Archive for December 2010
“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 if 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…..
I was just having a conversation with two sixth grade teachers this morning and the topic of long division came up. We had been talking about multiplication and how it would help children if we said, “groups of” instead of “times” when discussing multiplication. I made the comment that the same could be said for division, since multiplication and division are the inverse of each other. One of the teachers told me that she found it difficult to know exactly how to phrase a division problem when talking to her class. She wondered, in the problem 108 divided by 8 if she should say, “108 divided into 8 groups” or “108 divided into groups of 8″.
I think the confusion about division comes from the fact that there truly are two types of division but you don’t realize it in the “naked number” problem 108 divided by 8. It becomes apparent however, when we put the division into context. For example: We could be saying that 108 people will be divided onto 8 teams, which is a division partitive problem. When given this problem a child who is a direct modeler will make 8 groups and fair share out the 456 people onto those teams. In that instance we have divided the 108 people into 8 groups.
However, if the problem was 108 cookies were being distributed to people in packages containing 8 cookies and we were trying to determine how many people we could feed, then we have a division measurement problem. In that case, a child who is a direct modeler would take the 108 cookies and break them into groups of 8 and count the groups. Either way, you would arrive at the same answer, but the actions would be different.
The type of division problem that can be used to help students with base ten understanding is the division measurement problem. When the divisor is ten, or a power of ten, students can remove groups of ten from the dividend.
Just “Sum”thing to think about…
Some of the things that have been long accepted in the mathematics classroom, need to GO! They do nothing to help a student understand mathematics from a conceptual viewpoint. In this series, I will be talking about some of my pet peeves and things I would have outlawed if I was the “Queen of the Elementary/Middle School Math Universe”. (This is an imaginary title. Only a twisted math teacher would come up with or desire said title!)
In my opinion, the word “times” should no longer be used when teaching multiplication. It is eloquent when used in this opening sentence, “It was the best of times, it was the worst of times….”. It works well as the title for one of the world’s largest daily publications, “The New York Times”. It even has a place in the math classroom when we are teaching about its passage or how to read a clock. However, the word, “times” when it relates to multiplication, does not reflect any specific action and needs to be replaced with the phrase “groups of“. Students, especially those in the primary grades, are perfectly capable of multiplying, but are usually direct modelers. It helps them learn if they have an action to model. If we tell them they are putting candy into five groups of four, they can tell you that there would be 20 pieces of candy. If we tell them to take five times four, they have no idea what we are asking them to do.
The same can be said for division. If we ask students to take the 20 pieces of candy and break them into groups of four to share with their friends and we want to know how many children will receive candy, young children can model that action. However, if we merely ask them to do the problem 20 divided by four, we may or may not get the correct result.
Just “Sum”thing to think about!