Marilyn Paula Carlson

Arizona State University | ASU · School of Mathematical and Statistical Sciences

Mathematics education researchers describe decentering as a form of reflection consisting of the mental processes involved in constructing a model of another’s thinking (from the perspective of the other). Piaget first introduced the decentering construct to characterize children’s development from an egocentric state to awareness of other perspect...

Quantitative and covariational reasoning characterize essential ways of thinking for conceptualizing and representing growth patterns in pairs of quantities in dynamic situations. Despite the broad body of research documenting the central role of quantitative reasoning in students’ ability to construct meaningful function formulas and graphs relati...

We present our analysis of 254 Calculus I final exams from U.S. colleges and universities to identify features of assessment items that necessitate qualitatively-distinct ways of understanding and reasoning. We explore salient features of exemplary tasks from our data set to reveal distinctions between exam items made apparent by our analytical fra...

This article presents the results of our analysis of a sample of 254 Calculus I final exams (collectively containing 4,167 individual items) administered at U.S. colleges and universities. We characterize the specific meanings of foundational concepts the exams assessed, identify features of exam items that assess productive meanings, distinguish c...

Teachers' mathematical meanings impact their instructional practices and constitute their images of the mathematics they teach and intend students to learn (Thompson, 2013). While numerous studies have focused on K-12 teachers' mathematical knowledge for teaching (MKT) and mathematical meanings for teaching (MMT) (Thompson, 2013), few studies have...

This study explored two undergraduate precalculus students’ understandings of the idea of logarithm as they completed conceptually oriented exploratory lessons on exponential and logarithmic functions. The students participated in consecutive, individual teaching experiments that focused on Sparky – an animated mystical saguaro that doubled in heig...

Graduate student teaching assistants (GTAs) usually teach introductory level courses at the undergraduate level. Since GTAs constitute the majority of future mathematics faculty, their image of effective teaching and preparedness to lead instructional improvements will impact future directions in undergraduate mathematics curriculum and instruction...

We describe the Algebra and Precalculus Concept Readiness (APCR) instrument and the Calculus Concept Readiness (CCR) instrument, and the reasoning abilities and understandings they assess. We share several tasks from these instruments to illustrate specific reasoning abilities that are needed for understanding precalculus and ideas in introductory...

The Calculus Concept Readiness (CCR) instrument assesses foundational understandings and reasoning abilities that have been documented to be essential for learning calculus. The CCR Taxonomy describes the understandings and reasoning abilities assessed by CCR. The CCR is a 25-item multiple-choice instrument that can be used as a placement test for...

In this study, we developed a three-dimensional framework to classify post-secondary Calculus I final exams. Our Exam Characterization Framework (ECF) classifies individual exam items according to the cognitive demand required to answer the item, the representational context in which the item is asked, and the format of the item. Our results from u...

Mathematics educators and writers of mathematics education policy documents continue to emphasize the importance of teachers focusing on and using student thinking to inform their instructional decisions and interactions with students. In this paper, we characterize the interactions between a teacher and student(s) that exhibit this focus. Specific...

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...

This article reports findings from an investigation of precalculus students’ approaches to solving novel problems. We characterize the images that students constructed during their solution attempts and describe the degree to which they were successful in imagining how the quantities in a problem's context change together. Our analyses revealed tha...

The final exam in a mathematics course is one source of information about the nature and level of student learning that is expected in the course. In this study, a three-dimensional framework was developed to analyze post-secondary calculus I final exams in an effort to determine the skills and understandings that are currently being emphasized in...

The Calculus Concept Readiness (CCR) instrument is based on the broad body of mathematics education research that has revealed major understandings, representational abilities, and reasoning abilities students need to construct in precalculus level courses to be successful in calculus. The CCR is a 25-item multiple-choice instrument, and the CCR ta...

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 incl...

We introduce the sociomathematical norm of speaking with meaning and describe its emergence in a professional learning community (PLC) of secondary mathematics and science teachers. We use speaking with meaning to reference specific attributes of individual communication that have been revealed to improve the quality of discourse among individuals...

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 solv...

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 mathematician...

The chapters in this volume convey insights from mathematics education research that have direct implications for anyone interested in improving teaching and learning in undergraduate mathematics. This synthesis of research on learning and teaching mathematics provides relevant information for any mathematics department or any individual faculty me...

Project Pathways is a Math Science Partnership (MSP) providing professional development to high school math and science teachers in Phoenix metropolitan area school districts. The NSF funded project is using four semester-long courses and professional learning communities to enhance teacher pedagogy for improving mathematics and science learning an...

We examine the role of tasks that have the intended effect of teachers re-conceiving the mathematics they teach as comprising a coherent body of meaningful ideas. We ground our discussion in ideas of trigonometry and modular functions and draw from a professional development research project to illustrate our approach. In this project, many teacher...

A gender gap in science, a foundational subject area for engineering, begins to emerge in middle and early high school. This gap can negatively impact females' education and career decisions about science and engineering due to environmental and affective factors in the classroom. In order to identify, strategize, and address these gender equity is...

Over the past decade the first-time enrollment of females in undergraduate engineering has not increased and remains at about 20%, in spite of ongoing K-12 engineering gender diversity programs. The underlying cause for the decline is not cognitive ability or academic performance. Instead, the cause has sociocultural roots that create barriers to f...

This paper describes the problem-solving behaviors of 12 mathematicians as they completed four mathematical tasks. The emergent problem-solving framework draws on the large body of research, as grounded by and modified in response to our close observations of these mathematicians. The resulting Multidimensional Problem-Solving Framework has four ph...

A general discussion of learning theory and its relationship to instructional design and instructional practices is provided. This is followed by the presentation of two individual cognitive frameworks. A covariational reasoning framework that characterizes five mental actions of covariational reasoning is presented. A case is made for why the reas...

This chapter provides an illustration of how the models and modeling perspective can be used in collegiate mathematics education research and instruction. The models and modeling approach (chapter 1) provides instructional designers with a well-defined structure for creating curriculum. The curricular activities for this approach, referred to as mo...

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 o...

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 second semester calculus students' ability to reason about covarying qua...

This study investigated the effect of research-based curricular materials on the development of first semester calculus students' covariational reasoning abilities. It also explored the role of covariational reasoning in the development of the concepts of limit and accumulation. We found evidence that the covariation curriculum was effective and th...

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 m...

An exam measuring students' understandings of major aspects of the function concept was administered to high performing second semester calculus students at the completion of their course. Follow-up interviews were conducted with five students to gain more in-depth insights into the conceptions that motivated individual responses. Exam and intervie...

Thesis (Ph. D.)--University of Kansas, Special Studies, 1995. Includes bibliographical references (leaves [207]-[209]).

Reports the results of an investigation designed to help algebra teachers and curriculum developers by providing insights into how high-performing college algebra students (N=30) develop an understanding of major aspects of the concept of function. Results revealed that these high-performing students actually possessed weak understanding. Contains...

Isolation buffers mediocrity and hides high performers from those who might learn from their modeling, consultation, and coaching. When practice is deprivatized, teachers visit one another's classrooms to observe master teaching, to coach each other, to mentor, and to problem solve in the living laboratory of instructional space. —Garmston and Well...

This paper will use analyses of classroom discourse to characterize the effect that deliberate efforts within the projects' courses to have teachers speak meaningfully had on their conceptions of mathematics and mathematical explanations. Data from videotaped sessions of the courses indicate a synergy between their attempts to speak meaningfully, a...