Supporting Improvements in the Teaching of Precalculus Level Mathematics

Authors: Marilyn Carlson, Kay McClain

Analysis of pre- post- data revealed significant shifts in teachers' knowledge of concepts of function, proportion and rate of change over one semester of instruction. We have used variations of the PCA as a central tool for measuring change in teachers' knowledge over the functions course. The PCA measured teachers' overall gains in understanding function concepts, proportional reasoning, and covariational reasoning. The following table provided a breakdown of two cohorts of teachers' pre and post course scores by item content.

An item analysis of the PCA data indicates that teachers improved significantly on nearly every conceptual subcategory measured by the instrument. Broadly, these include assessments of

1. reasoning in, and across, multiple representations,

2. aspects of transition from an action view to a process view of function,

3. covariational reasoning and reasoning about rates of change,

4. linear and exponential functions, and

5. specific misconceptions related to these categories.

In addition positive shifts in teachers' beliefs about learning and teaching mathematics were realized over the first course and accompanying PLC sessions. However, observation of these teachers' practices initially revealed minor changes in their teaching practice. Our coding of the PLC video data also revealed that improving teaching practice towards more inquiry focused and conceptual rich instruction requires teachers to reconceptualize their instructional goals. Concurrently they must acquire knowledge and skills to reconstruct their lessons and assume a different role in their instruction. We have observed that most mathematics teachers confront many obstacles in their journey to improve their teaching practice. Some obstacles that make shifting teaching practice complex are: i) shifting their perspective of teaching as supporting students' construction of understandings and reasoning patterns is dramatically different from their current view of teaching as telling and showing. Realizing this shift requires more sustained interventions that help teachers acquire knowledge about knowing and learning content; ii) developing lessons that are inquiry based and conceptually focused also requires knowledge of learning and concepts that takes time to construct. iii) pacing their instruction in response to student learning is in conflict with the pressure they are under to "cover" content in a pre-described syllabus or textbook. These obstacles were revealed by analyzing teacher conversations during their PLC's and by observing teachers' classrooms.