"When I use a word," Humpty Dumpty said in rather a scornful tone,
"it means just what I choose it to mean — neither more nor less."
If someone says they 'understand' something, what does that mean? Elementary teachers often think explaining why means giving a cute rhyming 'reason',* a shocking thought to me. And a stark reminder that we each have our own ideas about the meanings of the most basic words.
In 1976, Richard Skemp wrote Relational Understanding and Instrumental Understanding (Mathematics Teaching, 77, 20–26, 1976). Here's the heart of it:
Stieg Mellin-Olsen**, of Bergen University, suggests that there are in current use two meanings of the word 'understanding'. These he distinguishes by calling them ‘relational understanding’ and ‘instrumental understanding’. By the former is meant what I have always meant by understanding, and probably most readers of this article: knowing both what to do and why. Instrumental understanding I would until recently not have regarded as understanding at all. It is what I have in the past described as ‘rules without reasons’, without realising that for many pupils and their teachers the possession of such a rule, and ability to use it, was what they meant by ‘understanding’.[The whole article is at Republic of Mathematics. You might want to read it now.]
Suppose that a teacher reminds a class that the area of a rectangle is given by A=L×B. A pupil who has been away says he does not understand, so the teacher gives him an explanation along these lines. “The formula tells you that to get the area of a rectangle, you multiply the length by the breadth.” “Oh, I see,” says the child, and gets on with the exercise. If we were now to say to him (in effect) “You may think you understand, but you don’t really,” he would not agree. “Of course I do. Look; I’ve got all these answers right.” Nor would he be pleased at our devaluing of his achievement. And with his meaning of the word, he does understand.
So when I ask my students to show me, with thumbs 'up, down, or sideways' whether they understand, they may not be telling me what I think I'm asking. Yikes!
How do we teach for 'relational understanding'? John Golden had his math ed students read the article and respond. And in this excellent article, Grant Wiggins says we can't teach this sort of thing, we can only design an environment that helps the students come to it on their own.
No, there is no way around it. If you want students to have meaningful learning experiences that culminate in transferable insight and know-how, then you have to lose time to gain it. You have to slow down the teaching to speed up the learning.
* In the experience of a wise friend who works with many elementary teachers, trying to help them improve their mathematical understandings.
** Also of interest, The Politics of Mathematics Education, by Mellin-Olsen.