Yesterday, we started investigating the remainder theorem. In their random groups, they divided f(x) = x² + 5x + 6 by x + 2, then by x + 3, then by x - 1. They also looked at f(-2), f(-3) and f(1). Then I asked what they noticed. Some really didn't notice much so I asked them to divide f(x) by (x+4) and find f(-4). They saw that the remainder from the division was the same as the value they calculated and that the value they were calculating was zero only if the divisor was a factor of f(x). (Note to future self: this was too scaffolded; fix for next year.)
So today we started with this: Find a factor of x³ + 5x² - 22x - 56. They were in new random groups of 3 and clearly did not make the connection to what they had started yesterday. Here is some of what I saw - lots to talk about!
We discussed our objective here - to factor this polynomial, which would allow us to sketch it. I asked something like "If something is a factor, what do we know?" The light bulb went on and they ran with that, finding at first one factor, then the remaining two using a variety of methods. Seeing that some were trying all integer values of x in f(x), I created a new question for them: Factor x³ + 6x² - 8x - 7. Those who had found all the factors to the previous question by systematically trying all integers starting at 0 soon got tired and asked if there was a better way (that was the point of the question). I suggested they look at the constant terms in their factors and the original polynomial. They were remarkably quick at putting those pieces together.
So, soon groups knew that only needed to try ±1 and ±7. They made more mistakes along the way (see below!), but there was progress. They found that they could not determine the other factors using the factor theorem, but had to divide by the first factor they had found. They understood the process and had ownership of it, having tried many paths that didn't take them where they wanted to go before figure out what would work consistently.
We didn't get through very many examples, but I firmly believe that it's better to work through one or two examples in depth, allowing students to find the pitfalls along the way and find their way out of them, than to spoon feed students a multitude of examples.
It's been ridiculously hot here the past few days so I feel somewhat incoherent and am not sure what point I was trying to make with this post anymore... I guess the takeaway I see is that using VNPS and VRG to let students explore and make mistakes is really powerful. I am really trying to get my students to do the thinking, not fall back on memorizing an algorithm, despite the fact that many think that is the best way to learn (ack!). I am trying to convince them that understanding the mathematics will take them so much further and be far more beneficial to them in the long run. That making attempts, some of which will fail, will prepare them for other times when they will struggle and not know where to start when solving a problem.
I'll end this with part of an e-mail I received from a student who graduated last year - I can't tell you how much it meant to me: "I have to say, I think your calculus class has been the most useful so far. The problem solving skills I learned in that class have taught me to set up equations and approach problems from a different point of view (in multiple classes... especially chemistry)! The self learning technique also helped because that's pretty much all I do now before I go to a lecture. "