About Developing Mathematical Ideas

The primary goal of Developing Mathematical Ideas (DMI) is to help teachers learn the mathematics content they are responsible for teaching. To this end, DMI asks teachers to make sense of the content, recognize where and how the content of their grade is situated in the trajectory of learning from kindergarten through middle school, build connections among different concepts, and analyze student thinking from a mathematical perspective. Through this work, teachers learn how to orient their instruction to specific mathematical goals and develop a mathematics pedagogy in which student understanding takes center stage.

Developing Mathematical Ideas is the product of decades of work in mathematics teacher professional development on the part of the authors.

When I began teaching teachers of mathematics in the mid-1980s, helping teachers develop a pedagogy that supported students’ mathematical understanding was a central goal for my colleagues and me. In order to accomplish this, we designed professional development sessions that gave teachers the experience of learners in a mathematics class structured to promote sense making. These sessions were generally organized around mathematical tasks at a level challenging to the teachers, giving them opportunities to verbalize their own thinking and work together as a group to move their ideas forward.

The teachers were very engaged by these sessions in which they examined such topics as the structure of the place-value system, rules of comparing fractions, features of geometric shapes, and properties of the operations. However, through watching the teachers in this context, I came to see the need for them to learn more mathematics content, a perception that was reinforced when I visited their classrooms to support their teaching during the year. I was not alone in this assessment. Many of the teachers themselves—especially elementary and middle school teachers—said that they needed to learn more content. They explained, however, that the traditional courses offered at the university would not help them. They did not need lectures in which an instructor demonstrated procedures for solving particular problem types. Instead, they wanted to take courses taught in the same manner we were teaching them to teach, courses that would help them actively engage with mathematical concepts and delve deeply into the mathematics content they were responsible for teaching. Having had the experience of understanding through the lessons we had taught, they were thirsty for more.

I began offering such courses in 1989, and indeed, they were well received; the classes were always oversubscribed. However, after a time I became aware that although the courses were satisfying teachers’ desire to understand mathematics content more deeply for themselves, their new knowledge did not necessarily carry over into their classrooms. In general, the teachers were not providing their students the same kind of opportunities to verbalize and develop mathematical ideas. I started to realize that there were additional skills that were part of the practice of teaching.

Among these skills were to elicit, hear, and analyze students’ ideas. To that end, my colleagues and I began assigning teachers the homework of recording events in their classroom and writing narrative pieces that captured their students’ mathematical thinking. Teachers told us that listening to and reflecting on their recordings and writing out their students’ words helped them attend to their students’ ideas in ways they never had done before. These assignments became the context for analyzing students’ understanding, situating student ideas in the context of the content to be taught, and considering teacher practices that would help move learning forward.

During that time, I was co-directing a project with Virginia Bastable and Susan Jo Russell that involved thirty-six teachers recording events from their classrooms. Through this writing, project staff and teachers investigated how mathematical ideas developed across the grades. Initially, I thought these records of classroom discussion would be used internally by the project. But after I had collected several rounds of episodes and had begun to group them according to topic, I could see that other teachers might use them profitably as the basis of case discussions. Soon Virginia, Susan Jo, and I began to envision casebooks, each addressing a central mathematical theme and organized into chapters, with each chapter focused on a subtopic. The resulting collection of cases illustrates how related mathematical ideas arise in different classroom contexts and develop over the grades. Used in conjunction with activities in which teachers dig directly into mathematics content, the cases provide a mechanism for teachers to apply their new depth of mathematical understanding to aspects of classroom practice.

When the Common Core State Standards (CCSS) were published in 2010, educators familiar with Developing Mathematical Ideas recognized it as an important support for CCSS implementation because it provides the kind of coherence and focus in the professional development of teachers of the elementary and middle grades to which CCSS also aspires. Focus is found in the selection, for each DMI module, of core mathematical ideas that underlie a key segment of mathematics content, while coherence comes from the careful analysis of how these core ideas connect to each other and are developed and applied by students across the grades. Furthermore, the cases at the heart of the materials illustrate students engaged in the Standards of Mathematical Practice specified in CCSS.

The current revision of Developing Mathematical Ideas is similar to the original version, but makes the connections to the Common Core State Standards explicit. The facilitator’s guide includes a list of content standards explored through the cases or mathematics activities of each session. The agendas provide opportunities for discussion of the practice standards, either through analysis of the cases or reflection on teachers’ own mathematical activity.

Deborah Schifter
September 2015