## Developing Mathematical Knowledge for Teaching in a Methods Course: The Case of Function

Source: *Journal of Mathematics Teacher Education, Volume 16, Issue 6*, December 2013, p. 451-482.

(Reviewed by the Portal Team)

This study describes changes in secondary mathematics teachers’ mathematical knowledge for teaching function through their engagement in a mathematics methods course teaching experiment.

**Method**

The functions course that is the focus of this study was a teaching experiment implemented as a graduate course for 21 prospective and practicing teachers at a large urban university in the United States.

Data were collected through a pre- and post-course written assessment of key mathematical and pedagogical ideas; two audiotaped interviews with each teacher—the first conducted near the beginning of the course and the second after the last class session; video records of class meetings as well as field notes and instructional artifacts for each meeting; and copies of teachers’ work.

The participants in the course showed growth in their ability to define function, to provide examples of functions and link them to the definition, in the connections they could make between function representations, and to consider the role of definition in mathematics and the K-12 classroom.

For example, data from the start of the course confirm previous findings—that while teachers are usually able to identify or provide examples of functions, they are less likely to give an accurate definition.

While most teachers were able to classify linear relationships as functions, they were less successful in determining whether more complex relationships were functions.

Data from the end of the course indicate significant growth in teachers’ abilities to provide a definition and a stronger ability to identify more exotic examples and to create non-examples of functions.

Furthermore, the data show that teachers entering the course could generate representations but were limited in their ability to connect them.

By the end of the course, the ability to fluently connect representations and to mobilize them in the service of making sense of a function situation was significantly stronger.

These data suggest that the course was worthwhile—while we might expect teachers to have strong common content knowledge (CCK) related to function, many did not at the start of the course.

Moreover, it was successful, in that teachers showed evidence of learning related to function. In particular, teachers showed changes in their specialized content knowledge (SCK), an aspect of mathematical knowledge for teaching that is linked to student achievement and for which there are few systematic opportunities for teachers to develop.

The authors designed the course with three key assumptions in mind: that solving tasks would provide opportunities to develop CCK, discussing solutions would provide opportunities to develop SCK, and considering the tasks as learners and teachers would connect CCK and SCK.

In this course, teachers learned about function in a variety of ways, from refining their own personal definition to expanding their repertoire with respect to examples to considering the ways in which their students might struggle with function concepts.

By focusing on function, teachers had opportunities to explore function in a much greater depth than a small collection of function activities within a more mathematically diverse course might afford.

This focus also supports work in the classroom; by focusing on one topic, teachers experience the sequencing of tasks and topics in ways that build a conceptual understanding, much in the way that they might design a curricular sequence in their own classroom.

Furthermore, the course activities provided teachers with opportunities to refine and elaborate those initial understandings.

Approaching mathematical tasks first as learners, sharing solutions, and then analyzing the tasks as teachers were consistently cited by teachers as an important feature of the course. This sequence provided teachers with opportunities to connect CCK and SCK around a particular aspect of function.

By engaging as a learner first and then as a teacher around the same task, teachers are more likely to notice finer-grained nuances related to teaching and learning around the mathematical idea, helping them to link their common and specialized content knowledge.

The authors argue that by building and linking common and specialized content knowledge related to function, teachers have taken a requisite first step in being able to engage their own students in deep and meaningful study of functions.

Moreover, the teachers in the course engaged in practices that served as a model for a conceptual, learner-centered inquiry into a complex mathematical topic.

Teachers engaged with function both as a learner and as a teacher, giving them the opportunity to both enhance their own mathematical knowledge and to consider issues of their own students’ learning.

By grounding these student-centered pedagogical practices in the context of learning mathematical content related to function, teachers experienced learning at the interface of theory and practice that is likely to impact not only their own understanding of the mathematics of function, but also the understandings they foster with in their own classrooms.