Most adults, including teachers, mistake showing students how to do a calculation with teaching them to understand math. An example of this appeared in the newspaper recently in an article mistakenly reporting how good a teaching practice was. It was taken from a program in some schools called “Number Literacy”. The concept behind Number Literacy is teaching kids to understand the concepts underlying numbers and symbols. This is a most laudable, and necessary, goal. But then the example given was one that simply taught a method for figuring out a particular answer. The method was not explanatory and not even all that useful, even when it is understood.
Fifth graders were asked to figure out what was one third of 48 dots on the board. (The dots were drawn laid out in rows of 10, making this particularly difficult, since you cannot manipulate the dots into an arrangement that is easier to see how to divide into three groups. Using objects, such as poker chips, small blocks, color tiles, or pebbles would make this much easier for students. ) One student got up and drew an outline around each of three groups of 16 dots, and said that 16 was one third of 48, which is correct. But many students did not know how he had figured that out, for seeing that an answer works is not necessarily to see how it is discovered. The teacher then showed her way to figure it out, by drawing circles around groups of three dots at a time and seeing how many circles like that there were in the set of 48 dots. It turned out, of course, there were 16 such groups of three dots. This was considered to be "explaining" division, and "explaining" how to find this answer. I want to say why that is not an explanation any child is likely to understand, even if they can use the method to get some answers. The method works to get an answer in simple cases and with relatively small numbers, but it doesn’t explain anything. It even makes it more confusing. Let me show why that is.
To begin with, take dividing a group of things by two – dividing them in half. If you ask a young child to divide a batch of M&M’s in half so that you and he can share them by each having the same amount, the child will typically start doling them out “one for you and one for me” and then “one for you and one for me” until he has divided them all between you and him. (If there are a lot of M&M's to begin with, some children will perhaps dole out two or four or five at a time.) Then if you ask how much you each have, he can count one of the piles. To make sure it actually worked, most children will count each of the two piles to see that it came out the way he thought it should.
Doling things out among the participants in this way is similar to dealing cards in a game where you have to deal out the whole deck as evenly as possible among the number of players. You keep doling (or dealing) out cards one at a time to each player until there are no more left or until there are not enough to make another full round. You don’t do what the teacher in the above lesson did which would be to divide the cards into piles, with each pile containing the same number of cards as the number of players in the game. For, say, four players her method would be to divide the deck of cards into groups of four cards per group, ending up with 13 piles of four cards each and that doesn’t do you any good. You need to end up with four piles of 13 cards each, not 13 piles of four cards each. And not only will it not work to separate cards that way, it is not obvious at all why four piles of thirteen cards gives you exactly the same number of cards as thirteen piles of four cards. It does, and adults know it does, but it is not easy to see why it does, not even for adults.
In the M&M's case above, no child and no adult would divide the M&M's into pairs and count how many pairs there are, because that is not an obvious or intuitive way to see how many M&M's there are in each of two groups. Similarly it is not obvious that when you are asked to find one third of 48, or to divide 48 into three equal groups, that you can do that by seeing how many groups of three things each there are. When you are asked to divide 48 into equal thirds, you are looking to end up with three equal groups. But if you divide up the 48 things into groups of three things each, you will end up with 16 equal groups. And 16 equal groups is not the same thing as 3 equal groups. So while the teacher's method gives an answer, it is not clear to students what that answer has to do with the original question. What the teacher says is true and it works, but it is a bad explanation because it does not explain anything, because it is counter-intuitive, and because it relies on knowing that division is in some sense the "opposite" of multiplication and that multiplication works numerically no matter which order you multiply the components. Adults have been taught that it does not matter in which order you multiply numbers – that 15 times 19 is the same as 19 times 15, and that is true, but it is not clear why it is true, and it does not mean that 15 bags of 19 things each is the same as 19 bags of 15 things each. It is the same total number of items, but it is not the same “thing”. If you invite 19 kids to a birthday party and want to give them each a bag with party treats in it, and you have a total of 285 pieces of candy, you had better make 19 bags of 15 each instead of 15 bags of 19 each.
[By the way, if you ask a child to divide something like Oreo cookies or poker chips evenly with you, many kids will use a different method -- making two equal height stacks. That is an interesting, and often true and useful, assumption on a child's part that there will be an equal number in equal heights. It would then be instructive to mix Oreo's from a package of regular ones and double-cream ones, since the double-cream ones will be thicker. Or one could mix regular M&M's and Peanut M&M's to let a child think about how to divide assortments of different things. This could lead to some interesting teaching and learning, some of which would stand the child in good stead later when he has to divide quantities like 12a's and 6b's in algebra, since that would be like dividing 12 regular and 6 peanut M&M's or 12 regular and 6 double-cream Oreos.]
The point of all this is that if you are truly trying to explain math to students (of any age) so that they really understand it, you cannot take shortcuts, you cannot assume that every true statement you make will have any explanatory value, and you cannot use just any form of representation of the numbers (such as immovable dots on paper instead of movable objects) . Unfortunately it is very difficult to realize when you are taking shortcuts or making assumptions that you really cannot explain. The above article in the paper is an excellent example of showing a method that really is likely to make no sense to a student, and that should make no sense to a student, but that looks like a good explanation to an adult. The reporter thought it really did explain division to kids and she really thought the kids understood the concept because they could repeat it.
But it is probably impossible for a teacher to explain the method itself
if a student were to be able to formulate the question “how can you find
out how many things are in three equal groups just by finding out instead
how many equal groups of three things there are?” The problem is
that most fifth graders would not be able to articulate that question.
And even if they did, most adults would not see what it is asking because
they don't see the difference between dividing by either number.
But being unable to articulate the question in the first place, most children
would only remain silent and figure there is something wrong with them
that they don’t understand and don’t even know what to ask. But that is
how students get totally lost by the time they get to algebra. At
best they would memorize the method, but the method is slow and cumbersome,
of little use, and no explanatory value. (Try using it to divide 12,333
by 3 or by 1/3.) Yet this is the kind of thing teachers and parents do
all the time in "explaining" math to children, unfortunately. And then
we wonder why children cannot do math. It is important to explain
math well to kids, but not every attempted explanation is a good one.
And some do more damage than would be done if they had not been given at