Schools are sometimes ridiculed for teaching different ways of doing multiplication. However, if this was so bad, then how can we teach applications of multiplication at all? Isn’t another method to do multiplication an easy application of a first method?
If a student can’t apply one method of multiplication to another, then how can he apply it to something else?
What allows us to apply multiplication or even addition to some application?
What allows it is that the application satisfies the laws of addition or multiplication so that it behaves the same way as addition or multiplication does.
So to understand that we can apply multiplication to a problem, we have to be able to show that the problem satisfies the properties of multiplication.
For whole numbers, this means showing that
n*0 = 0
n*m’ = n*m + n
For addition, we have to show that
x+0 = x
x+y’ = (x+y)’
For successor, we have to show that x’ satisfies the 5 Peano Axioms.
If we don’t show or even understand these proofs or that they are required, then an application is really applying rote learning.
One reason it is hard to apply one multiplication method to another or translate them, is that the definition of multiplication of natural numbers in terms of repeated addition is not formalized.
n*0 = 0
n*m’ = n+n*m
These two equations define multiplication in terms of addition. To apply this to area, we have to show that areas have this property, for whole lengths of the sides. This is a matter of intuitive geometric understanding at the earlier grades.
If we structure math ed using the Peano Axioms, we can understand why people can’t easily go from one method of multiplication to the next. The reason is they didn’t learn the definition of multiplication, or its properties. That is, they didn’t learn its structure. So they can’t apply the structure even to another method of multiplication, and recognize it has the same structure.
When we use the two equations to define multiplication, it is easy to verify or at least make plausible that another method of multiplication or an application such as area satisfy the same two rules.
This teaches abstraction and concepts. If this is not done for natural numbers, it is difficulty for the student to form any proper concept of fractions. Fractions are a mathematical structure with data and rules. The data are natural numbers and the rules are methods of manipulating natural numbers.
So if the understanding of structure of natural numbers is completely absent, fractions are learned by rote if at all.
We can tell if structure of natural numbers is absent by testing if the student can apply or relate one method of multiplication to another. If they can’t, they won’t understand the math structure of fractions either.
Rote learning experiences diminishing returns in math. As one ignores structure at one level, naturals, it becomes harder to learn the next level, fractions. After fractions, there are infinite decimals. There is also algebra. If showing two methods of multiplication have the same properties is not possible to the student, they will fail at algebra other than as rote learning. This shows up when they can’t do algebra word problems. They didn’t learn the structure of numbers let alone algebra, so they can’t translate a word problem to symbols.
Teaching math structure has to start with counting and relating counting to the 5 rules of counting by one, ie the Peano Axiom.