Instrumental and relational understanding in maths

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.

Kris pointed me towards Skemp’s original article on relational and instrumental understanding, as well as Willingham’s article on the value of what Willingham calls inflexible knowledge.

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!

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Name the steps

Teach Like a Champion Technique 13: Name the steps

Subdivide complex skills into component parts and build knowledge up systematically.

  • Give each step a name so that it can easily be recalled.
  • Key components of naming the steps:
    • Identify the (fewer than 7) steps with clear and crisp language and post on the wall.
    • Make them sticky by naming them and perhaps creating a mnemonic.
    • Build the steps with students.
    • Use two stairways – discussing both the current problem and a generalised form (both the problem and the process of solving these sorts of problems).

Teaching Mathematics Ch 8: Reflections on mathematical investigations

Maths investigations clearly have huge benefits; the example in the book was of a Happy Numbers investigation. The open-ended approach not only allows for differentiated approaches, but also encourages mathematical thinking and the development of process skills.

In planning for an investigative task in the classroom, I clearly need to plan carefully the scope of the task, and also the scaffolding I will provide to learners who are stuck, or are struggling for ideas. For example, I might include a list of approaches that students could take to get started on the task, as well as a ‘hints’ table if they are stuck.

Assessing performance on a task like this clearly requires a more considered approach than simply ticking correct answers and giving feedback. I need to consider what approaches a pupil has taken (which may not be explicit – thus I probably need to award marks for clarity of presentation as well as outcomes) and how they might improve their approach. However, this seems to accord closely with good practice in assessment anyway, so I don’t think it should put me off!

A brilliant AfL resource

Courtesy of the Guardian Teacher Network, this Assessment For Learning Toolkit is wonderful. It gives over 70 ideas for how to do AfL well in the classroom. Highlights for me are:

  • [an idea I’ve adapted] having a section in the front of every pupils’ exercise book kept aside for comments on homework and a dialogue between me and my pupil. The section would also have short- and long-term target-setting in a chart form, so that I and the pupil can see progress.
  • assessing mid-unit, not at the end, so any gaps can be addressed.
  • having a question box in the classroom, where pupils can place questions at the end of the lesson, to be answered the following lesson.
  • increasing wait-time, and encouraging thinking aloud.
  • using exemplar work that meets learning objectives, so pupils can see the kind of thing they need to produce.
  • using two stars and a wish for peer assessment
  • encouraging pupils to articulate their thinking alone, in pairs, written down etc. before answering.
  • bouncing answers around the room e.g. Jim what do you think of Emily’s answer?
  • explicitly discuss the importance of and benefits of collaboration (in a CAME lesson?)
  • question pupils using the phrase ‘why is x an example of y?’
  • explicitly discussing what makes a piece of work good (in a CAME lesson?)
  • Using the question stems: Why does…?; What if…?; How would you…?; Could you explain…?; What might…?

These ideas give me a shiver down my spine! I can’t wait to use them for real…

Easy-Hard-General

Another way to generalise: by asking pupils to make up an easy example of a type of problem the lesson is focusing on, a hard example, and then a general example. Asking them to make up a hard example can reveal what it is they find hard about the problem. I’m not yet sure what a formulating a general example would look like – something to come back to I suppose.

Generalisation and Another and Another

Next is an article in the importance of generalisation. It argues that the ability to generalise is fundamental to maths, and is also often a source of its excitement. This needs to be an explicit focus of the lesson. One challenge for me as a beginning teacher is how to make that generalisation explicit – something to come back to in my PDP I think.

Next comes Another and Another. Asking students to come up with examples is an excellent way to make them think about a concept, but they may not challenge themselves much that way. So asking for another and another is a good way to challenge.

I imagine this will be a very useful way to differentiate – by inviting students to make up their own example, it allows them to choose their level of challenge. And by asking stronger students later, this will increase the challenge for them.