## Tuesday, December 11, 2012

### A Must-Fix Misconception

I love the quote on the nrich.maths.org homepage (highlighted below)

I don't think it is any secret that division is the least understood of the four operations. There is often a lot of noise in the media (from people who have never had to teach math to elementary students) that students must learn long division. Right now (as I'm typing this) I'm trying to remember the last time that I had to do long division: it was way, way back. Most situations I can deal with using good number sense to get a good estimate. If an accurate answer is required, I might pull out a calculator (why bark yourself when you have a dog? as we say in Yorkshire). And if I have to do a bunch of such calculations, I'd use a spreadsheet.
But by focusing on long division, a more pressing concern is overlooked: students don't understand division. And learning the standard long division algorithm does not teach students (or adults) what division is. Students understanding of division should begin concretely by actually doing some division in the two most common situations: sharing and grouping.
For example, if these cubes represented 23 cookies:
and if these are shared between three people, then each person gets 7 cookies with two left over. This is a sharing situation:

If the cookies are to be put into packs of three then we will get 7 packs with two left over. This is a grouping situation:

Both situations, though different, are represented by the same number sentence (23 ÷ 3 = 7 with 2 remaining). Students must experience both sharing and grouping situations that involve remainders so that they develop a solid schema of division situations. Students can then develop from concrete representations of division to diagrammatic representation to symbolic representation. Not just jump straight in to long division.
There are also some other misconceptions that are ignored by focusing on long division. I asked some Grade 6s to complete the following two statements:
Multiplication makes numbers...
Division makes numbers ...

The whole class agreed that multiplication makes numbers bigger and that division makes numbers smaller. Rather than tell them that this was not necessarily true, I wrote down the following two questions:
8×10=…      8×2=…     8×1=...     8×0=…
After they gave me the answer to 8 times 1, I asked again "Does multiplication always make bigger?" Puzzled looks quickly gave way to smiles of recognition. "No, sometimes it makes the same!" was the agreement. After we agreed that 8 times zero is zero students were calling out "...and sometimes multiplication makes smaller." One student summarised it neatly thus:
"Multiplication can make bigger, stay the same or make smaller... it just depends on the number you multiply by."
I repeated this process for division using the following string:
10÷5=...      10÷2=...    10÷1=...    10÷½ =...
The class quickly agreed that division could also make bigger, stay the same, or make smaller depending on the number you divide by. Their huge misconception had been drawn out into the open and addressed. To emphasise why this is important, I should tell you that I have given this question to adults on many occasions:
½ x = 24
and sometimes as many as two-thirds of them have told me that the answer is 12. They go on to reason that to find x they have to do 24 divided by a half which gives 12 which makes sense since  division makes smaller!
We consolidated this learning by playing a game called Target. The Target game is very effective in getting students to understand that multiplying doesn’t always make a number bigger, and likewise, division doesn’t always make a number smaller. Students play this game in pairs with the use of one calculator. A starting number (e.g. 37) is chosen as is a target number (e.g. 100). The first player uses the calculator to multiply (only multiplication is allowed) the start number by any number to attempt to get the target number. If he or she gets 100 exactly or ‘100.something’ then that student wins. If not, the other student gets the calculator and tries to multiply the new number by another number to get the target number. Play continues this way until a winner is found. It is most engaging!

﻿