## Exploring the Mathematical Knowledge Needed for Teaching Teachers

Source: *Journal of Teacher Education, 65(4),* 303-314, 2014

(Reviewed by the Portal Team)

The purpose of this study is to analyze how particular mathematics teacher educators (MTEs) use knowledge in their practice.

Furthermore, this study also examines how they use this analysis as a tool for understanding the knowledge demands of work with preservice elementary teachers and how this knowledge is different from that required to teach K-12 students.

**Methods **

Data for this study come from a multimedia database of teaching and learning artifacts from five iterations of a university-based mathematics content course for preservice elementary teachers.

The authors illustrate different forms of knowledge observed across different mathematics teacher educators’ practice and discuss how the observed knowledge forms are different from knowledge used by K-12 teachers in their practice.

They describe, for example, how the MTEs exhibited knowledge of the nature of student errors and how that knowledge is used in practice with preservice teachers in ways that are different from how such knowledge is used by K-12 teachers.

Whereas K-12 teachers may have knowledge of student errors and potential reasons for the errors, they do not engage their students in explorations of student errors.

However, MTEs used their knowledge of the nature of student errors to connect preservice teachers’ exploration of the errors to teaching practice.

Specifically, the MTEs suggested an instructional move that preservice teachers could use in their future teaching to determine whether a student error is random or evidence of more systemic errors in students’ thinking.

By suggesting an instructional move that could be used in teaching, the MTE is connecting the mathematical work of analyzing student errors to teaching practice.

The MTE would have to know potential factors that may contribute to the development of the error, such as a lack of understanding of the role of place value in a multiplication algorithm, or due to a random misstep or number misplacement on the part of the student.

Thus, the MTEs are exhibiting knowledge about ways of recognizing and making explicit how analyzing student errors relates to teaching mathematics to students.

Finally, in another example, the MTE exhibited knowledge of research about student learning, and used that knowledge to connect research to preservice teachers’ content learning.

Teachers may also be knowledgeable of research about student learning.

However, teachers may use knowledge about research-based ways of supporting students’ understanding of the base-10 number system, for example, to directly inform their instructional interactions, either in their sequencing of instruction or guide their instruction in other ways.

However, the MTE did not directly use research about K-12 students’ learning to guide their own teaching of preservice teachers, but rather to make explicit the implications of research for preservice teachers’ current learning.

Thus, the MTE, as the example suggests, is exhibiting knowledge of mathematics education research in ways that connect to preservice teachers’ learning of content.

Taken together, these examples suggest that the mathematical knowledge needed for teaching teachers is different from that which teachers have to know to teach students.

In short, the forms of knowledge used by MTEs in this analysis included knowledge of certain concepts related to preservice teachers’ mathematics learning and how those concepts connect to teaching practice in K-12 classrooms.

This analysis also involved a process for identifying knowledge forms used by MTEs and thus fits well with one of the four features of the knowledge building process, namely, having multiple sources of innovation.

In the current analysis, having observations of different MTEs with different categories of expertise implementing the same tasks across different course iterations constitutes a source of innovation. Indeed, MTEs bring a host of resources to bear on their instruction and draw on different forms of knowledge in light of their expertise.

The authors argue that there needs to be more of a focus on understanding the knowledge drawn on by teacher educators as they teach content to preservice teachers.

This is critical as the field of teacher education not only lacks consensus about the content of many courses for preservice teachers but also lacks a system by which knowledge is communicated, shared, and improved.

Common language is crucial for systemwide improvement, yet there is currently no common language in the field of teacher education with which educators and researchers can engage in discussions of teacher educator practice.

Toward this end, the authors offer a potential approach for developing an evidence-based understanding of the knowledge demands of teacher educators’ work as they develop preservice teachers’ knowledge in ways needed for teaching.