Relational or Instrumental Understanding, that’s the question

Posted by France on Mar 1, 2016

The conundrum between success and failure could be down to this rhetoric question; has the mathematics ‘sold’ to us been based on relational or instrumental understanding? However, for parents and students, success can be measured by the grades achieved in final exams, irrespective of whether their understanding was relational or instrumental. Both Meyer and Skemp portrayed the past/current Maths curriculum as Dr Jekyll and Mr Hyde; two completely opposite characters living in the same body, thus creating a monster for those who perhaps would have been great at maths, but couldn’t cope with the method presented to them.

Meyer and Skemp warned us about our teaching method which could be compared to Mcdonald’s; you get what you want in a short period without taking into account of the long-term consequences. According to Skemp, “all they (students) want is some kind of rule for getting the answer. As soon as this is reached, they latch on to it and ignore the rest”. The danger with the instrumental understanding is that it gives you rules without reason, the inconvenience is when the reasoning aspects becomes relevant, anyone exposed to the ‘instrumental’ method of teaching would feel vulnerable, unproductive and disoriented. Meyer recommended that our method of teaching could be more realistic and using materials/resources that are relevant to students (cameras, mobile phones, etc.); “maths makes sense of the world and the vocabulary for our own intuition”

With regards to relational or instrumental understanding, the fact is more than half a million families in England will find out on Tuesday about their children’s secondary school places for this autumn and for any parent, this is serious business. So, success has some predominance. However, in recent years particularly in the UK, the Ark academies, the Harris Federation, Primary Advantage as well as numerous state, free, and independent schools are following the Singaporean Maths Mastery Programme which literally increases students mathematical ability and confidence without having to resort to memorising procedures to pass tests – making mathematics more engaging and interesting. When I read Skemp’s article I thought he was just crying in the desert, that the current method of teaching is suffering from Dissociative Identity Disorder, but it seems that things are moving in the right direction.

I believe that there is an amazing lesson from the animated part, Good Vibrations, the old man could relate to the vibrating paving slabs due to his illness called essential tremor, while others stumble and fall to the amusement of those working in the opposite building. Suppose the old man represent a teacher who teaches both methods and that all those who stumbled on the vibrating slab represent the instrumental understanding. The change in that specific environment had no effect on him as he was used to living in a shaky world, the vibration had a positive effect (- x – = +) on him, while a slight change had a negative effect on other pedestrians who didn’t even warn the others about the hazard. For wisdom sake, both methods are necessary.


  1. Charli
    2 March 2016

    I can’t believe I missed the link you have so beautifully articulated with the animation- thank you for mentioning this and helping me to understand the link. I love that you have brought in other eral life topical issues such as the Maths Mastery Programme. Your analogy with the short term sustainability of fast food as a comparison to the short term benefits of teaching an instrumental understanding. Do you think that there are practicalities of using multi-media would work with all current teachers?

    • fbontemps
      2 March 2016

      No I don’t think it would work with all teachers, perhaps knowing the students’ strengths and weaknesses could be a determinant factor in choosing which methods/resources/materials to use. I’ve just noticed that the topics that I’m least confident, i tend to rely more on the optical aspect. Instrumental understanding though it can achieve its intended role, is based on short-termism reminding me of the Bankers’s big mistake which contributed to the financial crisis in 2007.

  2. aw677
    2 March 2016

    I really enjoyed reading your article France. I love your Jekyll and Hyde analogy it really encapsulates the issues that a change in teaching methods could create. I also really enjoyed your interpretation of the animation at the end of Meyer’s talk, I think your observations are very perceptive and conclusion very apt. Thank you!

  3. Havva
    7 March 2016

    Hi France, I really enjoyed reading your think piece. I totally agree that it seems essential to have both relational and instrumental understanding – there is no harm in learning or teaching both methods.

  4. pepsmccrea
    8 March 2016

    Great to see lots of deep thinking going on here. You are grappling with the ideas presented and trying to make sense of them in our current context – including important links to Mastery Maths movement. Keep up the good writing!

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