1), 253-286.

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Mathematical Identity is the view and set of beliefs that one holds about oneself as a learner and user of mathematics. Mathematical identity can influence how a student relates to and engages with mathematics inside and outside the classroom. It is shaped by sociocultural ideas and stereotypes of who ‘does’ and is good at mathematics, attitudes of family and peers (many of which are negative), and former experiences with mathematics learning. Middle school age is a particularly vulnerable and critical time as students are negotiating their identities on several levels, some which could jeopardize a positive mathematical identity. A negative identity can end up compromising efforts and success in school mathematics at the very time when students are being challenged to transition to more abstract levels of thinking. Students who are marginalized due to racial or other stereotypes may struggle to envision themselves as mathematics learners when others have low expectations for them. Mathematical identity is a construct, along with agency and social justice, that supports empowerment as a learner of mathematics (Larson, 2016). Knowing what factors contribute to the development of identity can help a teacher create a classroom environment that nurtures positive mathematical identity. One component of mathematical identity involves what a person views mathematics is. If they view mathematics narrowly as a collection of algorithms and procedures to master, their identity as a learner of mathematics may be equally narrow compared to a more expansive view of mathematics as a study of relationships, structures, and problem solving strategies. Classroom pedagogical practices influence these views and shape what forms of knowledge students gain (Boaler & Green, 2000). Classrooms with practices that support a more expansive view of mathematics facilitate students having a more flexible knowledge of mathematics. Boaler (2002) proposed a ‘triangle’ relationship between knowledge, classroom practice, and identity. A teacher-centered classroom that does not allow opportunity for student voice and collaboration in mathematical thinking leaves little room for mathematical identity to develop. Students whose identities are otherwise growing as creative and social individuals may disconnect themselves from identifying with mathematics if an authoritative teaching approach conflicts with another identity that is important in their lives. Classroom practices that promote mathematics learning as a problem solving endeavor with student agency and opportunities to engage in mathematical discourse as they develop content knowledge are essential to identity development. They empower students as mathematical thinkers with positive mathematical identity who can establish a relationship with mathematics and the knowledge contained in that discipline. Boaler (2002) describes this as a disciplinary relationship, that is the relationship that a student develops with mathematics, with the interrelating components of identity, knowledge, and classroom practice. Also integral to mathematical identity is how students view their own abilities. Mindset, or how students perceive intelligence and their own abilities, can influence student motivation and achievement (Dweck, 2006). Students with a

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fixed mindset believe that people are born with a certain intelligence and have little control over it. Students with a growth mindset believe that their intelligence can be developed, thus their effort and experiences will bring them positive results. When teachers and students focus on effort and process over perceived ability, a growth mindset is nurtured, and positive mathematical identity is supported. The characterization of mathematics learning and participation as a racialized experience (Martin, 2006), and not culture-free, suggests that the development of mathematical identity grows in complexity when racialized experiences are examined. In a study of mathematically high-achieving African American college students, McGee (2015) proposed a framework to describe how these students straddled between “robust and fragile mathematical identities” (p. 599) as they strove to preserve their positive identities in the face of constant assault of low expectations due to racial bias. With a fragile mathematical identity, students expend energy to defend themselves against racial assumptions among teachers and others who assume they have low math ability. For instance, they work harder to achieve mathematically in order to disprove the unfair predictions. Their motivation is external, rather than for the love of mathematics and their own wellbeing. Students manifest a robust mathematical identity when they can define themselves rather than defend themselves against others’ definitions and expectations. They seek experiences that promote their interest in mathematics and sense of fulfillment in the pursuit of the discipline, thus strengthening their identity and preparing them to respond confidently from a higher vantage point when they are challenged by unfair assumptions. The robust mathematical identity was evident in the study when students believed that “their talents and abilities were greater than the stereotypes that agitated them” (McGee, 2015, p. 616). (KKM) Boaler, J. (2002). The development of disciplinary relationships: Knowledge, Practice, and identity in mathematics classrooms. For the Learning of Mathematics, 22(1): 42-47. Boaler, J., & Greeno, J. G. (2000). Identity, agency, and knowing in mathematics worlds. In Boaler, J. (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 171–200). Westport, CT: Ablex Publishing. Dweck, C. S. (2006). Mindset: The new psychology of success. New York, NY: Random House. Larson, M. (2016, September 15). A renewed focus on access, equity, and empowerment. Retrieved from https://www.nctm.org/News-and Calendar/Messages-from-thePresident/Archive/Matt-Larson/A-Renewed-Focus-on-Access,-Equity,-andEmpowerment/ Martin, D. B. (2006). Mathematics learning and participation as racialized forms of experience: African American parents speak on the struggle for mathematics literacy, mathematical thinking and learning. Mathematical Thinking and Learning, 8(3), 197-229. McGee, E. O. (2015). Robust and fragile mathematical identities: A framework for exploring racialized experiences and high achievement among black college students. Journal for Research in Mathematics Education, 46(5), 599-625.

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Mathematical Knowledge for Teaching (MKT) is the multi-faceted nature of knowledge needed by teachers to perform the work of mathematics teaching. Research on the knowledge base used by teachers has underscored the specialized nature of this knowledge that is specific to the teaching profession. Beginning with Shulman’s (1986) description of pedagogical content knowledge (PCK), researchers dug deeper into the complex nature of teacher knowledge and push past the notion that successful mathematics teachers needed only strong knowledge of mathematics to perform their job well. The study of teacher knowledge by Ball and her colleagues led to the construct of mathematical knowledge for teaching (MKT), defined as “the mathematical knowledge needed to carry out the work of teaching mathematics” (Ball et al., 2008, p. 395).

Ball et al.’s model (p. 403) for MKT is shown in the figure above, highlighting the various subcomponents of teacher knowledge needed by teachers such as: x x

x x x

Common content knowledge, defined as the “mathematical knowledge and skill used in settings other than teaching” (p. 399); Horizon content knowledge, which is “an awareness of how mathematical topics are related over the span of mathematics included in the curriculum” (p. 403); Specialized content knowledge, which is the knowledge predominantly used by teachers yet not needed outside of this profession; Knowledge of content and students, which teachers use to anticipate how students will think about content and plan instruction accordingly; Knowledge of content and teaching, which is the knowledge needed by teachers to plan instruction and sequence tasks to optimize student learning;

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Knowledge of content and curriculum, similar to Shulman’s (1986) curriculum knowledge, involves knowledge of how topics can be sequenced within a grade level and across grade levels as well as an understanding of the resources that can be used to assist student learning of the content. (SWD)

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407. Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.

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Mathematical Literacy, a term often used synonymously with other concepts such as quantitative literacy or mathematical knowledge, refers to the ability to understand, use, and interpret aspects of everyday living that require mathematical knowledge and skills. The term “mathematical literacy” is one among many terms used to describe the successful degree to which one uses mathematics stemming from real world settings. As part of its international assessments, PISA measures the reading, scientific, and mathematical literacy of students across the globe. To this extent, mathematical literacy is defined by PISA as “the capacity to identify, understand and engage in mathematics as well as to make well-founded judgements about the role that mathematics plays in an individual’s current and future life as a constructive, concerned and reflective citizen” (OECD, 2003, p. 20). Additionally, mathematical literacy is judged according to one’s capacity to use their mathematical knowledge and skills in real-world environments rather than in school settings. Finally, mathematical literacy goes beyond solving problems to capture the intrinsic qualities of those who can successfully navigate such scenarios, including self-confidence and curiosity (OECD, 2003). Although often used synonymously with the term quantitative literacy (QL), mathematical literacy is sometimes thought of as a higher level of knowledge and skill than QL, with QL being seen as a subset of quantitative literacy. Forman (1997) discusses the differences in these terms with mathematicians and mathematics educators, noting that the general consensus is that mathematical literacy is what is needed for advanced study in mathematics or a career in a STEM-related field. However, others use the term to describe what in essence is QL. Pugalee (1999) notes that the 2000 NCTM Standards advocate for mathematical literacy as “a societal need arising from increasing mathematical and technological influences that require quantitative understandings” (p. 19). Pugalee goes on to portray a model of mathematical literacy, closely mirroring the process standards from the NCTM standards. Jablonka (2003) points out that the terms “mathematical literacy” and “numeracy” (another synonym for QL) cannot be translated into many languages, and thus the term “mathematical illiteracy” is often used instead. Nevertheless, Jablonka added that the challenge in defining and measuring mathematical literacy lies in the fact that individuals use mathematics in very different settings, and should be thought of in functional terms according to how mathematics is being used in these situations. (SWD) Forman, S. L. (1997). Through mathematicians’ eyes. In L. A. Steen (Ed.), Why numbers count: Quantitative literacy for tomorrow’s America (pp. 161-172). New York, NY: College Board. Jablonka, E. (2003). Mathematical literacy. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.) Second international handbook of mathematics education (pp. 75102). Dordrecht, The Netherlands: Kluwer Academic Publishers. Organization for Economic Cooperation and Development OECD). (2003). Literacy skills for the world of tomorrow: Further results from PISA 2000. Paris, France: Author. Pugalee, D. K. (1999). Constructing a model of mathematical literacy. The Clearing House, 73(1), 19-22.

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Mathematical Modeling is the process of developing, testing, improving, and then using mathematical representations that exhibits a real-world situation. Pollak (2003) traces the growth of the term mathematical modeling to the School Mathematics Study Group (SMSG) work of the 1960s and 1970s. The concept has evolved since then, though it is often lumped together with other constructs such as word problems or applied mathematics. Mathematical modeling refers not only to types of problems but also to the process for solving such problems. Pollak (1997) outlines the steps often involved in mathematical modeling: 1. We identify something we want to know, do, or understand. 2. We select “objects” that seem important in the real-world question and study the relations among them. 3. We decide what to examine and what to ignore about the objects and their interrelations. 4. We translate this idealized version into mathematical terms, and obtain a mathematical formulation of it. 5. We identify the field(s) of mathematics that are needed, and bring to bear the instincts and knowledge of those fields. 6. We use mathematical methods and insights, and get results. 7. We return to the original field and obtain a theory of the idealized question. 8. We view the results through the lens of the original problem to assess the reasonableness of the solution (Pollak, 1997, p. 102). Mathematical modeling is an iterative process where the student is continually evaluating the model and the data being produced to verify whether the model is producing correct results or if changes are needed. A close but not equivalent term often used with mathematical modeling is “modeling mathematics”. Cirillo et al. (2016) note that concrete objects such as base-ten blocks and algebra tiles are often used to “model mathematics”, yet this differs from the mathematical modeling process outlined above. In fact, CCSSM provides several instances where students are to “model mathematics” with representations such as visual models for fractions, tiling for area and packing for volume, and arrays for multiplication. This contrasts with the Standard for Mathematical Practice 4, model with mathematics, and the Modeling strand in the High School standards, which connect mathematics to real-world situations and involve the modeling process described above (Cirillo et al., 2016). (SWD) Cirillo, M., Pelesko, J. A., Felton-Koestler, M. D., & Rubel, L. (2016). Perspectives on modeling in school mathematics. In C. R. Hirsch & A. R. McDuffie (Eds.), Mathematical modeling and modeling mathematics (pp. 3-16). Reston, VA: NCTM. Pollak, H. O. (1997). Solving problems in the real world. In L. A. Steen (Ed.), Why numbers count: Quantitative literacy for tomorrow’s America (pp. 91-105). New York, NY: College Board. Pollak, H. O. (2003). A history of the teaching of modeling. In G. A. Stanic & J. Kilpatrick (Eds.), A history of school mathematics (pp. 647-671). Reston, VA: NCTM.

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Mathematics Skills are the abilities needed to be successful in both learning mathematics and, more broadly, using mathematics as a knowledgeable person. Defining mathematics skills has long been a source of debate and discussion in mathematics education, as various groups often held different opinions on the knowledge and abilities students should possess. From the Committee of Ten report in 1894 through the standards era of the late 20th and early 21st centuries, professional organizations have offered their opinions on the mathematics skills important in K-12 education. Often the discussion centers around “basic skills”, which represent the most foundational abilities one should attain. In response to the “Back to Basics” movement of the 1970s, a number of efforts were made to codify what was meant by “basic skills”. Given the move to narrowly define basic skills and to center such discussions around students’ computational abilities, the National Institute of Education convened a conference in Euclid, OH, to bring experts together to broaden the scope of what comprised basic mathematical skills. The National Council of Supervisors of Mathematics (NCSM) produced a position paper outlining ten basic skills in mathematics: (1) Problem Solving; (2) Applying mathematics to everyday situations; (3) Alertness to the reasonableness of results; (4) Estimation and approximation; (5) Appropriate computational skills; (6) Geometry; (7) Measurement; (8) Reading, interpreting, and constructing tables, charts and graphs; (9) Using mathematics to predict; and (10) Computer literacy (NCSM, 1977). These efforts to expand basic skills provided the foundation for the standards movement that followed. In fact, many of the NCSM “basic skills” have parallel emphasis in the standards produced by NCTM that effectively describe the basic skills to be mastered at each grade level or in a grade band. These standards include skills stemming from one’s mathematical knowledge such as the development of computational abilities, number sense, or geometric knowledge (content standards) and those emerging from the mathematical processes students employ when doing mathematics such as problem solving, reasoning and communicating one’s thoughts (process standards). Although “basic skills” is most often used synonymously with mathematics skills, other terms or classifications have been used to discuss mathematics skills. Some have used the term “essential skills” (Dossey et al., 1997) to describe the goals of mathematics education. Others have broken down skills by age ranges and have examined “early skills” (Diezmann & Yelland, 2000) to define key abilities young children should develop and build prior to entering school. (SWD) Diezmann, C., & Yelland, N. J. (2000). Developing mathematical literacy in the early childhood years. In Yelland, N. J. (Ed.), Promoting meaningful learning: Innovations in educating early childhood professionals (pp. 47-58). Washington, DC: National Association for the Education of Young Children. Dossey, J. A., Peak, L., & Nelson, D. (1997). Essential skills in mathematics: A comparative analysis of American and Japanese assessments of eighth-graders. Retrieved from https://nces.ed.gov/pubs97/97885.pdf National Council of Supervisors of Mathematics (NCSM). (1977). Position statements on basic skills. Arithmetic Teacher, 25(1), 19-22.

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Meaningful Learning is learning with understanding. Over centuries of mathematics education, the primary view of mathematics was what Brownell (1947, p. 260) referred to as a “tool subject” – one of mechanical skills to be drilled by repetition, isolated facts to be memorized, and routine problems to be solved. Brownell advocated for meaningful learning, which he defined as “mathematical understandings” (p. 257) of relationships, pointing out that a narrow view of mathematics had not produced a mathematically literate population. “Without these meanings to hold skills and ideas together in an intelligible, unified system, pupils in our schools for too long a time have ‘mastered’ skills which they do not understand, which they can use only in situations closely paralleling those of learning, and which they … forget” (p. 260). In later decades, cognitive science provided support for Brownell’s ideas, explaining how memory works best with connected knowledge. Knowledge structured in relationships with multiple points for association allows for students to more easily connect with their existing knowledge and construct their own meaning. Such meaningful learning increases retention and transferability (Bransford et al., 2000) and increase confidence and perseverance in novel problem solving (Brownell, 1947; NCTM, 2000; Skemp, 1978). Hiebert & Carpenter (1992) further describe understanding: “A mathematical idea, procedure, or fact is understood thoroughly if it is linked to existing networks with stronger or more numerous connections” (p. 67). Skemp (1978) gives the name ‘relational understanding’ to the kind of understanding that is supported by knowledge of relationships and contrasts it with ‘instrumental understanding’ which is knowing how to perform a procedure but not knowing why it works. Of the six principles set forth by NCTM (2000), the Learning Principle highlights meaningful learning: “Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge” (p. 11). Stylianides and Stylianides (2007) emphasize the kind of curriculum that evokes learning with understanding including activities that promote conceptual understanding, problem solving, classroom discourse, and mathematical reasoning and justification. (KKM) Bransford, J., Brown, A., & Cocking, R. (2000). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press. Brownell, W. A. (1947). The place of meaning in the teaching of arithmetic. The Elementary School Journal, 47(1), 256-265. Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-97). Old Tappan, NJ: Macmillan. National Council of Teachers of Mathematics(NCTM). (2000). Principles and standards for school mathematics. Reston, VA: Author. Skemp, R. (1978). Relational and instrumental understanding. Arithmetic Teacher, 26(3), 9-15. Stylianides, A. J., & Stylianides, G. J. (2007). Learning mathematics with understanding: A critical consideration of the learning principle in the Principles and Standards for School Mathematics. The Mathematics Enthusiast, 4(1), 103-114.

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Mental Discipline (or mental discipline theory) refers to a theory of learning prevalent in the 1800s and early 1900s that centered on the idea that a subject such as mathematics is taught foremost as a mechanism to enhance one’s ability to think, and that by learning a subject such as mathematics, one’s mind is strengthened to learn other subjects. The theory of mental discipline was a major influence in mathematics education during the 1800s and early 1900s. At its most basic level, mental discipline likens learning of challenging subjects such as mathematics to physical exercise: just as physical exercise serves to strengthen the muscles of the body, so, too, does studying mathematics aid to strengthen the mind and improve people’s thinking (Stanic, 1986). This analogy dates back to Plato, who noted that those who were strong in mathematics were also strong in other subjects, while those who were weak became stronger with sustained study (Stanic, 1986). By training the mind in the study of mathematics, one would learn to become a better thinker, which would transfer to the learning of other subjects (Schoenfeld, 1992). The influence of this theory was seen throughout mathematics teaching and learning during the 1800s, as the primary pedagogy involved students learning a rule, viewing examples, and then practicing a number of problems in order to master the rule (NCTM, 1970). Textbooks during this time contained large sets of problems for students to work to strengthen their abilities. One of the most popular books was Warren Colburn’s First Lessons in Intellectual Arithmetic, which stressed the notion of mental arithmetic and, informed by Pestalozzi’s theory of individualized instruction, provided sequenced problems for students to develop rules for calculation. These problems and the approach to instruction were seen to exercise students’ mental faculties and support their mental development (NCTM, 1970). However, at the turn of the 20th century, educators and mathematicians began to question and challenge the theory of mental discipline. As the number of students attending high school surged between 1890 and 1940, meeting the needs of a changing population spurred discussion about changes needed to the school curriculum (Stanic, 1986). Researchers such as Thorndike (1924) conducted experiments that called into question the suitability of mental discipline, while other theories of learning – particularly behaviorism – began to take center stage (Schoenfeld, 1992). Though the theory of mental discipline has diminished over time, certain facets of this theory, including the role of mathematics in promoting and improving one’s thinking, still are studied and debated to this day. (SWD) National Council of Teachers of Mathematics (NCTM). (1970). A history of mathematics education in the United States and Canada. Washington, DC: Author. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook for research on mathematics teaching and learning (pp. 334-370). New York, NY: Macmillan. Stanic, G. A. (1986). Mental discipline theory and mathematics education. For the Learning of Mathematics, 6(1), 39-47. Thorndike, E. L. (1924). Mental discipline in high school studies. Journal of Educational Psychology, 15(1), 1-98.

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Mental Math is the process of performing calculations or computations in the brain and without the use of any tactile or visual tool and may involve rounding and estimation. Mental Math is a key component of Number Talks (Parrish, 2010), which are problems that are specifically designed to facilitate the use of number relationships, rounding and estimation, to make mental computations easier to perform. The subtraction problem, 300 - 249, could be solved mentally in many different ways using the relationship between 249 and 250. For example, 300 – 250 = 50 and 50 – 49 = 1 so the answer is 50 + 1 = 51. Case and Sowder (1990) characterized this as an example of the “nearness strategy” in their study of students’ computational estimation strategies. Other research has shown the importance of the connection between estimation and mental computation strategies and increased uses of technological devices such as calculators in reflecting on students making sense of their answers (Reys, 1984). Strategies for rounding have also been considered an important component of mental math strategies. Rules for rounding related to place value such as if the place value to the right is five or greater, round up, or less than five round down can facilitate mental math strategies (Threlfall, 2002). The processes involved in performing of mental computation strategies, as in the above example, also involve the use of properties of operations. The distributive and associative properties underpin this strategy and may also indicate evidence of a deeper understanding of mathematical ideas such as place value, i.e., 300 – (250 – 1) = 300 – 250 + 1 = 51. The term, mental math, is used in conjunction with flexibility, computation, and other research on how students invent strategies for computation problems (Thompson, 1999; Threlfall, 2002). Although there are a variety of ways that students might notate their invented strategies, many of these strategies originate without paper and pencil by using relational thinking (Carpenter et al., 2003; Empson & Levi, 2011). (LBK) Carpenter, T. P., Franke, M., and Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann. Case, R., & Sowder, J. T. (1990). The development of computational estimation: A neo-Piagetian analysis. Cognition and Instruction, 7(2), 79-104. Empson, S. B., & Levi, L. (2011). Extending children’s mathematics: Fractions and decimals. Portsmouth, NH: Heinemann. Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies, grades K-5. Sausalito, CA: Math Solutions. Reys, R. E. (1984). Mental computation and estimation: Past, present, and future. The Elementary School Journal, 84(5), 547-557. Thompson, I. (1999). Mental calculation strategies for addition and subtraction. Part 1. Mathematics in School, 28(5), 2-4. Threlfall, J. (2002). Flexible mental calculation. Educational Studies in  Mathematics, 50(1), 29-47.

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Metacognition is a higher order thinking process in which one analyzes, reflects upon, and monitors one’s own cognitive activity. Commonly referred to as “thinking about one’s own thinking”, metacognition includes awareness of strategy use, ability to gauge level of understanding, recognizing when to re-direct one’s thinking to a more productive approach, and knowledge of oneself as a learner. Metacognition contributes to strategic competence and adaptive reasoning, two of the five strands of proficiency for mathematics learning (NRC, 2001). Stated as a key finding on how people learn, “a metacognitive approach to instruction can help students learn to take control of their own learning by defining learning goals and monitoring their progress in achieving them” (Bransford et al., 2000, p. 3). Metacognition usually involves an internal dialogue, and students may not be aware of its importance unless a teacher draws attention to it. Formative assessment activities that make student thinking visible and encourage students to explain their reasoning helps them become more aware of their own thinking. Explicit teaching of metacognitive skills improves student achievement and ability to learn independently (Bransford et al., 2000). Teachers can model how they monitor their thinking as they engage in problem solving, verbalizing why they chose a certain strategy, noting when a strategy does not seem promising, and describing what they did next. Self-talk such as ‘does this make sense to me’ and ‘should I try another way’ should be encouraged. Students can be asked to analyze their own strategies, identify which led to dead ends, and explain how they knew when to re-direct their thinking. Schoenfeld (1992) observed high school and college students solving unfamiliar problems. The majority of solution attempts involved a quick read of the problem followed by a quick choice of strategy, with the rest of the time spent trying to make that strategy work, even if it was incorrect. He contrasted this to the attentive self-monitoring of a mathematician, who spent more time making sense of the problem, analyzing it and considering the feasibility of a variety of approaches, and quickly adapting when a solution path did not appear viable. However, when students participated in a problem solving course in which the instructor routinely asked them to explain their processes, why they made their decisions, and how they expected it would help with the solution, students increased their self-regulative talk and flexibility in problem solving. This explicit emphasis on metacognitive skills led to greater success with their problem solving by the end of the course. (KKM) Bransford, J., Brown, A., & Cocking, R. (2000). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press. National Research Council (NRC). (2001). Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. A. Grouws (Ed.), Handbook for research on mathematics teaching and learning (pp. 334-370). New York, NY: MacMillan.

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Misconceptions in mathematics are ideas based on faulty or incomplete reasoning that have not been properly conceptualized. When students’ mathematical thinking is based on misconceptions, predictable patterns of error are often revealed. Misconceptions can be the result of over-generalization or over-specialization of concepts, mistranslations, or limited conceptions (Ashlock, 1998). The term preconception or alternate conception is sometimes used to acknowledge the attempt to make sense of a concept despite any inconsistent reasoning. An example of a misconception is believing that multiplication always results in quantities that are larger than the factors being multiplied. To address this misconception, the student may need to reconceptualize multiplication as repeated groups, to see groups of numbers less than one whole (e.g. 3 x ¼ = ¼ + ¼ + ¼ = 3/4) revealing a smaller product. Simply correcting a student’s mathematical mistake may not be sufficient if the mistake is indicative of an error pattern based on a misconception. To undo the misconception, the teacher must offer ongoing conceptual experiences to convince students of the validity of the concept, with frequent formative assessments that continue the monitoring of student understanding (Keeley, 2011). Sometimes, the cognitive conflict that occurs when the result of an incorrect procedure does not make sense (e.g. ½ + ½ = 2/4 = ½) can motivate the student to seek resolution of the misunderstanding. Misconceptions “interfere with learning when students use them to interpret new experiences …. Students become emotionally and intellectually attached to their misconceptions because they have actively constructed them. Hence, they find it difficult to accept new concepts which are unfamiliar and dissimilar to their misconceptions” (Mohyuddin & Khalil, 2016, p. 135). Teachers need to approach instruction of new material by engaging students in exploratory activities that allow prior knowledge to be exposed and utilized, and misconceptions to be revealed and challenged. Exploration and discussion allow students to refine their thinking, which is a more productive goal than expecting students to replace their misconceptions by a teacher’s correction (Smith et al., 1994). Teaching with multiple representations of a concept will offer more connections to prior knowledge, while reinforcing prerequisite concepts that may have been misunderstood. (KKM) Ashlock, R. B. (1998). Error patterns in computation. Upper Saddle River, NJ: Prentice-Hall. Keeley, P. (2011). Uncovering student ideas: Formative assessment overview. Retrieved from http://uncoveringstudentideas.org/about/overview Mohyuddin, R. G., & Khalil, U. (2016). Misconceptions of students in learning mathematics at primary level. Bulletin of Education and Research, 38(1), 133-162. Smith, J. P., diSessa, A. A., & Roschelle, J. (1994). Misconceptions reconceived: A constructivist analysis of knowledge in transition. Journal of the Learning Sciences, 3, 115-163.

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Model-Eliciting Activities (MEA’s) are real-word problem solving scenarios in which solutions are not readily apparent and have sufficient complexity to warrant the construction of a conceptual model as a solution framework. Originally studying ‘thought-revealing’ activities to examine student thinking, researchers recognized that students would inevitably craft a mathematical model to represent the problem and its solution, thus prompting the term, model-eliciting activity. MEA’s are typically for small group engagement over the course of one or two class periods. The six principles for productive MEA’s (Lesh et al., 2000) are: 1. Model Construction Principle – Given a problem, students create a mathematical model to interpret, conceptualize, and explain the problem, and use it to predict based on underlying patterns within the model. Lesh et al. (2000) define model as a system that consists of elements, relationships among elements, operations that describe how the elements interact, and pattern or rules that allow a problem to be mathematicized. A mathematically significant model may go through several re-conceptualizations as students revise and refine throughout the problem solving process. 2. Reality Principle – A problem should be based on realistic situations and data with an authentic goal as if a company is in need of a solution, and has hired mathematically-capable students for that purpose. 3. Self-Assessment Principle – Working in problem solving teams, students assess their ongoing process to ensure a productive and applicable solution by comparing and testing a variety of ideas, evaluating strengths and weaknesses, selecting, revising and refining the most relevant and promising, and assessing and possibly adapting the final product. Acceptable solutions may go through several modeling cycles, thus continuous reflection is important. 4. Construct Documentation Principle – The development of the conceptual model (construct) for an MEA must be thoroughly documented. This demonstrates the students’ ability to articulate the process and rationale behind their model. 5. Construct Shareability and Reusability Principle – A productive MEA will elicit a model that can be generalized and applied to other problems, with a goal of having students develop generalized ways of thinking, and view mathematical models as conceptual tools that can be transported to new problems. 6. Effective Prototype Principle – MEA’s and their solutions can provide memorable contexts for discussing key mathematical ideas. Creating a model that is structurally significant yet has elegant simplicity, makes an effective prototype that is easily referenced when solving other problems. (KKM) Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thought-revealing activities for students and teachers. In A. Kelly, & R. Lesh (Eds.), Research design in mathematics and science education (pp. 591-646). Mahwah, NJ: Lawrence Erlbaum Associates.

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Multiple Embodiment refers to the instructional strategy of providing students with multiple concrete representations of a single concept to help build conceptual understanding. The multiple embodiment principle was proposed by mathematician and educational psychologist, Zoltan Dienes, as an instructional tool for developing mathematical understanding. Supportive of mathematics learning as an investigative endeavor akin to laboratory learning, Dienes suggests that students should interact with multiple concrete representations to explore and discover mathematical concepts, but that other factors need to be attended to for the use of manipulatives to result in conceptual understanding. “Multiple embodiment requires that mathematical concepts be developed in perceptually different situations” (Reys, 1972, p. 490). A mathematical concept can be more readily abstracted when students experience a variety of materials, models, and contexts that embody the same concept, and are challenged to recognize the structural similarities between them. For instance, if a student's experience with ¼ is limited to working with fraction circles and sectors of round pizza, a student may not recognize ¼ of a rectangular cake or ¼ of a dozen eggs. Having multiple embodiments of ¼ allows students to see the concept of ¼ as independent from the individual materials. Lesh et al. (1987) describe other principles set forth by Dienes that interrelate with the multiple embodiments principle: x

x

x

The Constructive Principle – as students experience a task using concrete materials, they construct their mathematical ideas not directly from the use of the materials, but by the organizational, operational, and relational systems that they impose on the materials as they explore. The mental activities associated with this allows the abstraction necessary to discover underlying mathematical structure. The Dynamic Principle – the mathematical structure behind the models becomes more apparent when the system undergoes change, and the transformation is recognized across other models. The Perceptual Variability Principle –the various models should be perceptually different including a variety of irrelevant attributes or distractors, different in each model. This necessitates the search for the significant features that are constant to all models and manipulations, allowing for the abstracting of mathematical concepts. “Collectively, models should illustrate all of the most important structural characteristics of the modeled system” (Lesh et al., 1987, p. 652). (KKM)

Lesh, R., Post, T., & Behr, M. (1987). Dienes revisited: Multiple embodiments in computer environments. In I. Wirsup & R. Streit (Eds.), Development in school mathematics education around the world (pp. 647-680). Reston, VA: National Council of Teachers of Mathematics. Reys, R. E. (1972). Mathematics, multiple embodiment, and elementary teachers. The Arithmetic Teacher, 19(6), 489-493.

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Multiplicative Reasoning is a type of mathematical thinking associated with multiplication and division situations. Multiplicative reasoning is sometimes contrasted with additive reasoning. “In contrast to additive reasoning, in which quantities of the same type are added or subtracted (e.g., 2 cookies and 3 cookies are 5 cookies altogether), multiplicative reasoning involves quantities of different types (e.g., 3 boxes with 4 cookies per box means 12 cookies altogether)” (Bakker et al., 2014, p. 60). Additive reasoning has been found to be useful in some multiplicative contexts involving whole number and fraction operations (Carpenter et al., 2015; Empson & Levi, 2011). More often, multiplicative reasoning has been associated with mathematical understandings elicited from proportional situations. Proportional situations involve the use of multiplicative reasoning strategies. Lamon (1993) categorized multiplicative strategies for solving proportion problems including preproportional, qualitative, and quantitative levels. Additional detailed analyses of both problem type structures and number combinations revealed students’ use of multiplicative reasoning strategies for solving proportion problems (Steinthorsdottir, 2006). Contextualized situations in which the quantities are represented in ways that support multiplicative reasoning have been found to facilitate students’ problem solving processes with fraction and proportion problems (Kent et al., 2002). Steinthorsdottir and Sriraman (2009) further analyzed students’ multiplicative reasoning strategies for solving proportion problems by delineating differences in ways that students respond to difficulty levels of within and between measure multiplicative relationships. (LBK) Bakker, M., van den Heuvel-Panhuizen, M., & Robitzsch, A. (2014). First-graders’ knowledge of multiplicative reasoning before formal instruction in this domain. Contemporary Educational Psychology, 39(1), 59-73. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., Empson, S. B. (2015). Children’s mathematics: Cognitively guided instruction (2nd ed.). Portsmouth, NH: Heinemann. Empson, S. B., & Levi, L. (2011). Extending children’s mathematics: Fractions and decimals. Portsmouth, NH: Heinemann Kent, L. B., Arnosky, J., & McMonagle, J. (2002). Using representational contexts to support multiplicative reasoning. In B. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios, and proportions: NCTM 2002 Yearbook (145-152). Reston, VA: NCTM. Lamon, S. J. (1993). Ratio and proportion: Connecting content and children's thinking. Journal for Research in Mathematics Education, 24(1), 41-61. Steinthorsdottir, O. B. (2006, July). Proportional reasoning: Variable influencing the problems difficulty level and one’s use of problem solving strategies. In Proceedings of the 30th conference of the international group of Psychology of Mathematics Education (Vol. 5, pp. 169-176). Steinthorsdottir, O. B., & Sriraman, B. (2009). Icelandic 5th-grade girls’ developmental trajectories in proportional reasoning. Mathematics Education Research Journal, 21(1), 6-30.

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National Assessment of Educational Progress (NAEP) is a nationally administered assessment given periodically to U.S. students in grades 4, 8 and 12 in multiple subjects (including mathematics) with the goals of not only measuring what students know and can perform at specific instances of time but also to examine student learning over time. NAEP was created in the 1960s as an effort to better understand what students were learning as a result of their schooling. The standardized tests of the period provided data for the comparison and categorization of students but offered little evidence regarding what students were actually learning. Francis Keppel, the U.S. Commissioner of Education from 1962 to 1965, led a series of conferences to discuss methods of collecting such data. As a result, the Exploratory Committee on Assessing the Progress of Education (ECAPE) was formed to create a program that would measure the knowledge, understanding, skill and attitudes of students at four age levels (9, 13, 17 and adults) for ten learning areas (Carpenter et al., 1978). The first NAEP assessments occurred in 1969 in the areas of science, writing, and citizenship, with the first mathematics exams occurring during the 1972-73 school year. Shortly after its inception, responsibility for NAEP was transferred to what is now the U.S. Department of Education. Presently the Commissioner of Education Statistics leads the NAEP program, and the National Assessment Governing Board designs the policy and assessment specifications for NAEP (NCES, 2017). Through 2017, NAEP has been administered in mathematics a total of 15 times, with the test given every other year since 2003. The assessment for adults was dropped, with the focus now aimed at students in the 4th, 8th, and 12th grades. In 2011-12, approximately 26,200 students completed the NAEP mathematics exam. NAEP undergoes only small, documented changes in order to stay roughly the same across administrations and provide a longitudinal picture of academic progress (NCES, 2017). Information on demographic, curricular, and instructional backgrounds from schools, students, and teachers are also collected to provide additional information and context to the results (Silver & Kenney, 2000). Over time, NAEP has been used to track academic progress and highlight areas of strength and concern. For example, the average math scores for 4th and 8th grade students steadily improved from the first administration in 1972-73 to the 201112 school year, while the scores for 12th grade students over the same period remained flat. The achievement gap between white and black students narrowed between 1972-73 and 2011-12 for all three grades. (NCES, 2016). (SWD) Carpenter, T., Coburn, T. G., Reys, R. E., & Wilson, J. W. (1978). Results from the first mathematics assessment of the national assessment of educational progress. Reston, VA: National Council of Teachers of Mathematics. National Center for Education Statistics (NCES). (2016). Reading and mathematics score trends. Retrieved from https://nces.ed.gov/programs/coe/pdf/coe_cnj.pdf National Center for Education Statistics (NCES). (2017). NAEP overview. Retrieved from https://nces.ed.gov/nationsreportcard/about/ Silver, E. A., & Kenney, P. A. (2000). Results from the seventh mathematics assessment of the national assessment of educational progress. Reston, VA: NCTM.

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NCTM Standards are the series of publications by the National Council of Teachers of Mathematics (NCTM) from 1989 to 2000 that outlined a vision for mathematics education through a series of goals for curriculum, teaching, and assessment. Beginning in the 1980s, NCTM took an increased leadership role in improving mathematics education through a series of publications designed around the use of standards to articulate reforms for school mathematics. The first publication was the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989), which articulates “a coherent vision of what it means to be mathematically literate” as well as develops “a set of standards to guide the revision of school mathematics curriculum and its associated evaluation toward this vision” (p. 1). The publication prescribes learning goals concerning various topics and skills for the K-4, 5-8, and 9-12 grade bands, as well as benchmarks to evaluate student learning and curriculum programs. Next came the Professional Standards for Teaching Mathematics (NCTM, 1991), which outlines goals for improving and evaluating mathematics teaching as well as the ongoing education and professional development of mathematics teachers. NCTM then published Assessment Standards for School Mathematics (NCTM, 1995), which provides guidance for assessment policies to support the reforms in teaching and learning specified in the prior publications. NCTM intended these documents to be part of an ongoing effort to reform the teaching and learning of mathematics, and for these documents to remain relevant, they would need to be revisited, studied, and revised. The revision process combined the efforts and vision of the three standards documents into the Principles and Standards for School Mathematics (NCTM, 2000), which outlines six over-arching principles (Equity, Curriculum, Teaching, Learning, Assessment, and Technology), learning expectations for five content areas (Number & Operations, Algebra, Geometry, Measurement, Data Analysis & Probability), and five processes (Problem Solving, Reasoning & Proof, Communication, Connections, Representation) for the PreK-2, 3-5, 6-8, and 9-12 grade bands. This publication served as an update to the previous standards and helped NCTM usher in a new period of reform for the beginning of the 21st century. There is some ambiguity to the term NCTM Standards due to the various iterations of NCTM standards publications. Some in the mathematics community use the term to refer to the collection of all four NCTM standards publications. However, most use the term to refer directly to Principles and Standards, the most recent standards from NCTM. (SWD) National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics (NCTM). (1991). Professional standards for teaching mathematics. Reston, VA: Author. National Council of Teachers of Mathematics (NCTM). (1995). Assessment standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: Author.

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New Math was the name popularized for the reform movement in the late 1950s and 1960s to radically change the way in which mathematics was taught, emphasizing structure to strengthen understanding of mathematical concepts. The New Math movement had its roots prior to the end of WW II, as NCTM called for mathematics education to adapt to a world that was rapidly changing. The U.S. government issued a similar call with concerns of a shortage of technical expertise needed for national defense. Math education in the 1950s was typified by emphasis on rote procedures, a shortage of qualified math teachers, 2/3 of students leaving math curriculum after one year of high school math, and many of the remaining 1/3 not prepared for college mathematics (Miller, 1990). The 1957 Soviet launch of the first satellite to orbit the earth provided evidence that U.S. math and science education had not kept pace with the Soviets and spurred the National Science Foundation to support reform curriculum. The NSFfunded School Mathematics Study Group, headed by math professor, Edward G. Begle, was the largest and most well-known reform effort, studying, creating, and implementing K-12 curriculum from 1958-1977. Multiple institutions, such as the University of Illinois Committee on School Mathematics headed by Max Beberman, produced a variety of curriculum materials as part of the movement, thus the New Math was not a single curriculum (Roberts & Walmsley, 2003). New topics were emphasized such as set theory, properties of operations, inequalities, non-decimal bases, modular arithmetic, matrices, deduction, and functions. The drive to deepen understanding of mathematics, better understand its structure, and encourage student discovery of concepts was believed to provide an energizing stimulus to mathematical learning (Fremont, 1967). The New Math curricula met with mixed success. Criticism grew in the 1960s as poorly-trained teachers were challenged to teach unfamiliar concepts in unfamiliar ways. High ability and college-bound students were having more success than others. Less emphasis was placed on basic computational skills and procedures, prompting critics to claim that students were not learning math. By the 1970s a back to basics movement arose and funding for projects ended. Although New Math was widely purported to be a failure, many ideas behind it endured, elevating the concept of functions to the core of mathematics curriculum, shifting focus to understanding over memorization, and offering calculus in high school. It provided a necessary shake up to a curriculum that had been static for centuries, and paved the way for NCTM to initiate a new reform movement, manifested in the 1989 NCTM publication of the Curriculum and Evaluation Standards for School Mathematics. (KKM) Fremont, H. (1967). New mathematics and old dilemmas. The Mathematics Teacher 60(7), 715718. Miller, J. W. (1990). Whatever happened to new math? American Heritage, 40(8). Retrieved from https://www.americanheritage.com /content/whatever-happened-new-math-0 Roberts, D. L., & Walmsley, A. L. E. (2003). The original new math: Storytelling versus history. Mathematics Teacher, 96(7), 468-473.

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Non-Anticipatory implies lack of prediction or expectation of a particular outcome. In mathematics education, non-anticipatory has been used to describe problem solving in which a student does not plan ahead in terms of strategy choice. Specifically related to fraction problems, young children initially use nonanticipatory strategies to solve equal-sharing problems (Empson et al., 2006; Hunt & Empson, 2015; Lewis et al., 2015). An example of a non-anticipatory strategy for sharing six brownies equally among nine children might be a student drawing a picture of six squares, dividing five of those into two equal pieces and giving a half to each of the nine children with one brownie and a half brownie leftover. Some young students might stop at that point and say that each child “gets a half of a brownie with 1 ½ brownies leftover for the teacher”. They may not try to partition the remaining 1 ½ squares or they may try fourths and eighths but would continue to have leftover amounts. As children gain confidence and understanding of fractions as quantities, they recognize the common factor of three in both six and nine and divide each of the six brownies into thirds so that each child gets 2/3 of a brownie. In contrast, anticipatory strategies involve planning and goal orientation in problem solving. Simon et al. (2016) describe anticipatory stages of learning as those that involve more abstraction of mathematical relationships. He contrasts the anticipatory stages with participatory stages for problem solving. He characterized the participatory stage as more concrete during early learning experiences involving sequential steps. Anticipatory stage occurs later as part of a reflective process on the ideas and steps of the participatory stage. This is similar to discussions of specific content areas such as students’ development of multiplicative coordination strategies from non-anticipatory strategies for solving equal sharing problems (Empson, et al., 2006). Non-anticipatory is also used in descriptions of advanced mathematics topics such as optimization problems (Gupta et al., 2017) and equations such as those associated with first-kind volterra equations (Lamm, 2000). (LBK) Empson, S. B., Junk, D., Dominguez, H., & Turner, E. (2006). Fractions as the coordination of multiplicatively related quantities: A cross-sectional study of children's thinking. Educational Studies in Mathematics, 63(1), 1-28. Gupta, V., Moseley, B., Uetz, M., & Xie, Q. (2017, June). Stochastic online scheduling on unrelated machines. In International Conference on Integer Programming and Combinatorial Optimization (pp. 228-240). Springer International Publishing. Hunt, J. H., & Empson, S. B. (2015). Exploratory study of informal strategies for equal sharing problems of students with learning disabilities. Learning Disability Quarterly, 38(4), 208220. Lamm, P. K. (2000). A survey of regularization methods for first-kind Volterra equations. In Surveys on solution methods for inverse problems (pp. 53-82). Vienna: Springer. Lewis, R. M., Gibbons, L. K., Kazemi, E., & Lind, T. (2015). Unwrapping students' ideas about fractions. Teaching Children Mathematics, 22(3), 158-168. Simon, M. A., Placa, N., & Avitzur, A. (2016). Participatory and anticipatory stages of mathematical concept learning: Further empirical and theoretical development. Journal for Research in Mathematics Education, 47(1), 63-93.

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Number Sense/Numeracy is the ability to make sense of mathematical concepts and procedures in a manner that makes sense to the learner. The term number sense evolved out of many discussions on what is truly meant by the term numeracy which was coined in 1959 by George Crowther and colleague in the report of the Central Advisory Council for Education (England) (McIntosh, Reys, & Reys, 1992). O’Donoghue (2002) states Crowther’s definition of numeracy as “the mirror image of literacy” (p. 47). This definition is open to interpretation, thus necessitating a more precise or descriptive definition of what is meant by numeracy, leading to the use of the phrase number sense. Thinking of numeracy using Crowther’s definition goes beyond the idea of focusing on number sense. Number sense is a commonly used phrase, yet its definition is rather complex and multifaceted. Number sense involves all the ways in which people make connections amongst numbers and the procedures they apply to numbers based upon their personal experiences (Howden, 1989). In the classroom setting, number sense is typically thought of as the ability of students to identify numerals and connect them with an appropriate corresponding quantity. This very minimal definition of number sense would limit numeracy to be just about numbers and possibly basic computations that can be performed with numbers. While it is true that students need to be able to complete basic mathematical computations, they must also be able to see the purpose of numbers in the world around them, beyond the classroom doors. Numbers are an all-encompassing part of everyday life, and the meaning of the numerals varies based upon the context in which they are situated. For example, in a list of needed materials, the item next to the numeral “1” does not mean that it is the most important or the first one that needs to be purchased; rather, the numeral is used a place marker within the list indicating that it is one of the materials that needs to be purchased. In other cases, the numeral “1” could be indicative of quantity, place value, monetary value, etc. (CCO) Howden, H. (1989). Teaching number sense. The Arithmetic Teacher, 36(6), 6-11. McIntosh, A., Reys, B. J., and Reys, R. E. (1992). A proposed framework for examining basic number sense. For the Learning of Mathematics, 12(3), 2-8. O’Donoghue, J. (2002). Numeracy and mathematics. Irish Mathematical Society Bulletin, 48, 4755.

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Numerical Estimation is typically defined as an approximate calculation. Siegel et al. (1982) described estimation as “a process that gives a rough solution to a problem in counting or measurement” (p. 211). Two different approaches to estimation are benchmark estimation and decomposition/recomposition (Siegel et al., 1982; Montague & van Garderen, 2003). With benchmark estimation, the approach uses a known standard to make an approximation or estimate. For example, the sum of 3/8 and 3/5 could be estimated by recognizing that both fractions are close to ½ so the sum would be close to one. The decomposition/recomposition approach involves using parts of a quantity or measure to estimate. For example, the area of a floor space can be estimated by adding up the number of one square foot tiles that fill up the room. Montague and Garderen (2003) further characterized the decomposition/ recomposition approach as the process by which pieces are of a similar size and a standard can be easily applied, as in estimating the area of a rectangular patio if one is using square-foot tiles. If the item cannot be easily divided into parts or the parts are of different sizes, then irregular decomposition occurs. Sowder and Wheeler (1989) studied estimation strategies of elementary school students. They found that younger children were more likely to estimate addends to find the sums, while older elementary children were more likely to do the exact computation and then round their answers. One of the most famous estimation problems was from the second mathematics assessment of the National Assessment of Education Progress (NAEP). This problem involved estimating the sum of 12/13 and 7/8. The study found that only 37 percent of 17 year olds and 24 percent of 13 year olds could correctly estimate the sum of these two fractions (Carpenter et al., 1980). This result set the stage for increased attention to students’ understanding of fractions over the subsequent decades. (LBK) Carpenter, T., Kepner, H., Corbitt, M., Lindquist, M., & Reys, R. (1980). Results and implications of the Second NAEP Mathematics Assessments: Elementary school. The Arithmetic Teacher, 27(8), 10-47. Montague, M., & Van Garderen, D. (2003). A cross-sectional study of mathematics achievement, estimation skills, and academic self-perception in students of varying ability. Journal of Learning Disabilities, 36(5), 437-448. Siegel, A. W., Goldsmith, L. T., & Madson, C. R. (1982). Skill in estimation problems of extent and numerosity. Journal for Research in Mathematics Education, 13(3), 211-232. Sowder, J. T., & Wheeler, M. M. (1989). The development of concepts and strategies us ed in computational estimation. Journal for Research in Mathematics Education, 20(2), 130-146.

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Pedagogical Content Knowledge (PCK) is the knowledge unique to teachers that connects their mathematical content knowledge with their knowledge of teaching to provide the most effective methods for teaching specific content and skills. The term Pedagogical Content Knowledge (PCK) is widely attributed to Lee Shulman, who in his 1985 presidential address to the American Educational Research Association (AERA) implored his colleagues to consider a more coherent theoretical framework for teacher knowledge and encouraged them to examine the relationship between general content knowledge and pedagogical knowledge (Shulman, 1986). Shulman offered the term PCK, which he described as the most useful forms of representation of those ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations – in a word, the ways of representing and formulating the subject that make it comprehensible to others …. Pedagogical content knowledge also includes an understanding of what makes the learning of specific topics easy or difficult: the conceptions and preconceptions that students of different ages and backgrounds bring with them to the learning of those most frequently taught topics and lessons. (p. 9) The concept of PCK pre-dates Shulman, as scholars such as John Dewey had urged teachers to “psychologize” the subject being taught to make the content more accessible to students (Grossman, 1990). Shulman’s definition of PCK underscores the important connection between content knowledge and pedagogical knowledge and its unique nature to mathematics teaching. PCK separates mathematics teachers from mathematicians or engineers in that a teacher’s understanding of mathematics needs not only to focus on its structure and applicability but also on how it can be organized and made understandable to those with novice abilities. Grossman (1990) breaks PCK into four components: (1) knowledge and beliefs about the purposes for teaching a subject at different grade levels; (2) knowledge of students’ understanding, conceptions, and misconceptions of particular topics in a subject matter; (3) knowledge of curriculum materials available for teaching particular subject matter; and (4) knowledge of instructional strategies and representations for teaching particular topics. Ball et al. (2008) outline PCK as one component of Mathematical Knowledge for Teaching (MKT), where PCK includes a teacher’s knowledge of (1) content and students; (2) content and teaching; and (3) content and curriculum. (SWD) Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407. Grossman, P. (1990). The making of a teacher: Teacher knowledge and teacher education. New York, NY: Teachers College Press. Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.

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Performance Based Assessments are authentic assessments in which students display the knowledge and skills they have acquired by performing tasks or presenting a product. Wiggins (1989) describes authentic assessments as “representative challenges within a given discipline. They are designed to emphasize realistic (but fair) complexity; they stress depth more than breadth. In doing so, they must necessarily involve somewhat ambiguous, ill structured tasks or problems” (p. 711). In addition to modeling such practices of the discipline, performance assessments commonly include open-ended and context-specific tasks, higher order thinking, student choice, teacher and public observation of student performance, individual or group assessment, and opportunities for self-assessment. Performance assessments are intended to reflect the complexities inherent in learning and understanding, rather than a more traditional test that may favor one’s ability for written communication or memorization. A performance assessment can provide a more equitable representation of student learning by allowing a wider range of skills and knowledge to be presented (Walker & Molisani, 2014). With equity in mind, Darling-Hammond (1994) emphasizes that performance assessment practices implemented by teachers should be supported at the policy level so that the teachers, students, schools, and community can benefit from a process of self-reflection, discussion, and growth stemming from information gathered about student learning. Darling-Hammond outlines how these assessments should be designed, implemented, and used: 1. Access to Educational Opportunity: Assessments are designed to survey possibilities for student growth, not for limitations. 2. Consequential Validity: Assessments are valid based on how they affect change in the teacher's implementation of appropriate learning activities and their responsiveness to a student's individual learning style and needs. 3. Transparency and Openness: The learning goals measured by the assessment must be made clear. 4. Self-Assessment: Students set step-by-step standards for themselves by which they can judge their work as they move forward in the learning process. 5. Socially Situated Assessment: Assessing students in differing situations allows for assessment of how students adapt. 6. Extend Tasks and Contextualized Skills: Assessments tasks are meaningful to the mathematical competency being assessed. 7. Scope and Comprehensiveness: Assessments attend to a wide range of performance and learning processes and the targeted content. (SJH & KKM) Darling-Hammond, L. (1994). Performance-based assessment and educational equity. Harvard Educational Review, 64(1), 5-30. Walker, E. T. & Molisani, J. S. (2014). Driving students to performance assessments: What students can do. Mathematics Teaching in the Middle School, 19(8), 468-476. Wiggins, G. (1989). A true test: Toward more authentic and equitable assessment. Phi Delta Kappan, 70(9), 703-713.

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Prior Knowledge is the collection of preexisting ideas, concepts, beliefs, and skills that students have before formal instruction of a topic begins. “Students come to the classroom with preconceptions about how the world works. If students’ initial understanding is not engaged, they may fail to grasp new concepts and information presented in the classroom” (Bransford et al., 2000, p. 14-15). According to Piaget, prior knowledge consists of schemata, organized blocks of knowledge, that need to be activated in order for new knowledge to be assimilated or accommodated (Piaget, 1952). The results of teaching new material may appear different from student to student depending on how they integrate new knowledge into their existing schemata. “Learning is enhanced when preconceived understandings are drawn out” (Fuson et al., p. 261). If the understandings are accurate, the foundation is fertile for learning activities that promote connecting new information. If students hold misconceptions, learning activities are needed that guide learners to reconceptualize. For example, students’ knowledge of fractions as part of a whole may interfere with their making sense of ratios. If a student views 2:3 as the fraction 2/3, building a physical model of the ratio of 2 red blocks and 3 blue blocks, then being challenged to find 2/3 within the model, may clarify the distinction. Numerous examples exist of how people use their own informal strategies to think mathematically in practical everyday applications yet do not perform well when similar problems are given more abstractly, as in a math class. “If there is no bridge between informal and formal mathematics, the two often remain disconnected” (Fuson et al., p. 219). Teachers must help bridge the gap by helping students bring deeper understanding, meaning, and appreciation to what they already know. Influenced by the work of Piaget and interested in the nature of meaning, Ausubel (1960) explains that meaningful learning requires individual sensemaking. New information only acquires meaning if it is processed through the lens of an individual’s existing cognitive structures. He proposed ‘advanced organizers’, such as graphic organizers, to provide an overview of the topic as a strategy to activate relevant prior knowledge and organize it in advance of the lesson. Similarly, anticipatory sets and student-centered exploratory activities engage and ready the learner to integrate new information. Such introductory activities and other preassessments are important to inform teachers of what students already know so that instruction can be modified accordingly. (KKM) Ausubel, D. P. (1960). The use of advance organizers in the learning and retention of meaningful verbal material. Journal of Educational Psychology, 51, 267-272. Bransford, J., Brown, A., & Cocking, R. (2000). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press. Fuson, K. C., Kalchman, M., & Bransford, J. D. (2005). Mathematical understanding: An introduction. In M. S. Donovan & J. D. Bransford (Eds.), How students learn mathematics in the classroom (pp. 217 – 276). Washington, DC: National Academies Press. Piaget, J. (1952). The origins of intelligence in children. New York, NY: International University Press, Inc.

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Problem Based Learning (PrBL) is a curricular approach in which ill-structured problems are posed and students are empowered to explore and articulate solutions, learning and utilizing mathematical content and practices along the way. A problem based approach challenges students to “make sense of mathematical situations for which no well-defined routines or procedures exist” (Erickson, 1999, p. 516). Larmer (2015) suggests typical steps of problem based learning as: x x x x x x

presentation of an ‘ill-structured’, multi-faceted problem problem definition or formulation generation of a ‘knowledge inventory’ (known and ‘need to know’) generation of strategies, representations, conjectures, generalizations, and solutions formulation of learning issues for self-directed and coached learning sharing of findings and solutions with explanations and justifications

Savery (2006) emphasizes the self-directed and self-regulated role of the student, the motivational aspect of engaging in inquiry where students are more invested in their own solutions, and the essential processes of collaboration, reanalysis, and reflection. Larmer (2015) describes problem based learning as a subset of project based learning and reports the tendency within math education to associate problem based as a more effective way to cover content standards than project based. He places both approaches under the broader category of inquiry based learning and acknowledges that variation in the use of these terms exists. For instance, a school in Boaler’s study (2002) described its curriculum as project based yet by recent definitions may more closely resemble problem based, as a tangible product or performance to a public audience was not an emphasis, nor were the problems always in a real-world context (see Project Based Instruction). The findings of this study, however, generally inform of the benefits of inquiry based learning, reporting that students who engaged in project or problem based inquiry had similar scores on procedural tasks as compared to the students from traditional math classes, but performed better on conceptual tasks, had better retention over time, greater ability to apply mathematics and transfer to other contexts, greater confidence and enjoyment as a learner of mathematics, and a broader view of what mathematics is. (KKM) Boaler, J. (2002). Experiencing school mathematics: Traditional and reform approaches to teaching and their impact on student learning. Mahwah, NJ: Lawrence Erlbaum Associates, Inc. Erickson, D. K. (1999). A problem-based approach to mathematics instruction. Mathematics Teacher, 92(6), 516-521. Larmer, J. (2015, July 13). Project-based learning vs. problem-based learning vs. X-BL. Retrieved from https://www.edutopia.org /blog/pbl-vs-pbl-vs-xbl-john-larmer Savery, J. R. (2006). Overview of problem-based learning: Definitions and distinctions. Interdisciplinary Journal of Problem-Based Learning, 1(1), 9-20. Retrieved from doi.org/10.7771/1541-5015.1002

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Problem Solving Heuristics are a general set of strategies and principles that could be used to solve a variety of mathematics problems. The mathematician, George Polya (1965, 2014), identified five strategies for solving problems (draw a picture, make a table, use an easier example, guess and check, look for a pattern) as well as four principles for successful problem solving (understand the problem, devise a plan, carry out a plan, and look back). Over the past decades, these heuristics and principles have been modified, expanded, and adapted to address changes in mathematics teaching and learning based on advances in cognitive science, technology, and curriculum. Most recently, the Common Core standards document included several of Polya’s original heuristics and principles in the eight Standards for Mathematical Practice including: make sense of problems and persevere in solving them, and look for and express regularity in repeated reasoning (NGA, 2010). Schoenfeld (1992) elaborated and further refined aspects of mathematics problem solving by discussing the influence of cognitive research, pedagogy, and curriculum on students’ problem solving approaches. He also synthesized different views of problem solving by describing problem solving in terms of teaching and learning goals, from completing routine exercises to the work of mathematicians involving problems of significant complexity. (LBK) National Governors Association Center for Best Practices (NGA), Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Author. Retrieved from http://www.corestandards.org/ Polya, G. (1965). Mathematical discovery: On understanding and teaching problem solving. New York, NY: John Wiley. Polya, G. (2014). How to solve it: A new aspect of mathematical method. Princeton, NJ: Princeton University Press. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook for research on mathematics teaching and learning (pp. 334-370). New York, NY: MacMillan.

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Problem Structure refers to the underlying operational and semantic aspects of a mathematics problem. Historically, problem structure has been used in mathematics education to describe components of word problems (Clement, 1982; De Corte & Verschaffel, 1987; Greer, 1997). From a curriculum perspective, mathematics textbooks were originally designed in a sequence based on covering operational aspects of problems. For example, addition was covered before subtraction, multiplication before division, etc. However, Greer (1997), for example, identified problems with an overemphasis on this approach. He described several scenarios in which students would extract the numbers from a word problem and perform several operations and choose the answer that they thought made the most sense. The operational components of mathematics problems are open to interpretation. For example, they could encompass the underlying operation (+, о, x, ÷) of a problem. Within this type of analysis, the underlying structure has been used to identify taxonomies of problems based on the unknown (Carpenter et al., 1999). For example, a Join Result Unknown problem is distinct from a Join Change Unknown or Join Start Unknown problem by whether the unknown is the total or one of the addends is unknown. The development of these types of structural taxonomies and cognitive science research refocused attention on the entire structure of a problem versus superficial features such as key words. Other research has addressed not only operational and structural issues but also the role of the unit as in the case of rational number concepts and constructs (Behr et al., 1992; Lamon, 2002). For example, Lamon (2002) described unitizing strategies as important for students’ developing proportional reasoning strategies. Other research has examined the differences between novice and experts in terms of interpreting and solving problems with various structures (Schoenfeld & Hermann, 1982). (LBK) Behr, M. J., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 296-333). Reston, VA: National Council of Teachers of Mathematics. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann. Clement, J. (1982). Algebra word problem solutions: Thought processes underlying a common misconception. Journal for Research in Mathematics Education, 13(1), 16-30. De Corte, E., & Verschaffel, L. (1987). The effect of semantic structure on first graders' strategies for solving addition and subtraction word problems. Journal for Research in Mathematics Education, 18(5), 363-381. Greer, B. (1997). Modelling reality in mathematics classrooms: The case of word problems. Learning and Instruction, 7(4), 293-307. Lamon (2002). Part-whole comparisons with unitizing. In B. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios, and proportions (pp. 79-86). Reston, VA: National Council of Teachers of Mathematics. Schoenfeld, A. H., & Herrmann, D. J. (1982). Problem perception and knowledge structure in expert and novice mathematical problem solvers. Journal of Experimental Psychology: Learning, Memory, and Cognition, 8(5), 484.

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Procedural Knowledge refers to the skills, understandings, and set of rules and conventions needed to utilize mathematical symbols and syntax to perform mathematical procedures. One of the five strands of mathematical proficiency calls for students to have procedural fluency, defined as an ability to accurately, appropriately, efficiently, and flexibly carry out procedures (NRC, 2001). The knowledge of procedures that leads to such fluency has been actively debated in the literature, with a consensus that procedural knowledge is much more than skills learned by rote and has important links with conceptual understanding. One component of procedural knowledge that is foundational to mathematics is the knowledge of the formal language, or symbol representation system, and syntax for expressing mathematics. A second type of procedural knowledge consists of the “rules or procedures for solving mathematical problems …. It is the clearly sequential nature of procedures that probably sets them apart from other forms of knowledge” (Hiebert & Lefevre, 1986, p. 7). Symbols develop meaning when they are connected to the conceptual knowledge they represent. Understanding the underlying concepts behind procedures make the procedures more meaningful, easier to retrieve, easier to apply in other contexts, and less prone to error (Hiebert & Lefevre, 1986). “It is an iterative process. Procedures are taught that can be supported by existing conceptual knowledge, and the conceptual knowledge base is extended to provide a basis for developing more advanced concepts” (Carpenter, 1986, p. 130). Describing a mutually dependent relation between conceptual and procedural knowledge, Baroody et al. (2007) suggest that adaptive expertise is attained when the two types of knowledge are well-connected. Skemp (1978) refers to students who have connected procedures to a conceptual base as having relational understanding, as opposed to instrumental understanding which enables only rote performance of a procedure. Yet “procedures can be known deeply, flexibly, and with critical judgment …. Procedural knowledge is valuable in and of itself, not solely because of its connections with and integration to conceptual knowledge” (Star, 2007, pp. 133-134). (KKM) Baroody, A. J., Feil, Y., & Johnson, A. R. (2007). An alternative reconceptualization of procedural and conceptual knowledge. Journal for Research in Mathematics Education, 38(2), 115-131. Carpenter, T. P. (1986). Conceptual knowledge as a foundation for procedural knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 113132). Hillsdale, NJ: Erlbaum. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1-27). Hillsdale, NJ: Erlbaum. National Research Council (NRC). (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, & B. Findell (Eds.). Washington, DC: National Academy Press. Skemp, R. R. (1978). Relational understanding and instrumental understanding. The Arithmetic Teacher, 26(3), 9-15. Star, J. (2007). Foregrounding procedural knowledge. Journal for Research in Mathematics Education, 38(2), 132-135.

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Productive Struggle is the intellectual effort students expend to make sense of mathematical concepts that are challenging but fall within the students’ reasonable capabilities. The fundamental characteristics of productive struggle in the process of learning and understanding have long been an area of concern and interest in mathematics education research. For John Dewey, the process of this struggle was essential for constructing deep understandings: The process begins with some perplexity, confusion, or doubt. It continues as students try to fit things together to make sense of them, to work out methods for resolving the dilemma. Deep knowledge of the subject is the fruit of the undertakings that transform a problematic situation into a resolved one. (Hiebert & Grouws, 2007, p. 388) Polya echoed many themes of Dewey’s philosophy by stating: A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings to play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. (Hiebert & Grouws, 2007, p. 388) In fact, many mathematicians and those who are concerned about how students interact with the discipline of mathematics have alluded to struggling with key ideas as a natural part of doing mathematics. The writing team for Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014) identifies the support of productive struggle in learning mathematics as one of the eight important teaching practices (Warshauer, 2015). Hence, determining what kinds of activities or tasks that support productive struggle has also been a significant area of research in mathematics education. Vygotsky’s learning theory regarding the Zone of Proximal Development (ZPD) provides a guide for answering the question of what types of activities support the productive struggle in learners. It does so because the ZPD is often understood to be the space within which a student’s struggle is likely to be beneficial and productive. Henningsen and Stein (1997) sought to investigate which classroom factors support or hinder students’ engagement in high-level mathematical thinking and reasoning. They found the following aspects of Vygotsky’s learning theory that supported students maintaining high-level thinking and reasoning: 1. 2. 3. 4. 5.

Tasks build on students’ prior knowledge; Scaffolding; Appropriate amount of time for students to being doing activity; Sustained pressure for explanation and meaning; High-level performance modeled by someone else.

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It is not only the kind of tasks that a student does that can support productive struggle through an activity but also the kind of support the student receives from the teacher that allows the student to continue to struggle productively. Warshauer (2014) found that teachers had four “typical” responses to students who were struggling: telling, directed response, probing guidance, and affordance. Telling involves the teacher providing the student more than sufficient information to overcome their current struggle, hence lowering the cognitive demand from what was intended. Directed guidance refers to when the teacher eliminates the possibilities for what students could do by having them focus on the teachers’ thinking and not their own. This is done by breaking down problems into smaller pieces or altering the problems so that instead of being algebraic they became numerical. This type of response ultimately lowered or maintained the cognitive level desired compared to what was desired. Probing guidance is the process of reverting to the student’s thinking by building on their original thinking through asking for reasons and justifications of their work. Probing guidance led to a maintained level of cognitive demand when compared to what was originally intended. Lastly, the affordance type of teacher response is characterized by the teacher giving the student enough or more time to attend to their own thinking while also motivating them to continue to work on their task, which ultimately led to a maintained or increased level of cognitive demand from what was originally expected. (SJH) Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroombased factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524-549. Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on student learning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371-404). Charlotte, NC: Information Age Publishing. Warshauer, H. K. (2014). Productive struggle in middle school mathematics classrooms. Journal of Math Teacher Education, 18(4), 375-400. Warshauer, H. K. (2015). Strategies to support productive struggle. Mathematics Teaching in the Middle School, 20(7), 390-393.

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Professional Development (PD) refers to the continued training and educational experiences in which teachers participate that promote ongoing teacher learning and skill development. As new knowledge regarding mathematics teaching and learning is gained, it is necessary to consider how that knowledge can serve to inform and influence teacher practice. Teachers, as with any profession, must continue to learn and adapt to changing student needs. Teacher PD and teacher learning provide settings in which teachers can continue to improve their teaching and gain knowledge. To some, PD is seen as “inservice” days during the school year, where teachers are brought together to learn about a specific topic. Others may see PD as graduate coursework, where hours are used to earn an advanced degree or to satisfy requirements for continuing education (Guskey, 2000). These, however, are but a few of the experiences that may qualify as PD, which Guskey (2000) defines as “those processes and activities designed to enhance the professional knowledge, skills, and attitudes of educators so that they might, in turn, improve the learning of students” (p. 16). Loucks-Horsley et al. (2010) provide a list of PD strategies that could be used to promote teacher learning, including lesson study, curriculum topic study, content courses, action research, coaching, and mentoring. DarlingHammond et al. (2017) analyzed 35 studies of PD and found that for PD to be effective, it must include some or all of the following facets: (1) be content focused; (2) incorporate active learning embedded in adult learning theory; (3) support collaboration in job-embedded contexts; (4) use models and modeling of effective practice; (5) provide coaching and expert support; (6) offer opportunities for feedback and reflection; and (7) is of sustained duration (p. 1). There is a vast array of PD programs for mathematics teacher training. The Cognitively Guided Instruction (CGI) PD uses research on student learning to inform instructional practices for elementary teachers. The work surrounding CGI has sprouted PD opportunities for primary teachers (grades 3-5) called Extending Children’s Mathematics focused on student learning of fractions and decimals, and for middle grades teachers (grades 6-8) called Thinking Mathematically. Other PD programs center around the use of a particular curriculum (e.g., the Connected Mathematics Project workshops held at Michigan State University), while others grow from a funding agency that promotes connections across disciplines (e.g., the U.S. Department of Education’s Math-Science Partnerships (MSP), or the National Science Foundation STEM-C program, which promote connections between the STEM disciplines and computer science). (SWD) Darling-Hammond, L., Hyler, M. E., & Gardner, M. (2017). Effective teacher professional development. Learning Policy Institute Research Brief. Retrieved from https://learningpolicyinstitute.org/sites/default/files/product-files/ Effective_Teacher_Professional_Development_BRIEF.pdf Guskey, T. R. (2000). Evaluating professional development. Thousand Oaks, CA: Corwin Press. Loucks-Horsley, S., Stiles, K. E., Mundry, S., Love, N., & Hewson, P. W. (2010). Designing professional development for teachers of science and mathematics (3rd ed.). Thousand Oaks, CA: Corwin Press.

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Professional Organizations in Mathematics Education are the various groups of stakeholders – university faculty and students, state policymakers, K-12 teachers and curriculum specialists, etc. – tasked with providing guidance to the field as new knowledge is created and new issues are confronted. In the late 1800s, as the field of mathematics education began to grow in terms of both its prominence in K-12 education as well as the number of stakeholders working in this area, professional organizations began to emerge as the venue by which ideas and research could be shared. Today these organizations provide leadership on a variety of issues, from the composition of the school mathematics curriculum to the education and preparation of mathematics teachers. These organizations provide members an outlet by which to express their thoughts, share research, and learn from other members regarding advances in mathematics education. These professional organizations include the following: National Council of Teachers of Mathematics (NCTM): Founded in 1920, NCTM is the world’s largest organization devoted to mathematics education, with over 80,000 members. NCTM has provided leadership to the mathematics education field in a number of ways, most notably in the development of a series of standards documents (see NCTM Standards) as well as by holding a number of conferences each year. In addition, NCTM publishes five practitioner and research-based journals (Journal for Research in Mathematics Education; Teaching Children Mathematics; Mathematics Teaching in the Middle School; Mathematics Teacher; Mathematics Teacher Educator, which is a joint effort with AMTE). National Council of Supervisors of Mathematics (NCSM): Founded in 1968, NCSM provides a venue for leaders in mathematics education to further their professional growth and to discuss issues pertaining to mathematics teaching and learning. NCSM provides leadership to the field by hosting an annual conference for members and publishing the Journal of Mathematics Education Leadership. Association of Mathematics Teacher Educators (AMTE): Founded in 1991, AMTE aims to support and improve mathematics teacher education both at the preservice and inservice level. AMTE has provided leadership in this arena in a number of ways, including the publication of the Standards for Preparing Teachers of Mathematics document, hosting an annual conference, and jointly publishing the Mathematics Teacher Educator with NCTM. Psychology of Mathematics Education (PME): PME began in 1976 as an organization devoted to the psychological components of mathematics teaching and learning. PME works to provide venues for researchers to discuss research and share scientific studies pertaining to work in mathematics education. PME hosts an annual international conference and has commissioned a number of research publications. In addition, regional organizations such as the North American

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chapter of the PME offer venues for researchers to convene and discuss or collaborate on mathematics education research. American Educational Research Association (AERA): Founded in 1916, AERA is comprised mostly of education researchers working to improve various disciplines through research and scholarly work. AERA spans a number of disciplines beyond mathematics education. However, AERA also provides guidance to the field through an annual conference as well as the publication of six research journals (Educational Researcher; American Educational Research Journal; Educational Evaluation and Policy Analysis; Review of Educational Research; Review of Research in Education; and Journal of Educational and Behavioral Statistics, which is a joint effort with the American Statistical Association). International Commission on Mathematical Instruction (ICMI): Founded in 1908, ICMI is a worldwide institution focused on improving mathematics teaching and learning P-16. Currently a commission of the International Mathematical Union, ICMI is charged with hosting the International Congress on Mathematics Education (ICME) every four years. ICMI also hosts a number of other conferences and produces publications such as the ICMI Bulletin. Research Council on Mathematics Learning (RCML): Founded in 1974, RCML brings together educators to discuss and share ideas regarding the learning of mathematics. RCML initially focused on bringing together researchers with similar interests in order to form collaborative working teams to study issues related to mathematics teaching and learning. RCML hosts an annual conference, publishes a newsletter (Intersection Points) for its membership, and publishes Investigations in Mathematics Learning. Mathematical Association of America (MAA): Founded in 1915, the MAA is predominantly focused on research in mathematics, particularly at the undergraduate level. However, MAA has also provided leadership during its history for issues pertaining to mathematics education, including professional development for its membership. MAA provides leadership through an annual conference and a number of journal publications. American Mathematical Society (AMS): Founded in 1888, the AMS is one of the oldest professional organizations devoted to mathematics and mathematics education. It is mostly comprised of mathematicians devoted to mathematics research and training, though as with the MAA, the AMS has also provided leadership on issues related to mathematics education. The AMS hosts an annual conference with the MAA and publishes a large number of journals for its membership. (SWD)

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Program for International Student Assessment (PISA) is an examination given every three years to 15-year olds in more than 70 countries around the world in order to measure students’ reading, mathematics and science knowledge. The Program for International Student Assessment (PISA) was initiated in 2000 as a way to measure student achievement and assess various educational practices on an international scale. Coordinated by the Organization for Economic Cooperation and Development (OECD) and led by the National Center for Education Statistics in the U.S., the test measures reading, mathematics, and scientific knowledge for a representative sample from each country, with the major domain of study rotating between these three subjects in each cycle (NCES, 2017). The sixth administration of PISA was in 2015, where over half a million students in 72 countries took the two-hour, computer-based test (OECD, 2017). In addition to the subject-specific assessment, PISA also provides data on interdisciplinary skills, such as financial literacy and collaborative problem solving. In the 2015 PISA that focused on science, the U.S. performed below average in mathematics in comparison to the other countries that participated. The U.S. average score in mathematics fell 11 points from the 2012 PISA test, on which the U.S. also performed below average. The U.S. rankings for science and reading were average and did not represent significant change from past PISA administrations (OECD, 2016). The 2003 and 2012 PISA tests focused on mathematics, and in 2012 the U.S. ranked 27th out of the 34 OECD countries in this category. Results from 2012 PISA indicated that just over 1 in 4 U.S. students do not reach the PISA baseline for mathematics proficiency, and that the U.S. has a below-average share of top performers. The results also show that U.S. students are strong in interpreting mathematical results and working with algebraic concepts yet struggle in performing high-cognitive demand tasks of translating, interpreting and solving real world problems, and in working with geometry-related topics (OECD, 2012). Analysis of additional background features of the U.S. education system suggested that SES background had a significant impact on U.S. student performance, and that U.S. education spending did not necessarily translate to better performance (OECD, 2012). (SWD) National Center for Education Statistics (NCES). (2017). Overview of PISA. Retrieved from https://nces.ed.gov/surveys/pisa/ Office for Economic Cooperation and Development (OECD). (2012). Programme for international student assessment (PISA) results from PISA 2012: United States. Retrieved from https://www.oecd.org/unitedstates/PISA-2012-results-US.pdf Office for Economic Cooperation and Development (OECD). (2016). Key findings from PISA for the United States. Retrieved from https://www.oecd.org/pisa/PISA-2015-United-States.pdf Office for Economic Cooperation and Development (OECD). (2017). What is PISA? Retrieved from http://www.oecd.org/pisa/aboutpisa/

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Project Based Learning (PBL) is a curricular approach in which students engage in projects of a complex nature driven by an authentic question or problem, as a medium for learning standards-based content and developing inquiry skills. “PBL is the central framework upon which the teaching and learning of core concepts is built, not a supplementary enrichment activity …. Students are pulled through the curriculum by a driving question or authentic problem that creates a need to know the material” (Markham et al., 2003, p. viii, p. 5). Although PBL implementation in classrooms may vary, a comprehensive model has been developed as a set of guidelines to emulate, called “Gold Standard PBL” (Larmer et al., 2015). The essential project design elements of this model further clarify how PBL is distinct from what may be known as ‘doing a class project’: x x x x x x x

Challenging problem or question – a driving question of appropriate challenge that intrigues and focuses the inquiry process in a purposeful pursuit Sustained inquiry – extended, often cyclic, process of asking and investigating questions, finding resources and applying information Authenticity – real-world context, tasks and tools, and personal impact that promote relevance and meaningfulness to students Student voice and choice – autonomy in planning, investigating, and decisionmaking as a necessary key to motivation that supports sustained inquiry Reflection – students reflect on what they have learned and still need to learn, quality of work, and obstacles encountered, then modify accordingly Critique and revision - students give, receive, and use feedback and formative assessment to improve their process and products Public product – the project results in a product to be shared publicly by presenting the results of their work to people beyond the classroom

In the early 20th century, John Dewey advocated experiential learning as a meaningful and motivating approach to learning. As cognitive research emerged providing evidence for how people learn, PBL gained theoretical footings (Krajcik & Blumenthal, 2005). PBL has been found to improve the skills demanded by our technological society, such as problem solving, critical thinking, and collaboration, along with academic results of equal or better achievement and a higher quality of learning (Markham et al., 2003). (KKM) Krajcik, J., & Blumenfeld, P. (2005). Project-based learning. In R. Sawyer (Ed.), The Cambridge handbook of the learning sciences (pp. 317-334). Cambridge, England: Cambridge University Press. Larmer, J., Mergendoller, J., & Boss, S. (2015). Setting the standard for project based learning. Alexandria, VA: Association of Supervisors of Curriculum Development. Markham, T., Larmer, J., & Ravitz, J. (2003). Project Based learning: A guide to standards-focused project based learning. Novato, CA: Buck Institute for Education.

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Proportional Reasoning describes strategies and thinking used to solve proportion problems. Proportional reasoning strategies are potentially evident in young children’s strategies for solving multiplication and division problems. For example, a problem like “How many apples would each child get if there are 15 apples and five children and you want each child to have the same number of apples?” Children as young as kindergarten may solve that problem by using what they know about multiplication and say, “the answer is three apples for each child since five groups of three is fifteen”. While this problem type structure is often called partitive division (Carpenter et al., 2015), it has also been characterized as an associated sets proportion problem (Lamon, 1993). Tournaire and Pulos (1985) synthesized a variety of problem types associated with proportions. Lamon (1993) further narrowed this list to four problem types based on semantic features. One of the important transitions in the development of proportional reasoning is students’ recognition that quantities are compared multiplicatively as opposed to additively. Some have also characterized this as absolute versus relative comparisons (Langrall & Swafford, 2000). Lamon (1993) described additive strategies as non-constructive in solving proportion problems. Other research has noted that even when there is a whole number multiplicative relationship either between measures or within measures in a proportion problem, some students attempt to solve the problem using erroneous additive strategies (see Nielsen et al., 2016, for a recent middle school example). The use of quantitative reasoning strategies to find the fourth value when the other three values of a proportion are known involves either employing the means-extremes property or using another multiplicative strategy to determine the unknown. More recent research has identified aspects such as number combinations in determining students’ success and level of strategy used to solve the proportion problem (Steinthorsdottir & Sriraman, 2009). (LBK) Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., Empson, S. B. (2015). Children’s mathematics: Cognitively guided instruction (2nd ed.). Portsmouth, NH: Heinemann. Lamon, S. (1993). Ratio and proportion: Connecting content and children's thinking. Journal for research in mathematics education, 24(1), 41-61. Langrall, C. W., & Swafford, J. (2000). Three balloons for two dollars: Developing proportional reasoning. Mathematics Teaching in the Middle School, 6(4), 254-261. Nielsen, L., Steinthorsdottir, O. B., & Kent, L. B. (2016). Responding to student thinking: Enhancing mathematics instruction through classroom based professional development. Middle School Journal, 47(3), 17-24. Steinthorsdottir, O. B., & Sriraman, B. (2009). Icelandic 5th-grade girls’ developmental trajectories in proportional reasoning. Mathematics Education Research Journal, 21(1), 6-30. Tourniaire, F., & Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational Studies in Mathematics, 16(2), 181-204.

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Quantitative Literacy (QL), also called Numeracy, is the ability to fluently work with numbers and data when solving real-world problems. QL entails interpreting, reasoning with, and constructing arguments involving quantitative information. As the world around us becomes more complex and saturated with data, the ability to understand, comprehend and use quantitative information becomes more important to an educated citizenry (Madison & Steen, 2008). This skill set, often referred to in the U.S. as QL and in other parts of the world as numeracy, is seen as so important in today’s world that Steen (2004) argues QL reflects a more sophisticated “third R” (reading, writing, and arithmetic) needed for modern living. Madison and Steen (2008) note that “the need for a high level of quantitative literacy is an American characteristic, reinforced by individual freedoms, economic competitiveness, and the lack of economic safety nets” (p. 2). Although the role of “functional” or “practical” mathematics has long been discussed in the mathematics curriculum, the origins of modern-day QL can be traced back to the 1959 Crowther Report in the UK and the use of the term “numeracy”, which was used to describe the mathematical analog of “literacy”. However, QL became more common (and its definition more robust) after the 1982 Cockroft Report Mathematics Counts characterized numerate citizens as those with “the ability to use mathematics in everyday life and to understand and appreciate information presented in mathematical terms” (Madison & Steen, 2008, p. 3). Since the 1980s, a number of researchers and professional organizations have urged the continued development of QL as a “habit of mind” (Karaali et al., 2016) that supersedes the study of mathematics and statistics into all subjects of the school curriculum. The importance of QL has led to a rich effort across the U.S. in providing high school and college students greater opportunities to sharpen their QL skills in preparation for civic and economic life (Steen, 2004). At its very heart, QL requires mathematical knowledge yet is different from mathematics. Steen (1999) argues that QL requires not only basic mathematical skills such as arithmetic, percentages, simple algebra, and data analysis, but also more sophisticated concepts and skills not typically taught in school mathematics such as estimating tolerances and errors, drawing statistical inferences appropriately, and presenting data-based arguments using technology. Steen urges that a quantitatively literate person needs to hone this habit of mind in multiple settings, and that the true test of one’s QL capabilities lies in their ability to use these skills in a variety of contexts. (SWD) Karaali, G., Villafane Hernandez, E. H., & Taylor, J. A. (2016). What’s in a name? A critical review of definitions of quantitative literacy, numeracy, and quantitative reasoning. Numeracy, 9(1). Madison, B. L., & Steen, L. A. (2008). Evolution of numeracy and the national numeracy network. Numeracy, 1(1). Steen, L. A. (1999). Numeracy: The new literacy for a data-drenched society. Educational Leadership, 57(2), 8-13. Steen, L. A. (2004). Achieving quantitative literacy: An urgent challenge for higher education. Washington, DC: Mathematical Association of America.

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Quantitative Reasoning (QR) refers to the process of using one’s knowledge and skills to make sense and solve problems involving quantitative information. The terms “quantitative literacy” (QL) and “quantitative reasoning” (QR) are often used synonymously by researchers, referring to the ability to work with, think about, and use quantitative information to solve everyday problems. However, others draw a distinction between the two terms. Karaali et al. (2016), in reviewing the various uses of these terms, use the National Numeracy Network’s perspective to differentiate between the two ideas, noting that QL refers to the “comfort, competency, and ‘habit of mind’ in working with numerical data” while QR alludes to the “higher-order reasoning and critical thinking skills needed to understand and to create sophisticated arguments supported by quantitative data” (p. 16). Karaali et al. (2016) state that QR is what those who are quantitatively literate are able to do: “QR seems to be more of a process, while QL is a state” (p. 9). Since QR refers to higher-order thinking skills, the development of QR typically precedes QL and begins in elementary grades as students study relationships between numbers. Thompson (1993) defines QR as “the analysis of a situation into a quantitative structure – a network of quantities and quantitative relationships” (p. 165). He notes that numbers and numeric relationships are secondary to the relationships that exist between the quantities situated in a context. As young students begin working with quantities, Smith and Thompson (2007) advocate for an early emphasis on developing children’s QR skills – the ability to comprehend, reason about and work with complex ideas and quantitative relationships – as a method to strengthening students’ early algebra learning. This focus on higherorder thinking would translate to students’ arithmetical and algebraic reasoning and provide a basis for algebra through problem solving situations. As is the case with QL, QR involves mathematical knowledge yet goes beyond mathematics. Grawe (2011) provides a definition of QR as “the habit of mind to consider both the power and limitations of quantitative evidence in the evaluation, construction, and communication of arguments in public, professional and personal life” (p. 41), and outlines four basic facets of QR: (1) QR requires command of mathematical skills; (2) QR involves application in context; (3) QR involves communication; and (4) QR involves a habit of mind in comparison to a set of topics and skills outlined in a curriculum. These distinctions highlight the higher-order reasoning skills needed for developing QR. (SWD) Grawe, N. D. (2011). Beyond math skills: Measuring quantitative reasoning in context. New Directions for Institutional Research, 2011(149), 41-52. Karaali, G., Villafane Hernandez, E. H., & Taylor, J. A. (2016). What’s in a name? A critical review of definitions of quantitative literacy, numeracy, and quantitative reasoning. Numeracy, 9(1). Smith, J. P., & Thompson, P. W. (2007). Quantitative reasoning and the development of algebraic reasoning. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 95-132). New York, NY: Erlbaum. Thompson, P. W. (1993) Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics, 25(3), 165-208.

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Radical Constructivism is a theory where those who identify with being (or who are labeled) “radical” do not adhere to the traditional theory of what constitutes knowledge/truth but strictly orient around a theory of knowing, where learners construct their own knowledge which is unique to their personal experiences. Historically, epistemology has held knowledge as independent of the knowingsubject. In fact, there are two common requisites held by many philosophers regarding knowledge: (1) “Whatever we would like to call ‘true knowledge’ has to be independent of the knowing subject”; and (2) “a fully structured and knowable world exists and that is the business of the cognizing human subject to discover what that structure is” (von Glasersfeld, 1990, p. 21). It is these requisites that Ernst von Glasersfeld, who is widely credited for the theory of radical constructivism (Mueller, 2007), believes that are inconsistent to what radical constructivists truly believe. von Glasersfeld attributes much of the theoretical underpinnings of this theory to Jean Piaget, and that if one were to comprise Piaget’s works into one coherent theory, the radical constructivist theory would emerge. He found from Piaget’s work the following main principles: 1. Knowledge is not passively received through the senses or by communication. 2. Knowledge is actively built up by the cognizing subject. 3. The function of cognition is adaptive, in the biological sense of the term, trending towards fit or viability. 4. Cognition serves the subject’s organization of the experiential world, not the discovery of an objective ontological reality (von Glasersfeld, 1990, p. 22). Von Glasersfeld believes that the first two principles are what is radical about radical constructivism, and that if one were to hold strongly to these principles, as a constructivist should, one would be forced to transition from a theory of knowledge – some entity existing outside of the learner – to a theory of knowing – something that the learner is actively building up that is unique to his or her personal experience. von Glasersfeld agreed that this theory of knowing needed to be “offered as a post-epistemological perspective” (p. 19), where the traditional idea of truth or knowledge is dead and that one can manage without this notion of truth. The challenge to this approach to knowledge is that it is the antithesis to the historical requisites that are attached to the traditional theory of knowledge. It is this split from the traditional understanding of truth and knowledge that cannot be taken lightly by the constructivist and is why von Glasersfeld (1990) believes that this type of constructivism “needs to be radical” (p. 19). (SJH) Mueller, K. (2007). What is radical about radical constructivism? In R. Glanville, & A. Riegler (Eds.), The importance of being Ernst (pp. 239-261). Vienna, Austria: Echoraum. von Glasersfeld, E. (1990). An exposition of constructivism: Why some like it radical. In R. B. Davis, C. A. Maher, & N. Noddings (Eds.), Monographs of the Journal for Research in Mathematics Education, 4, 19-29. Reston, VA: NCTM.

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Reification is the process of representing something that is abstract as a material or concrete thing. Sfard (1991) explicitly defined the term reification in mathematics as “an ontological shift – a sudden ability to see something familiar in a totally new light” (p. 19). The idea of reification has been used to describe students’ learning of a wide variety of topics, from numbers and integers to functions and advanced mathematics topics in calculus (Cobb, 1994; Sfard & Linchevski, 1994; Gravemeijer & Doorman, 1999). These uses of the term reification primarily focus on the shift in understanding mathematics as processes to understanding the same topics as objects. Functions is a topic that is often used to illustrate reification (Sfard, 1991; Slavit, 1997). For example, a linear function f(x) = 3x + 1 is initially considered by students as a process of “input-output” in order to graph the line in a coordinate plane. Reification occurs when students understand functions as arbitrary relations that meet the requirement of univalence (Even, 1993). Reification is considered an important component of developing understanding of mathematics (Sfard, 1994; Portnoy et al., 2006). (LBK) Cobb, P. (1994). Where is the mind? Constructivist and sociocultural perspectives on mathematical development. Educational Researcher, 23(7), 13-20. Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept. Journal for Research in Mathematics Education, 24(2), 94-116. Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational Studies in Mathematics, 39(1-3), 111-129. Portnoy, N., Grundmeier, T. A., & Graham, K. J. (2006). Students’ understanding of mathematical objects in the context of transformational geometry: Implications for constructing and understanding proofs. The Journal of Mathematical Behavior, 25(3), 196-207. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 136. Sfard, A. (1994). Reification as the birth of metaphor. For the Learning of Mathematics, 14(1), 4455. Sfard, A., & Linchevski, L. (1994). The gains and the pitfalls of reification – the case of algebra. In P. Cobb (Ed.), Learning mathematics (pp. 87-124). Dordrecht, The Netherlands: Springer. Slavit, D. (1997). An alternate route to the reification of function. Educational Studies in Mathematics, 33(3), 259-281.

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Relational Thinking describes student uses of number relationships, either explicitly or implicitly, within their computation strategies (Empson & Levi, 2011). The phrase relational thinking is a broad term that has been developed in contexts beyond number relationships (Jones, 2009). Jones (2009) uses the term “relational thinking” to describe an open approach to networking space relationships within human geography. However, the more common mathematical references build on the research on children’s thinking and approaches to equality and number relationships (Carpenter et al., 2003). Studies have examined both students’ use of relational thinking strategies and mathematics teachers’ awareness of these types of strategies (Stephens, 2006; Stephens & Wang, 2008). The foundation of the number relationships is the set of properties of operations that apply to all real numbers. For example, young students who determine that 7+4 = 11 by adding seven and three to get 10 and then adding one more to get 11 are intuitively applying the associative property of addition. Often, the implicit application of these types of properties enhances students’ abilities to become fluent with their number facts for single digit numbers. Upper elementary grades students apply these properties to multi-digit operations as well as their strategies with fraction and decimal computations. Examples involving the use of the distributive property of multiplication are prevalent as students decompose numbers, multiply them in smaller chunks and find the sum of those chunks (i.e., 20 x 15 = 20 x 10 + 20 x 5 = 200 + 100 = 300). (LBK) Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Heinemann: Portsmouth, NH. Empson, S. B., & Levi, L. (2011). Extending children’s mathematics: Fractions and decimals. Portsmouth, NH: Heinemann. Jones, M. (2009). Phase space: geography, relational thinking, and beyond. Progress in Human Geography, 33(4), 487-506. Stephens, A. C. (2006). Equivalence and relational thinking: Preservice elementary teachers’ awareness of opportunities and misconceptions. Journal of Mathematics Teacher Education, 9(3), 249-278. Stephens, M., & Wang, X. (2008). Investigating some junctures in relational thinking: A study of year 6 and year 7 students from Australia and China. Journal of Mathematics Education, 1(1), 28-39.

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Representational Fluency involves the individual’s ability to make meaningful interpretations across multiple forms of a concept. Multiple representations of mathematical concepts have been identified and explored for decades. Dienes (1969) advocated for children’s experiences with representations such as base 10 materials that mirror the structure of place value prior to experiences with those same concepts with numbers. Kaput (1992) contributed the importance of technology in considering multiple representations of concepts including numerical, tabular, and graphical. From the early discussions of multiple representations, the idea of representational fluency was used as the basis for studying students’ learning of a variety of mathematical concepts. Lesh (2000) described learning mathematics using real-life situations and technology such as spreadsheets as ways to build representational fluency with mathematical quantities and operations. Nathan and Kim (2007) focused on the use of multiple representations to build students’ representational fluency with algebraic reasoning. Representational fluency has also been described in terms of learning with more advanced topics and beyond mathematics (Santos & Thomas, 2001; Warfa et al., 2002, etc.). Representational fluency has also been associated with teacher understanding of mathematics content (Bowers & Stephens, 2001). (LBK) Bowers, J. S., & Stephens, B. (2011). Using technology to explore mathematical relationships: A framework for orienting mathematics courses for prospective teachers. Journal of Mathematics Teacher Education, 14(4), 285-304. Dienes, Z. P. (1969). Building up mathematics. London, England: Hutchinson Educational. Kaput, J. J. (1992). Technology and mathematics education. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 515-556). Reston, VA: National Council of Teachers of Mathematics. Lesh, R. (2000). Beyond constructivism: Identifying mathematical abilities that are most needed for success beyond school in an age of information. Mathematics Education Research Journal, 12(3), 177-195. Nathan, M. J., & Kim, S. (2007). Pattern generalization with graphs and words: A cross-sectional and longitudinal analysis of middle school students' representational fluency. Mathematical Thinking and Learning, 9(3), 193-219. Nathan, M. J., Stephens, A. C., Masarik, D. K., Alibali, M. W., & Koedinger, K. R. (2002). Representational fluency in middle school: A classroom study. In Proceedings of the twentyfourth annual meeting of the North American chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 462-472). Columbus, OH: ERIC Clearinghouse for Science, Mathematics and Environmental Education. Santos, A. G. D., & Thomas, M. O. (2001). Representational fluency and symbolisation of derivative. In Proceedings of the Sixth Asian Technology Conference in Mathematics (pp. 282-291). Suh, J. M., Johnston, C., Jamieson, S., & Mills, M. (2008). Promoting decimal number sens e and representational fluency. Mathematics Teaching in the Middle School, 14(1), 44-50. Warfa, A. R. M., Roehrig, G. H., Schneider, J. L., & Nyachwaya, J. (2014). Role of teacher-initiated discourses in students’ development of representational fluency in chemistry: A case study. Journal of Chemical Education, 91(6), 784-792.

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Representations are the various ways or tools students use to symbolize their thinking about a particular mathematical concept. Mathematical concepts can be represented in a variety of ways depending upon the context in which they are situated. “Using different representations is like examining a concept through a variety of lenses, with each lens providing a different perspective that makes the picture (concept) richer and deeper” (NCTM, 2014, p. 25). There are five basic ways of representing mathematics: symbols, graphs, pictures and diagrams, words, and number patterns (Preston & Garner, 2003; van de Walle, Karp, & Bay-Williams, 2013). In the elementary classroom, representations typically begin with the manipulation of objects which is quickly followed by pictorial representations in which students attempt to mimic their manipulations. The simplistic pictures and drawings are later enhanced with words to describe the mathematics. As students delve further into mathematics, their need for other, potentially more complex representations becomes apparent. Although there is no prescribed order of how they are presented, students should become familiar with the representations and initially use ones that make sense to them. x

x

x

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Pictures and Diagrams: Initially students utilize self-created pictures and diagrams as a means to illustrating their mathematical thinking when solving problems. Words: Student explanations of their work may or may not contain appropriate mathematical language, but should move toward using correct language as students' mathematical knowledge increases. Graphs: These are visual representations of mathematics and mathematical thinking. Not all mathematics problems will be best represented with graphs, yet in some instances, graphs may be the most appropriate representation depending on the context of the problem. Symbols: Symbols are used once students are able to think about the mathematics they are doing symbolically. The symbolic representations are a very common way to notate thinking when solving problems. Number Patterns: Patterns exist within mathematics, therefore making notice of them important. Tables are typically used to record information; therefore, they are a tool that allows for students to notice patterns inherently present in the mathematics. (CCO)

National Council of Teachers of Mathematics (NCTM). (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. Preston, R. V., & Garner, A. S. (2003). Representation as a vehicle for solving and communication. Mathematics Teaching in the Middle School, 9, 38-43. Van de Walle, J. A., Karp, K. S., and Bay-Williams, J. (2013). Elementary and middle school mathematics: Teaching developmentally (9th ed.). Upper Saddle River, NJ: Pearson.

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Response to Intervention (RtI) is a multi-tiered model used to provide academic for students, utilizing appropriate interventions as needed. Response to Intervention (RtI) dates back to the late 1970s. RtI is based on Stanley Deno’s modification model which is a data-based approach used to identify students with learning disabilities so that teachers can use appropriate interventions based on student progress (Coleman et al., 2006). After the passage of the No Child Left Behind Act, RtI has become more general and inclusive of all students and has become more mainstreamed with the need to ground teaching practices and interventions in a research base (Gresham & Little, 2013). The major premise of RtI is that early intervention can help ensure the mathematical success of all students by providing appropriate interventions, even referrals when necessary. RtI is an approach to providing struggling students with necessary support before they fail as opposed to waiting for them to fail and get further behind in the curriculum (van de Walle et al., 2016). RtI is a three-tiered model. The first level is the most inclusive, as it consists of the core instructional program and is implemented by the teacher. Approximately 85% of the students are successful with the instruction as provided. The second level is a targeted intervention for students who did not make satisfactory progress with the core instruction. Approximately 10% of the students will benefit from this intervention strategy. The third level is intensive intervention which is best suited for less than 5% of the students who were not successful with the core instructional program. For students not making progress after participating in the third level of intervention, consideration for specialized services must be made, with additional testing as needed, with referral decisions being made about how students respond to the interventions. With the RtI model, frequent assessments are taken to determine student success. Depending on the level of student achievement, students can be given targeted help and support to help them understand the content in the curriculum. As students succeed, their targeted intervention and support can decrease, and vice versa, they can receive targeted interventions and support can increase, so as to ensure success. (CCO) Coleman, M. R., Buysse, V., and Neitzel, J. (2006). Recognition and response: An early intervening system for young children at-risk for learning disabilities. Chapel Hill, NC: The University of North Carolina at Chapel Hill, FPG Development Institute. Gresham, R. H., & Little, M. E. (2013). RTI and mathematics: Practical tools for teachers in K-8 classrooms. Upper Saddle River, NJ: Pearson. Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2016). Elementary and middle school mathematics: Teaching developmentally (9th ed.). New Jersey: Pearson.

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Responsive Teaching has been described as “a type of teaching in which teachers’ instructional decisions about what to pursue and how to pursue it are continually adjusted during instruction, rather than determined in advance, in response to children’s content-specific thinking” (Jacobs et al., 2015, p. 2). The core of responsive teaching is that many aspects of instructional decisionmaking are “in-the-moment” based on observations of students’ written approaches and verbal descriptions of their mathematical strategies for solving problems. For example, Jacob et al. (2015) developed a model to study the teaching and learning of fractions based in part on professional noticing of children’s thinking about fractions. Their findings involved categories of noticing such as attending to how well students are able to make sense of the story contexts designed to elicit and extend students’ understandings of fraction concepts. Responsive teaching has been described in a variety of mathematics classrooms and grade bands. Dyer and Sherin (2016) analyzed the responsive teaching strategies of two secondary mathematics teachers to explore how they listened and responded to students’ thinking and mathematical descriptions. Empson (2014) examined second grade students’ learning of base 10 concepts through a responsive teaching approach. Kiefer et al. (2014) examined the effects of responsive teaching strategies on middle school students in a variety of content areas. Responsive teaching has been associated widely with teaching and learning mathematics and science concepts across the grade spectrum (Robertson et al., 2015). (LBK) Dyer, E. B., & Sherin, M. G. (2016). Instructional reasoning about interpretations of student thinking that supports responsive teaching in secondary mathematics. ZDM, 48(1-2), 6982. Empson, S. B. (2014). Responsive teaching from the inside out: teaching base ten to young children. Investigations in Mathematics Learning, 7(1), 23-53. Jacobs, V. R., Empson, S. B., Krause, G. H., & Pynes, D. (2015). Responsive teaching with fractions. In annual meeting of the Research Conference of the National Council of Teachers of Mathematics. Boston, MA. Kiefer, S. M., Ellerbrock, C., & Alley, K. (2014). The role of responsive teacher practices in supporting academic motivation at the middle level. RMLE Online, 38(1), 1-16. Retrieved from https://files.eric.ed.gov/fulltext/EJ1039613.pdf Robertson, A. D., Scherr, R., & Hammer, D. (Eds.). (2015). Responsive teaching in science and mathematics. New York, NY: Routledge.

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Rigor generally refers to the level of cognitive demand, complexity, and overall challenge required of students during the process of learning. Rigor is used to describe a number of entities in mathematics education. The term has often been used to describe curriculum, standards, and overall learning experiences provided to students. Schmidt (2008) uses the term “rigorous” in examining the level of mathematics standards in comparison to those of other high-achieving TIMSS countries: In the middle grades, the rest of the world is teaching algebra and geometry. The U.S. is still, for most children, teaching arithmetic … other countries outperform us in the middle and upper grades because their curricular expectations are so much more demanding, so much more rigorous. (p. 23) Joftus and Berman (1998) note that more demanding or rigorous standards should not be confused with more difficult standards, as standards should be consistent for all students. They define rigor in mathematics standards so that “standards should require all students, at the appropriate grade level, to learn the essential concepts and skills of mathematics at the level of sophistication” (p. 9) as outlined by the NAEP frameworks and the NCTM Standards publications. In terms of curricular materials, rigor is often used to describe the levels of cognitive complexity required of students. Hess et al. (2009) used two existing models for describing academic rigor – Bloom’s Taxonomy and Webb’s Depth-ofKnowledge (DOK) model – to develop a measure for cognitive rigor associated with classroom tasks. In this sense, rigor is used similarly to the term cognitive demand as used by researchers to describe the level of mathematical thinking and reasoning needed to solve a problem (Smith & Stein, 1998). Rigor is also used to describe demanding or strict circumstances, or a rigidity in one’s work, and therefore is used to describe standards for what constitutes mathematical proof (i.e., mathematical rigor) or the methods used to conduct research (methodological rigor). Kliener (1991) notes the standards of rigor for mathematical proof have evolved over time, given the tools available to mathematicians as well as the modes of communication used at the time. (SWD) Hess, K. K., Jones, B. S., Carlock, D., & Walkup, J. R. (2009). Cognitive rigor: Blending the strengths of Bloom’s taxonomy and Webb’s depth of knowledge to enhance classroom-level processes. [Technical Report]. Retrieved from ERIC database. (ED517804). https://files.eric.ed.gov/fulltext/ED517804.pdf Joftus, S., & Berman, I. (1998). Great expectations? Defining and assessing rigor in state standards for mathematics and English language arts. Washington, DC: Council for Basic Education. Kleiner, I. (1991). Rigor and proof in mathematics: A historical perspective. Mathematics Magazine, 64(5), 291-314. Schmidt, W. H. (2008). What’s missing from math standards? Focus, rigor, and coherence. American Educator, 32(1), 22-24. Smith, M. S., & Stein, M. K. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3(5), 344-350.

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Rote Learning is the memorization of information, usually by repetition, with little or no context or understanding of underlying relationships. Memorization, repetition, and practice play important roles in gaining automaticity in mathematics so that students can easily access prerequisite knowledge while solving more complex problems. However, rote learning is not synonymous with memorization. Typically associated with memorization, for example, is the learning of basic multiplication facts. “Acquiring proficiency with single-digit computations involves much more than rote memorization” (NRC, 2001, p. 194). Relational strategies are recommended to assist with committing these facts to memory (NGA, 2010). Likewise, rather than memorize by rote the formula for the area of a rectangle, students can attach meaning to the formula by visualizing the length as the number of square units in one row multiplied by the number of rows. Ideally, such information is eventually retrieved automatically for applications without considering the conceptual base each time, but the initial sense-making helps with retention and transfer of the knowledge (NRC, 2001). When information is organized in chunks of related ideas, it takes up less room in one’s working memory, helping to prevent cognitive overload, and connects more readily to existing networks of information in long term memory, which makes it easier to retrieve (Eggen & Kauchak, 2010). Hiebert and Lefevre (1986) explain: Rote learning … produces knowledge that is notably absent in relationships and is tied closely to the context in which it is learned. The knowledge that results from rote learning is not linked with other knowledge and therefore does not generalize to other situations; it can be accessed and applied only in those contexts that look very much like the original. (p. 8) Procedures, particularly with a sequential nature, can be learned by rote. “Certainly children can learn procedures by rote without relating them to any appropriate form of conceptual knowledge, and some invention appears to occur strictly within the context of procedural knowledge” (Carpenter, 1986, p. 120). However, the procedures that are weakly linked to conceptual knowledge are more prone to error. (KKM) Carpenter, T. P. (1986). Conceptual knowledge as a foundation for procedural knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 113132). Hillsdale, NJ: Erlbaum. Eggen, P. D., & Kauchak, D. (2010). Educational psychology: Windows on classrooms. Upper Saddle River, NJ: Pearson. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1-27). Hillsdale, NJ: Erlbaum. National Governors Association Center for Best Practices (NGA), Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Author. National Research Council (NRC). (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, & B. Findell (Eds.). Washington, DC: National Academy Press.

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Scaffolding (or Instructional Scaffolding) is the support and guidance provided by a teacher or knowledgeable adult to a student as the student navigates through a problem solving situation. The idea behind instructional scaffolding is widely attributed to the psychologist Jerome Bruner and was first used by Wood, Bruner & Ross (1976) as a way to describe the supports that an adult can provide a learner. Scaffolding is a process involving “the adult ‘controlling’ those elements of the task that are initially beyond the learner’s capacity, thus permitting him to concentrate upon and complete only those elements that are within his range of competence” (p. 90). Bruner and his colleagues outlined how instructional scaffolding might proceed: 1. Recruitment: the instructor elicits the student’s interest in the problem and highlights the requirements of the task. 2. Reduction in degrees of freedom: the instructor simplifies the task by reducing the number of steps to reach a solution. 3. Direction maintenance: the instructor keeps the student in pursuit of a specific objective. 4. Marking critical features: the instructor highlights or emphasizes the relevancy of certain features of the task. 5. Frustration control: the instructor reduces stress from working the problem. 6. Demonstration: the instructor models an idealized form of a solution or strategy (Wood et al., 1976, p. 98). The ultimate goal is to remove the instructor support so the student can work without assistance (Anghileri, 2006). This goal is closely related to Vygotsky’s Zone of Proximal Development (ZPD), as scaffolding addresses the nature of the guidance needed for a student to progress in one’s ZPD (Wood & Wood, 1996). This assistance can take on various levels of complexity and differs by student. Anghileri (2006) proposes three levels of scaffolding teachers may use. At a foundational level, a teacher attends to environmental provisions such as the organization of the classroom, task structure, and tools or manipulatives needed, to provide support for the student. As the student works, a more sophisticated level occurs, as the teacher explains aspects of the tasks, reviews and highlights important pieces of the task, and restructures the task as needed. At the highest level, the teacher supports conceptual thinking by the student by facilitating classroom discourse, making connections to other concepts, and creating representational tools that model the phenomenon under investigation. (SWD) Anghileri, J. (2006). Scaffolding practices that enhance mathematics learning. Journal of Mathematics Teacher Education, 9, 33-52. Wood, D., Bruner, J. S., & Ross, G. (1976). The role of tutoring in problem solving. Journal of Child Psychology and Psychiatry, 17, 89-100. Wood, D., & Wood, H. (1996). Vygotsky, tutoring and learning. Oxford Review of Education, 22(1), 5-16.

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Sense-Making refers to a process by which a person constructs personal meanings and internalize the phenomena they experience (Weinberg, Wiesner, & FukawaConnelly, 2014). Robust disciplinary knowledge in content was, at one time, considered the only necessary qualification for an instructor. That idea ignored how alienating the learning experience itself can be for students when engagement and idiosyncratic sense making are suppressed (Klein, 2002). It is necessary to find the importance of understanding the ways students make sense of aspects of mathematics and how their sense-making practices influence what they might learn. (Weinberg, Wiesner, & Fukawa-Connelly, 2014). A teacher must discover how the students make sense of problems. Psychological views of learning and advancements in educational study have informed mathematics and numeracy education. Such views support reasoning processes as foundational to the construction of knowledge and its application. The learning process is productive, though it can also be productive in the construction of identity (the learner’s sense of self as a legitimate participant in numerate practices) (Klein, 2002). For example, from a constructivist standpoint, previous experiences create new schema in the sense making process (Holmes, 2005). Varying educational theories speculate how students make sense of what they learn. There are numerous strategies aimed to aid in the sense-making process for students. These strategies are as diverse as the students for which they are intended (Rahman, Scaife, Yahya, & Jalil, 2010). For instance, one study, there was a statistically significant improvement in the learners’ competence in solving word problems when learning occurred in a discussion-based classroom (Webb & Sepeng, 2012). Supporting students in making sense of their studying and empowering them in their understanding is a critical aspect of their learning. Additionally, the potential for productive, meaningful and positive learning is relevant to the student’s life, and the way in which the students analyze and interpret situations they encounter, both physically and academically, are advanced by sense making activities. (Ingerman & Booth, 2004). (SRF) Holmes, M. P. (2005). Schema learning: Experience-based construction of predictive action models. In Advances in Neural Information Processing Systems (pp. 585-592) Ingerman, A., & Booth, S. (2004). Making sense of studying physics. Physics Education. arXiv preprint physics/0401052 Klein, M. (2002). Teaching mathematics in/for new times: A poststructuralist analysis of the productive quality of the pedagogic process. Educational Studies in Mathematics, 50(1), 6378. Rahman, F. A., Scaife, J., Yahya, N. A., & Jalil, H. A. (2010). Knowledge of diverse learners: Implications for the practice of teaching. International Journal of Instruction, 3(2), 83-96. Webb, P., & Sepeng, P. (2012). Exploring mathematical discussion in word problem-solving. Pythagoras, 33(1), 1-8. Weinberg, A., Wiesner, E., & Fukawa-Connelly, T. (2014). Students’ sense-making frames in mathematics lectures. The Journal of Mathematical Behavior, 33, 168-179.

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Situated Learning (Cognition) is a theory that asserts that individual learning is determined by environmental factors. Brown, et al., (1989) described several aspects of situated cognition including cognitive apprenticeship, authentic learning environments, enculturation, and learning tools. They advocated for an epistemology or framework that supports the study of learning in classrooms. Lave and Wenger (1991) connected the idea of situated learning to legitimate peripheral participation (or LPP), in which learners make sense of new concepts through gradual and continual immersion in the learning environment. Theories of situated cognition build on social cultural views of learning described by Vygotsky (1978). These theories were contrasted with views of individual learning common with information processing (Ausubel, 1977). Research studies of classroom-based learning of mathematics have incorporated situated cognition (Boaler, 1999, 2000; Cobb & Whitenack, 1996; Ladson-Billings, 1997; Moschkovich, 2007). For example, Moschkovich (2007) utilized situated cognition to analyze students’ interactions and discourse patterns in making sense of geometry concepts. Cobb and Whitenack (1996) incorporated situated learning theories as part of their framework for analyzing mathematics classroom interactions. (LBK) Ausubel, D. P. (1977). The facilitation of meaningful verbal learning in the classroom. Educational Psychologist, 12(2), 162-178. Boaler, J. (1999). Participation, knowledge and beliefs: A community perspective on mathematics learning. Educational Studies in Mathematics, 40(3), 259-281. Boaler, J. (2000). Exploring situated insights into research and learning. Journal for Research in Mathematics Education, 31(1), 113-119. Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational researcher, 18(1), 32-42. Cobb, P., & Whitenack, J. W. (1996). A method for conducting longitudinal analyses of classroom videorecordings and transcripts. Educational Studies in Mathematics, 30(3), 213-228. Ladson-Billings, G. (1997). It doesn't add up: African American students' mathematics achievement. Journal for Research in Mathematics Education, 28(6), 697-708. Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge, England: Cambridge University Press. Moschkovich, J. (2007). Examining mathematical discourse practices. For the Learning of Mathematics, 27(1), 24-30. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological functions. Cambridge, MA: Harvard University.

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Social Constructivism is a branch of constructivism which focuses on the social nature of cognition and posits that the learner constructs knowledge based on social interaction and one’s understanding based upon these interactions. Social constructivism is a branch of constructivism that doesn’t allow for the separation of learning from the social context where it was formed. Based on the work of Vygotsky, at its core it is centered on the idea that the learner’s “construction of knowledge is the product of social interaction, interpretation and understanding” (Adams, 2006, pp. 245). It is argued by this position that all cognitive abilities and functions are products of social interactions and are created through the process of integration and assimilation into a community of practice. During this process, the learner does not simply construct their own knowledge, but rather their knowledge is co-constructed with the others in the community of practice. Hence, the consensus between the others in the community and the learner becomes the criterion used to judge what is truth and what is not. This view of learning and knowledge has had many profound implications for teaching and, in fact, is the foundation for many instructional approaches used today. One of those approaches is the situated learning approach which puts into practice the basic tenants of social constructivism. Herrington and Oliver (1990) sought to define the critical attributes of the situated learning approach and develop a framework for successful implementation. They found that situated learning environments: 1. Provide authentic contexts that reflect the way knowledge will be used in real life 2. Provide authentic activities 3. Provide access to expert performances and the modeling of processes 4. Provide multiple roles and perspectives 5. Support collaborative construction of knowledge 6. Promote reflection to enable abstractions to be formed 7. Promote articulation to enable tacit knowledge to made explicit 8. Provide coaching and scaffolding by the teacher at critical times 9. Provide for authentic assessment of learning within the tasks (Herrington & Oliver, 2000, p. 4, emphasis in original text). From the list above, it can be seen that this approach is motivated from the viewpoint of social constructivism. There is an emphasis on socially authentic lessons mixed with a collaborated effort to create knowledge with the other learners and the more knowledgeable other in the community of practice, namely the teacher. (SJH) Adams, P. (2006). Exploring social constructivism: Theories and practicalities. Education 3-13, 34, 243-257. Herrington, J. A. (2000). An instructional design framework for authentic learning environments. Educational Technology Research and Development, 48(3), 23-48.

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Socio-Cultural Learning Theory (SCLT) posits that the development of a child’s psychological abilities begins with the child’s social interaction with adults, peers, and objects within their native culture through their shared language and symbolic frameworks. Before these abilities become implemented as a part of the child’s personality, the child experiences the language and frameworks in their world through interactions with people around him or her, which are gradually absorbed and transformed within the child, resulting in the child’s psychological development. Built from the work of Lev Vygotsky, SCLT is based on the notion that the social setting is an integral piece to the learner’s development. According to Vygotsky’s work, the learning and development process has three key themes: the culture, the language, and the zone of proximal development (ZPD). SCLT and the Culture: Unlike many of the theories of his time, Vygotsky viewed the origins of human intelligence and development in the society or culture. A culture, Vygotsky theorized, is created and formed with the use of tools and symbols and these constructions are what separates humans from animals. The development of the learner begins when they are able to “internalize” the tools and symbols that are used within that culture. In fact, Vygotsky (1978) stated the following in his book, Mind in Society: Any function in the child’s cultural development appears twice, or on two planes. First it appears on the social plane, and then on the psychological plane. First it appears between people as an interpsychological category, and then within the child as an intrapsychological category. This is equally true with regard to voluntary attention, logical memory, the formation of concepts, and the development of volition … [I]t goes without saying that internalization transforms the process itself and changes its structure and functions. Social relations or relations among people genetically underlie all higher functions and their relationships. (p. 57) Vygotsky placed a much greater emphasis on how social interactions and factors influence the child’s development. When compared to other psychologists of his time such as Freud, Skinner, and Piaget, all of whom stressed how a child’s interactions and explorations influenced development, Vygotsky differed in that he stressed that the social interaction was the essential piece that catalyzed development while also being a necessity to reach full cognitive development. SCLT and Language: Vygotsky believed that all children are born with inherent mental limitations; however, each culture provides certain tools for intellectual growth. One of these tools is their language and the customs that surround the language. To learn the language, it is theorized that the learner must go through three stages of language development: (1) social speech, (2) private speech, and (3) inner speech (Lantolf & Thorne, 2000).

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1. Social Speech – This stage usually occurs for the child under the age of three or when the child is not capable of transcribing his or her thoughts for others to understand. Since the child’s thoughts, in every aspect, are basic, the child uses his or her speech to express simple thoughts of displeasure, being content, being discontent, or of hunger. The purpose of the language used here is to control others to satisfy needs and/or desires of the child. 2. Private Speech – If the purpose of social speech was to control others, the purpose of inner speech is to control the learner’s self. This stage usually occurs between the ages of 3 and 7 or when the child begins to elicit words more clearly and is able to form complete sentences that have more sense and thought attached to them. This can be seen as a running commentary as the child is going about their own activities. 3. Inner Speech – The third and final stage of development occurs when the child starts moving into adulthood and is able to use the language to direct his thinking and the behavior associated with the thinking. This type of speech is usually internal and is characterized by being able to do mental calculations, being able to analyze a situation from different angles, and being able to form an opinion or an argument without needing to verbalize. SCLT and the Zone of Proximal Development: Vygotsky believed that full cognitive development could not be had without social interaction (Doolittle, 1997). The zone of proximal development (ZPD) is the embodiment of that theory. Given the social nature of this theory, the ZPD captures the difference between what a student can individually in relation to what that same student could learn with the help of someone with greater knowledge. (SJH) Doolittle, P. E. (1997). Vygotsky’s zone of proximal development as a theoretical foundation for cooperative learning. Journal on Excellence in College Teaching, 8(1), 83-103. Lantolf, J. P., & Thorne, S. L. (2000). Sociocultural theory and second langauge learning. In J. P. Lantolf (Ed.), Sociocultural theory and second language learning (pp. 197-221). Oxford, England: Oxford University Press. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.

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Sociomathematical Norms was a term originally created as part of an interpretive framework to account for student beliefs, values, and contributions specific to a mathematics class/lesson that also included classroom social norms and classroom mathematical practices (Cobb & Yackel, 1996; Voigt, 1995). The distinction between social norms and sociomathematical norms within this framework delineates aspects of students’ participation in a mathematics class (Cobb & Yackel, 1996; McClain & Cobb, 2001; Yackel & Cobb, 1996). Notating and verbally describing their invented strategies for solving a problem is contrasted with more generic normative practices such as grouping arrangements or mechanisms for participating in small group or whole class discussions. For example, students would likely discuss the use of base 10 materials for whole number and place value concepts by groups and powers of 10 in a mathematics class rather than a focus on the color and/or shape of the materials for a nonmathematical purpose. Analysis of mathematics lessons and classroom based studies with a focus on sociomathematical norms has illuminated and detailed aspects of students’ thinking about mathematical processes and conventions. For example, Yackel and Cobb (1996) describe the sociomathematical norms established by the teacher in a second grade classroom to elicit and further students’ understanding of place value concepts. Researchers and mathematics educators across the grade spectrum have incorporated aspects of sociomathematical norms to analyze mathematical aspects of classroom practice including collegiate level classrooms (Dixon et al., 2009; Fukawa-Connelly, 2012). (LBK) Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31(3-4), 175-190. Dixon, J. K., Andreasen, J. B., & Stephan, M. (2009). Establishing social and sociomathematical norms in an undergraduate mathematics content course for prospective teachers: The role of the instructor [Monograph]. Scholarly Practices and Inquiry in the Preparation of Mathematics Teachers (pp. 43-66). San Diego, CA: Association of Mathematics Teacher Educators. Fukawa-Connelly, T. (2012). Classroom sociomathematical norms for proof presentation in undergraduate in abstract algebra. The Journal of Mathematical Behavior, 31(3), 401-416. McClain, K., & Cobb, P. (2001). An analysis of development of sociomathematical norms in one first-grade classroom. Journal for Research in Mathematics Education, 32(3), 236-266. Voigt, J. (1995). Thematic patterns of interaction and sociomathematical norms. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 163-201). Hillsdale, NJ: Lawrence Erlbaum Associates. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458-477.

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Spatial Thinking involves “the ability to generate, retain, and manipulate abstract visual images. At the most basic level, spatial thinking requires the ability to encode, remember, transform, and match spatial stimuli” (Lohman, 1979, p. 127). Spatial tasks involve mentally generating images in two and three dimensions, visualizing their orientation in space and their relation to other objects, and can be as commonplace as catching a ball or navigating with a map. Consider the mental process as one parallel parks a car: the driver knows his location, pictures the size and shape of the car, considers how it relates to the size and shape of the parking spot, and visualizes a path for getting there. On a grander scale, the famed Watson and Crick, co-discoverers of the structure of DNA, were able to interpret Rosalind Franklin’s 2-dimensional X-rays of DNA to visualize and create a 3-D model, the double helix. Lohman (1979) identifies three major factors of spatial ability: Spatial Relations – being able to mentally rotate an object and recognize the result Spatial Orientation – being able to imagine what a rendering of an object looks like from a different view Spatial Visualization – being able to envision the resulting 3D figure from folding a 2D net (an outline of the fold lines) “Mathematics is a central subject in which spatial thinking is needed, because space provides a concrete grounding for number ideas, as when we use a number line, use base 10 blocks, or represent multiplication as area” (Newcombe, 2010, p. 34). Spatial ability is positively related to mathematical achievement, as well as to interest and success in STEM fields (Wai et al., 2009). Boys tend to outperform girls on some spatial tasks, particularly mental rotations (Voyer et al., 1995), but research resoundingly states that spatial ability can improve with practice of spatial tasks and that the improvement is durable and transferable to other tasks (Uttal et al., 2013). Thus, the potential exists for all to access the spatial reasoning skills necessary for success in mathematics and other STEM fields. Because of this, the literature commonly advocates for K-12 curriculum to offer more explicit opportunities for students to engage in spatial reasoning tasks. (KKM) Lohman, D. F. (1979). Spatial ability: A review and reanalysis of the correlational literature. Aptitude Research Project, Technical Report No. 8. Palo Alto, CA: Stanford University School of Education. Newcombe, N. S. (2010). Picture this: Improving math and science learning by improving spatial thinking. American Educator, 34(2), 29-35. Uttal, D. H., Meadow, N. G., Tipton, E., Hand, L. L., Alden, A. R., Warren, C., & Newcombe, N. S. (2013). The malleability of spatial skills: A meta-analysis of training studies. Psychological Bulletin, 139, 352–402. Voyer, D., Voyer, S., & Bryden, M. P. (1995). Magnitudes of sex differences in spatial abilities: A meta-analysis and consideration of critical variables. Psychological Bulletin, 117(2), 250270. Wai, J., Lubinski, D., & Benbow, C. P. (2009). Spatial ability for STEM domains: Aligning over 50 years of cumulative psychological knowledge solidifies its importance. Journal of Educational Psychology, 101, 817-835.

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Strands of Mathematical Proficiency refer to the five components (strands): conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition that make up mathematical proficiency. The five strands of mathematical proficiency emerged in 2001 from the National Research Council (NRC) as a report about the learning of mathematics in school settings as a guide to best practices in mathematics teaching through eighth grade. In 1998, the NRC established the Committee on Mathematics Learning to address concerns of students not achieving at high levels in mathematics (NRC, 2001). One key finding is that K-8 is a shallow curriculum, typically being described as an inch deep and a mile wide, indicating that students are exposed to numerous ideas, but do not receive enough experiences with all topics to create proficiency in all areas (NRC, 2001). NRC coined and described the phrase “mathematical proficiency” as a means of capturing their interpretation of what is means for anyone to successfully learn mathematics (NRC, 2001). The five strands of mathematical proficiency provided in the NRC’s Adding it Up report are conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition, which are similar to the conceptual model used in mathematics in Singapore (Kilpatrick, 2009). These strands are often referred to when formulating goals and standards for curriculum, as well as being used in framing instructional materials. These strands or components, as they are sometimes referred, should be thought of as interconnected and interdependent. Therefore, in teaching, the strands should not be taught as individual skills, but should be taught simultaneously in order to help students achieve mathematical proficiency (Hiebert et al., 2003). NRC describes each of the strands in the following manner:

x x x x x

conceptual understanding – comprehension of mathematical concepts, operations, and relations procedural fluency – skill in carrying our procedures flexibly, accurately, efficiently, and appropriately strategic competence – ability to formulate, represent, and solve mathematical problems. adaptive reasoning – capacity for logical thought, reflection, explanation, and justification productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy (NRC, 2001, p. 116). (CCO)

Hiebert, J., Morris, A. K., Glass, B. (2003). Learning to learn to teach: An “experiment” model for teaching and teacher preparation in mathematics. Journal of Mathematics Teacher Education, 6, 201-222. Kilpatrick, J. (2009). The mathematics teacher and curriculum change. PNA, 3(3), 107-121. National Research Council (NRC). (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, & B. Findell (Eds.). Washington, DC: National Academy Press.

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Subitizing, a term coined by Kaufman et al. (1949) is an important mathematical skill which aids in the development of number concepts. Subitizing refers to the ability of looking at a group of objects and instantly seeing how many objects are in the group. Initially, subitizing was thought of as a precursory skill to counting, but with time has become viewed as more of a “basic” skill, even more so than counting (Clements, 1999, p. 400). Whereas counting traditionally focuses on the unit and measurement focuses on the whole, subitizing focuses on the whole as well as individual units. Although Kaufman et al. first coined this term, the idea of subitizing expanded to two types: perceptual and conceptual. Perceptual subitizing remains most consistent with the original definition of subitizing. The role of perceptual subitizing, making units to count, is an automated activity for most adults. However, it has also been used to help young children consolidate quantitative aspects of whole numbers (Clements, 1999; MacDonald & Shumway, 2016). For example, MacDonald and Shumway (2016) described activities with dot cards to help children gain automaticity with five dots and the numeral five. Conceptual subitizing plays an advanced role in counting and number sense so as to help children develop abstract number and arithmetic strategies. MacDonald and Shumway (2016) describe conceptual subitizing as recognizing the composite unit from two subsets. For example, if three dots are shown on one die or domino and a four is shown on another die or domino, a child who recognizes quickly that the sum is 7 using (dot) number relationships such as 3 + 3 + 1 = 7 would be considered to have conceptual subitizing skills. Others describe this type of skill with numbers as derived facts (Carpenter et al., 1999). There are varying levels of difficulty of subitizing activities that progress as students develop their mathematical skills in primary grades and beyond. They are exposed to a variety of spatial arrangements of sets including linear, circular, and scrambled, listed in respective difficulty level and number size. Development of these skills in the younger grades enhances their prerequisite skills for later mathematics topics. (CCO & LBK) Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann. Clements, D. H. (1999). Subitizing: What is it? Why teach it? Teaching Children Mathematics, 5(7), 400-405. Kaufman, E. L., Lord, M. W., Reese, T. W., and Volkmann, J. (1949). The discrimination of visual number. The American Journal of Psychology, 62(4), 498-525. MacDonald, B. L., & Shumway, J. F. (2016). Subitizing games: Assessing preschoolers' number understanding: Reflect and discuss. Teaching Children Mathematics, 22(6), 340-348.

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Task Analysis consists of breaking a complex task or problem into a series of smaller and more easily managed steps. The idea of task analysis of mathematics problems has been around for decades. Resnick et al. (1973) described hierarchies of tasks used to facilitate children’s learning of early number concepts. These hierarchies were utilized to organize problems in a sequence that built on previous content. For example, counting out of order would follow counting in order. Radatz (1979) conducted an error analysis on various types of mathematics problems. Following sequential and error analysis of mathematics tasks, the focus shifted to level of complexity of tasks. Stein and Lane (1996) studied student performance on high level versus low level tasks. They found that students who had engaged with tasks that involved more complex reasoning outperformed students who only engaged in lower level mathematics problems involving simple procedures. The NCTM Standards documents (1989; 1991) for mathematics curricula shifted to contextualized and real-life situations to enhance learning and improve student achievement and interest in mathematics. Revelations in cognitive science also influenced shifting views on organizing and sequencing mathematics tasks (Carpenter & Moser, 1984). Problem type taxonomies were derived from studying individual students’ strategies. Boston and Smith (2009) used task analysis to make the distinction between low cognitive demand and high cognitive demand in mathematics tasks in secondary mathematics. Within this dichotomy, aspects such as strong conceptual connections and self-monitoring were considered components of high cognitive mathematics tasks. They found that teachers who participated in their professional development sessions and attended more closely to the features of mathematics problems improved their use of high cognitive tasks as a result of participation in their program.(LBK) Boston, M. D., & Smith, M. S. (2009). Transforming secondary mathematics teaching: Increasing the cognitive demands of instructional tasks used in teachers' classrooms. Journal for Research in Mathematics Education, 4(2), 119-156. Carpenter, T. P., & Moser, J. M. (1984). The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education, 15(3), 179-202 National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics (NCTM). (1991). Principles and standards for school mathematics. Reston, VA: Author. Radatz, H. (1979). Error analysis in mathematics education. Journal for Research in Mathematics Education, 10(3), 163-172. Resnick, L. B., Wang, M. C., & Kaplan, J. (1973). Task analysis in curriculum design: A hierarchically sequenced introductory mathematics curriculum. Journal of Applied Behavior Analysis, 6(4), 679-709. Stein, M. K., & Lane, S. (1996). Instructional tasks and the development of student capacity to think and reason: An analysis of the relationship between teaching and learning in a reform mathematics project. Educational Research and Evaluation, 2(1), 50-80.

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Teacher Noticing involves intentional awareness within the context of instruction and learning. “Central to this view is the idea that noticing is a collection of practices designed to sensitize oneself so as to notice opportunities in the future in which to act freshly rather than automatically out of habit” (Mason, 2011). In contrast to general observation in the classroom, teacher noticing is specific and involves in-the-moment decision making. In mathematics classrooms, one type of specialized teacher noticing described as professional noticing of children’s mathematical thinking characterizes the expertise involved in attending to the nuances in children’s verbal descriptions and written representations of their problem-solving strategies (Jacobs et al., 2010). Professional development programs such as Cognitively Guided Instruction (CGI) (Carpenter, et al., 2000) support the development of teacher noticing by engaging teachers in learning about differences in how students solve problems and levels of sophistication represented by their strategies. Research on teacher noticing has suggested there are different levels of expertise in terms of what teachers notice and how they notice student thinking (van Es, 2011). When viewing video cases of mathematics lessons, teachers initially notice more general aspects of the lesson, such as classroom management and pedagogical strategies. Over time, they begin to attend to more explicit details about how students are approaching and solving problems (van Es, 2011, pp. 146147). (LBK) Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (2000). Cognitively Guided Instruction: A research-based teacher professional development program for elementary school mathematics. Research report. Madison, WI: NCISLA, Wisconsin Center for Education Research, University of Wisconsin. Jacobs, V. R., Lamb, L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41, 169-202. Mason, J. (2011). Noticing: Roots and branches. In M. G., Sherin, V. R. Jacobs, & R. A Philipp (Eds.), Mathematics teacher noticing (pp. 35-50). New York, NY: Routledge. van Es, E. A. (2011). A framework for learning to notice student thinking. In M. G., Sherin, V. R. Jacobs, & R. A Philipp (Eds.), Mathematics teacher noticing (pp. 134-151). New York, NY: Routledge.

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Technological and Pedagogical Content Knowledge (TPACK) is a conceptual framework that describes three interrelated components of specialized teacher knowledge necessary for successfully integrating technology into classroom instruction: content, pedagogy, and technology.

The TPACK Framework and Its Components. http://tpack.org (Reproduced by permission of the publisher, © 2012 by tpack.org)

Teachers with pedagogical content knowledge, PCK (Shulman, 1986) know how to organize and represent content using instructional strategies that make learning accessible to students. The use of technology as an instructional strategy has such unique complexities to effectively implement in a classroom, that Mishra and Koehler (2006) introduced technology as a third component to create the theoretical framework of TPACK. TPACK encourages the thoughtful pedagogical use of technology for teaching content, and reminds teachers that simply adding technology to a lesson will not guarantee student understanding. A focus on any one of the three components individually is not sufficient for technology to reach its potential in classroom instruction. Koehler et al. (2013) explain that TPACK “is an emergent form of knowledge that goes beyond all three ‘core’ components … it is an understanding that emerges from interactions among content, pedagogy, and technology knowledge” (p. 16).

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To maximize the learning in an activity, a teacher’s knowledge must lie in the intersection of the three components (Browning & Garza-King, 2010). For instance, when using a graphing calculator, teachers need to be familiar with what the calculator does and how the various features and applications work. They must also have algebraic content knowledge regarding the relationship between equations and their graphs. How teachers facilitate the development of algebraic concepts when students use graphing calculators is where TPACK is most relevant. Teachers must devise a suitable activity that guides students to see multiple representations and to explore the effects on a graph when changing elements of an equation. An awareness of difficulties that students typically encounter and a consequent plan to scaffold learning are necessary to ensure the lesson does not get derailed. (KKM) Browning, C. A. & Garza-King, G. (2010). Graphing calculators as tools. Mathematics Teaching in the Middle School, 15(8), 480-485. Koehler, M. J., Mishra, P., & Cain, W. (2013). What is technological pedagogical content knowledge (TPACK)? Journal of Education, 193(3), 13-19. Mishra, P., & Koehler, M. J. (2006). Technological pedagogical content knowledge: A new framework for teacher knowledge. Teachers College Record, 108 (6), 1017-1054. Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.

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Trends in Mathematics and Science Study (TIMMS) is an international study that periodically compares student achievement in mathematics and science in over 60 countries. TIMSS is sponsored by the International Association for the Evaluation of Educational Achievement (IEA). The U.S. portion of the study is conducted by the National Center for Education Statistics (NCES), a division of the Department of Education, typically targeting 4th and 8th grade math and science classes every four years, and less frequently, 12th grade advanced math and physics. A random sample of schools, weighted to be representative of the U.S. population, is chosen, followed by classrooms chosen randomly to take the 4th or 8th grade assessments. Individual 12th graders are randomly selected for the 12th grade assessment. In addition to math and science content questions, students and teachers respond to a questionnaire investigating attitudes, instructional practices, and home and school life. Curriculum materials are also analyzed. The Third International Study of Mathematics and Science occurred in 1995, the most extensive international study of its time, and was to become widely cited in the move to reform mathematics education in the U.S. The results ranked U.S. schools below many other developed countries, confirming what documents such as A Nation at Risk had been warning in the previous decade. The study included a video analysis of classrooms which described the U.S. math classroom as focused on procedures over higher-level thinking, compared to Japanese schools where teaching conceptual understanding was the goal. The study reported that U.S. teachers have more classes and less planning time than high-ranking countries, and the teachers were aware of reform recommendations but not typically implementing them (NCES, 2018). In a report analyzing and reacting to the study, the U.S. curriculum was described as a “splintered vision” and lacking coherence. “Our curriculum, textbooks, and teaching are ‘a mile wide and an inch deep.’ … This preoccupation with breadth rather than depth, with quantity rather than quality, probably affects how well U.S. students perform in relation to their counterparts in other countries” (Schmidt et al., 1997, p. 1) as students have less opportunity to deepen understanding, explore complex ideas, and make connections that can improve mathematical reasoning skills. Subsequent TIMMS assessments have shown signs of improvement. Data from TIMMS reports are available at nces.ed.gov/timss. Although the TIMMS acronym was first used for the widely-publicized Third International Mathematics and Science Study, the IEA currently refers to the ongoing study as Trends in Mathematics and Science Study, using the same acronym. (KKM) National Center for Education Statistics (NCES). (2018). Trends in International Mathematics and Science Study (TIMMS). Retrieved from https://nces.ed.gov/timss/ Schmidt, W. H., McKnight, C. C., & Raizen, S. A. (1997). A splintered vision: An investigation of U.S. science and mathematics education. U.S. National Research Center for the Third International Study of Mathematics and Science Study. Retrieved from http://www2.phy.ilstu.edu/pte/310content/splintered_vision.pdf

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Van Hiele Levels of Geometric Thinking describe five levels of understanding through which students progress in the learning of geometry. In their 1950s dissertations, Pierre van Hiele proposed the theory of levels and Dina van Hiele-Geldof proposed the experiences that students need to have to advance to higher levels. Having worked with geometry students who struggled with formal proof, they hypothesized that students did not have the necessary prior experiences to support geometric thinking at that level. Their research sought to understand what conceptual understanding was needed, and how that insight into geometry develops. Their findings resulted in the description of levels of geometric thinking and laid the groundwork for others to investigate geometry learning and pedagogy, testing and confirming the theory of levels and adding to the literature (Fuys et al., 1988). The van Hieles’ numbering of the levels begins at level 0, as used below. However subsequent literature used a 1-5 numbering and attached names to each level (Usiskin, 1982). Level 0 (Recognition) – Students recognize and name shapes based on a holistic sense of how they appear. Properties are secondary to appearance thus a square that has been rotated 45 degrees may no longer be recognized as a square. Level 1 (Analysis) – Students describe properties and recognize classes of shapes. Yet the properties are not ordered, so a square is not viewed as a sub-class of a rectangle, with no awareness of the minimal characteristics for defining a shape. Level 2 (Ordering, Informal Deduction) – Students can logically order and interrelate properties of shapes using if-then statements. They can formulate definitions considering necessary and sufficient conditions and understand class inclusions. They have an increased ability to make and follow informal arguments. Level 3 (Deduction) – Students understand the structure of definitions, postulates, and theorems and can utilize them to construct a formal deductive proof. The significance of proof is appreciated. Level 4 (Rigor) – At the level of a mathematician, axiomatic systems are analyzed. Pierre Van Hiele (1959/2004) describes the levels as sequential, with progression to the next level more dependent on instructional experiences than age. (KKM) Fuys, D., Geddes, D., & Tischler, R. (1988). The van Hiele model of thinking in geometry among adolescents [Monograph]. Journal for Research in Mathematics Education. Reston, VA: National Council of Teachers of Mathematics. Usiskin, Z. (1982). Van Hiele levels and achievement in secondary school geometry. Retrieved from http://ucsmp.uchicago.edu/resources/van-hiele/ van Hiele, P. (1959/2004). The child’s thought and geometry. In T. P. Carpenter, J. A. Dossey, & J. L. Koehler (Eds.), Classics in Mathematics Education (pp. 61-65). Reston, VA: National Council of Teachers of Mathematics.

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Zone of Proximal Development (ZPD) is a concept developed by Russian psychologist Lev Vygotsky and refers to the difference between what one can learn without any assistance versus what one can learn with the assistance of a knowledgeable other. Lev Vygotsky (1896-1934) was a Russian philologist and psychologist widely credited as a pioneer in the sociocultural theory of learning and development, which works to describe the role of society and culture in human development. Vygotsky began his career in psychology after the 1917 Russian Revolution, working in a period where great debate was occurring regarding human cognitive development. Vygotsky often referred to a “crisis in psychology” (Vygotsky, 1978, p. 5) in regards to the competing theories of behaviorism and Gestalt psychology that were prominent at this time and devoted his work to creating a fusion of these theories. As a result of his study and work, Vygotsky began to articulate a theory of human intellectual development centered on a student’s ability to learn how to use tools and signs familiar in one’s culture through the interactions with other students and adults that in turn help to socialize the learner into their culture (Doolittle, 1997). Vygotsky stressed the importance of internalization, where a learner experiences a phenomenon during social interaction and that through internalizing the event, a transformation occurs that pushes the individual to a later form of development. To Vygotsky, the process of development was greatly situated in one’s society and culture (Vygotsky, 1978). A critical component of sociocultural theory is that of the zone of proximal development (ZPD). Vygotsky posited that an individual has limits to what he/she can learn, and that one’s potential for growth can only occur within this zone. On one end of the spectrum lies the Level of Actual Development, where a student can perform tasks independently without the guidance of a knowledgeable other. On the other end of the zone lie tasks that are completely out of the realm of possibility for the learner, regardless of the assistance that can be provided. What lies between these two markers is the ZPD, what Vygotsky defined as “the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers” (Vygotsky, 1978, p. 86). It is in this realm that Vygotsky states that development and learning could occur. The ZPD is not static, meaning that as a student continues to learn, tasks that were located within the ZPD will move to the Level of Actual Development (where assistance would not be needed), or that tasks that were out of the realm of the student may move into one’s ZPD. The ZPD can therefore be used to describe both the student’s current developmental state as well as where the student may be headed in the future (Vygotsky, 1978). (SWD) Doolittle, P. E. (1997). Vygotsky’s zone of proximal development as a theoretical foundation for cooperative learning. Journal on Excellence in College Teaching, 8(1), 83-103. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.