Marilyn P. Carlson

Applying Covariational Reasoning While Modeling Dynamic Events: A Framework and a Study

The article develops the notion of covariational reasoning and proposes a framework for describing the mental actions involved in applying covariational reasoning when interpreting and representing dynamic function events. It also reports on an investigation of high-performing 2nd-semester calculus students’ ability to reason about covarying quantities in dynamic situations. The study revealed that these students were able to construct images of a function’s dependent variable changing in tandem with the imagined change of the independent variable, and in some situations, were able to construct images of rate of change for contiguous intervals of a function’s domain. However, students appeared to have difficulty forming images of continuously changing rate and could not accurately represent or interpret inflection points or increasing and decreasing rate for dynamic function situations. These findings suggest that curriculum and instruction should place increased emphasis on moving students from a coordinated image of two variables changing in tandem to a coordinated image of the instantaneous rate of change with continuous changes in the independent variable for dynamic function situations.

Foundational Reasoning Abilities that Promote Coherence in Students' Function Understanding

The concept of function is central to undergraduate mathematics, foundational to modern mathematics, and essential in related areas of the sciences. A strong understanding of the function concept is also essential for any student hoping to understand calculus – a critical course for the development of future scientists, engineers, and mathematicians. Since 1888, there have been repeated calls for school curricula to place greater emphasis on functions (College Entrance Examination Board, 1959; Hamley, 1934; Hedrick, 1922; Klein, 1883; National Council of Teachers of Mathematics, 1934, 1989, 2000). Despite these and other calls, students continue to emerge from high school and freshman college courses with a weak understanding of this important concept (Carlson, 1998; Carlson, Jacobs, Coe, Larsen & Hsu, 2002; Cooney & Wilson, 1996; Monk, 1992; Monk & Nemirovsky, 1994; Thompson, 1994a). This impoverished understanding of a central concept of secondary and undergraduate mathematics likely results in many students discontinuing their study of mathematics. The primarily procedural orientation to using functions to solve specific problems is absent of meaning and coherence for students and has been observed to cause frustration in students (Carlson, 1998). We advocate that instructional shifts that promote rich conceptions and powerful reasoning abilities may generate students’ curiosity and interest in mathematics, and subsequently lead to increases in the number of students who continue their study of mathematics. This article provides an overview of essential processes involved in knowing and learning the function concept. We have included discussions of the reasoning abilities involved in understanding and using functions, including the dynamic conceptualizations needed for understanding major concepts of calculus, parametric functions, functions of several variables, and differential equations. Our discussion also provides information about common conceptual obstacles to knowing and learning the function concept that students have been observed encountering. We make frequent use of examples to illustrate the ‘ways of thinking’ and major understandings that research suggests are essential for students’ effective use of functions during problem solving, and that are needed for students’ continued mathematics learning. We also provide some suggestions for promising approaches for developing a deep and coherent view of the concept of function.

THE MATHEMATICAL BEHAVIOR OF SIX SUCCESSFUL MATHEMATICS GRADUATE STUDENTS: INFLUENCES LEADING TO MATHEMATICAL SUCCESS

This study investigated the mathematical behavior of graduate students and the experiences that contributed to their mathematical development and success. Their problem-solving behavior was observed while completing complex mathematical tasks, and their beliefs were assessed by administering a written survey. These graduate students report that a mentor, most frequently a high school teacher, facilitated the development of their problem solving abilities and continued mathematical study. The mentors were described as individuals who provided challenging problems, encouragement, and assistance in learning how to approach complex problems. When confronted with an unfamiliar task, these graduate students exhibited exceptional persistence and high confidence. Their initial problem solving attempts were frequently to classify the problem as one of a familiar type, and they were not always effective in accessing recently taught information or monitoring their solution attempts, but were careful to offer only solutions that had a logical foundation. These results provide numerous insights into the complexities of using and extending ones mathematical knowledge and suggest that non-cognitive factors play a prominent role in a students mathematical success.

The Precalculus Concept Assessment: A Tool for Assessing Students’ Reasoning Abilities and Understandings

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 include a strong understanding of ideas of rate of change and function, a process view of function, and the ability to use covariational reasoning to examine and represent how two covarying quantities change together. This taxonomy guided the PCA development and now provides the theoretical basis for interpreting and reporting PCA results. A critical element of PCAs design was to identify the constructs essential for learning calculus and to employ methods to assure that PCA items are effective in assessing these constructs. We illustrate the role that cognitive research played during both the design and validation phases of the PCA instrument. We also describe our Four-Phase Instrument Development Framework that articulates the methods used to create and validate PCA. This framework should also be useful for others interested in developing similar instruments in other content areas. The uses of PCA are described and include (a) assessing student learning in college algebra and precalculus, (b) comparing the effectiveness of various curricular treatments, and (c) determining student readiness for calculus.

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.

High school teacher change, strategies, and actions in a professional development project connecting mathematics, science, and engineering

Project Pathways, an NSF Math Science Partnership professional development project, uses four semester-long courses and professional learning communities (PLCs) with the goal of enhancing teacher knowledge, skills and practice. The unifying concept of function is applied to promote conceptual competence in core content subjects and key problem solving processes. Modules integrating math, science, and engineering are delivered in team-based studio labs complemented by associated PLCs. The research question here is, ldquoWhat is the effect of a function-driven joint high school math/science teacher based professional development project on teacher change, strategies, and actions?rdquo The relevance is that it addresses issues about student math and science achievement and the STEM pipeline. Teacher change was evaluated using qualitative analysis of post-class question responses for five factors: creating a math/science teacher culture of collaboration; deepening content understanding by use of function; integrating math, science and engineering; developing inquiry strategies and materials and; promoting metacognition on student thinking for effective learning. For 27 responses, 24 showed positive change shown by shifts for one or more of five factors. Overall, the project created function-infused courses linked with multifaceted, synergistic PLCs that facilitated teacher change, strategies, and actions for improved practice.

Making the Connection: Promoting Effective Mathematical Practices in Students: Insights from Problem Solving Research

Mathematicians and mathematics educators have been curious about the processes and attributes of problem solving for over 50 years. As mathematics teachers at any level of education, we want to know what teaching practices we can employ to help our students develop effective problem solving abilities. This curiosity has led to numerous investigations of the attributes and processes of problem solving. In this chapter, we describe insights from a study we conducted of the mathematical practices of 12 research mathematicians. We believe these insights are useful to teachers striving to promote mathematical practices in students at all levels—from first-grade mathematics to beginning algebra, calculus, and abstract algebra. Our chapter begins by inviting you to work a problem that our research study posed to 12 mathematicians and to reflect, as they did, on your own problem solving behavior as you attempt to solve this problem. In inviting you to work this problem, our intent is to raise your awareness of the processes, emotions, knowledge, heuristics, and reasoning patterns that you use when working a novel problem. Our research suggests that by reflecting on our own mathematical practices, instructors can become more attentive to the development of problem solving attributes in students (Bloom, 2004). This exercise should make the remaining sections of our chapter more meaningful. In particular, it is our hope that our description of the Multidimensional Problem Solving Framework is more accessible. After describing how one of our subjects attempted the same problem, our chapter provides an overview of the research literature on problem solving in mathematics. We then describe our own study in more detail and conclude with suggestions for developing students’ problem solving practices. We believe that this chapter illustrates the importance of exploring the mathematical practices and behaviors that lead to mathematical proficiency. We also believe that the chapter explicates a way of thinking about problem solving that posits a reflexive relationship between the development of students’ content knowledge and their mathematical practices.

The Evolution of an Interdisciplinary Collaborative for Pre-Service Teacher Reform

The Arizona Collaborative for Excellence in Preparation of Teachers is a large National Science Foundation funded project aimed at revising science and mathematics pre-service courses at a large public university in the South-western United States. This chapter describes the collaborations of a community of university faculty in reforming a block of five pre-service mathematics and mathematics education courses. Through a series of workshops and ongoing dialogue, both the instructional delivery and curriculum for these pre-service courses has shifted to student-centred classrooms with inquiry, concept development and problem solving as central themes. The chapter provides information about the process and products of these reforms, with a major focus on providing specific insights into the role of research in guiding the curricular and instructional philosophies and decisions.