What topics should be studied in a methods course? History may help provide an answer.

### Dale M. Shafer

The twentieth century has witnessed the establishment of numerous commissions and committees whose purpose has been to study the optimum preparation that colleges and universities should provide future teachers of secondary school mathematics. Generally these study groups recognized the existence of two basic problems. The first problem was to determine the type of mathematics courses the future teachers should study while attending college, and the second was to determine how the individual can best be prepared pedagogically to teach mathematics. In response to the first problem, the groups were quite diligent in outlining their recommendations concerning an appropriate sequence of mathematics courses and the content of those courses. In response to the second problem, however, the groups either omitted making recommendations or made them in very general terms. The structure of the methods course in mathematics has, therefore, remained undefined in comparison with the content courses in mathematics.

### Alfinio Flores and Carmina M. Brittain

During their first mathematics methods course, many prospective elementary teachers confront their previous conceptions about mathematics and its teaching for the first time. This juncture makes the course important in their evolution as teachers of mathematics. Prospective teachers in a mathematics methods course must develop the ability to reflect on their actions, beliefs, knowledge, and attitudes. Writing in a mathematics methods course fosters reflection in a natural way; it serves as a tool for documentation, analysis, and discussion to help prospective teachers internalize what they learn and reach new levels of comprehension. At the same time, what teachers in training write gives teacher educators a window into their reflection and growth process.

### Michael D. Steele and Amy F. Hillen

In the majority of secondary mathematics teacher preparation programs, the work of learning mathematics and the work of learning to teach mathematics are separated, leaving open the question of when and how teachers integrate their knowledge of content and pedagogy. We present a model for a content-focused methods course, which systematically develops a slice of mathematics content in the context of typical methods course activities. Three design principles are posited that undergird the design of such a course, addressing the nature of the mathematics content, the sequencing and design of activities, and the ways in which the course addresses the needs of diverse learners. Data from an instantiation of one such course is presented to illustrate the ways in which the course design framed teachers' opportunities to learn about both content and pedagogy.

### Alfinio Flores and Carmina Brittain

For more than a decade, several authors have highlighted the benefits to students of writing to learn mathematics. Writing is an important component of communication in the classroom. As *Principles and Standards for School Mathematics* (NCTM 2000) notes, “Writing in mathematics can also help students consolidate their thinking because it requires them to reflect on their work and clarify their thoughts about the ideas developed in the lesson” (p. 61). Teachers probably will not use this tool, however, unless they have had the experience themselves of writing in relation to mathematics. This article presents a brief review of the benefits of students writing to learn mathematics. In the second part of the article, we invite the reader to consider another possible use of writing: as a tool to help preservice teachers reflect on their own growth as they learn to teach mathematics. We discuss some of the benefits that writing has for prospective teachers and present examples of preservice elementary teachers' writing that were collected in several one-semester undergraduate mathematics methods courses that the first author taught. The second author participated as a student in one of the courses. In a second article to be published in this journal, we will focus on the process of writing and writing for an audience.

### Tiffany G. Jacobs, Marvin E. Smith, Susan Swars Auslander, Stephanie Z. Smith and Kayla D. Myers

Teacher preparation programs face increasing demands to demonstrate the competencies of prospective teachers in their programs, while maintaining a focus on developing high-leverage instructional practices in their methods coursework. This study used a mixed methods approach to examine how the implementation of a simulated edTPA elementary mathematics task influenced prospective teachers' experiences in an elementary mathematics methods course. The course curriculum featured cognitively guided instruction (CGI) as the exemplar for understanding and implementing problembased, cognitively oriented pedagogy. Our findings indicate that elements of both CGI and the simulated edTPA mathematics task worked synergistically to enhance opportunities for CGI-type lesson enactment, support productive changes in beliefs, and contribute to the prospective elementary teachers' (PTs) preparation.

### Charles H. D'augustine

A criticism made frequently by students in colleges of education is that their methods classes are not sufficiently related to work with children.

### Aidin Amirshokoohi and Daniel P. Wisniewski

Key elements can enhance teacher candidates' understanding, interest, and confidence with learning and teaching mathematics while decreasing their math-related anxiety and fear.

### Clarence Phillips

Colleges and Universities have the responsibility of training prospective elementary teachers in the teaching of elementary mathematics. These teachers must be trained so that they can teach elementary school mathematics for meaning and understanding. In addition to training prospective elementary school teachers, the colleges and universities also have the responsibility of acquainting teachers in-service with the “newer” approach in teaching elementary mathematics. In this article, I will use “elementary teachers” as including both groups.