In this series of posts, we looked at various embodiments of multiplicative structure. Each involves ideas about multiplication. How are these ideas learned?
From reading this blog, you will have gathered that I mean the question “how is an idea learned?” in a particular way. I don’t mean how did the teacher tell them, and did they remember. I also don’t mean if they can apply the rule, or recite the rule. I mean something more along the following lines: if you observe them work on problems in the right problem domain, what ideas does their work reflect, and how do these ideas grow and develop over time? Is there a process by which a person gives up an old idea, an old way of thinking about something, and try out a new idea and how do they deal with the dislocation of it? Is there a sequence, a logic, by which these ideas become their own? Do they trust their ideas, and do they have a sense of the situations in which these ideas apply?
In that context, for example, I regard it as significant That I encounter lots of kids who don’t seem to have a strong and stable sense of what multiplication is but who are completely solid and resourceful in their idea of “double”. In other words, if it is common for kids to have a strong understanding of what “double” means and at the same time have a weak understanding of what multiplication is, then I think it useful to look at the notion of “double” as a building stone or as a predecessor idea for multiplication – not just in the sequence in which a modern child learns it, but perhaps “double” is a precursor for the very idea of multiplication in a historical sense as well.
So let’s look at the child’s idea of “double”. Though you and I know that “double” is related to multiplication, that in fact “double” is multiplication by two, the idea of “double” – if it precedes the idea of multiplication – is not based on that. Quite the contrary. We’re looking at “double” even as a precursor to “triple”, at least we won’t presume that a child who has the idea of “double’ is necessarily also fluent in the idea of “triple” or sees complete parallels between the two.
“Double” may be an action, or it may be a relationship. “Can you make me a pile of blocks that has double the number of blocks in this pile?” would be an example of an action; looking at a stack of papers and contemplating the relationship between the number of sheets in the stack and the number of sides (pages) in the stack would be an example of a relationship. There are lots of examples of that precise kind of relationship: the number of faces and the number of cheeks; the number of bunnies and the number of ears; the number of kids and the number of thumbs. Similar ones would be: the number of dimes and the number of nickels worth the same amount; the number of wheels you’d need to make bicycles from a given number of bicycle frames.
On the left, in the picture, we use a box to depict “double” as an action: we imagine a number going in, and another number coming out, somewhat like a Coke machine. In a Coke machine, coins go into the slot, and a can of Coke comes out at the other end. On the right, we show a table to depict the relationship between frames and wheels. The number of frames on the left corresponds to a number of wheels on the right.
Kids often think of “double” as addition. To get the number of wheels, you can skip-count by 2. Or you can put down a number of wheels that matches the number of frames, and then do that again.
If each person gets two cookies, you can act that out by giving the first person two cookies, then the next person two cookies, and so on; or you can give each person a cookie, and after each person has had their first cookie you can come back and give them another cookie in the second round. These scenarios are different; yet each involves a joining or adding of piles of cookies.
Though I haven’t adduced any real evidence, I do think there’s truth in saying that kids typically understand ‘double’ before they understand other kinds of multiplication; and I think the same principle applies even stronger with regard to ‘half’ and other kinds of division. My earlier post on percentages gives clear evidence for the latter.
I think adults, too, often relate to certain multiplicative structures as addition. In the USA it is customary, and in some cases enforced by law, to pay double the usual rate when a worker works overtime on weekends; at other times, pay is described as “time and a half”. This formulation for a multiplier of 1.5 reflects the way people usually think of it, which is, that they get paid extra, and that the extra pay is for half the hours worked overtime. Similarly, we talk about “amount plus tax”, and for good reason, where the relationship really is multiplicative: amount times 1.1. And where tips are customarily given as a percentage of the bill, e.g. 15%, we talk about “amount plus tip”, and again the total amount we are out of pocket is related to the amount of the bill by multiplication: amount times 1.15.
Though I think the case can be made that each of these situations is thought of as an addition, that is not the case I want to make here. The case I want to make is a simpler one, that each of these situations (even if thought of as a multiplicative relationship) is thought of as distinct. Like “double”, they are as a box with one input and one output. Sales tax: amount-without-tax goes in, amount with tax comes out. Tipping: amount-before-tipping goes in, amount-after-tipping comes out. Overtime: Amount-earned-without-regard-for-overtime goes in, amount-actually-received comes out. A similar thing (though we haven’t used the example before) for discounts: for a 35% discount, there is a box where the amount-before-the-discount goes in, and the amount-after-the-discount comes out.
Each of these boxes, with a single input and a single output, looks different from a multiplication box, which would have two inputs and one output. This is worth looking at in more detail in a later post.