Mathematical voices

    Published: 27 May 2016

    Rachel Rayner is a primary mathematics adviser for Herts for Learning.

    How many times have you heard the following?

    ‘I only really understood maths once I started teaching it.’

    We all recognise the importance of subject knowledge in teaching any subject. Many of our schools in Hertfordshire have been engaging with our advisory team to discover what that really means for mathematics. Together, we have wrestled with the fact that the subject knowledge we were taught ourselves may not have translated into the deep conceptual knowledge and understanding we are committed to exploring alongside our charges. That leaves us with personal knowledge gaps that we need to fill.

    Who remembers the rule: ‘Yours is not to reason why, just invert and multiply,’ when dividing fractions? In training, we often ask delegates to wrestle with the understanding behind the rule. Most are amused and horrified to find that they could find the answer (if they remembered the rule); but not really explain or represent it. Furthermore, which of us really understood the different models and contexts for fractional development such as the discrete set model, the continuous model and the area model? How many of us were explicitly taught these structures ourselves and were allowed time to explore and compare them? So it is hardly surprising if we are finding the shift in landscape a challenge.

    This is in contrast to teachers from the often-lauded and high performing jurisdictions who, studies show, were more likely to have developed deep conceptual understanding in their own education which allows them to more easily plan to teach this with their pupils. Of course, as with anything in education, there are many other contributing factors affecting how we teach and how pupils learn mathematics.
    Currently, teachers are bombarded with information about how they should be teaching mathematics. But it is important to consider that without secure subject knowledge, this could be ineffective. The impact of the concrete, pictorial and abstract approach, for example, is compromised if the teacher is not sure which models and representations best exemplify the structure being explored.

    But amidst this, many of our Hertfordshire teachers are rising to this challenge and embracing the change. Take for example one teacher, Maria, working in Year 6 with a group of children who began the year with a range of needs and achievements. Working alongside her pupils and with the support of her subject leader, Maria has transformed the way she views teaching and learning of mathematics. How can we tell? Her pupils’ voices sing through the work in their books. Their understanding, their ways of seeing, their misconceptions are all available and celebrated.

    This is in distinct contrast to books where the child’s mathematical voice is absent. These are often only a replication of what the teacher has shown the pupils. And let’s not get started on the jumble of worksheets, stored in a folder, that are disconnected from the rest of the child’s journey. Maria’s increasing subject knowledge has allowed her to promote the experiences key to developing understanding.


    Y6 Book showing pupils representations

    Here, one year-six child’s journey in dividing fractions by whole numbers is exposed. Maria can see that he uses a sharing model to solve them having used Cuisenaire to support the concrete to pictorial stage. His own representations strengthen his conceptual understanding leading towards implementing the rule, understanding what happens to the numerator, the denominator, when and why.


    And listen to the learning that he grappled with here.


    Maria’s books are only one of many fantastic examples our advisers see that show how teachers are exploring their own subject knowledge alongside the pupils, acknowledging to themselves that, as ever, our job is to learn too. To our great delight and overwhelmingly, our teachers are reporting back to us that they are really enjoying teaching mathematics again now, and the pupils are too.

    Here at HfL, our advisory team are unwavering. Everything hinges on teachers having the right kind of subject knowledge, which will continue to be reflected in our training, support and blogs over the year.

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