Task 2 – The rules are confusing

Posted by Ray on Feb 28, 2016

The video and article were very relevant, and well timed, for me because they link directly to my recent experiences in university and in observing lessons in school. We are benefiting from relational* learning at university and are mostly observing instrumental** teaching in school.

Last week, at Hastings Academy, I watched real life examples of problems that can be caused by using an instrumental teaching approach. We observed 2 lessons that used this method, by different teachers, and both of them had issues. I was particularly surprised by a boy in a year 10 algebra class. He quickly progressed, ahead of most of the class, from solving simple problems ( such as a + 4 = 15 ) through a group of slightly harder ones ( such as 4a + 3 = 2a + 9 ) and then onto ones that involved brackets ( such as 3 ( a – 2 ) = a + 4 ). After some independent work time, the class was brought back together to work on the slightly harder category of question. The boy found it impossible to switch back from the “solving equations with brackets mode” and to find the “rule” he had used for the slightly harder questions. This demonstrated, to me, that he didn’t actually understand what he was doing and would struggle to solve this type of problem in the future. In the other example, the teacher gave a formula that she openly copied from a piece of paper and made no attempt to explain where it came from.

The video had some great ideas for a relational teaching approach to maths lessons. Let the class explore ideas of how to solve a real problem, such as steepest part of a ski chair lift or time to fill a water tank, and to identify for themselves the information they would enable them to calculate an answer. It seems so much better than leading the student like sheep through a prescribed step of simple operations that they are less likely to learn from.

The article*** gave a balanced view of the pros and cons of each teaching approach but the author admitted his clear preference for relational teaching. As someone new to the teaching profession, and keen on promoting a proper understanding of maths, it was interesting to hear an opinion that it would be easier for me to embrace a relational approach to teaching than someone with years of experience using an instrumental style. The challenge may be in convincing more senior staff that a potentially “radical” approach is appropriate in their school environment.

The most likely objection to adopting a relational teaching approach is that it initially takes longer to learn a topic and to therefore be assessed on it. The benefits really come in the longer term as it is easy to revise, or re-learn, something that you understood thoroughly before. It would be great, as a secondary maths teacher, if key stages 1 and 2 adopted a relational teaching approach.


Relational teaching is based on building an understanding of what to do and why.

** Instrumental teaching is based on learning rules without necessarily understanding the reasoning behind the rules.

*** I thought the musical analogy in this article was great and liked the fact that it aligned with the Jo Boaler views from task 1. It also amused me that the relational approach to music teaching used instruments and the instrumental approach didn’t, highlighting another point that was made about how language can be misleading.


  1. aw677
    1 March 2016

    An interesting piece Ray, I agree much of what we have seen in a real life situation has erred on the side of instrumental learning, but I do see sign’s that this is changing. The introduction of Maths Mastery immediately springs to mind. At this stage our experience of this fledgling idea is extremely limited and everyone is yet to see how successful this is in real life, but baby steps…. I also agree that if we want to adopt a new approach to learning Maths then perhaps this should be implemented from key stage 1, from the ground up so to speak.

  2. jat32
    4 March 2016

    Hi Ray
    I found your reflections on the classroom experience really interesting. I do wonder what motivates teachers to want to move to an instrumental teaching mode….is it easier, less time consuming, satisfies the pupils because they can answer the question? One of the lessons I was in at SLA was taught in a relational way and then when the learners could understand what was going on they developed the rule which they then used to apply to a variety of problems, almost like they had ‘earned’ the right to use the rule as they had got their understanding. But the pupils were asking why they had to go through all that thinking and not allowed just to apply the rule from the start!

  3. Fintan Donnellan
    5 March 2016

    Hi Ray! I know that I’ve argued in my own think-piece that there are some merits to the more traditional instrumental approach, but like you, I think we should move more towards a relational approach. I say ‘move towards’ because I don’t think we should completely cut ties with a system that has developed a love of math for some people (if only a few). However, if teachers can adopt an approach that will engage more students in the subject, then who can argue with that?! Teachers can be encouraged to this, but they need to have the freedom to do this and I don’t think that’s the case. I think Skemp mentions in his article that students are being taught too many different areas of math. Teachers have a very limited time frame to cover all these areas, so it’s no wonder that instrumental teaching and learning have become commonplace. I think we need to provide teachers with a curriculum that can accommodate relational understanding.
    By the way, that is funny that an instrumental approach to teaching doesn’t use instruments. I didn’t pick that up in my reading XD. Apologies for the extra long comment.

  4. pepsmccrea
    8 March 2016

    Lots of great thinking here. You have clearly taken on board the ideas and have made some strong links with your recent school experience. Good to see a balanced approach too, looking at the realities and the downsides of the ideas too.

    + nice point about instruments and instrumental!

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