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"To truly engage in mathematics is to become curious and intrigued about regularities and patterns, then describe and explain them. A focus on the behavior of the operations allows students starting in the familiar territory of number and computation to progress to true engagement in the discipline of mathematics." -Susan Jo Russell, Deborah Schifter, and Virginia Bastable Algebra readiness: it's a topic of concern that seems to pervade every school district. How can we better prepare elementary students for algebra? More importantly, how can we help all children, not just those who excel in math, become ready for later instruction? The answer lies not in additional content, but in developing a way of thinking about the mathematics that underlies both arithmetic and algebra. Connecting Arithmetic to Algebra invites readers to learn about a crucial component of algebraic thinking: investigating the behavior of the operations. Nationally-known math educators Susan Jo Russell, Deborah Schifter, and Virginia Bastable and a group of collaborating teachers describe how elementary teachers can shape their instruction so that students learn to: *notice and describe consistencies across problems *articulate generalizations about the behavior of the operations *develop mathematical arguments based on representations to explain why such generalizations are or are not true. Through such work, students become familiar with properties and general rules that underlie computational strategies-including those that form the basis of strategies used in algebra-strengthening their understanding of grade-level content and at the same time preparing them for future studies. Each chapter is illustrated by lively episodes drawn from the classrooms of collaborating teachers in a wide range of settings. These provide examples of posing problems, engaging students in productive discussion, using representations to develop mathematical arguments, and supporting both students with a wide range of learning profiles. Staff Developers: Available online, the Course Facilitator's Guide provides math leaders with tools and resources for implementing a Connecting Arithmetic to Algebra workshop or preservice course. For information on the PD course offered through Mount Holyoke College, download the flyer.
Fostering Algebraic Thinking is a timely and welcome resource for middle and high school teachers hoping to ease their students' transition to algebra.
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.
Like algebra at any level, early algebra is a way to explore, analyse, represent and generalise mathematical ideas and relationships. This book shows that children can and do engage in generalising about numbers and operations as their mathematical experiences expand. The authors identify and examine five big ideas and associated essential understandings for developing algebraic thinking in grades 3-5. The big ideas relate to the fundamental properties of number and operations, the use of the equals sign to represent equivalence, variables as efficient tools for representing mathematical ideas, quantitative reasoning as a way to understand mathematical relationships and functional thinking to generalise relationships between covarying quantities. The book examines challenges in teaching, learning and assessment and is interspersed with questions for teachers’ reflection.
Originally published: Boston: Houghton Mifflin, 1987.
A hallmark of much of the research on children’s thinking in the 1970s had been the focus on explicit content domains. Much of this research had been represented by an eclectic collection of studies sampled from a variety of disciplines and content areas. However, in the few years before this publication, research in several content domains has begun to coalesce into a coherent body of knowledge. Originally published in 1982, the chapters in this work represent one of the first attempts to bring together the perspectives of a variety of different researchers investigating a specific, well defined content domain. This book presents theoretical views and research findings of a group of international scholars who are investigating the early acquisition of addition and subtraction skills by young children. Together, the contributors bring a blend of psychology, educational psychology, and mathematics education to this topic. Fields of interest such as information processing, artificial intelligence, early childhood, and classroom teaching and learning are included in this blend.
Accessible Algebra: 30 Modules to Promote Algebraic Reasoning, Grades 7-10 is for any pre-algebra or algebra teacher who wants to provide a rich and fulfilling experience for students as they develop new ways of thinking through and about algebra. The book includes 30 lessons that identifies a focal domain and standard in algebra, then lays out the common misconceptions and challenges students may face as they work to investigate and understand problems. Authors Anne Collins and Steven Benson conferred with students in real classrooms as the students explained what problem-solving strategies they were using or worked to ask the right questions that would lead them to a deeper understanding of algebra. Each scenario represents actual instances of an algebra classroom that demonstrate effective teaching methods, real-life student questions, and conversations about the problems at hand. Accessible Algebra works for students at every level. In each lesson there are sections on how to support struggling students, as well as ways to challenge students who may need more in-depth work. There are also numerous additional resources, including research articles and classroom vignettes.