The First EDGS 646 Assignment

(30% of overall assessment)

This part of the first assignment involves the preparation of a 1500 to 2000 word essay on Topic A OR Topic B (but NOT BOTH). The submission should be word-processed, and APA style should be used. It should be with me by April 15, 1996. (30% of assessment)

TOPIC A: "How can a teacher wishing to teach through a problem-posing and problem-solving approach make best use of the Working Mathematically section of the national Mathematics Profile?"

Comment specifically on the potential role of technology in helping students to "think and work mathematically."

OR

TOPIC B: Read the first two chapters of Polya, Krutetskii and the Restaurant Problem, focusing on the distinctions between intuitive, inductive, and deductive thinking. Read also the following paper, a copy of which is attached to the Subject Guide:

Clements, M.A. (1986). Personally speaking. Vinculum, 23 (3), 3 - 5.

The so-called 'circle and spots' problem brings out the weakness of arguing merely from pattern. This kind of reasoning is called induction. Most people, on seeing the sequence: 1, 2, 4, 8, 16, . . . would be prepared to bet anything that the next term after 16 is 32. But the 'circle and spots' provides a straightforward situation suggesting that 31 can be appropriate. Clearly, then, in the absence of a given rule, there is no strong logical reason for 32 to be preferred to 31 as the next term in the sequence.

The sequence 1, 2, 4, 8, 16, 31, ... arises from the counting of regions. However, if the number of chords joining spots are counted, then the sequence 1, 3, 6, 10, 15, ... is generated, and the nth term of this sequence is n (n-1)/2. The terms of the sequence are fairly predictable, and it is not difficult to justify, by deduction, that the nth term is n (n-1)/2. A typical deductive argument might be:

If there are n spots, then each spot can be joined to (n-1) other spots. But for every two spots there is just one chord. Hence the number of chords is n (n-1)/2.

It is interesting to reflect on the power of such deductive thinking in our Western culture. Mathematically trained people could not contemplate the possibility that if there were 100 spots (say) on a circle, then there would not be (100 x 99)/2 = 4950 chords.

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Write an "essay" on the strengths and weaknesses of deductive argument in mathematics, and implications for mathematics teachers (at any level). You may find it useful to provide examples, and to refer to approaches used by mathematics textbook authors.

Comment briefly on the extent to which mathematics teachers should encourage learners to use (a) intuitive reasoning, (b) inductive reasoning, and (c) deductive reasoning.

You might also consider the extent to which modern technology (hand-held calculators and computers, etc) may serve to facilitate the engagement of students in inductive and deductive reasoning activities.