Students Conceptions Of Variables And Their Uses For Generalization Of Mathematical Patterns

This book synthesizes research findings on patterns in the last twenty years or so in order to argue for a theory of graded representations in pattern generalization. While research results drawn from investigations conducted with different age-level groups have sufficiently demonstrated varying shifts in structural awareness and competence, which influence the eventual shape of an intended generalization, such shifts, however, are not necessarily permanent due to other pertinent factors such as the complexity of patterning tasks. The book proposes an alternative view of pattern generalization, that is, one that is not about shifts or transition phases but graded depending on individual experiences with target patterns. The theory of graded representations involving pattern generalization offers a much more robust understanding of differences in patterning competence since it is sensitive to varying levels of entry into generalization. Empirical evidence will be provided to demonstrate this alternative view, which is drawn from the author’s longitudinal work with elementary and middle school children, including several investigations conducted with preservice elementary majors. Two chapters of the book will be devoted to extending pattern generalization activity to arithmetic and algebraic learning of concepts and processes. The concluding chapter addresses the pedagogical significance of pattern learning in the school mathematics curriculum. ​

This book highlights new developments in the teaching and learning of algebraic thinking with 5- to 12-year-olds. Based on empirical findings gathered in several countries on five continents, it provides a wealth of best practices for teaching early algebra. Building on the work of the ICME-13 (International Congress on Mathematical Education) Topic Study Group 10 on Early Algebra, well-known authors such as Luis Radford, John Mason, Maria Blanton, Deborah Schifter, and Max Stephens, as well as younger scholars from Asia, Europe, South Africa, the Americas, Australia and New Zealand, present novel theoretical perspectives and their latest findings. The book is divided into three parts that focus on (i) epistemological/mathematical aspects of algebraic thinking, (ii) learning, and (iii) teaching and teacher development. Some of the main threads running through the book are the various ways in which structures can express themselves in children’s developing algebraic thinking, the roles of generalization and natural language, and the emergence of symbolism. Presenting vital new data from international contexts, the book provides additional support for the position that essential ways of thinking algebraically need to be intentionally fostered in instruction from the earliest grades.

Awesome Algebra: Looking for Patterns and GeneralizationsWe do not usually think of algebra as a topic for the elementary mathematics classroom. However, algebra is one of the five major content strands outlined by the National Council of Teachers of Mathematics (NCTM) in Principles and Standards for School Mathematics (2000). Looking for patterns, extending a pattern, making a generalization about a pattern -- all are part of algebraic thinking. So we talk about algebraic thinking or reasoning as opposed to the formal study of algebra. In our Project M unit Awesome Algebra: Looking for Patterns and Generalizations, students are encouraged to study patterns and determine how they change, how they can be extended or repeated and/or how they grow. They then move beyond this to organize the information systematically and analyze it to develop generalizations about the mathematical relationships in the patterns. There is a strong focus on mathematical discourse revolving around how to verbalize a generalization. During Awesome Algebra: Looking for Patterns and Generalizations students will be encouraged to use the idea of a variable as they think about how to represent a rule. This will help them become aware of the usefulness of a variable when representing a generalization. Our emphasis on number patterns is designed to challenge mathematically talented students by encouraging them to take a new look at basic number concepts, that is, arithmetic from an algebraic perspective. Students will become better estimators and give have effective tools to perform computation mentally. We hope that the experiences and discussions in the unit will provide a rich context for introducing students to algebraic thinking and strengthen their reasoning and communication skills. Student Mathematician's Journal The Student Mathematician's Journal is a unique feature of every unit in the Project M: Mentoring Mathematical Minds series, encouraging students to communicate in writing. It includes the student worksheets from each lesson. In these journals we ask students to reflect on what they have learned and write about it; in effect, they are working and acting like real mathematicians when they do this. Components used to teach this module: Awesome Algebra Teacher Guide (0-7575-2331-5)Awesome Algebra Student Mathematician's Journal Awesome Algebra: Looking for Patterns and GeneralizationsWe do not usually think of algebra as a topic for the elementary mathematics classroom. However, algebra is one of the five major content strands outlined by the National Council of Teachers of Mathematics (NCTM) in Principles and Standards for School Mathematics (2000). Looking for patterns, extending a pattern, making a generalization about a pattern -- all are part of algebraic thinking. So we talk about algebraic thinking or reasoning as opposed to the formal study of algebra. In our Project M unit Awesome Algebra: Looking for Patterns and Generalizations, students are encouraged to study patterns and determine how they change, how they can be extended or repeated and/or how they grow. They then move beyond this to organize the information systematically and analyze it to develop generalizations about the mathematical relationships in the patterns. There is a strong focus on mathematical discourse revolving around how to verbalize a generalization. During Awesome Algebra: Looking for Patterns and Generalizations students will be encouraged to use the idea of a variable as they think about how to represent a rule. This will help them become aware of the usefulness of a variable when representing a generalization. Our emphasis on number patterns is designed to challenge mathematically talented students by encouraging them to take a new look at basic number concepts, that is, arithmetic from an algebraic perspective. Students will become better estimators and give have effective tools to perform computation mentally. We hope that the experiences and dis

The concept of an algebraic variable is both important in its own right and foundational for higher levels of math, but many students struggle to comprehend its meaning and purpose, demonstrating a variety of misconceptions about the interpretation of a variable and algebra's relation to arithmetic. Common educational practices fail to support a substantial portion of students in connecting their intuitive cognitive capabilities to the formal external representations (i.e., symbolic notation) of algebra, depriving these students of understanding how and why variables are used, as well as their relevance in solving real-world problems. Previous attempts at improving students' understanding of variables have focused on schematic induction across varied concrete examples or the generalization of relational thinking from arithmetic. While these efforts are important, the approaches do not fully elucidate the purpose of using formal symbols (e.g., letters) to represent unknown numbers. I posit that the clearest way to demonstrate the purpose of symbolic variables is through students' formulation and attempted solution of mathematical problems where multiple unknowns must be represented (and distinguished from each other), such as in a system of equations word problem. Guided by principles from cognitive psychology and educational research, I formulate a framework for encouraging and supporting students' intuitive discovery of the concept of variable using purpose-driven contrast comparisons, active learning techniques such as constructive struggling with intuitive hints, and contextual facilitation of students' natural problem solving for meaningful, concrete tasks. Through this process, variables representations are introduced progressively, first by using more interpretable word equations and later by abbreviating word phrases into letter symbols. I implemented this framework into novel multimedia educational materials, which were iteratively piloted and revised, and then experimentally tested with middle and high school students against a more traditionally structured control version of the materials and a baseline condition. The results from this experimental testing suggest that students who were encouraged to infer the purpose of a variable before its formal representation was introduced went on to provide more correct answers to analogous problems on a post-test given 1-3 weeks later.

Algebra is the gateway to college and careers, yet it functions as the eye of the needle because of low pass rates for the middle school/high school course and students’ struggles to understand. We have forty years of research that discusses the ways students think and their cognitive challenges as they engage with algebra. This book is a response to the National Council of Teachers of Mathematics’ (NCTM) call to better link research and practice by capturing what we have learned about students’ algebraic thinking in a way that is usable by teachers as they prepare lessons or reflect on their experiences in the classroom. Through a Fund for the Improvement of Post-Secondary Education (FIPSE) grant, 17 teachers and mathematics educators read through the past 40 years of research on students’ algebraic thinking to capture what might be useful information for teachers to know—over 1000 articles altogether. The resulting five domains addressed in the book (Variables & Expressions, Algebraic Relations, Analysis of Change, Patterns & Functions, and Modeling & Word Problems) are closely tied to CCSS topics. Over time, veteran math teachers develop extensive knowledge of how students engage with algebraic concepts—their misconceptions, ways of thinking, and when and how they are challenged to understand—and use that knowledge to anticipate students’ struggles with particular lessons and plan accordingly. Veteran teachers learn to evaluate whether an incorrect response is a simple error or the symptom of a faulty or naïve understanding of a concept. Novice teachers, on the other hand, lack the experience to anticipate important moments in the learning of their students. They often struggle to make sense of what students say in the classroom and determine whether the response is useful or can further discussion (Leatham, Stockero, Peterson, & Van Zoest 2011; Peterson & Leatham, 2009). The purpose of this book is to accelerate early career teachers’ “experience” with how students think when doing algebra in middle or high school as well as to supplement veteran teachers’ knowledge of content and students. The research that this book is based upon can provide teachers with insight into the nature of a student’s struggles with particular algebraic ideas—to help teachers identify patterns that imply underlying thinking. Our book, How Students Think When Doing Algebra, is not intended to be a “how to” book for teachers. Instead, it is intended to orient new teachers to the ways students think and be a book that teachers at all points in their career continually pull of the shelf when they wonder, “how might my students struggle with this algebraic concept I am about to teach?” The primary audience for this book is early career mathematics teachers who don’t have extensive experience working with students engaged in mathematics. However, the book can also be useful to veteran teachers to supplement their knowledge and is an ideal resource for mathematics educators who are preparing preservice teachers.

In Greek geometry, there is an arithmetic of magnitudes in which, in terms of numbers, only integers are involved. This theory of measure is limited to exact measure. Operations on magnitudes cannot be actually numerically calculated, except if those magnitudes are exactly measured by a certain unit. The theory of proportions does not have access to such operations. It cannot be seen as an "arithmetic" of ratios. Even if Euclidean geometry is done in a highly theoretical context, its axioms are essentially semantic. This is contrary to Mahoney's second characteristic. This cannot be said of the theory of proportions, which is less semantic. Only synthetic proofs are considered rigorous in Greek geometry. Arithmetic reasoning is also synthetic, going from the known to the unknown. Finally, analysis is an approach to geometrical problems that has some algebraic characteristics and involves a method for solving problems that is different from the arithmetical approach. 3. GEOMETRIC PROOFS OF ALGEBRAIC RULES Until the second half of the 19th century, Euclid's Elements was considered a model of a mathematical theory. This may be one reason why geometry was used by algebraists as a tool to demonstrate the accuracy of rules otherwise given as numerical algorithms. It may also be that geometry was one way to represent general reasoning without involving specific magnitudes. To go a bit deeper into this, here are three geometric proofs of algebraic rules, the frrst by Al-Khwarizmi, the other two by Cardano.

This book discusses the relationships between mathematical creativity and mathematical giftedness. It gathers the results of a literature review comprising all papers addressing mathematical creativity and giftedness presented at the International Congress on Mathematical Education (ICME) conferences since 2000. How can mathematical creativity contribute to children’s balanced development? What are the characteristics of mathematical giftedness in early ages? What about these characteristics at university level? What teaching strategies can enhance creative learning? How can young children’s mathematical promise be preserved and cultivated, preparing them for a variety of professions? These are some of the questions addressed by this book. The book offers, among others: analyses of substantial learning environments that promote creativity in mathematics lessons; discussions of a variety of strategies for posing and solving problems; investigations of students’ progress throughout their schooling; and examinations of technological tools and virtual resources meant to enhance learning with understanding. Multiple perspectives in the interdisciplinary fields of mathematical creativity and giftedness are developed to offer a springboard for further research. The theoretical and empirical studies included in the book offer a valuable resource for researchers, as well as for teachers of gifted students in specialized or inclusive settings, at various levels of education.