Keith R. Leatham

Vital directions for mathematics education research

Preface.- Reflections on a Portrait of our Field.- Making Progress in U.S. Mathematics Education: Lessons Learned-Past, Present and Future.- The Constantly Underestimated Challenge of Improving Mathematics Instruction.- In the Absence of Meaning.- The Need for Theories of Conceptual Learning and Teaching of Mathematics.- Intellectual Need.- The False Dichotomy in Mathematics Education between Conceptual Understanding and Procedural Skills: An Example from Algebra.- Needed: Critical Foxes.- Where are the Foxes in Mathematics Education?.

Pre-service Secondary Mathematics Teachers' Beliefs about the Nature of Technology in the Classroom

This study investigated pre-service secondary mathematics teachers’ (PSTs) beliefs about teaching mathematics with technology, the experiences in which those beliefs were grounded, and the organizational structure of those beliefs. In particular, this article reports on research that was designed to investigate questions beyond that of whether technology can profitably be used before mathematical content has been mastered (the question about technology most commonly reported in the literature). Beliefs were defined as dispositions to act and were viewed through what is referred to as the sensible system framework. Through the qualitative research methodology called grounded theory, four PSTs were purposefully selected and studied. The primary dimensions of the PSTs’ core beliefs with respect to technology, referred to as their beliefs about the nature of technology in the classroom, were beliefs about the availability of technology, about the purposeful use of technology, and about the importance of teacher knowledge of technology.RésuméCette étude porte sur les idées préconçues des futurs enseignants des mathématiques au secondaire quant à l’enseignement des mathématiques par le moyen des technologies, les expériences sur lesquelles ces idées se fondent et le système grâce auquel ces idées sont maintenues. En particulier, nous analysons les résultats de recherches réalisées dans le but d’aller au-delà des idées les plus courantes dans la littérature sur les technologies, à savoir si les technologies peuvent être utilisées de façon profitable avant que les contenus mathématiques aient été maîtrisés. Ces idées ont été définies comme une disposition à agir, et sont étudiées dans un cadre que nous appelons système sensé. Au moyen d’une méthodologie de recherche qualitative appelée théorie fondée, quatre futurs enseignants ont été choisis comme objet de notre étude. Les principales dimensions des idées fondamentales de ces futurs enseignants sur les technologies, que nous avons appelées idées sur la nature des technologies dans la salle de classe, portaient sur la disponibilité des technologies, leurs différentes utilisations pratiques ainsi que l’importance des connaissances qu’en ont les enseignants.

Attributes of Instances of Student Mathematical Thinking that Are Worth Building on in Whole-Class Discussion

ABSTRACT This study investigated attributes of 278 instances of student mathematical thinking during whole-class interactions that were identified as having high potential, if made the object of discussion, to foster learners’ understanding of important mathematical ideas. Attributes included the form of the thinking (e.g., question vs. declarative statement), whether the thinking was based on earlier work or generated in the moment, the accuracy of the thinking, and the type of thinking (e.g., sense-making). Findings illuminate the complexity of identifying student thinking worth building on during whole-class discussion and provide insight into important attributes of these high potential instances that could be used to help teachers more easily recognize them. Implications for researching, learning, and enacting the teaching practice of building on student mathematical thinking are discussed.

The Structure of Student Teaching Can Change the Focus to Students’ Mathematical Thinking

This paper describes our efforts to change the focus of our student teaching experience by altering the structure of that experience. We provide evidence that the restructuring accomplished its purposes and, in so doing, addressed a number of problems with the traditional structure it replaced. In particular, we achieved less focus on issues of classroom management and student behavior, more focus on students’ mathematics, and substantial opportunity to grapple with the elicitation, interpretation and use of student mathematical thinking during class discussion. Although there is still room for improvement, our model provides an existence proof that the focus of the student teaching experience can indeed be altered and improved.

From a Framework to a Lens: Learning to Notice Student Mathematical Thinking

Teaching is a complex endeavor that necessarily requires teachers to attend to some activities and ignore others. This case study focuses on prospective teachers’ learning to notice student mathematical thinking. We frame our view of noticing with the professional noticing framework (Jacobs, Lamb, & Philipp, in Journal for Research in Mathematics Education 41:169–202, 2010), and our view of student mathematical thinking with the MOST analytical framework (Leatham, Peterson, Stockero, & Van Zoest, in Journal for Research in Mathematics Education 46:88–124, 2015). We share evidence that a research experience that focused prospective teachers in a sustained, intense experience focused on articulating student mathematical thinking through focused video analysis influenced their ability to notice in-the-moment student mathematical thinking during their student teaching experience.

Noticing Distinctions Among and Within Instances of Student Mathematical Thinking

In this chapter, we argue that there are two critical aspects of noticing student mathematical thinking: noticing within an instance of student thinking and noticing among instances of student thinking. We use the noticing literature to illustrate these distinctions. We then discuss how the MOST Analytic Framework analysis provides structure and guidance for noticing both within and among instances, and illustrate the complex interaction of these two types of noticing through the analysis of an excerpt of classroom dialogue. We conclude by offering the perspective that studies of noticing must go beyond placing value on student mathematical thinking to discriminating among instances of student thinking based on their potential to be used to support students’ understanding of important mathematics.

The Instructional Quality of Mathematics Student Teachers in the United States and Japan: The Possible Impact of the Structure of Student Teaching

In this chapter we explore the instructional quality of four US student teachers in a novel student teaching structure. To overcome some of the common problems associated with student teaching documented in the research literature, we adapted a student teaching structure commonly used in Japan. We evaluate the instructional quality of the lessons by using the Mathematical Quality of Instruction (MQI) video coding protocol. We compare the instructional quality to a sample of Japanese student teachers and to a large sample of lessons from six large US school districts, utilizing the Measures of Effective Teaching (MET) study. We also illustrate the quantitative findings with vignettes from US and Japanese student teaching. The results show that given the right support and structure, student teachers in the USA can implement lessons that are similar in quality to Japanese student teachers and much richer than typical US mathematics instruction.