Group Discussion on Skemp

We had a lovely discussion about Skemp's article in the garden the other day. Going through the questions really quickly...

1) Are the two kinds of understanding distinct/separable?

We pretty much all agreed at the outset that the two types of understanding are both distinct and separable. That being said, near the end of the discussion, there was talk of both types of understanding as lying on opposite sides of a spectrum. While I think there is some truth to this with regards to how one might approach teaching a math topic (more relationally, or more instrumentally), I believe these two types of understanding exist independently of each other, and what's more accurate to reality is that we all have varying degrees of each type of understanding depending on the topic.

2) Is there a "best" order to teach them?

At first, we were discussing how teaching instrumental math is a great way to introduce a topic, and then after students learn the "how to's" it's best to develop more relational ideas. However, we concluded that this is not necessarily always the case, and different topics probably lend themselves more to beginning with one approach over the other. Near the end of the discussion we briefly toyed with the possibility that we might be limiting ourselves in thinking in terms of only two types of understanding - very exciting.

3) What kind of activities promote each one?

This was the point we spent the least amount of time discussing... But if I remember correctly, we briefly touched on how traditional-style worksheets generally reinforce instrumental understanding, while discussions can be more conducive to promoting relational understanding.

4) How to assess understanding?

Similar to the thoughts on the previous question, we discussed how traditional forms of assessment (tests and worksheets, say) can be easily employed to assess instrumental understanding. On the other hand, relational understanding might be better assessed through discussion, or through some well-designed questions. For example, you might design a question for which a learned algorithmic approach would lead to an error. Probably if a student has only ever solved equations of the form x + a = b (which had maybe been taught/learned as "one-step equations"), they would need some relational understanding to solve even something like 10 - x = 4.

Shoutout to Andy, Jasmine, Jacob, Krystal, and Raymond for making it a positive space for discussion.