Chris Rasmussen

Effects of Standards-Based Mathematics Education: A Study of the Core-plus Mathematics Project Algebra and Functions Strand

oping student ability to solve algebraic problems when those problems are presented in realistic contexts and when students are allowed to use graphing calculators. Conventional curricula are more effective than the CPMP curriculum in developing student skills in manipulation of symbolic expressions in algebra when those expressions are presented free of application context and when students are not allowed to use graphing calculators.

Locating starting points in differential equations: a realistic mathematics education approach

The paper reports on ongoing developmental research efforts to adapt the instructional design perspective of Realistic Mathematics Education (RME) to the learning and teaching of collegiate mathematics, using differential equations as a specific case. This report focuses on the RME design heuristic of guided reinvention as a means to locate a starting point for an instructional sequence for first-order differential equations and highlights the cyclical process instructional design and analysis of student learning. The instance of starting with a rate of change equation as an experientially real mathematical context is taken as a case for illustrating how university students might experience the creation of mathematical ideas. In particular, it is shown how three students came to reason conceptually about rate and in the process, develop their own informal Euler method for approximating solution functions to differential equations.

When the Classroom Floor Becomes the Complex Plane: Addition and Multiplication as Ways of Bodily Navigation

In this article we contribute a perspective on mathematical embodied cognition consistent with a phenomenological understanding of perception and body motion. It is based on the analysis of 4 selected episodes in 1 session of an undergraduate mathematics class. The theme of this particular class session was the geometric interpretation of the addition and multiplication of complex numbers. On the basis of these episodes, the article examines 2 conjectures: (a) The mathematical insights developed by an individual or a group are expressed in and constituted by perceptuo-motor activity, and (b) the learning of mathematical ideas is shaped in nondeterministic ways by the setting or learning environment.

Capitalizing on advances in mathematics and k-12 mathematics education in undergraduate mathematics: An inquiry-oriented approach to differential equations

This paper provides an overview of the Inquiry-Oriented Differential Equations (IO-DE) project and reports on the main results of a study that compared students’ beliefs, skills, and understandings in IO-DE classes to more conventional approaches. The IO-DE project capitalizes on advances within mathematics and mathematics education, including the instructional design theory of Realistic Mathematics Education and the social negotiation of meaning. The main results of the comparison study found no significant difference between project students and comparison students on an assessment of routine skills and a significant difference in favor of project students on an assessment of conceptual understanding. Given these encouraging results, the theoretical underpinnings of the innovative approach may be useful more broadly for undergraduate mathematics education reform.

The calculus student: insights from the Mathematical Association of America national study

In fall 2010, the Mathematical Association of America undertook the first large-scale study of postsecondary Calculus I instruction in the United States, employing multiple instruments. This report describes this study, the background of the students who take calculus and changes from the start to the end of the course in student attitudes towards mathematics and intention to continue in mathematics.

Reasoning Using Particulate Nature of Matter: An Example of a Sociochemical Norm in a University-Level Physical Chemistry Class.

In college level chemistry courses, reasoning using molecular and particulate descriptions of matter becomes central to understanding physical and chemical properties. In this study, we used a qualitative approach to analyzing classroom discourse derived from Toulmins model of argumentation in order to describe the ways in which students develop particulate-level justifications for claims about thermodynamic properties. Our analysis extends the construct of sociomathematical norms to a chemistry context in order to describe disciplinary criteria for reasoning and justification, which we refer to as sociochemical norms. By examining how whole class and small group discussions shape norms related to reasoning, we provide suggestions for teaching practices in inquiry-oriented settings.

Women 1.5 Times More Likely to Leave STEM Pipeline after Calculus Compared to Men: Lack of Mathematical Confidence a Potential Culprit

The substantial gender gap in the science, technology, engineering, and mathematics (STEM) workforce can be traced back to the underrepresentation of women at various milestones in the career pathway. Calculus is a necessary step in this pathway and has been shown to often dissuade people from pursuing STEM fields. We examine the characteristics of students who begin college interested in STEM and either persist or switch out of the calculus sequence after taking Calculus I, and hence either continue to pursue a STEM major or are dissuaded from STEM disciplines. The data come from a unique, national survey focused on mainstream college calculus. Our analyses show that, while controlling for academic preparedness, career intentions, and instruction, the odds of a woman being dissuaded from continuing in calculus is 1.5 times greater than that for a man. Furthermore, women report they do not understand the course material well enough to continue significantly more often than men. When comparing women and men with above-average mathematical abilities and preparedness, we find women start and end the term with significantly lower mathematical confidence than men. This suggests a lack of mathematical confidence, rather than a lack of mathematically ability, may be responsible for the high departure rate of women. While it would be ideal to increase interest and participation of women in STEM at all stages of their careers, our findings indicate that if women persisted in STEM at the same rate as men starting in Calculus I, the number of women entering the STEM workforce would increase by 75%.

An Example of Inquiry in Linear Algebra: The Roles of Symbolizing and Brokering

Abstract In this paper we address practical questions such as: How do symbols appear and evolve in an inquiry-oriented classroom? How can an instructor connect students with traditional notation and vocabulary without undermining their sense of ownership of the material? We tender an example from linear algebra that highlights the roles of the instructor as a broker, and the ways in which students participate in the practice of symbolizing as they reinvent the diagonalization equation A = PDP−1.

Key Mathematical Concepts in the Transition from Secondary School to University

This report from the ICME12 Survey Team 4 examines issues in the transition from secondary school to university mathematics with a particular focus on mathematical concepts and aspects of mathematical thinking. It comprises a survey of the recent research related to: calculus and analysis; the algebra of generalised arithmetic and abstract algebra; linear algebra; reasoning, argumentation and proof; and modelling, applications and applied mathematics. This revealed a multi-faceted web of cognitive, curricular and pedagogical issues both within and across the mathematical topics above. In addition we conducted an international survey of those engaged in teaching in university mathematics departments.

A Framework for Characterizing Students' Thinking about Logical Statements and Truth Tables.

While a significant amount of research has been devoted to exploring why university students struggle applying logic, limited work can be found on how students actually make sense of the notational and structural components used in association with logic. We adapt the theoretical framework of unitizing and reification, which have been effectively used to explain the types of integrated understanding required to make sense of symbols involved in numerical computation and algebraic manipulation, to investigate students’ conceptualizations of truth tables and implication statements. We put forth a two-dimensional space consisting of two continua as a framework to analyse the degree to which students’ thinking is compartmentalized or unified. Results indicate that students tend to treat the constituent pieces that make up these mechanisms independently without an understanding of each as a whole or an integrated view of the two together. This fragmented treatment is contrasted with the instructors unified view of both truth tables and implication statements.