Task 2 – Patient Problem Solving

Posted by Martha on Mar 1, 2016

Relational and instrumental understanding are particularly relevant to maths teaching. Maths is seen by many as difficult and maybe even pointless. Perhaps this is because those people don’t have a relational understanding of many maths topics and fail to make the links between maths in the classroom and maths in ‘real’ life. The problem probably stems from some teachers only having procedural knowledge and therefore only passing on that part of their understanding to their students. That’s why I’m very glad I’m doing this course! Deepening my knowledge and relational understanding of maths and hopefully in the future passing that knowledge onto my students.

One problem with teaching for relational understanding is time constraints. Is there enough time in the 3 hours per week* to teach every topic so that students are learning relationally? Schools have policies where you have to cover certain amount topics in a given time**. They then have an end of term test on those topics, sometimes not coming back to them until much later, by which time they’ve probably forgotten it. So, in reality, wouldn’t teaching for relational understanding be quicker and more beneficial, as teachers wouldn’t have to keep re-teach things? I think it’s important for links within maths to be made as this will help students grasp concepts. Maybe some kind of assessment for relational understanding needs to be developed?

Instrumental understanding of math can see students getting good marks in exams (apart from the odd trick questions) but won’t have much benefit after the exams. Relational understanding will help children learn and apply their mathematical knowledge to real life situations, therefore becoming more useful. Helping people to realise that maths isn’t pointless! If you teach for relational understanding aren’t students likely to get a better mark anyway?

When I do my private tutoring, I’m getting paid on the basis that I’ll help the students with their maths (hopefully!). Their short term aim is to get a good GCSE in maths so they can go on to do whatever they wish. I went through SOHCAHTOA with one girl for a test she had and by the end of the session she could label a right angled triangle correctly and find a missing angle or length. Did she understand why? Probably not. Another girl was struggling with factorising quadratics, partly because she didn’t get the point and didn’t know what a quadratic was. I showed her what a quadratic function looked like and that by factorising you can solve for x. It then made much more sense to her.

Problem solving links back to the first think piece. Group work and asking questions and figuring things out are important – not being spoon fed information. If you work something out for yourself, you will have a greater understanding of it. It will be embedded deeper in your memory and easier to recall. A student gets 10/10 in a test, great! But what’s the point, if by the next week the student has forgotten what they’ve learnt or can’t link what they’ve learnt to other problems? I think text books are outdated, especially when we have so many other resources at our fingertips.

*In most schools I’ve been to students have 3, one hour maths lessons per week.

**Another problem with time is the amount of time you spend with a class before they leave school. In year 11, maths lessons are geared towards getting the GCSE. To strip back the students knowledge and make sure they really understood the basics wouldn’t be feasible. Where as if you start with a year 7 class you have more time to build up that relational understanding. I liked the idea of the maths mastery at Hastings Academy for the year 7’s.

– I just want to point out that I loved maths at school – and the way it was taught to me was probably very instrumental. I liked getting the answer right. It was encouraging and made me want to do more, so I don’t think all is lost when it comes to instrumental understanding. 


  1. aw677
    1 March 2016

    I really liked this article and agree with what you are saying at a fundamental level. I also loved maths at school and I was most definitely taught in a prescriptive, instrumental way – you are right, there is something very satisfying about getting a question correct. I guess at some level it is this that has to be addressed, we need to find a balance between relational understanding and the joy of just getting something right.

  2. jat32
    4 March 2016

    Hi Martha
    I enjoyed hearing about your reflections on your personal experience and agree that time is, for me, one of the biggest challenges to this approach. I also thought that you brought home the understandable aim of young people to want to pass/do well in their exams and can see that relational understanding at this point in an education career would just be met with just tell me how to get the answer right!

  3. fbontemps
    5 March 2016

    Thanks Martha, absolutely right instrumental understanding seems to get you the answer quickly and it gives more confidence too. Similarly, I am grateful for the opportunity to refresh, reassess as well as reevaluate my instrumental understanding during this course. I am now convinced that relational understanding is indispensable to achieve the goal of being a good teacher. Many thanks!!!

  4. pepsmccrea
    8 March 2016

    Interesting and balanced response. You make some important observations around testing, enjoyment and time. Good to see you making links with previous thinkpieces too.

Leave a Reply

You must be logged in to post a comment.