MACMTL Quantitive Study

In 2005, a team of graduate students under the direction of Pat Campbell and Anna Graeber began an extensive research study to address a fundamental question about teacher content and pedagogical knowledge, namely does it affect student learning? Working on the premise that a characterization of teachers’ mathematical content and pedagogical knowledge could only be represented by direct measures, the project team initially conducted an extensive literature review to identify components for a teacher-knowledge framework as well as released assessment items. This framework was then intersected with Delaware, Maryland and Pennsylvania curriculum standards for school mathematics (Grades 4-5 and 6-8) to define understandings associated with teaching the Grades 4-8 school mathematics content in those three states, along with domains specifying pedagogical content knowledge. A second literature review addressed teachers’ beliefs about mathematics teaching and learning, as well as literature addressing students’ mathematical dispositions and teachers’ awareness of or efforts to influence those perceptions.

Building on these preliminary reviews, the research team then established a definition of teachers’ mathematical content knowledge as well as a definition of the distinct, but linked, construct of pedagogical content knowledge.  Subsequently the team developed over 400 multiple-choice assessment items based on these frameworks and definitions that were then screened and vetted by mathematics educators, mathematicians, and school-district mathematics supervisors prior to piloting by over 150 practicing or pre-service teachers. Following completion of classical test theory procedures, two 120-item assessments were developed (80 content and 40 pedagogical items for mathematics teachers of Grades 4-5 and another set of 120 items for teachers of middle-grades mathematics). The research team also developed a beliefs and awareness survey consisting of 40 Likert-format items addressing teachers’ beliefs about mathematics teaching and learning (30 items) and the degree to which teachers claimed to be aware of or to gather explicit information about their students’ mathematical dispositions (10 items).

The teacher knowledge assessments and the beliefs and awareness survey were administered to 266 upper-elementary and 193 middle-grades mathematics teachers from 23 school districts across Maryland, Delaware, and Pennsylvania. Teachers also supplied information on their professional background and instructional assignments. Administrators in the cooperating school districts then provided the project with individual, anonymous student standardized mathematics achievement and demographic data linked to the participating teachers.

Because the teacher assessment measures differed by grade band and because the student data reflected a hierarchical structure with students nested within a teacher, the upper-elementary and middle-grades data were separately analyzed using hierarchical linear modeling.

Findings indicated that, after controlling a variety of teacher-level variables, the content knowledge of upper-elementary teachers did affect their students’ performance on standardized mathematics achievement tests. But this relationship did not hold for teachers’ pedagogical content knowledge. In addition, those Grade 4 and 5 teachers with high content knowledge scores who also scored highly on the survey items measuring their perceived awareness of student mathematics dispositions had students with significantly higher mathematics achievement than did other teachers. Middle-grades results were stronger as the mathematical content and pedagogical content knowledge of middle-grades math teachers predicted their students’ mathematics achievement with and without teacher-level control variables in the analytic model. Further, interactions were identified. Middle-grades mathematics teachers with high content knowledge or high pedagogical content knowledge who strongly held beliefs supporting modeling solutions with instruction organized for incremental mastery had significantly higher student mathematics achievement that those teachers who did not demonstrate this combination. However, those middle-grades teachers with low mathematical content knowledge or low pedagogical content knowledge who held these beliefs had significantly lower student mathematics achievement. One implication of this finding is that limiting instructional materials to a sequential demonstrate/model-guided practice-independent practice routine will not compensate for a middle-grades teacher’s weak understanding of mathematics content and pedagogy.

A secondary analysis noted that those middle-grades teachers who had earned semester credits in mathematics at or above the level of calculus had significantly higher scores on both the mathematical content and the pedagogical content knowledge assessments. Similarly, middle-grades teachers who had completed at least one mathematics education course (generally a mathematics methods course) had significantly higher scores on both the mathematical content and pedagogical content knowledge assessments.  Neither of these findings was identified for the upper-elementary teachers.

Although upper-elementary teachers, as compared to middle-grades teachers, were less likely to believe that students should struggle with problems prior to teacher intervention, more mathematically knowledgeable upper-elementary teachers were likely to hold this belief more strongly, as compared to less mathematically knowledgeable upper-elementary teachers. This relationship between teacher knowledge and teacher beliefs was not identified for teachers of students in the middle grades. In contrast, across both the upper-elementary and middle-grades, teachers who believed that mathematics instruction should be characterized by teacher modeling and demonstration of symbolic procedures followed by student practice were more likely to have lower mathematical knowledge. This may mean that teacher education efforts supporting ambitious mathematics teaching with a focus on addressing students’ mathematical understanding will be more effective if the teacher enhancement efforts integrate and target mathematics content and pedagogy, with an intent to influence both teacher knowledge and teacher beliefs.