### May 16, 1997

Today we will talk about one specific development in algebra that occurred during the same time period we've been discussing. It is due to the mathematician Viete, who was a lawyer in France at the time. Viete does have a connection to Copernicus, being one of his annotators. He was also interested in calendar reform and late in his life got into trouble for publishing a revision of the Gregorian calendar without permission. He didn't think much of Copernicus.

To grasp his contribution to algebra it is useful to understand some basic ideas about cognition. The issues surrounding Viete's contribution to algebra illuminate some of those surrounding Copernicus' contribution to astronomy.

The circle: write two representations of a circle on board: a geometric picture and an equation:

x2 + y2 = 1.

a. Question: Are both of these examples "circles" in the same sense? What are the differences between them? To be more precise, which model is a more accurate description of a circle? Which one is more satisfying? Which example would you give to a 1st grader? Which one would you give to a 10th grader? Why?

b. Suppose you want to "put a circle of radius 2 at the point "A". What would be the difference between this and using a picture description in an algebra description?

Discussion of what a "schema" is and what a "cognitive map" is.

• a. Cognitive map is a set of associations with a given idea.

Example: picture of circle, pictures of other shapes, distance, size, class of geometrical objects
form cognitive map.

• b. Schema: cognitive map plus a set of relationships plus methodology

Examples: previous cognitive map; relationships such as larger, smaller, similar, congruent, far or near; a methodology, as in "you can place a circle of radius 2 at the point A", "you can copy the
object in a different place", "you can cut it up and reassemble it"; all these taken together is
a schema.

• c. Euclid's "Elements" represents a complete schema into which the picture of the circle fits very well as a representation of the circle. The representation "x2 + y2 = 1" doesn't fit at all well into this schema.

• d. "To understand something means to assimilate it into an appropriate schema." The Psychology of Learning Mathematics, Skemp. You might think of a schema as a small written version of one of Holton's themata.

Exercise: Divide class into three groups of 2 or 3.

Group 1: Explain what 3x = 6 means to a third grader.

Group 2: Explain what "y = 2x, therefore y + 3 = 2x + 3" means to a ninth grader.

Group 3: Explain what "all equations of the form y3 = mx2 + b" means to a college freshman who can't quite remember algebra.

Students will have 10 minutes to prepare before they present explanations of their respective schemas to the class. The basics of each explanation can be written on the board.

Go over the three schema weaving people from each group into the discussion. Students should be careful to note any evidence of the three different schema.

Beginning questions:

a. What metaphors are used and which are critical to our understanding of schema?
b. Is anyone uncomfortable with the third one?
c. The third schema is Viete's major contribution to algebra.

Discussion Questions:

a. What can you do in one schema that you can't do in the others? Is there a hierarchy?

b. Where does, "to restore", belong?

c. What sort of schemas do you think the Greeks had?

d. Discuss the historical development of the following schemas: rhetorical; Diophantine or syncopated from 250 a.d.; Vietan from the 16th century; Descartes.

e. How do people's schemas change?

f. What do Holton and Kuhn'have to say about schemas?

g. What is cognitive dissonance?

h. In Eon Harper's psychological study, "Ghosts of Diophantus," what was the source of the cognitive dissonance that led Viete to a new schema?

I. What inspired Copernicus to create a schema? (This is exactly the kind of challenging question history of science and math claims to answer.)

Epilogue: Why French lawyers? Remember the dual inheritance of Greek and Arabic math in the 16th century. There was a real stylistic difference between the two? Viete is engaged in the reconciliation of the two systems. He is putting a notational and philosophical framework under all of Arabic and other ancient algebra. Of course, he claimed that he was just rediscovering material due to Diophantus, who knew everything Viete knew but liked to keep it secret in order that no one would know how he solved all his problems. Here is the most tantalizing fact. At precisely the same time in France, the French legal system had two contradictory sets of laws. One was feudal law (e.g., anything the local king says goes, no private property and so forth), and the other was Roman common law (law that is above the local ruler, private property, etc.). To reconcile these, a legal system evolved based on the idea of legal precedent, where one particular case stood as an example generalizing to a whole class of cases. A whole class of cases--does that sound familiar? Perhaps the idea of it as a legal precedent was a major thema for Viete. Indeed, Holton would ask, why wouldn't it be so?

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