
Proving in the Elementary Mathematics Classroom addresses a fundamental problem in children’s learning that has received relatively little research attention: Although proving and related concepts (e.g., proof, argumentation, conjecturing) are core to mathematics as a sensemaking activity, they currently have a marginal place in elementary classrooms internationally. This book takes a step toward addressing this problem by examining how the place of proving in elementary students’ mathematical work can be elevated through the purposeful design and implementation of mathematics tasks, specifically proving tasks. In particular, the book draws on relevant research and theory and classroom episodes with 8–9yearolds from England and the United States to examine different kinds of proving tasks and the proving activity they can help generate in the elementary classroom. It examines further the role of elementary teachers in mediating the relationship between proving tasks and proving activity, including major mathematical and pedagogical issues that can arise for them as they implement each kind of proving task in the classroom. In addition to its research contribution in the intersection of the scholarly areas of teaching/learning proving and task design/implementation, the book has important implications for teaching, curricular resources, and teacher education. For example, the book identifies different kinds of proving tasks whose balanced representation in the mathematics classroom and in curricular resources can support a rounded set of learning experiences for elementary students related to proving. It identifies further important mathematical ideas and pedagogical practices related to proving that can be studied in teacher education.

Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane moves around the outside of a convex shape according to a scheme that is reminiscent of ordinary billiards. This book provides a combinatorial model for orbits of outer billiards on kites. The book relates these orbits to such topics as polytope exchange transformations, renormalization, continued fractions, corner percolation, and the Truchet tile system. The combinatorial model, called “the plaid model,” has a selfsimilar structure that blends geometry and elementary number theory. The results were discovered through computer experimentation and it seems that the conclusions would be extremely difficult to reach through traditional mathematics. The book includes an extensive computer program that allows readers to explore the materials interactively and each theorem is accompanied by a computer demonstration.

The history of mathematics starts in earnest with one of Pythagoras’ most important proofs, the Pythagorean theorem. This proof was the first link in a chain of groundbreaking ideas, all interconnected with each other, that turned mathematics into an “art of the mind.” The chain continues to be extended today. There would be no computers, science, engineering, or philosophy without Pythagoras’ legacy. This book sketches an outline of that legacy by presenting and discussing ten of the greatest ideas in the mathematical chain. Its aim is to illustrate why mathematics can be designated an intellectual art, a creative enterprise that mirrors any art, from music to painting. Pythagoras actually connected music and mathematics into a theory of the world called the Harmony of the Spheres. The book is intended for a general audience, and especially those who may think that mathematics is uninteresting or boring. Each of its ten chapter ends with five exploratory puzzles that will allow readers to become engaged in some of the ideas treated in the chapter without any technical knowledge. This will allow readers to use this book as well as a collection of fairly easy math problems.

How to Free Your Inner Mathematician: Notes on Mathematics and Life offers readers guidance in managing the fear, freedom, frustration, and joy that often accompany calls to think mathematically. With practical insight and years of awardwinning mathematics teaching experience, DAgostino offers more than 300 handdrawn sketches alongside accessible descriptions of fractals, symmetry, fuzzy logic, knot theory, Penrose patterns, infinity, the Twin Prime Conjecture, Arrows Impossibility Theorem, Fermats Last Theorem, and other intriguing mathematical topics. Readers are encouraged to embrace change, proceed at their own pace, mix up their routines, resist comparison, have faith, fail more often, look for beauty, exercise their imaginations, and define success for themselves. Mathematics students and enthusiasts will learn advice for fostering courage on their journey regardless of age or mathematical background. How to Free Your Inner Mathematician delivers not only engaging mathematical content but provides reassurance that mathematical success has more to do with curiosity and drive than innate aptitude.

This book examines computer aided assessment (CAA) of mathematics in which computer algebra systems (CAS) are used to automatically establish the mathematical properties of expressions provided by students in response to questions. In order to automate such assessment, the relevant criteria must be encoded. This is not so simple. Even articulating precisely the desired criteria forces the teacher to think very carefully indeed. Hence, CAA acts as a vehicle to examine assessment and mathematics education in detail and from a fresh perspective. For example, the constraints of the paperbased formats have affected what we do and why. It is natural for busy teachers to set only those questions which can be marked by hand in a straightforward way. However, there are other kinds of questions, e.g., those with nonunique correct answers, or where assessing the properties requires the marker themselves to undertake a significant computation. It is simply not sensible for a person to set these to large groups of students when marking by hand. And yet such questions have their place and value in provoking thought and learning. Furthermore, we explain how, in certain cases, these can be automatically assessed. Case studies of existing systems will illustrate this in a concrete and practical way.

Equivariant cohomology is concerned with the algebraic topology of spaces with a group action, or in other words, with symmetries of spaces. First defined in the 1950s, it has been introduced into Ktheory and algebraic geometry, but it is in algebraic topology that the concepts are the most transparent and the proofs are the simplest. One of the most useful applications of equivariant cohomology is the equivariant localization theorem of Atiyah–Bott and Berline–Vergne, which converts the integral of an equivariant differential form into a finite sum over the fixed point set of the group action, providing a powerful tool for computing integrals over a manifold. Because integrals and symmetries are ubiquitous, equivariant cohomology has found applications in diverse areas of mathematics and physics. Assuming readers have taken one semester of manifold theory and a year of algebraic topology, the book begins with the topological construction of equivariant cohomology, then develops the theory for smooth manifolds with the aid of differential forms. To keep the exposition simple, the equivariant localization theorem is proven only for a circle action. An appendix gives a proof of the equivariant de Rham theorem, demonstrating that equivariant cohomology can be computed using equivariant differential forms. Examples and calculations illustrate new concepts. Exercises include hints or solutions, making this book suitable for selfstudy.

Students in the sciences, economics, social sciences, and medicine take an introductory statistics course. And yet statistics can be notoriously difficult for instructors to teach and for students to learn. To help overcome these challenges, Gelman and Nolan have put together this fascinating and thoughtprovoking book. Based on years of teaching experience the book provides a wealth of demonstrations, activities, examples and projects that involve active student participation. Part I of the book presents a large selection of activities for introductory statistics courses and has chapters such as ‘First week of class’ with exercises to break the ice and get students talking; then descriptive statistics, graphics, linear regression, data collection (sampling and experimentation), probability, inference, and statistical communication. Part II gives tips on what works and what doesn’t, how to set up effective demonstrations, how to encourage students to participate in class and to work effectively in group projects. Course plans for introductory statistics, statistics for social scientists, and communication and graphics are provided. Part III presents material for more advanced courses on topics such as decision theory, Bayesian statistics, sampling, and data science.

This book is a guide to the scholarship of teaching and learning (SoTL) for scientists, engineers, and mathematicians teaching at the collegiate level in countries around the world. It shows instructors how to draw on their disciplinary knowledge and teaching experience to investigate questions about student learning. It takes them all the way through the inquiry process beginning with framing a research question and selecting a research design, moving on to gathering and analyzing evidence, and finally to making the results public. Numerous examples are provided at each stage, many from published studies of teaching and learning in science, engineering, or mathematics. At strategic points, short sets of questions prompt readers to pause and reflect, plan, or act. These questions are derived from the authors’ experience leading many SoTL workshops in the United States and Canada. The taxonomy of SoTL questions—What works? What is? What could be?—that emerged from the SoTL studies undertaken by the Carnegie scholars provides a useful framework at many stages of the inquiry process. The book addresses the issue of evaluating and valuing this work, including implications for junior faculty who wish to engage in SoTL. The authors explain why SoTL should be of interest to science, technology, engineering, and mathematics (STEM) faculty at all types of institutions, including faculty members active in traditional STEM research. They also give their perspective on the benefits of SoTL to faculty, to their institutions, to the academy, and to students.

Curricular Resources and Classroom Use examines the use of curricular resources, that is, the different kinds of materials (digital or physical) that teachers use in or for their teaching (textbooks, lesson plans, etc.). These resources have a significant influence on students’ opportunities to learn. At the same time, teachers play a crucial role as interpreters and users of curricular resources, so there is a complex relationship between curricular resources and their classroom use. Research thus far has mostly focused on developing approaches for studying either particular curricular resources or their classroom use. This book aims to bridge these highly related programs of research by describing, comparing, and exemplifying new research approaches for studying curricular resources and their classroom use, as well as the complex interplay between the two. This book exemplifies the approaches in the area of mathematics, but the approaches can be more broadly applicable and be used in isomorphic ways in other subject areas (science, history, etc.). As issues concerning curricular resources and the classroom use of such resources are of interest to researchers, curriculum developers (such as textbook authors), and teacher educators in many countries, this book is addressed to a broad international audience. In addition to providing implications for research, this book has implications for curriculum development and teacher education. Specifically, this book deepens understanding of how curriculum developers can better exploit the potential of curricular resources to support classroom work, and how teacher educators can better support teachers to use curricular resources in the classroom.

Seventyfive years of the study of matroids has seen the development of a rich theory with links to graphs, lattices, codes, transversals,0020and projective geometries. Matroids are of fundamental importance in combinatorial optimization and their applications extend into electrical and structural engineering. This book falls into two parts: the first provides a comprehensive introduction to the basics of matroid theory, while the second treats more advanced topics. It contains over 700 exercises, and includes proofs of all of the major theorems in the subject. The last two chapters review current research and list more than eighty unsolved problems along with a description of the progress towards their solutions.