Source: *Journal of Mathematics Teacher Education, Vol. 16, No. 4,* August 2013, p. 245-268

(Reviewed by the Portal Team)

In this article, the author describes the development of a series of tasks designed to investigate and measure teachers’ mathematical knowledge for teaching geometry and measurement.

The author presents three design features for rich, open-response items that assess mathematical knowledge for teaching:

• Tasks are grounded in the context of teaching

• Tasks, as a set, measure aspects of and relationships between common and specialized content knowledge related to geometry and measurement

• Tasks capture nuances of teacher knowledge beyond correct and incorrect answers, including specific misconceptions and the identification of vectors for change

**Research context and process**

The six tasks were created as a part of the design of a Masters-level content-focused methods course, taught at a mid-sized public university in the eastern United States.

The six tasks are:

1. Tangrams task measured teachers’ common content knowledge (CCK) in knowing that partitioning a figure into pieces and rearranging them can change the perimeter but not the area.

2. Area of a Parallelogram task assessed teachers’ CCK regarding the common misconception that a fixed area implies a fixed perimeter and that a changing area implies a changing perimeter.

3. Fence in the yard task requires teachers to generate length/width pairs and calculate perimeter and area and to recognize that a constant perimeter does not imply a is designed to capture aspects of teacher’s abilities to describe the relationships between length, perimeter, and area.

4. Minimizing Perimeter Lesson Planning task asks teachers to engage in solving the task themselves, and to consider the ways in which they might plan a lesson for a middle school class using this task.

5. The Big Ideas task asked teachers identify the ideas that they felt middle grades students should learn related to two-dimensional geometry, including length, perimeter, and area.

6. The Considering Formula Use task is designed to assess the reasons why teachers might select one formula for calculating the area of a rectangle over another.

The set of six two-dimensional geometry and measurement tasks embody these design features and illustrate the ways in which the tasks are grounded in the context of teaching, capture nuanced teacher performance, and measure common and specialized content knowledge.

The examples of teacher performance on these tasks illustrate the ways in which the tasks can differentiate teacher performance.

Moreover, teacher work on the tasks provides important windows into the connections between common and specialized content knowledge in teaching.

The tasks illustrate the important connections between common and specialized content knowledge and the ways in which CCK can influence how teachers make use of specialized content knowledge (SCK).

For example, teachers with stronger abilities to describe the relationships between length, perimeter, and area clearly on the Fence in the Yard Task were more likely to use multiple representations in their response.

The ability to use multiple representations and to understand the ways in which those representations make aspects of the mathematics salient implicates a wider range of pedagogical choices they might make in their classrooms.

This strong SCK is more likely to provide teachers with observable pedagogical tools and practices to support the development of students’ understanding of this relationship in their own classroom.

Given that these tasks are designed for use with secondary teachers, it is anticipated that teachers will approach these items with relatively strong CCK.

Teachers are likely to be able to calculate perimeter and area and make basic connections between length, perimeter, and area; the data from the items bore that hypothesis out at some level.

However, items that measured those aspects of CCK from multiple angles and that provided opportunities to mobilize both CCK and SCK in the service of the same task revealed important nuances and connections.

The Minimizing Perimeter task also revealed an important relationship between CCK and SCK and teachers’ ability to write goals for a mathematical lesson.

Teachers with stronger mathematical performances on the task were better able to write more specific goals for the use of the Minimizing Perimeter task with students.

This finding has particularly important implications for supporting prospective teachers in building capacity to plan for, teach, and reflect on strong, conceptually based mathematics lessons.

The three design features exemplified by these tasks—grounded in the context of teaching, measuring CCK and SCK, and capturing nuanced performance—are intended to be of use to researchers interested in investigating other aspects of mathematical knowledge for teaching.

These items provide us with tools to assess teacher knowledge in geometry and measurement at the secondary level.