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

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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.

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.

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

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)

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)

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.

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.

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.

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

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”.

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)

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.