Researchers are currently investigating how calculus students understand the basic concepts of first-year calculus, including the integral. However, much is still unknown regarding the "cognitive resources" (i.e., stable cognitive units that can be accessed by an individual) that students hold and draw on when thinking about the integral. This…

This study investigates Chinese and U.S. teachers' construction and use of pedagogical representations surrounding implementation of mathematical tasks. It does this by analyzing video-taped lessons from the Learner's Perspective Study, involving 15 Chinese and 10 U.S. consecutive lessons on the topic of linear equations/linear relations. We…

This study examined prospective middle grade mathematics teachers' knowledge of algebra for teaching with a focus on knowledge for teaching the concept of function. 115 prospective teachers from an interdisciplinary program for mathematics and science middle teacher preparation at a large public university in the USA participated in a survey. It…

In this paper two 10th graders having an accumulated experience on problem-solving ancillary to the concept of area confronted the task to find Pick's formula for a lattice polygon's area. The formula was omitted from the theorem in order for the students to read the theorem as a problem to be solved. Their working is examined and emphasis is…

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 (nonradical) constructivism, was that children, in suitable carefully arranged circumstances, can build their own knowledge of mathematics. The experiment,…

In the present study we explore changes in perceptions of our class of prospective mathematics teachers (PTs) regarding their mathematical knowledge. The PTs engaged in problem posing activities in geometry, using the "What If Not?" (WIN) strategy, as part of their work on computerized inquiry-based activities. Data received from the PTs'…

This article focuses on spontaneous knowledge-building in the field of "the arithmetic "of" the child." The aim is to investigate the conceptual progress of fifteen children during their early school years in the compulsory school. The study is based on the epistemology of radical constructivism and the methodology of…

We report a case study that explored how three college students mentally represented the knowledge they held of inferential statistics, how this knowledge was connected, and how it was applied in two problem solving situations. A concept map task and two problem categorization tasks were used along with interviews to gather the data. We found that…

The purpose of this study is to analyze the relationship between student behaviors and the growth of mathematical ideas (using the Pirie-Kieren model). This analysis was accomplished through a series of case studies, involving middle school students of varying ability levels, who were investigating a combinatorics problem in after-school…

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…

In this paper we examine the roles that metonymy may play in student reasoning. To organize this discussion we use the lens of a structured derivative framework. The derivative framework consists of three layers of process-object pairs, one each for ratio, limit, and function. Each of the layers can then be illustrated in any appropriate context,…

The purpose of this paper is to re-examine the relationships between mathematical problem solving and learning mathematics. After introducing a diagram representing the notion of problem solving, four types of problem solving approaches used in mathematics classrooms will be distinguished according to which aspect of that diagram is attended to.…

This paper illustrates the role of a "Thinking-about-Derivatives" task in identifying learners' derivative conceptions and for promoting their critical thinking about derivatives of absolute value functions. The task included three parts: "Define" the derivative of a function f(x) at x = x[subscript 0], "Solve-if-Possible" the derivative of f(x) =…

Describes one possible approach to solving the following problem: given a pentagon ABCDE in which triangles ABC, BCD, CDE, DEA, and EAB all have area=1, is the area of the pentagon determined? (AIM)

Profiles a middle school mathematics teacher and examines her use of two problems from a pilot version of a 6th grade unit developed by a mathematics curriculum project. Reports that problem-solving-oriented curricula provide opportunities for students to make mathematics connections and leads to student confusion and uncertainty. Contains 35…