Have you ever met anyone who attracted your attention because he had the right idea, but the more you got to know how he arrived at that idea, the less attracted you felt?
All our lives, we learn how important it is to be correct, to have the right answer. You gotta have the right answer to make good grades in school, to nail that interview, to be accepted by your peers and your families and your supervisors, .... But too many people think that an education is merely a sequence of milestones at which you demonstrate that you know the right answer. That view of education is unfortunate.
Here’s a little trick that will help me demonstrate. I’m sure you already know how to “cancel” factors in fractions, like I showed in my Filter Early post, to make division simpler. Like this:But, did you know that you can do this, too?
You never knew you could do that, did you?
Well, that’s because you can’t. Canceling the nines produces the right answer in this case: 95/19 is in fact 5/1. But the trick works only in a few special cases. It doesn’t, for example, work here:
Canceling digits like this is not a reliable technique for reducing fractions. (Here’s a puzzle for you. For how many two-digit number pairs will this digit-canceling trick work? What are they? How did you figure it out?)
The trick’s problem is precisely its lack of reliability. A process is reliable only if it works every time you use it. Incomplete reliability is the most insidious of vices. If you have a tool that never works, you learn quickly never to depend upon it, so it doesn’t hurt you too badly. But if you have a tool that works sometimes, then you can grow to trust it—which increases the stakes—and then it really hurts you when it fails.
Of course, you can make a partially reliable tool useful with some extra work. You can determine under what limited circumstances the tool is reliable, and under what circumstances it isn’t. Engineers do it all the time. Aluminum is structurally unreliable in certain temperature ranges, so when a part needs to operate in those ranges, they don’t build it out of aluminum. In some cases, a tool is so unreliable—like our cancel-the-digits trick—that you’re better off abandoning it entirely.
So, if your student (your child) were to compute 95/19 = 5/1 by using the unreliable cancel-the-digits method, should you mark the problem correctly solved? It’s the right answer; but in this case, the correctness of the answer is actually an unfortunate coincidence.
I say unfortunate, because any feedback that implies, “you can reduce fractions by canceling digits,” helps to create a defect in the student’s mind. It creates a bug—in the software sense—that he’ll need to fix later if he wants to function properly. That’s why showing your work is so important for students. How can someone evaluate your thinking if all you show is your final answer?
Being a good teacher requires many of the same skills as being a good software tester. It’s not just about whether the student can puke out the right answers, it’s whether the process in the student’s mind is reliable. For example, if a student is prone to believing in an unreliable trick like cancel-the-digits, then a test where all the problems submit nicely to that trick is a really bad test.
Likewise, being a good student requires many of the same skills as being a good software developer. It’s not just fitting your mind to the problems in the book; it’s exploring how the things you’re learning (both code path and data) can help you solve other problems, too. Being a good student means finding out “Why?” a lot. Why does this work? Does it always work? When does it not work?
Clear thinking is more important than the right answer. Certainly you want the right answer, but knowing how to find the right answer is far more important. It’s the difference between having a fish and knowing how to catch more.