MTE 598
Research in
Undergraduate Mathematics Education I
Fall 2002
Discussion Questions
September 3 - Knowing and Learning
Mathematics
Schoenfeld, A. (1987). Cognitive Science and Mathematics Education: An
Overview, In Cognitive Science and Mathematics Education, 1-31.
What is “productive thinking?”
What is the relation of pseudo-analytical thinking to
Wertheimer’s examples? What do these students understand?
How is mathematical conceptual structure different from the
behaviorist and gestaltist perspectives? Does Skinner’s
approach rely
on a grasp of this structure? If so, in what sense?
On page 8, Schoenfeld characterizes some mathematics education
research as over-interpreting statistical results. What is the
nature of the gap between the statements “verbal ability is an
important aspect of problem solving performance” and “scores on tests
of verbal ability correlated highly with scores on a problem-solving
test.”
What is a “process model” of Student’s understanding?
In the section on the cognitive science approach, notice the
assumption that all thinking can be modeled by a computer program.
(This is not accidental – much of the origins of the field was
intertwined with the advent of computers which served as a sort of
“existence proof” for their descriptions of
cognition.) What are the corresponding implicit assumptions about
the nature of thought and knowledge?
Schoenfeld notes that competent mathematicians employ strategies
such as getting rid of “nasty” terms when working nonstandard algebra
problems. What might happen if we teach students such methods?
Schoenfeld gives several examples of the problem solving strategy
“look at special cases” in various contexts. What is the gap
between being able to implement each of these separately and being able
to employ the strategy generally and flexibly to new problem
situations? What might be the role of “integration” and
“restructuring?” (See p. 16)
What does Shoenfeld mean by the word ‘‘trained” on page 19?
Schoenfeld, A. (2000). Purposes and Methods of Research in Mathematics
Education. Notices of the American Mathematical Society, 47,
641-649.
Schoenfeld notes that evidence and argument in mathematics
education research are different than in mathematics. What are
some of these differences and how might they affect the way one
perceives the enterprise of research in mathematics education?
Schoenfeld notes that a common question from mathematicians to
mathematics education researchers is, “Tell me what works in the
classroom.” What might you say to a mathematician who asks you this
question?
Vinner, S. (1997). The Pseudo-Conceptual and the Pseudo-Analytical
Thought Processes in Mathematics Learning. Research Studies in
Mathematics, 34, 97-129.
What are some elements of the standard didactic contract assumed
by students when they enter a mathematics classroom? Why do you
think these specific expectations have become norms? What are
some ways that a teacher can establish the expectation that an answer
“I don’t know” with actual mathematical thought behind it is more
valued than a “correct answer” supported by no mathematical thought?
What is the difference between a “misconception” and
“pseudo-conceptual” or “pseudo-analytical” behavior?
What social needs might pseudo-analytical behavior fill?
What intellectual needs might it fill?
Describe circumstances when a lecture is likely to produce
pseudo-analytical behavior and when it is likely to produce analytical
behavior.
Describe circumstances when group activity is likely to produce
pseudo-analytical behavior and when it is likely to produce analytical
behavior.
Draw a larger version of the diagram in Figure 1 and label
its parts with meaningful phrases (like “mathematical problem” instead
of “X”). Construct a similar diagram for the pseudo-analytical
approach
to a problem outlined on page 114. In terms of the area example
given, describe the differences in these processes and what a student
misses
conceptually if they are allowed to only work pseudo-analytically.
On page 117, an incorrect solution to the equation x2-5x+6=2
is given. Explain why this solution is not correct. Why
does this procedure produce an actual solution (x=4) even though
the method was invalid? Main question: What mathematical
understandings are
necessary to see the error in this procedure and to be able to work the
problem correctly with understanding?
What implications does this article have for analyzing research
data?