Clyde says: > You are declaring that children learn arithmetic most naturally via the mathematics of *quantities" which have whole-number numerators. In that much you surely are correct. But much of your rhetoric and logic will be simplified if you rework it all through the context of quantities.
Robert responds: >This is false by definition. This would be like saying that children learn to read most naturally by reading prescriptions off medicine bottles. At the heart of arithmetic is number (not quantity) and operation. When we say 3 apples plus 4 apples, this is a physics problem, that involves math.
I agree with Clyde - children learn to count, and even to count perhaps in pure numbers. Its a game, with its own rhythm, like skipping rope. But expressions like "how many" beg for an object: "how many apples". Beginning arithmetic such as addition has always been greatly aided by grounding in real world objects. That's how children begin to understand expressions like 3 + 5; they relate it to a real-world situation, like "3 apples + 5 apples".
Multiplication again is aided by relating to real world situations. I have 12 cars that need 4 tires each. Note the asymmetry that is captured by the tradition terminology - 12 (cars) is the multiplier and 4 (tires each) the multilplicand (most natural mapping, though their are others.)
As learning progresses people get used to simply dealing with the expressions as entities in their own right without caring too much how they map. That's been called the "symbolic" level of understanding, I see it as a "meta" level.
I'd like to combine this with some observations about variables and language. In a prior post Robert Hansen question whether children can know the "significance" of variables like "x". I didn't quite know what to make of that position, but I'll guess he meant that they are not ready for contemplating algebraic expressions as entities in their own right. If so I agree, but I also agree with Clyde that they certainly can understand "length x width".
What are variables anyway? What is their significance?
(This is not about what to teach, but about what to keep in mind) One way of thinking about those questions is to get very basic and general. A variable can be viewed as any symbol that works by indicating a "to one" relationship lurking between values. So, for "length x width", we expect that in a context in which that can be made concrete, we have values: length->value1, and width->value2.
Note the similarity to words indicating roles, like mother or father. Children easily and intuitively use these words, and knows that in any concrete instance of having a mother we have mother->(Mrs. so-and-so).
One could argue that's really all that variables are - but by relating them to our normal and intuitive (hard-wired in some sense) use of language we see they are actually quite pervasive and absolutely essential to expressing thought. Children use "variables" all the time.
(Note we also have connections here to the modern conception of "function", Kirby's "dot" notation, etc.)
Does this way of looking at variables have anything to do with Clydes insistence that children learn via "quantities"? I think so.
- --- Robert Hansen says: >Also note, 3x2 is the multiplication of two numbers. The first factor (3) is labeled the multiplier, the second (2) the multiplicand and the result (6) the product. That is simply a matter of convention.
I don't think its "convention" at all - the traditional terms can be seen in a couple different ways where they make more sense: (1) in mapping to real world situations for simple multiplication type word problems, and (2) from the fact that in many situations the "multiplicand" need not be a pure integer at all. If we teach multiplication straight away as a symmetrical (commutative) operation between two numbers (which is not how the ancients thought of it) then those terms make little sense. What does not make sense for me, though, is the teaching multiplication devoid of such contexts where those terms are helpful.