|Feedback and Questions on Categories
Based on our brief discussion about categories at the 10/24/06 Polycom meeting and the documents that PSU sent to UGA, we have tried to understand and analyze the two proposed category systems called Process categories and Drawing-on-mathematics. We have not tried to use either system to analyze Situations yet. We did try to look at the Process categorization and to reduce the grain size of the categories. We have attached questions about the categories and the process of creating categories. We have also attached a list of subcategories (mathematical activities) that add to the description of Process categories.
Questions about Categories
Why and how are we creating categories or criteria?
1. Where does the process of analyzing situations using categorical systems (or other analysis) fit in our ultimate goal of creating a framework for the Mathematical Knowledge of Teaching at the Secondary level?
We think that the purpose of creating categories is to analyze the Situations. (You have stated this in your introduction to the 10/24/06 document). What is the purpose of this analysis and how does it fit into the larger set of goals. For example, would this analysis become a template for creating situations? Would it be used to describe the set of situations in contrast to individual situations? Would the analysis identify missing constructs and repetitious constructs in the set of situations? How does creating an analytical system help us? How does analyzing the Situations help us?
2. Are the categories drawn from brainstorming (e.g. literature, practice, discussion, other classification systems) or are they drawn from the current Situations (e.g. components, experience in creating situations) or a combination?
If part of the role of the analysis of the Situations is to validate the Situations as representing secondary curriculum (e.g. algebra, geometry, calculus, probability), NCTM standards (e.g. process standards, content standards), mathematical activities (e.g. defining, generalizing, proving), or something else, it seems like it would not be appropriate to base the analysis system on the Situations. If part of the role of analysis is to dissect and analyze what we have seen in practice and the created foci, it may be important that the categories come from the Situations. What should be the relationship between the categories and the Situations?
Where would we find specific constructs?
3. Would Polya’s distinctions between “presenting mathematics” and “doing mathematics” be useful in our category schemes? How would these constructs fit within the 2 proposed systems of “process” and “drawing-on-mathematics.”? We may also want to look at his distinctions among deductive reasoning, inductive reasoning, and plausible reasoning which includes strategies like using analogies. One of the most difficult parts of the process is deciding what to prove. How does this fit?
4. Where is the construct of “abstracting”? There is a critical distinction between abstracting and generalizing. Is this distinction made explicit in the categories?
5. Where are the constructs of “representing” and “visualizing”? A category of Representing might have subcategories of symbolizing and visualizing.
6. How would the process categories accommodate discussing, refuting, and conjecturing? Some of these words are in the definitions of various categories but there seems to be a bigger idea here of using discourse.
7. Are you thinking of creating categories or a structure for the Drawing-on-mathematics list of questions?
The list of 15 items seems to beg for a structure (and more items).
8. Does the Symbolic Working category refer only to algebraic symbols?
We are still puzzled by the Symbolic Workings category. It not only seems to be quite different from the other categories, but it is not very useful because it is so inclusive. We seem to need a distinction between using symbols and talking/thinking about symbols.
9. We need to think about and discuss definitional and nondefinitional generalization. How does this relate to the definition category?
Situations Group at UGA –feedback on Process Categories
This is our first attempt to provide sub-categories for the process standards. This list is not meant to be exhaustive and is open for interpretation, debate, and modification.
Categories and Sub-divisions
Process Standard: Proving/Justifying
Proof by Construction/Geometric Proof
Proof by Induction
Proving the Contrapositive
Proof by Exhaustion when dealing with a finite number of cases
Proof by Contradiction
Generating an appropriate number of examples to convince
Use of graphs to verify domain and range
Checking solutions to equations
Using visualization to explain
Agreeing on a definition
Refutation in classroom discourse
Using Always/Sometimes/Never statements
Using plausible reasoning (e.g. analogy) (Polya)
Process Standard: Defining
Using standard geometric definitions
Using standard algebraic definitions, including formulas
Identifying characteristics and properties of a mathematical object
Exclude, Generalize, Replace, Add to a known object
Process Standard: Generalizing
Process Standard: Symbolic Working
Manipulation of objects in a software program like GSP
Symbolic Manipulation in an algebraic sense
Different visualizations of an object
Representations of geometric objects