The topic of multiple representations of functions is important in secondary school mathematics curricula, yet many students leave high school lacking an understanding of the connections among these representations. Research results are presented from a study in which students' understandings of the connections between algebraic and graphical representations of functions were examined. Responses from 178 students, enrolled in 1st-year algebra through calculus, revealed an overwhelming reliance on algebraic representations, even on tasks for which a graphical representation seemed more appropriate. The findings indicate that for familiar routine problems many students have mastered the connections between the algebraic and graphical representations; however, such mastery appeared to be superficial at best.

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## Student Understanding of the Cartesian Connection: An Exploratory Study

### Eric J. Knuth

## Secondary School Mathematics Teachers' Conceptions of Proof

### Eric J. Knuth

Recent reform efforts call on secondary school mathematics teachers to provide all students with rich opportunities and experiences with proof *throughout* the secondary school mathematics curriculum—opportunities and experiences that reflect the nature and role of proof in the discipline of mathematics. Teachers' success in responding to this call, however, depends largely on their own conceptions of proof. This study examined 16 in-service secondary school mathematics teachers' conceptions of proof. Data were gathered from a series of interviews and teachers' written responses to researcher-designed tasks focusing on proof. The results of this study suggest that teachers recognize the variety of roles that proof plays in mathematics; noticeably absent, however, was a view of proof as a tool for learning mathematics. The results also suggest that many of the teachers hold limited views of the nature of proof in mathematics and demonstrated inadequate understandings of what constitutes proof.

## Understanding Connections between Equations and Graphs

### Eric J. Knuth

General consensus holds within the mathematicseducation research community that functions are among the most important unifying ideas in mathematics (Romberg, Carpenter, and Fennema 1993). In fact, “it can be argued that functions form the single most important idea in all of mathematics, at least in terms of understanding the subject as well as for using it” (Dubinsky 1993, 527). Further, the introduction of algebraic and graphical representations of functions can be seen as a crucial moment in mathematics learning and represents “one of the earliest points in mathematics at which a student uses one symbolic system to expand and understand another” (Leinhardt, Zaslavsky, and Stein 1990, 2).

## Fostering Mathematical Curiosity

### Eric J. Knuth

Problem solving is an important component of learning mathematics, and that topic continues to receive significant attention in recommendations for school mathematics (NCTM 2000). Yet despite Brown and Walter's (1990) contention that problem posing is an integral part of problem solving, “problem posing is almost always overlooked in discussions of the importance of problem solving in the curriculum” (Silver, Kilpatrick, and Schlesinger 1995, p. 15).

## Proof as a Tool for Learning Mathematics

### Eric J. Knuth

Proof is considered to be central to the discipline of mathematics and the practice of mathematicians. Yet its role in secondary school mathematics has traditionally been peripheral at best; the only substantial treatment of proof is limited to geometry. According to Wu (1996, p. 228), however, the scarcity of proof outside of geometry is a misrepresentation of the nature of proof in mathematics.

## A Call for Postdoctoral Positions in Mathematics Education

### Elise Lockwood and Eric Knuth

In many STEM-related fields, graduating doctoral students are often expected to assume a postdoctoral position as a prerequisite to a faculty position, yet there is no such expectation in mathematics education. In this commentary, the authors call on the mathematics education research community to consider the importance of postdoctoral fellows and make the case that prioritizing postdoctoral positions could afford mutual benefits to the postdocs, to faculty mentors, and to the field at large.

## Unpacking the Nature of Discourse in Mathematics Classrooms

### Eric Knuth and Dominic Peressini

The role of discourse, although always central in education and learning, is receiving increased attention in classrooms today as mathematics educators strive to better understand the factors that lead to increased learning.

## The Role of Tasks in Developing Communities of Mathematical Inquiry

### Dominic Peressini and Eric Knuth

Examines students' responses to a mathematically rich task and explores approaches for using such tasks to foster inquiry that engages children in mathematical practice.

## Characterizing Students’ Understandings of Mathematical Proof

### Eric J. Knuth and Rebekah L. Elliott

For many secondary-level mathematics teachers and students, the notion of mathematical proof often conjures up images of mathematical rigor in the form of two-column proofs. Yet in recent years, many mathematics educators have been reexamining the nature of mathematical proof in the secondary curriculum.

## Fostering Mathematical Curiosity: Highlighting the Mathematics

### Eric J. Knuth and Blake E. Peterson

In “Fostering Mathematical Curiosity” (Knuth 2002), Eric Knuth discusses the idea of problem posing as a means of fostering students' mathematical curiosity. A mathematician colleague, after reading the article, commented that a significant amount of the mathematics in the discussion of the various problems and their solutions had been “left out” —and he was right. The author's primary intent in writing that article, however, was to illustrate “what it might mean to engage students in problem posing and how teachers might begin to create classroom environments that encourage, develop, and foster mathematical curiosity” (Knuth 2002, p. 126), not to discuss in detail the mathematics underlying the solutions to the problems posed.