Jon R. Star, Ph.D.
Assistant Professor of Education
Harvard University, Gutman Library 442, 6 Appian Way, Cambridge, MA 02138 USA
Tel: 617.496.2511
Fax: 617.496.3095
Jon_Star@Harvard.edu
Funded Grants, Research Projects, & Research Interests
Mathematical Transitions Project, Michigan State University
Project RAPPS, Lehigh University
Statement of Research Interests
My research primarily concerns the development of mathematical understanding. My experiences as a middle and secondary school mathematics teacher initially motivated my interest in this topic; my graduate school training and dissertation research helped me to situate mathematical understanding within related work in the educational, psychological, and philosophical literatures. I explore mathematical understanding and its development in undergraduate and graduate students, as well as preservice and inservice mathematics teachers.
My interests in mathematical understanding lie at the boundary of the fields of education and psychology. By virtue of my training and research experience in cognitive, educational, and developmental psychology, I am well versed in the psychological literature relating to student cognition, including skill acquisition, problem solving, and strategy choice. However, to questions of student cognition I also bring the experiences and perspectives of a classroom educator, having taught mathematics full-time for six years prior to returning to graduate school. I feel my interdisciplinary training and background is a particularly good match for addressing compelling issues in mathematics education, particularly relating to the development of mathematical understanding.
In the sections below, I elaborate more on several themes which characterize my current research.
Theme One: The Nature of Mathematical Understanding
The nature of mathematical understanding has been a contentious issue in mathematics education since the formal beginnings of this field almost a century ago. In the current incarnation of the debate around this topic (in the form of the ‘math wars’), questions about mathematical understanding are particularly strident. A vastly over-simplified depiction of the current controversy pits advocates of what has been termed “conceptual understanding” versus advocates of “procedural skill.” Those on the side of conceptual understanding believe that too many students merely develop rote knowledge of mathematical procedures; to avoid this outcome, “reform” curricula and pedagogy focusing explicitly on conceptual understanding and de-emphasizing procedural skill are advocated. Opponents of this view agree that rote knowledge is to be avoided and that conceptual understanding is a goal, but they believe that a focus on skill is a necessary component of (and perhaps precursor to) conceptual understanding.
What has been interesting to me about this debate is the reluctance of the field to discuss definitions and operationalizations of contentious terms such as conceptual understanding and procedural skill. Despite the fact that these terms are omnipresent (in both the research and lay literatures), little effort has been devoted to articulating and problematizing their definition and use. I believe that establishing dialogue in the field about the ways these terms are used would be constructive in moving the conversation about mathematical understanding forward, particularly with respect to the relationship between skill and understanding. To do so, in my work I draw upon this issue’s rich and long history in the field of mathematics education (Brownell, 1945; Hiebert, 1986), as well as psychological (e.g., Gagné, 1983; Skemp, 1987) and philosophical (e.g., Ryle, 1949; Scheffler, 1965) theorists whose work is relevant.
I am particularly interested in procedural skill. There has been little research on skill development among mathematics educators, particularly in the last 20 years. However, the acquisition of procedural skill has been extensively and recently studied in cognitive science, where skill does not carry the political baggage that it does in some mathematics education circles. Psychologists know a lot about the process by which skills are learned. However, many psychological studies of skill acquisition were conducted in laboratory settings, and their applicability to classroom environments is perhaps questionable. Much work remains to be done in this area, both in bringing the findings of the psychological literature on skill acquisition to the classroom (where appropriate) and in initiating new programs of research on the process by which skills are learned in classroom environments. I feel that my training as a psychologist, as well as my experience and training as a mathematics educator, puts me in a strong position to make a significant contribution to the field of mathematics education in the area of procedural skill.
My ideas about the nature of mathematical understanding, particularly with respect to procedural skill, pervade all three themes of my research. However, I have two published papers in the Journal for Research in Mathematics Education that deal most explicitly with this topic (Star, 2005, 2007).
Theme Two: The Development of Procedural Flexibility
Building on and making use of my interest in the epistemology of mathematical understanding, the bulk of my work in the past three years has focused on exploring one component of understanding, which I refer to as procedural flexibility. In my research, I define a flexible problem solver in mathematics as one who (a) possesses knowledge of multiple strategies, and (b) makes use of a range of strategies at any given time, depending on which strategies are more useful in a given situation. I argue that flexibility is a critical component of (and perhaps lies at the core of) mathematical understanding; a flexible solver not only knows several ways to complete problems but also picks and chooses among known approaches, based on his knowledge of which strategies work best on particular problems. In my research, I am interested in learning more about this important outcome, including identifying instructional conditions that reliably lead to its development.
My dissertation (Star, 2001), was an initial attempt to study procedural flexibility. I worked with 6th and 7th grade students as they learned to solve linear equations in algebra; I was interested in their discovery of strategies for solving problems in this domain and if and how flexibility emerged.
Study 2 from my dissertation resulted in a recent paper in Contemporary Educational Psychology (Star & Seifert, 2006). In this study, I worked with 36 7th grade students on linear equation solving. I experimentally evaluated an instruction intervention that I hypothesized would led to greater flexibility. In this intervention, which I refer to as the “alternative ordering” treatment (Star, 2001), students are asked to re-solve previously completed problems, but using a different ordering of steps. I hypothesized that the process of generating multiple, different approaches to solving a problem, and subsequently reflecting on the differences between the multiple strategies, would cause students to become more flexible problem solvers. My results indicated that the treatment was successful.
Building on this initial research, I conducted a study where 134 6th grade students participated in five one-hour problem-solving sessions on linear equation solving. In order to replicate the results from my thesis studies, the alternative ordering treatment was again evaluated. And as additional treatment (as part of a 2 x 2 design), some students were provided explicit, direct instruction on equation solving strategies. My results confirmed the effectiveness of the alternative-ordering task on improving students’ flexibility. In addition, explicit strategy instruction was also found to significantly improve flexibility. A paper describing some of the results of this study (Star & Rittle-Johnson, 2007) is published in Learning and Instruction.
In 2005 I was awarded a $1 million grant from the Institute for Education Sciences to continue my research on the development of mathematical flexibility. Along with my collaborator Dr. Bethany Rittle-Johnson, at Vanderbilt University, I am conducting several studies that evaluate the effectiveness of a particular instructional intervention, the use of contrasting cases, on the development of procedural flexibility and conceptual understanding. This collorabation has so far produced a 2007 paper in the Journal of Educational Psychology (Rittle-Johnson & Star, 2007).
In 2007 I began a collection with Dr. Kristie Jones Newton, at Temple University, to continue to explore instructional strategies that promote the development of procedural flexibility.
Theme Three: The Influence of Curriculum on Mathematical Understanding
A third theme of my work concerns the influence of instruction and curriculum on the development of mathematical understanding. The epistemological dispute over the nature of mathematical understanding in the ‘math wars’ is manifest in intense battles over curriculum, where different curricula are predicated on (and convey to students) presumably different perspectives on the nature of mathematical understanding. In collaboration with Jack Smith at Michigan State University, and with funding from the National Science Foundation (to Smith), I explored students’ experiences in middle school, high school, and college mathematics classes, as they traverse between different types of mathematics curricula and pedagogy. Our approach was qualitative; we sought to learn more about students’ perceptions of what it means to know, do, and understand mathematics in various curricular and instructional environments. In addition to developing broader knowledge of students’ experiences and perceptions in different curricular environments, our work also sought to develop better methods of evaluating the impact of curricula on the development of student understanding. The project, which we call the Mathematical Transitions Project, currently has four papers in print (Star & Hoffmann, 2005; Star & Smith, 2006; Smith & Star, 2007; Star, Smith, & Hoffmann, 2007).