Draga Vidakovic

Understanding the Limit Concept: Beginning with a Coordinated Process Scheme.

Many authors have provided evidence for what appears to be common knowledge among mathematics teachers: The limit concept presents major difficulties for most students and they have very little success in understanding this important mathematical idea. We believe that a program of research into how people learn such a topic can point to pedagogical strategies that can help improve this situation. This paper is an attempt to contribute to such a program. Specifically, our goal in this report is to apply our theoretical perspective, our own mathematical knowledge, and our analyses of observations of students studying limits to do two things. First, we will reinterpret some points in the literature and second, we will move forward on developing a description, or genetic decomposition, of how the limit concept can be learned. In discussing the literature, we will suggest a new variation of a dichotomy, considered by various authors, between dynamic or process conceptions of limits and static or formal conceptions. We will also propose some explanations of why these conceptions are so difficult for students to construct. In describing the evolution of a genetic decomposition for the limit concept, we will give examples of how we used our analysis of interviews of 25 students from a calculus course to make appropriate modifications.

Relationship between Students’ Understanding of Functions in Cartesian and Polar Coordinate Systems

Abstract The present study was implemented as a prelude to a study on the generalization of the single variable function concept to multivariate calculus. In the present study we analyze students’ mental processes and adjustments, as they are being exposed to single variable calculus with polar coordinates. The results show that there appears to be a relation between students’ definition of function and their success (or lack of it) in transferring this definition to the polar representation. This is seen, in particular, with the transference of misconceptions.

Conceptions of Area: In Students and in History

Bronislaw Czarnocha, Ed Dubinsky (edd@cs.gsu.edu), Sergio Loch (sloch@gvc.edu), Vrunda Prabhu (vprabhu@williamwoods.edu), and Draga Vidakovic (draga@cs.gsu.edu) have origins in five different countries: Poland, the United States, Brazil, India, and Serbia. They are all members of the collaborative research group, RUMEC—Research in Undergraduate Mathematics Education Community. They are currently conducting research in topics in calculus, linear algebra, set theory and cooperative learning. Their training and professional experience is in the field of physics, mathematics, and mathematics education, and they all share a keen interest in the improvement of undergraduate mathematics education.

Modules as Learning Tools in Linear Algebra

Abstract This paper reports on the experience of STEM and mathematics faculty at four different institutions working collaboratively to integrate learning theory with curriculum development in a core undergraduate linear algebra context. The faculty formed a Professional Learning Community (PLC) with a focus on learning theories in mathematics and curriculum development. Beginning with an online reading seminar, faculty paired with at least one other STEM faculty member at each institution to develop modules collaboratively with each other and with the group as a whole through regular project meetings. Two representative modules from this collaboration are presented with student and faculty feedback. A link is provided to other modules. We consider this a proof-of-concept paper with broader applicability beyond linear algebra to other mathematics and even STEM courses. We focus on linear algebra because it is widely taught in undergraduate mathematics programs and lends itself easily to a variety of applications and geometric analyses. We intend to continue to apply this approach to other content domains. We chose a particular theoretical framework, Action-Process-Object-Schema (APOS), because the mathematics education faculty in this group had considerable experience in the development and use of the framework.

APOS Theory: Use of Computer Programs to Foster Mental Constructions and Student’s Creativity

According to Piaget, the root of all intellectual activity is reflective abstraction. In this context, mathematical creativity arises through students’ abilities to make reflective abstractions. Considering that reflective abstraction is the main premise of APOS Theory, the theory provides a theoretical tool to guide the development of instruction that supports mathematical creativity. The letters that make up the acronym—A, P, O, S—represent the four basic mental structures—Action, Process, Object, Schema—that an individual constructs as he or she reflects on and reorganizes content in coming to understand a mathematical concept. Much of the instruction that involves the application of APOS Theory has been delivered using the ACE Teaching Cycle, a lab-oriented pedagogical approach that facilitates collaborative activity within a computer environment (programming and/or dynamic). The letters that make up the acronym—A, C, E—represent the three components of a pedagogical cycle—Activities, Classroom Discussion, Exercises—that facilitate reflection and collaboration. Numerous studies have demonstrated the efficacy of this approach when applied to the teaching and learning of a variety of mathematical topics at the elementary, secondary, and collegiate levels. We illustrate this with a description of instruction for the topics of cosets, infinite repeating decimals, and slope. To introduce these examples, we provide a brief overview of APOS theory with all its components in the context of learning the concept of function. Opportunities for development of mathematical creativity are emphasized throughout the entire chapter.

Calculus students’ understanding of the vertex of the quadratic function in relation to the concept of derivative

ABSTRACT The purpose of this study was to gain insight into 30, first year calculus students’ understanding of the relationship between the concept of vertex of a quadratic function and the concept of the derivative. APOS (action-process-object-schema) theory was applied as a guiding framework of analysis on student written work, think-aloud and follow up group interviews. Students’ personal meanings of the vertex, including misconceptions, were explored, along with students’ understanding to solve problems pertaining to the derivative of a quadratic function. Results give evidence of students’ weak schema of the vertex, lack of connection between different problem types and the importance of linguistics in relation to levels of APOS theory. A preliminary genetic decomposition was developed based on the results. Future research is suggested as a continuation to improve student understanding of the relationship between the vertex of quadratic functions and the derivative.

Vectors, change of basis and matrix representation: onto-semiotic approach in the analysis of creating meaning

In a previous study, the onto-semiotic approach was employed to analyse the mathematical notion of different coordinate systems, as well as some situations and university students’ actions related to these coordinate systems in the context of multivariate calculus. This study approaches different coordinate systems through the process of change of basis, as developed in the context of linear algebra, as well as the similarity relationship between the matrices that represent the same linear transformation with respect to different bases.