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

Supported in part by

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.