M. Mcgowen

William Rainey Harper College, United States · Department of Mathematical Sciences

We discuss and examine a numerical indicator—the individual gain—of students’ engagement and mathematical growth in relation to an instructor’s course aims and goals. The individual gain statistic assesses the fractional amount an individual student improves initial-test to final- test. We argue that an initial-test score and a final-test score, if...

New understandings of the functioning of human brains engaged in mathematics raise interesting questions for mathematics educators. Novel lines of research are suggested by neuroscientific findings, and new light is shed on some longstanding issues in mathematics education.

The purpose of this study was to delineate differences between students who made substantive gains and those who made only limited gains over the course of a semester. The divergent performance of two students, one with high gain and one with low gain, that occurred over a semester was examined. We found that the student with high gain—initial-test...

Widespread emphasis on developing students' algorithmic competency and symbol manipulation has resulted in students failing to think analytically and critically. If students are not encouraged to think flexibly about arithmetic and algebra in school, then this needs to be addressed by developmental courses and tasks designed to change the procedura...

Divergent thinking (DT) is the ability to take a topic and generate multiple connections and associations (Guilford, 1967; Nusbaum & Silvia, 2011). The Divergent Haiti seminar was designed to facilitate the development of DT in students at a university in Jeremie, Haiti. The seminar exposed participants to a set of individual, group, and class exer...

We examine a measure of individual student gain by preservice elementary teachers, related to Richard Hakes use of mean gain in the study of reform classes in undergraduate physics. The gain statistic assesses the amount individual students increase their test scores from initial test to final test, as a proportion of the possible increase for each...

In this paper we study the difficulties resulting from changes in meaning of the minus sign, from an operation on numbers, to a sign designating a negative number, to the additive inverse of an algebraic symbol on students in two-year colleges and universities. Analysis of the development of algebra reveals that these successive meanings that the s...

In 1998, the mathematics department at a large midwestern suburban college initiated a longitudinal study designed to improve the effectiveness of the mathematics curriculum. The goal was to assess, revise, and restructure the curriculum to meet the needs of students for the 21st century and the needs of the various disciplines served by the Depart...

While the general notion of ‘metaphor’ may offer a thoughtful analysis of the nature of mathematical thinking, this paper suggests that it is even more important to take into account the particular mental structures available to the individual that have been built from experience that the individual has ‘met-before.’ The notion of ‘met-before’ offe...

Incorporating the AMATYC standards, the DeMarois/McGowen/Whitkanack series takes an active-learning approach to Introductory Algebra and Intermediate Algebra, focusing on collaborative work both in and out of class. The problem-solving skills presented in these texts encourage independent learning and emphasize connections among mathematical ideas...

We examine the use of formative assessment as a tool to assist teachers of mathematics to become more mindful developers of curricula. We focus on instructional design that is based on careful examination of student answers to questions. Empirical studies have shown the effectiveness of formative assessment for students, and recent theoretical work...

This paper examines the questions: Who are the undergraduate students who enroll in precalculus courses? What courses do students take after completing a precalcu- lus course? These questions are addressed by an analysis of enrollment in mathe- matics courses at two- and four-year colleges and at universities over the past twenty years, followed by...

The narrow approach on showing students how to use a rule to get the answer has failed many students which often results in limiting their mathematical vision. An outcome of the NSF-funded conference, “Rethinking the Preparation for Calculus,” was the realization that we needed to rethink the mathematical experiences of students in courses below ca...

Since 1980, increasing numbers of students are repeating their high school mathematics courses as undergraduates and enrollment in the developmental courses has continued to grow. In Fall 2000, more than three million students were enrolled in undergraduate mathematics courses taught in departments of mathematics. Thirty-one percent of these studen...

Acknowledgement of the need to address the special nature of mathematics knowledge needed by teachers has profound implications for those who teach mathematics courses at institutions of higher learning. Various reports urge the establishment of essential partnerships and collaborative efforts among the various groups and institutions involved in t...

For several years we have been interested in pre-service teachers' memory for mathematical episodes. Partly this is because memory is such a vital aspect of mathematical problem solving. Long-term declarative memory is the sort of memory involved when a person talks, writes, draws, or otherwise consciously represents their recollections. We examine...

This paper explores how college students understand ideas of functions, and which representations are productive for them in promoting their ability to work flexibly across representations. The study used pre- and post-test scores, and triangulations via student self evaluations, to generate a hypothesis related to flexible thinking and success in...

We detail the mathematical growth of a pre-service elementary teacher of the period of a semester. This student had an unusual combination of very poor basic computational skills and a capacity for higher level problem solving involving algebraic notation. We discuss the extent to which one or two semesters of mathematics content, however well-stru...

In this article we address the question: “What are the implications for the preparation of prospective elementary teachers of “early algebra” in the elementary grades curriculum? As we discuss this question it will become clear that part of our answer involves language aspects of algebra: in particular, how a change in pre-service teachers’ attitud...

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 conce...

The MET recommendations present a vision of mathematics instruction for future teachers which requires a careful rethinking of what constitutes an appropriate and useful preparation. The goal is to have students develop mathematical habits of mind —developing careful reasoning and mathematical “common sense” in analyzing conceptual relationships an...

Built on AMATYC and NCTM standards, this book takes an active learning approach, focusing on collaborative for small groups. Independent learning is encouraged as problem-solving skills illustrate connections among mathematic ideas as readers discover and build on concepts explored as a group, as the book focuses on learning in a social context. Ea...

We examine the question of whether the introduction and use of the function machine representation as a scaffolding device helps undergraduates enrolled in a developmental algebra course to form a rich, foundational concept of function. We describe students' developing understanding of function as an input/output process and as an object, tracing t...

The concept of function is considered as foundational in mathematics. Yet it proves to be elusive and subtle for st udents. In this paper we suggest that a generic image that can act as a cognitive root for the concept is the function box. We see this not as a simple pattern-spotting device, but as a concept that embodies the salient features of th...

We describe students'developing understanding of function as an input/output process and as an object by tracing the internalization of the function machine concept as it relates to representations of functions We examine whether function machines serve as a cognitive root for the function concept for undergraduates enrolled in a developmental alge...

This reform algebra curriculum project had four main goals: 1. to develop, pilot, and revise Beginning and Intermediate Algebra curriculum and assess-ment materials based on the concept of function as an organizing lens; incorporating reported results of research about how students learn algebra and the use of technology; 2. to develop an implement...

The major focus of this study is to trace the cognitive development of students throughout a mathematics course and to seek the qualitative differences between those of different levels of achievement. The aspect of the project described here concerns the use of concept maps constructed by the students at intervals during the course. From these map...

The fragmentation of strategies that distinguishes the more successful elementary grade students from those least successful has been documented previously. This study investigated whether this phenomenon of divergence and fragmentation of strategies would occur among undergraduate students enrolled in a remedial algebra course. Twenty-six undergra...

A reaction to Underwood Dudley's discussion of the issue of whether mathematics is necessary for citizens of the United States to function in the world of work.

We report the use of mathematics education research activities designed to reflect the course objectives of broadening and deepening pre-service teachers' understanding of the complexities of teaching and learning. These activities provided experiences which resulted in changes in their attitudes about mathematics. The activities and their impact o...

This article reports the results of a longitudinal study which compared the performance of college students taught introductory algebra using non-traditional materials to the performance of students taught using traditional materials and methods and their success in subsequent courses. The non-traditional materials focused on the concept of functio...

The purpose of this study was to investigate whether students classified as 'developmental' or 'remedial' could, with a suitable curriculum, demonstrate improved capabilities in dealing flexibly and consistently with ambiguous notation using various representations of functions. The stability of their responses over time was also a subject of study...