Sierpinska, A. (1992). On Understanding the Notion of Function,
The
Concept of Function: Aspects of Epistemology and Pedagogy, MAA
Notes, Vol. 25, 25-58.
- Sierpinska suggests that "Both syntheses and discriminations are
needed to fully understand the concept of function." In what
sense are they both required related to the same ideas. That is,
what are aspects of functions that students need to be able to, at the
same time, regard
as both similar and different?
- On pp. 42-44, Sierpinska discusses the perception of proportion
as a "priveleged relationship." In explicating the positive
aspects of this epistemological obstacle, she cites David Bohm's
characterization of proportion as the root of rational thought (a view
first laid out by Aristotle, by the way). He goes on to discuss,
rationality/irrationality, mathematicle structure, order, and
coherence. Compare and contrast Bohm's views with Piaget's
characterization of conceptual structure as modeled by the
algebraic group.
- Sierpinska claims that "In order to be ready to accept these
examples ["strange functions" included in the Dirichlet definition] as
examples of functions one has to be sufficiently mature in the
mathematical culture to see the role of definitions in mathematics as
binding logically and not as descriptions of certain aspects of an
object otherwise known by senses or insight." What are some
possible views students might have about the meaning or role of
definition in mathematics that might be problematic? What
influence might the introduction of specific examples play in this?
What is the role of prototypes in students' understanding, and
what is their relationship to definition?
- On pp. 54-55, Sierpinska applies Aristotle's decomposition of
"coming into being" to the function concept. In her analysis, she
relates questions 1-4 to the mathematical structure of functions but
relates question 5 to the creation of the function concept.
In the context of our class, wrestling with the nature of the
origin of concepts, this is a far more interesting issue. How
could you apply questions 1-4 to the
creation of the function concept, as well?
- Explain how an epistemological obstacle may have positive, as
well as negative, aspects.
See the class diagram here
Carlson, M. (1998). A Cross-Sectional Investigation of the Development
of the Function Concept,
Research in Collegiate Mathematics
Education III, Conference Board of the Mathematical Sciences,
Issues in Mathematics Education Volume 7; American Mathematical
Society, 114-163.
- What technique did Carlson use to draw conclusions about the
long-term development of function concepts without conducting a
long-term study? What are the advantages and disadvantages of this
approach?
- What steps did Carlson take to validate her coding of students
responses?
- Throughout her paper, Carlson makes several claims about the
nature of students understanding of function relative to the APOS
framework. For each of these claims, find specific statements
made by students in the excerpts that either support the claim or
suggest an alternative possibility.
- Carlson lists several things that high performing college
algebra and calculus students can, and cannot, do (pp. 33-34).
Can these
claims be correlated to the epistemological obstacles and acts of
understanding in Sierpinska's framework? How (or why not)?
