Department of Mathematics Education

Comprehensive Examination, Winter 1997

INSTRUCTIONS: Respond to

Section 1. (

1. Recent statements of standards for school mathematics include an emphasis on building connections in mathematics.

a. Briefly define or describe what building connections in mathematics might entail.

b. Identify some goals relating to building connections that you would adopt as a teacher of school mathematics. Explain why you would adopt these goals as opposed to others.

c. Describe the features (e.g., content, organization, teaching methods, activities) of a secondary school curriculum that emphasizes the goals you have identified.

2. Your department head has asked you to prepare a position paper for your department as you plan changes in the geometry curriculum. (OPTION: Substitute a different content for geometry, e.g., algebra, statistics, consumer mathematics, probability, etc.)

a. What are the main goals of the high school geometry curriculum.

b. To what extent do you agree with the NCTM Standards regarding geometry topics that should receive increased or decreased attention?

c. What role will technology play in you revised curriculum?

d. What forms of assessment, if any, should your department be using to assess geometry learning?

Section 2. (

1. Choose three topics related to the concept of variable that are difficult for students in Grades 7 to 9. Identify common misconceptions or problems that students are likely to have. Suggest teaching activities that would promote understanding.

2. Consider a unit of mathematics instruction you have prepared (such as the one for EMT 669) that uses technology. Briefly (in a couple of paragraphs) describe the unit. Then:

a. Discuss some key instructional assumptions you make about the use of technology.

b. Discuss the classroom interaction and student response you expect, or try to develop, with mathematics instruction impacted by technology. If appropriate, you might want to contrast this instruction with your mathematics instruction without technology.

Section 3. (

1. Discuss the implementation of technology in high school mathematics instruction. What do you envision? What are the significant barriers to implementation? What can be done to overcome these barriers?

2. Select a topic in mathematics for which you would like to develop materials incorporating the Geometer's SketchPad. Outline what you would do and illustrate, as best you can in this limited time, one or two examples of student investigations and your expectations.

3. Given an angle in a plane with vertex O and a point P in the interior of the angle. Take a line through P intersecting the sides of the angle at points A and B. Determine a construction for the line to minimize the area of the triangle AOB. Include a proof.