"Thinking about the expression "rote learning has made me come to realise it is a bit oxy-moronic. For example, how many times did you have to learn 5 x 5 is 25? I would probably only once. When you were sat there in primary school with your abacus, block or what ever and you gathered in 5 lots of 5 or whatever you did to find it was 25 that was when you "learnt" it. Everything you after that is reinforcing the memory you made such as repetition of the times table or flicking the beads again".
I am thinking that someone who has learnt 5 x 5 = 25 might be stumped when asked to calculate 25 divided by 5. "Learning" something like this could mean anything from being able to repeat it faultlessly to being able to know it in lots of different ways and different contexts.
I know I have quoted these two things from John Holt before in other threads, so apologies to those for whom this is unnecessary repetition, but I have found these very helpful in thinking about doing and understanding.
But pieces of information like 7 x 8 = 56 are not isolated facts. They are parts of the landscape, the territory of numbers, and that person knows them best who sees most clearly how they fit into the landscape and all the other parts of it.
The mathematician knows, among many other things, that 7 x 8 = 56 is an illustration of the fact that products of odd and even integers are even, that 7 x 8 is the same as 14 x 4 or 28 x 2 or 56 x 1; that only these pairs of positive integers will give 56 as a product; that 7 x 8 is (8 x 8) – 8, or (7 x 7) + 7, or (15 x 4) – 4; and so on.
He also knows that 7 x 8 = 56 is a way of expressing in symbols a relationship that may take many forms in the world of real objects; thus he knows that a rectangle 8 units long and 7 units wide will have an area of 56 square units.
But the child who has learned to say like a parrot, “Seven times eight is fifty-six” knows nothing of its relation either to the real world or to the world of numbers. He has nothing but blind memory to help him. When memory fails, he is perfectly capable of saying that 7 x 5 = 23, or that 7 x 8 is smaller than 7 x 5, or larger than 7 x 10. Even when he knows 7 x 8, he may not know 8 x 7, he may say it is something quite different.
And when he remembers 7 x 8, he cannot use it. Given a rectangle of 7cm. by 8cm., and asked how many 1sq. cm. pieces he would need to cover it, he will over and over again cover the rectangle with square pieces and laboriously count them up, never seeing any connexion between his answer and the multiplication tables that he has memorized.
(From “How Children Fail” by John Holt, page 110.)
“I feel I understand something if and when I can do some, at least, of the following:
1. State it in my own words;
2. Give examples of it;
3. Recognise it in various guises and circumstances;
4. See connections between it and other facts or ideas;
5. Make use of it in various ways;
6. Foresee some of its consequences;
7. State its opposite or converse.”
(From “How Children Fail” by John Holt.)