Alien Encounters - Questions and Exercises for Act 2

Author(s):

Gavin Hitchcock

What cultural and historical conclusions may we draw from the length and style of Saunderson's lecture, especially in view of the undoubted popularity of his lectures in his time?

What are the four conclusions reached by Saunderson?

What assumptions does he make in his first demonstration of these four conclusions? What assumptions does he make in his second, briefer, demonstration?

Compare Saunderson's two demonstrations from the point of view of mathematical soundness, and from the point of view of teaching effectiveness.

How do Saunderson's methods compare with those used in school texts today? Do we think of our students as people who "expect a demonstration?"

How does Euler demonstrate each of the four conclusions of question 1?

What are Euler's hidden (implicit, unacknowledged) assumptions? Would any of these follow from the assumption of the multiplicative cancellation law for real numbers (positive and negative)?

What do you think of the suggestion that the "exceptional didactic quality" of Euler's Algebra might be attributable "to the fact that it was dictated by the blind author through a relatively untutored domestic?" (See ref.19.) In simpler words, this is saying that the message gets through to us because it had to be made clear enough to get through to Euler's secretary first!
In the writing of school textbooks today, what should be the role of professional mathematicians, professional educationists, practising teachers, pupils/students?

What justification does Laplace give for considering "affected" or "signed" numbers? Does he, in your opinion, offer a real proof of the law of signs?

Does Laplace go any further than Euler in allowing a letter such as a to stand for a negative number. What didactical lessons are there? (Consider classroom responses to the question: Is -a positive?)

Look at the different teaching methods of Saunderson, Euler and Laplace. Which one in your opinion communicates best? Which one shows the most awareness of what is appropriate to his students - of what they can receive ?