Vinner S. & Dreyfus, T. (1989). Images and Definitions for the Concept of Function. Journal for Research in Mathematics Education, 20, 356-366.

• Describe concept image and concept definition. Illustrate this with an example of your own understanding of a concept from advanced mathematics. What is meant by compartmentalization?
  

• What is the difference between an identification and a construction task? What information does asking one or the other in a research project provide?
  

• Give a description of each of the definition categories observed (correspondence, dependence relation, rule, operation, formula, and representation).
  

• Give a descriptiopn of each of the aspects of concept images observed (one valuedness, discontinuity, split domain, and exceptional point).  How were these aspects of concept images used?
  

• What evidence do Vinner and Dreyfus give to support their claim that concepts are acquired over time and not in one single step?
  

Monk, S. (1992). Students’ Understanding of a Function Given by a Physical Model: The concept of function, aspects of epistemology and pedagogy, The Concept of Function: Aspects of Epistemology and Pedagogy, MAA Notes, 25, 175-194.

• Monk distinguishes between pointwise and across-time questions. What is the difference between such questions and why do students find across-time questions more difficult. That is, what additional (or different) is required cognitively to answer an across-time question?

• What are the similarities and differences between Monk's description of pointwise vs. across-time understandings and actions and processes in the APOS framework?
  

• What is a blurred concept?  What is iconic-translation?  How do each of these contribute to students difficulties in addressing across-time questions?  Why do the same issues not surface for pointwise questions?

• What technique did Monk use to distinguish between the effects of blurred concepts and iconic translation in students' work?
  

• Why does Monk introduce a physical model to represent functional situations to the subjects of his study?

• In Monk's study, how were students' classification with respect to the pointwise vs. across-time distinction triangulated? In what sense did this distinction have predictive power?  

• On  page 189, Monk says, "there appear to be no explicit visual features of the model that these students [who have drawn a 'sagging graph'] are simply importing into their graphs."  He is wrong.  Re-examine the physical situation and find a visual feature might be cuing the students to draw a sagging graph.
  

Monk, S. & Nemirovsky, R. (1994). The Case of Dan: Student Construction of a Functional Situation Through Visual Attributes, CBMS Issues in Mathematics Education, 4, 139-168.

• In episode 1, why do you think Dan might be making a connection between fast and straight on one hand and slow and wavy on the other? How is this different from iconic translation?
  

• In episode 2, why does Dan think that he must first push then pull on the bellows? Is this an example of iconic translation?  Why or why not?
  

• In what ways is Dan's use of visual-graphical information detrimental?  In what ways is it powerful?
  

• Dan eventually concludes that producing this graph is impossible.  How is it, then, that his grasp of the situation is almost immediate upon being shown how to produce the graph?
  

• In what sense is a Piagetian notion of structure embodied in these tasks?
  

• What evidence do Monk and Nemirovsky give to support their claim that learning is a process of gradual refinement, not replacement of misconceptions?