Epsilon sandwiches is a great speech, with advice that I wish I could follow—but it pre-supposes students for whom the referenced very easy proofs are indeed very easy conceptually, and the struggle is only to put those concepts into words. I can believe that this is true of students in a UPenn first analysis course, but it is not true at the less prestigious university where I teach students who are encountering proofs for the first time (well before analysis)—and I have long struggled with how to break down this two-step complication into separate manageable steps with students for whom, say, it is still a real challenge to understand (in the context of proving facts about sums and products of even numbers) why 2(x + y) = 2x + 2y is true, but 2(xy) = (2x)(2y) is not. If anyone knows how to adapt Wilf's advice to such students, then I—and they, in my fall class!—will be grateful to hear it!
Thanks for the suggestion, which I think is exactly the right way to address this particular issue.
Indeed, now that I've learned from a few times teaching the course that it always comes up, I will address it. But, if I spend my time building up basic-algebra proficiency at this level, then I'm never going to get into the meat of proving things. Perhaps the answer is to view the very act of writing your argument as a proof—and I like that perspective (because it genuinely is!); but, if I were to break that down to its full details, then I would have to write 2(x + y) = (x + y) + (x + y) = x + (y + (x + y)) = x + (y + (y + x)) = x + ((y + y) + x) = x + (x + (y + y)) = (x + x) + (y + y) = 2x + 2y, and writing down proofs of that sort risks alienating students who are eager for a conceptual picture, and guarantees giving the wrong idea of what sort of activity proving is. (And it also risks confusing why later I will say that, for a silly example, π(x + y) = πx + πy "by the distributive law"—I can't think of multiplication by pi as repeated addition, so I must cite the distributive law; and then why didn't I just do that before?)
This has confused me a bit about the US education system. Shouldn't this be taught in highschool? Where I live if someone wants to study maths in university, and they haven't done the B2 math track, the university would simply not accept them into enroll into that major. The student would have to get a highschool degree that includes the maths B2 track, if they're under 18 they could simply return to highschool, or if they're older then there's adult (night) schools.
> This has confused me a bit about the US education system. Shouldn't this be taught in highschool? Where I live if someone wants to study maths in university, and they haven't done the B2 math track, the university would simply not accept them into enroll into that major. The student would have to get a highschool degree that includes the maths B2 track, if they're under 18 they could simply return to highschool, or if they're older then there's adult (night) schools.
It's not that they haven't been taught this; it's that they haven't conceptualised it, so that 2(x + y) = 2x + 2y to them is just a meaningless rule, and there's no particular reason why it should be true but not 2(xy) = (2x)(2y), or 2^(x + y) = 2^x + 2^y, or whatever. Of course ideally one would only have students in the course who have conceptualised this, but that's not how it happens, and it's not fair for me to teach the course to the students I wish I had.
You can find proofs by making analogies in almost any direction, but it helps to know which direction leads towards solid intuition the student already has and can build on, rather than just leading to more abstract nonsense that's equally unfamiliar.
In the case of having no intuition about algebraic manipulation, you can suggest a geometric interpretation to connect to intuition that's more likely to be there. For 2xy != 2x2y, draw two xy rectangles and one 2x by 2y rectangle.
Now the students all see the problem. Now they just have to connect the geometric intuition back to the algebra. This helps motivate the algebraic rules and shows why they must be what they are. Just the idea that geometric intuition exists -- that you can solve problems by putting pictures together in your head -- this isn't something every incoming freshman already consciously knows they have as a technique always available to them.
(This is just a wordy re-telling of Polya's "Draw a figure", from How to Solve It; if you haven't read, drop everything and get a copy.)
