“Don’t ask why, just invert and multiply”, talk about an arcane procedure. Maybe it’s just because I’m old, but I really can remember this being said in the math classroom. I’m not sure teachers actually use the phrase anymore, and I hope not, but I know they use the procedure.
The saying itself insinuates that one has no need to understand the rationale for dividing a fraction or dividing with fractions in the first place; one merely has to perform a trick to find the correct answer. The problem is, with no understanding, a student has no idea if the answer they compute is reasonable. Some students are even confused by the procedure itself and invert the wrong number in the problem. Again, they have no way to know whether their answer makes sense. The other misconception this creates for students is that they can change the operation in a problem for no apparent reason.
This is another one of the things I think needs to change in the math classroom. I have made it a mission to help teachers understand the concept of dividing fractions in order to help their students do the same. The first step in the process is to help teachers understand a real-world application for division of fractions. I do this by giving them the “naked number” problem 3 1/2 divided by 3/8 and asking them to quickly estimate an answer, write it on a sticky note, and put it in their pocket (where it will no doubt be washed and disintegrate in their washing machine and no one will ever see what they wrote). I never allow much time for this because most adults have the natural inclination to compute the answer using the invert-and-multiply procedure rather than to estimate. This signals a lack of benchmarking or estimation skill when it comes to fractions.
I then ask teachers to think of a real-world reason that could be solved with this fraction problem. The next step in the process is to have them illustrate the problem and its solution. Upon examining the steps taken to solve the problem, one discovers the fact that it was never necessary to invert anything.
So, if you’re still reading this post and you’re up for a challenge, try it for yourself. Take less than ten seconds (really, no more than ten seconds) and estimate an answer to the problem 3 1/2 divided by 3/8. Then think of a reason you would ever need to divide 3 1/2 by 3/8 in the first place? And what would it look like if you did? In my next post, I’ll reveal my problem, the illustration, and my rationale for banishing inverting and multiplying from the math class. I might even explain why inverting and multiplying works in the first place.
Just “Sum”thing to do in your spare time…..