*"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[The whole article is at Republic of Mathematics. You might want to read it now.]andtheir teachersthe possession of such a rule, and ability to use it, was what they meant by ‘understanding’.

Suppose that a teacher reminds a class that the area of a rectangle is given byA=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.

Yes, yes, yes!! We see this every day as teachers if we stop kidding ourselves. We give them the formulas and tell them what to do, those who study to regurgitate get that shiny A on the test -- everyone is happy. Sam Shah recently wrote a post on asking kids to write about what they really understand relationally. So I'm reminded to make this happen more with my kids. But all this takes time. And isn't it worth the time?

ReplyDeleteThank you, Sue.

Yep. I just hope I can practice what I'm preaching.

ReplyDeleteGreat post! I agree completely, but sometimes the biggest resistance comes from students themselves. You have to actively and transparently deconstruct their meaning of the word 'understanding' and convince them to come with you down the path of understanding as you mean it and show them the benefit of this approach. But I really love the quote, "You have to slow down the teaching to speed up the learning." Something for me to work on, for sure.

ReplyDeleteYep. In

ReplyDeleteThe Teaching Gap, James Stigler wrote: The scripts for teaching in each country appear to rest on a relatively small and tacit set of core beliefs about the nature of the subject, about how students learn, and about the role that a teacher should play in the classroom." (page 87)Changing that script takes some serious work.