[I transfer this post from my old blog.]
John Baldwin writes:
The associative law can only work on two applications of the plus sign. We generalize it to say we can regroup any sequences of additions. It is quite plausible that this is a distinction one should not make for teachers or at least the question should be at what level you want them to be aware of it. The second problem is that when minus signs are interspersed this gets more complicated. Since associativity fails for subtraction some further rules are required.
Indeed, notice that (3 -2) – 1 is not the same as 3 – (2-1). Subtraction is not associative!
This is a classical example of a “fuzziness” in mathematics teaching, a phenomen especially noticeable at earlier stages of school education. Very frequently, children are placed in position when they have to figure out rules which have not been made to them explicit. Of course, children learn the grammar of their mother tongue exactly that way, by absorption. Unfortunately, by the time they are taught mathematics, their natural ability to extract grammar rules from adult’s speech is already significantly suppressed.
A few days after I wrote the above text, I had to correct a student in my (university) mathematics class who was trying to apply the associativity law to conditional statements in Propositional Logic and write
p –> (q –> r) = (p –> q) –> r
–which is of course wrong, and exactly for the same reason why (3 -2) – 1 is not the same as 3 – (2-1).
This is a very delicate point indeed. How explicit should we be in formulating formal mathematical rules in (early) mathematics teaching? In teacher’s training?
I am more and more inclined to think that, in mathematics teaching, the famous quote from John von Neumann
In mathematics you don’t understand things. You just get used to them.
can be usefully reformulated:
In (early) mathematics teaching, you do not explain the rules. You just follow them, and let your students get used to that.
Admitedly, it is a contentious thesis. Peter McB in his comment to my previous post, Mathematics of Finger-Pointing, takes the position which is very different from John Baldwin’s. I quote Peter McB:
The examples of student “errors” shown on the post you linked to are, in fact, examples of the medieval-craft nature of pure mathematics, as noted 30 years ago by the computer scientist, Edsger Dikstra. The discipline still has no systematic, agreed, industrial-strength, notion of semantics: everything is still ad hoc. For instance, in some cases it is correct to cancel the symbol “n” above and below a fraction line; in other cases (when, for example, the symbol “n” is embedded inside a “sine” function), it is not. Why there is difference here is never explained, but bright students somehow master it implicitly.
As you can see, I am not in total agreement with Peter McB and will try to list some arguments in support of my thesis:
- (Young) children are exceptionally good at picking up implicit rules if adults strictly follow these rulles.
- Sometimes, formally stated rules could only confuse children.
- On the other hand, ability to explicate informal or undisclosed rules is one of the most important mathematical skills — and children should be trained in that.
- In the classroom, mathematical rigour is not children’s responsibility; it should manifest itself in the teacher’s self-discipline and consistency in following undisclosed (for a time being) rules.
- Therefore (and I differ on this point from John Baldwin) teachers, in the ideal world, should have very explicit understanding of the true nature of mathematical conventions.
I would be most happy to hear the readers’ opinions on the matter. Meanwhile, I am thinking about some further posts where I will try to give some examples of “leading by example” in teaching. The first of these posts, however, is likely to be about the first of famous “3 Rs”: Reading.