Department of Mathematics Education
Master of Education
Comprehensive Examination, Winter 1997
INSTRUCTIONS: Respond to one question in Section 1, one question
in Section 2, and one question in Section 3. Allow yourself about
3 hours total for the three responses (if you take longer, let us know with
Section 1. (Choose only one.)
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
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
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,
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. (Choose only one.)
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
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
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. (Choose only one.)
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.