Fascinating exchange with @Kris_Boulton about his post on simplifying area. This post is in part a response to that post, in part me trying to make sense of the relational and instrumental understanding debate.
Starting with the caveat that I don’t yet have any experience of teaching instrumental or relational understanding in the maths classroom, the exchange has stimulated lots of thoughts. Pulling them all together, I’m left with a few conclusions:
1. Relational understanding, whilst the ideal, is rarer and more difficult than Skemp implies.
…even relational mathematicians often use instrumental thinking.
This quote from Skemp sums it up for me – however valuable relational understanding is (and I do think it is very valuable), it just isn’t realistic for mathematicians in many day-to-day applications, be that as simple as working out the area of a trapezium or as complex as factorising quadratics that don’t have integer solutions. The instrumental approaches offered by the respective formulae are vital for mathematicians operating at all levels.
2. BUT in the case of area, a drive for simplicity actually leads me towards starting with relational understanding not instrumental understanding
In the simplifying area post Kris highlights an interesting approach to reduce the complexity of the formulae for area of four basic 2D shapes to three words: ‘length times height’. Keeping in mind the caveat above, it seems to me the approach works really well, and if anything promotes relational understanding more effectively than the four formulae.
But the point I want to focus on is whether it would be even simpler to start with relational understanding, and then teach the instrumental approach.* Perhaps its the naivete of the beginning teacher, but my instinct is to take an approach where students (who are confident in how to calculate the area of a rectangle) mark length and height on card rectangles, then cut them up to make the other three basic 2D shapes.
This approach seems engaging, manageable and most importantly gets at the ‘why’ of both the formulae for area and Kris’ simpler ‘length times height’. Subsequently teaching ‘length times height’ offers a handy shorthand for the relational approach, and is hopefully more meaningful as a result of previously exploring the relationship between each shape and the rectangle. Of course, revisiting and revising both the relational approach and the instrumental approach will aid both recall and understanding.
3. Inflexible knowledge is in a different category to instrumental understanding
Willingham makes a great case for inflexible knowledge – it seems clear that inflexible knowledge is a necessary step on the road to relational understanding. But it seems to me that inflexible knowledge is quite distinct from the knowledge that underpins instrumental understanding. Inflexible knowledge is narrow but meaningful. Instrumental understanding can be useful, may be quite broad, but isn’t meaningful in the way I understand the term. An example of the former might be knowing why the area of a rectangle is ‘length times height’, but not being able to use that to investigate the area of a triangle because I can’t visualise the relationship between the two shapes. An example of the latter might be using the ‘length times height’ rule to work out the area of a whole range of shapes, without understanding why it works.
As a result Willingham’s article isn’t, for me, a case for promoting instrumental understanding. It’s a case for promoting a particular, narrow form of relational knowledge.
Putting all this together my current thinking is that relational understanding is often tricky to promote. In some specific cases a mathematics teacher is justified in retreating from this ideal and instead fostering a narrower form of knowledge, at least initially. However, wherever possible this narrower form ought to be inflexible knowledge rather than instrumental knowledge.
I’m just beginning my teaching career and feel as though I’m stretching the limits of my own knowledge with this post – any comments on anything I’ve written would be really valuable for my own learning!
*As an aside, I am aware that this is almost exactly the approach that Kris describes as ‘doomed to failure’. Eek!