This is important. In all other teaching, we present students with examples of the best: the best literature, the best art, the best physics experiments. We ought to do the same in math as well. Giving kids models of the sort of reasoning that actually goes into deductive argument can really engage them (and form them intellectually).

> There are as many fractions as whole numbers. Our intuition screams that this is crazy. Of course there are more fractions! Only a deductive proof can convince people that infinity is stranger than we give it credit.

There’s a lovely paper by Calkin and Wilf about this: http://www.math.upenn.edu/~wilf/website/recounting.pdf

I’ve often used it with enterprising calculus classes, as it’s short, surprising and, written in a readable style. It includes a couple of different patterns of reasoning to support one conclusion.

We generally spend five or six hours reading it together in class (as one might do with a tricky text in a literature class). There are vocabulary quizzes interspersed, and finally the students work in groups to construct presentations to make the argument accessible to seventh graders, which they then give.

Some examples of these kinds of situations where deductive reasoning becomes useful to answer a question one might have:

Example 1:
The Bridges of Koninsberg – I introduced this problem to a group of 9th graders in inner city Brooklyn. Three weeks later, there were still pockets of kids coming up to me (from all over the school) claiming that they had found a solution to the problem. When I finally relented and showed them Euler’s clever proof that no solution was possible, they were genuinely interested in the proof. It mattered to them that we could show it was not possible so that they could lay the problem to rest. I think the hundreds of years that people attempted to solve this problem before Euler came around suggest that inductive reasoning wasn’t very useful here.

Example 2:
Come up with a formula that will give the maximum number of pieces with n number of straight slices of the circle (source: http://mathforum.org/library/drmath/view/55283.html). This problem is easily stated, and the first few patterns are so nice, that students are usually fooled into thinking they have the pattern, and so they will often start filling in their tables of values before they check their results carefully. It leads to a situation where inductive reasoning fails quickly, suggesting that deductive reasoning would be useful. I don’t know how interesting students find this problem though; probably not very.

Example 3:
There are as many fractions as whole numbers. Our intuition screams that this is crazy. Of course there are more fractions! Only a deductive proof can convince people that infinity is stranger than we give it credit.

For instance: “The diagonals of a rectangle are congruent.” If we show students some non-rectangles, and have them notice that the diagonals aren’t congruent, then it makes sense to wonder, “Okay — what is it about having *right angles* that makes the *diagonals* congruent? That is weird and sort of unexpected.”

Launch!

http://researchinpractice.wordpress.com/2010/05/07/pattern-breaking/

and

http://researchinpractice.wordpress.com/2010/07/12/pattern-breaking-ii/

If anyone has a few examples of these, that would be really useful the need for proofs in my opinion.

I got the impression from ‘The Lament’ that his approach to “nailing that point where the seemingly-true hasn’t yet turned into the obviously-true” involved having a huge magazine of interesting problems to pose to students (or individuals) as needed. In ‘Measurement,’ Lockhart seems to be opening up his bag of tricks and he’s quickly becoming one of my biggest idols.

I’d also like that add that an appreciation for history seems to be a recurring theme here with Harel (via Dan):

“intellectual perturbations that are similar to those that resulted in the discovery of new knowledge”

With Serra (via Dan):

“The Great Pyramid was built in 3900 B.C. by rules based on practical experience: Euclid’s system did not appear until 3,600 years later. It is quite unfair to expect children to start studying geometry in the form that Euclid gave it. One cannot leap 3,600 years of human effort so lightly!”

with Dan:

“a perturbing moment experienced by so many athematicians before them”

And with Lockhart (sorry, no quotes… But his work I’ve seen is strewn with historical annecdote and reference).

I’ve been kicking myself for not having a stronger foundation in the history of math ever since I heard about the Bridges of Konningsburg this summer. I attended a session on math history just yesturday and learned this gem: the symbol e (maybe the value also, my notes were kinda furious) was first used by Euler in a paper called ‘Meditation on an experiment made recently on the firring of cannon.”

…If you involve cannon balls, is it even possible to setup a boring Act 1? (and now we need another two-column proof!)

I barely have time to sneak in some If-Then statements and the converse with Pink lyrics . . .

Where there is desire
There is gonna be a flame
Where there is a flame
Someone’s bound to get burned
But just because it burns
Doesn’t mean you’re gonna die
You’ve gotta get up and try, and try, and try