Teachers must be prepared to recognize valid alternative representations of arithmetic problems. Challenging examples involving mixed fractions and division are presented along with teacher's discussion from a professional development workshop. (Contains 6 figures and 1 table.)

In the late seventies, Guy Brousseau set himself the goal of verifying experimentally a theory he had been building up for a number of years. The theory, consistent with what was later named (non-radical) constructivism, was that children, in suitable carefully arranged circumstances, can build their own knowledge of mathematics. The experiment,…

This article communicates findings from a year-long constructivist teaching experiment about the relationship between four sixth-grade students' multiplicative structures and their construction of improper fractions. Students' multiplicative structures are the units coordinations that they can take as given prior to activity--i.e., the units…

This paper concerns computer aided assessment (CAA) of mathematics in which a computer algebra system (CAS) is used to help assess students' responses to elementary algebra questions. Using a methodology of documentary analysis, we examine what is taught in elementary algebra. The STACK CAA system, http://www.stack.bham.ac.uk/, which uses the CAS…

The aims of this study were to investigate whether the executive functions, inhibition, shifting, and updating, are distinguishable as latent variables (common factors) in children aged 9 to 12, and to examine the relations between these executive functions and reading, arithmetic, and (non)verbal reasoning. Confirmatory factor analysis was used…

History of mathematics occupies itself describing processes of growth and development, whereas philosophy of mathematics is concerned with questions of justification. Both play an essential role within the educational context. But there is a problem because genuine historical studies necessitate ever greater particularity whereas mathematics and…

The addition and subtraction of integers is the first major avenue, and roadblock, to student success in learning algebra. This article describes a hands-on activity using a regular deck of playing cards that facilitates class discussion and helps students overcome this initial roadblock. (Contains 14 figures.)

This article describes one way to introduce fraction, decimal, and percent equivalents by linking student-created Op Art with examples of twentieth-century abstract painting. (Contains 7 figures.)

Learning mathematics has traditionally been thought of as a sequential progression. Children learn to count to 10, then to 20, and then to 100. They learn to add without regrouping and then with regrouping. The authors teach addition before multiplication and the two-times table before the six-times table. They usually teach division as a separate…

Presents a defense of traditional teaching methods during the progressive education movement. Reprint from April 1931 "Elementary School Journal." (JMB)

This textbook is a course in mental arithmetic that is essentially based on the object-lesson plan. It is believed that the pupil in the primary school can more readily acquire a familiarity with numbers from the use of objects and pictures than from an uninterrupted drill on the tables. It also includes suggestions for teachers how to use the…

This textbook aims to present the mathematical sciences in clear, everyday language for apprentices, students and mechanics--willing, industrious, anxious to study at home, yet unable to give time or money for professional teaching. It is trusted that this effort to place before English-speaking students a treatise on arithmetic and algebra, freed…

This textbook is second in a series of arithmetic books that apply the principles discovered by the psychology of learning, by experimental education, and by the observation of successful school practice, to the teaching of arithmetic. Parts One and Two are intended for grades five and six respectively. Part One provides for mastery of the four…