I suspect that EHK had not been exposed enough to the word problems that would connect the formal addition to something familiar to a child, would give her some informal ways of thinking about addition, rather than in terms of counting to 7 and then continue counting 5 more times to get to 7+5. Such a sipmle problem as “there are 2 baskets of apples, 5 apples in one and 7 apples in the other, how many apples are in two baskets together?” would make clear that addition is commutative. Thinking about addition in terms of adding lengths (like putting one stool on top of the other, how high the top of the stool on top will be?) would also help.

My diagnosis: too many topics and factoids taught in isolation, too much mechaniccal drilling, too few informal explanations, too few connections with other topics and subjects, too few word problems. After all, the bulk of our brain is made of connections, that’s what makes us smart. Teaching should be easy on facts and heavy on connections between the facts, on explanations of these facts, on letting students think about what they learn and letting them find their own explanations and connections, on figuring things out. The most interesting thing about something new is usually not the fact itself, but some innovative way to get to it, some novel way to figure it out. Unfortunately it’s not what is discussed in most of the articles and textbooks, especially in mathematics, the true springs of discovery and understanding are obscured by dry formalism and heavy terminology.

And now we are wondering why education, the way it is, doesn’t work so well. It’s like cutting out a big part of a frog’s brain and then wondering why it doesn’t jump. Calculus, by the way, is a prime example of this phenomenon.

To EHK: I see I guessed it right.

At one time of my life I was involved in running maths competitions for children and the selection procedure for FMSh, Physics and Matheatics School at Novosibirsk University (a preparatory boarding school). There was a whole genre of interview problems, designed and selected to pick particular traits of mathematical thinking in the interviewees. There was an interesting subgenre of problems on “hidden counting”.

For example: a rectangle 19 by 99 is divided, by lines parallel to its sides, into equal squares. How many of these squares does a diagonal of the rectangle intersect?

To solve this problem, it could be useful remember that “to count” means, in the initial meaning of this world, “to count by ones”.

Elementary mathematics contains a number of hidden structures; usually they sit in ambush exactly in dangerous points where lower leve concepts and techniques are integrated into a higher level context, like counting by one into addition. Children normally jump over such stfethes of white water as salmons over a waterfal, by process of reification (you can find more on that in my book). For a teacher, it is not enough to understand psychological difficulties experienced by a child in reification, the teacher should also be able to understand mathematical difficulties involved and hidden dangers awaiting his/her pupils.

The situation you describe does crop up at university level though: I recall when I was an undergraduate going to my tutor for help on an algebra problem. He replied: “look, I’ve not done any algebra since I was a undergrad! you know much more about it than me”. It was even worse for a course on knot theory, most of which he was (understandably for someone who was an undergraduate in the 70s) completely unaware of.

To Mgccl: 4th grade calculus? What are they going to do with it? Isn’t it better to develop some problem solving skills?