Using a mixed methods approach, we explore a relationship between students' graph reasoning and graph selection via a fully online assessment. Our population includes 673 students enrolled in college algebra, an introductory undergraduate mathematics course, across four U.S. postsecondary institutions. The assessment is accessible on computers,…

This study examined college calculus instructors' preferences in solving two calculus tasks to examine college calculus instructors' use of important cognitive roots in understanding derivatives of function. Our results showed that only one instructor consistently uses cognitive roots while other instructors either focus on algebraic methods or…

When establishing connections among representations of associated mathematical concepts, students encounter different difficulties and successes along the way. The purpose of this study was to uncover information about and gain greater insight into how student processes connections. Pre-calculus students were observed and interviewed while…

In the subject matter of functional relationships, a student's ability to translate from one form of representation to another is seen as a central competence. In the course of the HEUREKO project (heuristic work with representations of functional relationships and the diagnosis of mathematical competencies of students), a theoretical competence…

Today, beginning algebra in the high school setting is associated more with remediation than pride. Students enroll by mandate and attend under duress. Class rosters in this "graveyard" course, as it is often referred to, include sophomores and juniors who are attempting the course for the second or third time. Even the ninth graders…

This study investigates cognitive difficulties and mathematical ideas that are related to making connections among representations. A three-week intervention was designed and implemented to help prospective secondary mathematics teachers develop understanding of big ideas that are critical to connection of representations in algebra. This study…

This article describes the development of the Precalculus Concept Assessment (PCA) instrument, a 25-item multiple-choice exam. The reasoning abilities and understandings central to precalculus and foundational for beginning calculus were identified and characterized in a series of research studies and are articulated in the PCA Taxonomy. These…

Reports on how students managed some technical aspects of graphics calculators as they used them to study graphs of straight lines and parabolas. Identifies three common student difficulties: (1) tendency to be unduly influenced by the jagged appearance of graphs; (2) poor understanding of the zoom operation; and (3) limited grasp of the processes…

Documents what high school students showed when asked to work on tasks that involved the use of various representations. Indicates that few students made proper connections between representations and the initial situation or problem on their own and exhibited competence in procedures such as expressing an area or solving quadratic equations, but…

This paper is structured in two sections. The first examines views of mathematical abstraction in two broad categories: empiricist and dialectical accounts. It documents the difficulties involved in and explores the potentialities of both accounts. Then it outlines a recent model which takes a dialectical materialist approach to abstraction in…

Presents evidence that there exists much confusion regarding the connection between the algebraic and geometric aspects of slope, scale, and angle. Participants responded to a simple but non-standard task concerning the behavior of slope under a non-homogeneous change of scale. Analysis of the responses reveals two main approaches termed…

The presentations address a variety of difficulties occurring in students' solutions of equations and inequalities, and suggest different reasons for these difficulties. When analyzing students' performances, [BB] and [TTT] mention students' tendencies to make irrelevant connections between equations and inequalities as a problematic phenomenon.…

This study investigates relationships between differences in students' achievement and differences obtained by a technique held to yield a reasonably valid representation of students' memory structures of a particular subject matter. Also investigated was the nature of the flow of information between curriculum materials, teachers, and students.…

This paper speaks to the challenges, risks, and rewards of teaching about linear functions in a technology-rich environment from a constructivist perspective by describing a Foundations of Algebra class, which was designed and taught to junior high school students in such an environment in the summer of 1991. The design and implementation of the…

This study investigates middle school mathematics students' views and interpretations of graphical representations as they use graphs to answer algebraic questions--specifically, questions that require them to extrapolate information from graphs. From data gathered in videotaped interviews, students' verbal responses were analyzed as well as any…