Facets and layers of function for college students in beginning algebra

The first mathematics course for approximately 53 percent of U.S. community college students is a developmental algebra course. Many such students appear to be severely debilitated by their previous encounters with mathematics. Due to numerous misconceptions that dictate against a traditional course, a "reform" beginning algebra course, with function as the unifying concept, was designed. Since there is little research on this population to justify such a approach, the key research question for this thesis becomes: Can adult students who arrive at college having had debilitating prior experiences with algebra acquire at least a process level understanding of function through appropriate instructional treatment? Answering this question provides crucial information for future curricular design in the area of developmental mathematics at the college level. The theoretical framework considers different aspects that make up the function concept, taking critical account of several current theories of multiple representations and encapsulation of process as object to build a view of function in terms of different facets (representations) and different layers (of development via procedure, process, object, and procept). Ninety-two students at four community colleges completed written function surveys before and after a "reform" beginning algebra course. Twelve students, representing all four sites, participated in task-based interviews. Comparison of pre- and post-course surveys provided data indicating statistically significant improvement in student abilities to correctly interpret and manipulate function machines, two-variable equations, two-column tables, two-dimensional graphs, written definitions and function notation. The students were divided into three categories (highly capable, capable, and incapable) based on their demonstrated understanding of function. Using the interviews, visual profiles for students in each category were developed. The profiles indicate that the development of the concept image of function in such students is complex and uneven. The cognitive links between facets is sometimes nonexistent, sometimes tenuous, and often unidirectional. The highly capable demonstrated some understanding across all facets while the incapable indicated understanding of the more primitive facets, such as colloquial and numeric, only. Profound differences were noted particularly in the geometric, written, verbal, and notation facets. Overall, the target population appeared able to develop a process layer understanding of function, but this development was far from uniform across facets and across students

Symbols and the bifurcation between procedural and conceptual thinking

Symbols occupy a pivotal position between processes to be carried out and concepts to be thought about. They allow us both to d o mathematical problems and to think about mathematical relationships. In this presentation we consider the discontinuities that occur in the learning path taken by different students, leading to a divergence between conceptual and procedural thinking. Evidence will be given from several different contexts in the development of symbols through arithmetic, algebra and calculus, then on to the formalism of axiomatic mathematics. This is taken from a number of research studies recently performed for doctoral dissertations at the University of Warwick by students from the USA, Malaysia, Cyprus and Brazil, with data collected in the USA, Malaysia and the United Kingdom. All the studies form part of a broad investigation into why some students succeed yet others fail

Facets and layers of the function concept

This paper considers different aspects that make up the function concept, taking critical account of several current theories of multiple representations and encapsulation of process as object to build a view of function in terms of different facets (representations) and different layers (of development via process and object). An interview technique is used to determine the profile of students according to this view. Facets and Layers of the Function Concept The function concept has been a major focus of attention for the mathematics education research community over the past decade. (See Dubinsky & Harel, 1992, for example.) Schwingendorf et al (1992) contrast the vertical development of the concept in which the process aspect is encapsulated as a function concept and the horizontal development relating different representations. We refer to these as depth and breadth respectively (noting that increasing depth here means higher levels of cognitive abstraction) and investigate the way in which the student’s concept image of function can be described in terms of these two dimensions

Beginning Algebra Students' Images of the Function Concept Phil DeMarois

This paper describes research focussed on the feasibility of using the function concept as a core idea in developmental mathematics. This research considers different aspects that make up the function concept, building a view of function along both breadth (facets or representations) and depth (layers) dimensions. Pre- and post-course surveys along with task-based interviews are used to build a profile of developmental algebra students' concept image of function