Content and Process in a Year-Long Capstone Sequence for

Secondary Teachers

Dale Oliver

Humboldt State University

Collaborators

• Phyllis Chinn

• Beth Burroughs

• Sharon Brown

• PMET

Humboldt’s 1-year “sequence”

Fall of the Junior Year

School Mathematics from an Advanced Viewpoint I (Content for Teaching)

Spring of the Senior Year

School Mathematics from an Advance Viewpoint II (Process for Teaching)

“Bookends for upper division work.”

10 years in Development

• Spring 1995, the Senior Capstone– A Call for Change (1991)– CA Math Framework (1992) – CA Commission on Teacher Credentialing (1993)

• Fall 2005, The Junior Cornerstone– CBMS MET Report (2001) – CA Framework (2000)– CA Commission on Teacher Credentialing (2001)

The Ideal Californian Math Teacher (CCTC, December 2001):

Candidates for Single Subject Teaching Credentials in mathematics …

– develop, analyze, draw conclusions, and validate conjectures and arguments.

– create multiple representations of the same

concept.

CCTC, 2001…

– know the interconnections among mathematical ideas,

– use techniques and concepts from different domains and sub-domains to model the same problem.

– communicate their mathematical thinking clearly and coherently to others, orally, graphically, and in writing, through the use of precise language and symbols.

CCTC, 2001…

– solve routine and complex problems … while demonstrating an attitude of persistence and reflection in their approaches.

– analyze problems through pattern recognition and the use of analogies.

– formulate and prove conjectures, and test conclusions for reasonableness and accuracy.

CCTC, 2001…

– select and use different representational systems (e.g., coordinates, graphs).

– understand the usefulness of transformations and symmetry to help analyze and simplify problems.

– make mathematical models to analyze mathematical structures in real contexts.

Our Question in the 1990’s

What would it take to achieve these learning outcomes for teachers without a capstone course?

After 5 years of our first capstone

Fall 2000, Spring 2001

1) Suppose that on a round trip you drive at 30 mph on the way out and 60 mph on the way back. What is your average speed?

2) On a trip with your friend, you drive 30 mph for a certain length of time, and then your friend drives 60 mph for the same length of time. What is your average speed?

The student responses

13 out of 13 fifth-year math credential students answered, “45 mph is the average speed in each problem.”

15 out of 15 senior math majors answered, “45 mph is the average speed in each problem.”

Our question in the 2000’s

How can we include more content relevant to school mathematics without a second capstone course?

Goals

The goals of the courses are to enable students to• construct a deeper understanding of school

mathematics• build connections among the mathematical

areas they have studied and between undergraduate mathematics and school mathematics

• develop their understanding of mathematics as an integrated discipline

• strengthen their oral and written communication skills in mathematics.

Content and Process

Under what conditions is

Ax2+Bxy+Cy2+Dx+Ey+F=0

an equation for a pair of lines?

A linear algebra standard in the CA Framework: “Students … can compute the inverse to 2 x 2 and 3 x 3 matrices using row reduction methods...”

Write a detailed explanation of an algorithm that students might follow to complete the computation. Your detailed explanation should indicate not only how the algorithms works, but why the algorithms works.

Fall of the Junior Year (Content)

Elementary Functions and EquationsRate, Mixture, Averaging ProblemsQuadratic Equations and functionsHarmonic, Arithmetic, and Geometric Mean

Complex NumbersReal NumbersComplex Number GeometryRiemann Sphere

TrigonometryFunction AlgebraLimitsEpsilon-Delta Definitions

Resources

• Usiskin, Peressini, Marchisotto, & Stanley, Mathematics for High School Teachers: An Advance Perspective, Pearson (2003)

• Hahn, Liang-Shin. Complex Numbers and Geometry. MAA (1994)

What do students do?

Problems/projects from the Usiskin Text

Analyze problems

Extend Problems

Make Connections

For example

How many ounces of a 90% alcohol solution need to be mixed with 5 ounces of 50% alcohol solution to create an 80% alcohol solution?

Spring of the Senior Year (Process)

• Multiple Representations — appreciating, recognizing, and using the structure of mathematics to gain conceptual insight through transformations of objects or ideas in one context to an alternative context.

• Algorithmic Thinking — developing, interpreting, and analyzing algorithms to develop procedural insight into the nature of mathematics.

• Mathematics and Real-World Applications — building connections between real-world situations and mathematics, solving problems, and investigating the limitations of mathematical models.

…

• Mathematical Argumentation — creating and interpreting reasons and reasoning to communicate mathematical structures, relationships, and connections (along with responses to the “why” questions of school mathematics.

• Variation — studying how change in one parameter is associated with change in another parameter to gain insight into dynamic systems

Resources

Many and Varied, including– Conferences– Colleagues– School Math Texts– NCTM PSSM– CA Framework– MET Document– Journals

What do Student’s do?

Journal Writing

Problems-of-the-Week

Projects

Sample Journal

Skim the CA Framework sections on standards and classroom considerations for Grades 7 though high school for concepts related to functions and variation.

Select four areas of conceptual understanding regarding function or variation that students should achieve during these grades. For each, write down the concept and briefly describe a sample activity that could be used to teach the concept.

Sample Problem of the Week

What is the smallest square that can be formed on a 5 peg by 5 peg Geoboard using more than one rubber band? Generalize your result to an n peg by n peg Geoboard. Compute the area of this smallest square that you find, clearly showing the methodology.

Sample Project

A section of roadway is not in full use due to repair work. The repairs mean that single lane traffic only is permitted alongside the repair section. Normally there is a single lane of traffic flowing in each direction.Due to budget shortfalls (there have been lots of slides this year!) Caltrans is not able to use “flaggers” to control the traffic, but must instead use temporary traffic lights to control the traffic through the repair area. The problem is to decide on the traffic light settings so that the most efficient passage of vehicles is achieved.

Initial Reactions from Students• Excited to be taking courses directly

related to their future teaching careers.• Dismayed at how difficult it is to work

toward an understanding of “elementary” mathematics.

• Not prepared to work collaboratively.• Too much writing.• Too much knowledge assumed by the

instructors.

Positive Indications

High level of student satisfaction reported during and after their student teaching.

Math department evaluation of student writing proficiency in a mathematical context meets expectations.

Feedback from cooperating teachers and hiring school districts indicates relevant math preparation.