Recently in calculus, we’ve been working hard on u-substitution to solve integrals. Integrals are not intuitive. They are motivated (area under the curve), they are justified (the anti-derivative), and we try to play around with them until we “get them.” But when it comes down to it, you can’t really capitalize on much intuition when introducing them.
In my class, the most we did with honing intuition was to ask basic questions like these:, , , , , and .
To solve them, students had to “work backwards” from their derivative knowledge. They were just guessing and checking to see what functions they could take the derivative of to get, say, . (In this case, it was ). They figured out some general strategies on their own, and I validated them.
And then we moved on to u-substituion, to solve integrals like and .The hardest part of u-substitution is picking a good u which actually simplifies the integral. (In the first integral, and in the second integral above, .) The question for students is: why? It would be a mistake to assume that any explanation you give them will make sense (initially). For the second integral, saying–
“Well, because I see that square root on the bottom, I immediately think of inverse sin, because we know . Hence, I see that we should pick a u which makes the integral take this form, and hopefully replacing dx with something in terms of du will cancel out the numerator.”
–will never work. You can say it until you’re blue in the face, but until they try out a whole bunch of possible us which get them nowhere, until they’ve seen enough similar questions like this, until they try to articulate for each integral why they chose that particular u, then they won’t gain intuition. You can’t force it. It comes through familiarity.
This is one of those topics where I think new math teachers probably dread introducing (and if they aren’t concerned about it, they are better people than I am). Students invariably will be confused. And what’s worse (from the student perspective) is that there isn’t a procedure or routine to get the answer. It involves guessing (a u). Educated guessing, but guessing none the less.
A few students have been freaking out — they just don’t “get it.” But I promise them that if they keep at it, it will work out. But they need patience, practice, and to reflect on each step of what they’re doing (why am I picking this particular u?). I’ve designed my lesson plans around these three elements: we’re going slowly, we’re doing tons of problems, and students need to be able to articulate some reason why they chose a specific u. I want them to know that this is a process they need to go through to come out better on the other side.
As Math Stories eloquently puts it:
Learning is not easy. If it was, we wouldn’t need schools and teachers. My kids frequently say that something I’ve taught them is easy. That’s once they’ve learned to do it – it’s the reward at the end of the process. While they’re learning it, they whine and complain and get headaches and have to use the bathroom and everything else imaginable. But that knowledge can only be integrated into their heads by experience, by wrestling with the fundamentals, by trying it out and repeating it and seeing how it works together with other things they know. They may eventually learn shortcuts that make things easier for them, leaving the methods they cut their teeth on in the dust. There are many teachers who will jump straight to the easiest methods, because that’s how the students will end up having to apply it in the real world. But jumping ahead, providing easy technological solutions for things they would otherwise struggle with, is just robbing them of an opportunity to really really learn.
Except for that one unnecessary word (“technological”), this quotation is exactly how I feel now. I will probably show it to my kids too.