Randolph A. Philipp

The role of a children's mathematical thinking experience in the preparation of prospective elementary school teachers

Abstract Historically, content preparation and pedagogical preparation of teachers in California have been separated. Recently, in integrating these areas, many mathematics methodology instructors have incorporated childrens thinking into their courses, which are generally offered late in students’ undergraduate studies. We have implemented and studied a model for integrating mathematical content and childrens mathematical thinking earlier, so that prospective elementary school teachers (PSTs) engage with childrens mathematical thinking while enrolled in their first mathematics course. PSTs’ work with children in the Childrens Mathematical Thinking Experience (CMTE) may enhance their mathematical learning. Preliminary study results indicate that the sophistication of CMTE students’ beliefs about mathematics, teaching, and learning increased more than the sophistication of beliefs held by students enrolled in a reform-oriented early field experience and that experiences considering childrens mathematical thinking provided PSTs with increased motivation for learning mathematics.

Conceptions and practices of extraordinary mathematics teachers

Abstract Four teachers were identified as “extraordinary“ teachers of mathematics. Data gathered from interviews, tests on content knowledge, discussions during a series of seminars, and one observation in each teachers classroom were used in summarizing characteristics of these teachers under three headings: (a) their mathematical preparation and their content knowledge of mathematics: (b) their conceptions about mathematics, about learning, about teaching, about the roles of teachers and of students, and about the assessment of learning; and (c) their teaching practices. Within each of these three areas, many commonalities were found among these teachers. This article will describe these commonalities and make inferences about the conditions and support necessary for teachers to begin to adopt beliefs and practices necessary to sustain reform of school mathematics.

Leveraging Structure: Logical Necessity in the Context of Integer Arithmetic

ABSTRACT Looking for, recognizing, and using underlying mathematical structure is an important aspect of mathematical reasoning. We explore the use of mathematical structure in children’s integer strategies by developing and exemplifying the construct of logical necessity. Students in our study used logical necessity to approach and use numbers in a formal, algebraic way, leveraging key mathematical ideas about inverses, the structure of our number system, and fundamental properties. We identified the use of carefully chosen comparisons as a key feature of logical necessity and documented three types of comparisons students made when solving integer tasks. We believe that logical necessity can be applied in various mathematical domains to support students to successfully engage with mathematical structure across the K–12 curriculum.

‘Negative of my money, positive of her money’: secondary students’ ways of relating equations to a debt context

We interviewed 40 students each in grades 7 and 11 to investigate their integer-related reasoning. In one task, the students were asked to write and interpret equations related to a story problem about borrowing money from a friend. All the students solved the story problem correctly. However, they reasoned about the problem in different ways. Many students represented the situation numerically without invoking negative numbers, whereas others wrote equations involving negative numbers. When asked to interpret equations involving negative numbers in relation to the story, students did so in two ways. Their responses reflect distinct perspectives concerning the relationship between arithmetic equations and borrowing/owing. We discuss these findings and their implications regarding the role of contexts in integer instruction.

Commentary on Section 3: Research on Teachers’ Focusing on Children’s Thinking in Learning to Teach: Teacher Noticing and Learning Trajectories

This commentary reviews the four chapters in this section on tools and techniques to support teacher learning. By focusing on students’ mathematical thinking, the authors of these four chapters studied the complexities associated with preparing prospective teachers to teach, and supporting practicing teachers in making the kind of changes that will lead to ambitious teaching. Each of the four chapters is briefly summarized and is placed within the literature on decomposition of practice. Authors of three of the chapters place their studies in the context of mathematics teacher noticing children’s mathematical thinking, and the author of the fourth chapter studies the effect of learning trajectories on teachers’ learning. The contributions made by each study to the literature are highlighted, and suggestions for moving the field forward are presented.

Students’ Thinking About Integer Open Number Sentences

We share a subset of the 41 underlying strategies that comprise five ways of reasoning about integer addition and subtraction: formal, order-based, analogy-based, computational, and emergent. The examples of the strategies are designed to provide clear comparisons and contrasts to support both teachers and researchers in understanding specific strategies within the ways of reasoning. The ability to categorize strategies into one of five ways of reasoning may enable teachers to organize knowledge of student thinking in ways that are useable and accessible for them and provide researchers with sufficient information about the strategies and ways of reasoning such that they can reliably build on this work.

Examining Student Thinking Through Teacher Noticing: Commentary

With the growing research base on teacher noticing has come a similar expansion of methodologies used to measure teacher noticing. The six chapters in this section reflect a range of methodologies, and this commentary is organized around three methodological considerations showcased in the chapters: (a) adoption of a conception of teacher noticing, (b) design of data-collection tools, and (c) choice of data-analysis lenses.