Matthew S. Winsor Ph.D. The University of Texas at El Paso mwinsor@utep

1. The Transformation of Secondary Preservice Teachers’ Mathematical Content Knowledge in a Capstone Course Matthew S. Winsor Ph.D. The University of Texas at El Paso mwinsor@utep.edu

2. Supported in part by

3. Capstone Course Initial Study • Did a Capstone experience increase students’ mathematical content knowledge? • Is students’ increased mathematical content knowledge evident in their lesson plans? • Currently I am interested describing the types of changes that happen in students’ mathematical content knowledge.

4. Student 1 (Doug) • Did not use any textbook as a resource for lesson plans. Relied on own prior knowledge. • Mathematical content knowledge was weak (pre-test, participation in study sessions, self admission in interview.

5. Doug’s pre-lesson plan • Can we solve y = 3x? • Generate a table • Get ordered pairs • Plot a graph • “y depends on x” • Effect of changes in parameters on graph

6. Doug’s post lesson plan • Function Machine • Each input only has one output • Functions do not only involve numbers, can involve relations such as assigning people to their hair color (assuming they have one hair color)

7. Student 2 (Sabrina) • Did not use any textbook as a resource for lesson plans. Relied on own prior knowledge. • Students’ math knowledge solid (pre-test, participation in class)

8. Sabrina’s pre-lesson plan • Objectives: • Understand a function is a relation via graphs • Understand that if f(x) = z and f(y) = z then x = y (which is the definition of a 1-1 function) • Uses several graphs to provide examples and non-examples of functions.

9. Sabrina’s post lesson plan • Objectives: • A function is a relation but not all relations are functions • Understand Domain and Range • Functions can be represented in more than one way.

10. Sabrina’s post lesson plan • Activity 1: Has students create several relationships form own life. Some relationships will be functions others will not. • Uses mappings to represent functions.

11. Sabrina’s post lesson plan • Helps students understand domain and range via mappings. • More set theoretical • Anticipates student responses. • Activity 2 and 3 similar to pre lesson plan

12. Observations • Changes in Doug’s lesson plan came as a result of changes in his mathematical content knowledge. • Post lesson plan had a large shift away from procedural to more conceptual. • Perhaps Doug had the opportunity to fill in his own “mathematical gaps”.

13. Observations • Changes in Sabrina’s lesson plan were partially due to changes in mathematical content knowledge (e.g. erroneous definition of function(confused function and 1-1 function). • Most changes were a result of Sabrina thinking about what examples would be most effective to teach students with. (PCK)

14. Observations • Doug’s knowledge was not sufficient to allow him to think about student learning and what examples were most effective. • Sabrina’s knowledge was strong enough to allow her to focus on student learning and effective methods for teaching.