December 2022
What is my objection to the ‘small steps’ narrative that seems to be currently dominating KS1 and EY mathematics teaching?
Firstly, what am I hearing? There seems to be a focus on accumulating and developing skills and knowledge one lesson at a time, by taking children – all together – on a step-by-step journey through the material. And that by doing this, we are making learning more accessible for children and straightforward for us to teach.
“The way we communicate about the world shapes our experience of the world (..) This is particularly clear in mathematics because unlike most other subjects, mathematics is not about material objects but about abstractions and our only access to them are through language.” (Gíslason, 2022)
The beauty of mathematics is in its complexities and connections. All children need to experience this. Mathematics only becomes straightforward to teach if the focus is reduced to developing memorised skills and knowledge.
This step-by-step narrative seems to be leading to – sorry, but I am going to say it – quite ridiculous – demands on practitioners to produce learning progressions for all areas of the curriculum. The daftest one I have seen is how to teach children step-by-step to use a paint brush.
All learning is messy. Breaking complex mathematics into a lesson-by-lesson hierarchy implies that all children’s learning progresses in the same way and at the same rate.
It doesn’t.
It implies that every child is taking the same thing away from a lesson or mathematics session.
They are not.
Whilst mathematics content can be arranged into a broad hierarchy, and some things depend on us knowing some, earlier, ideas, this doesn’t mean presenting material hierarchically is the most effective way to either learn or to teach mathematics (Coles and Sinclair, 2022).
What is the alternative?
We flip things. Whilst we might have a rich problem up our sleeves to end a teaching sequence, try starting a sequence with an open task, supporting children in their thinking. This provides a valuable opportunity for us to assess the knowledge and understanding different children bring to the situation, and for the children to enjoy sharing and building on others’ ideas. We can build effective teaching upon these observations. Starting with the teaching often leads to us having to ‘mop up’ those who don’t understand, or results in us moving so slowly that it is difficult to retain engagement (Kuchmann 2017).
We vary things. I have no problem with all children working on the same mathematical idea at the same time. If this is approached sensitively, I have witnessed it as a powerful way to build a strong learning community enjoying mathematics together (1). But these moments are not the entirety of the mathematical experiences we offer our children and they need to include children’s interests and explorations as well as our own, along with a real recognition that independent thought is both valuable and important.
We embrace complexity. Finding things difficult at times is obviously bound to happen and it is important to experience such difficulties mathematically. The EYFS statutory Characteristics of Effective Teaching and Learning make it clear how important it is to build resilience in the face of difficulties: https://birthto5matters.org.uk/overview-characteristics-of-effective-learning-and-areas-of-learning-and-development/. Smoothing over likely bumps is not teaching our children how to tackle mathematics. Together reviewing how different children have tackled some maths and giving them time to have another go is crucial in helping our children grown into confident, capable mathematicians.
We familiarise ourselves with learning trajectories. Developmentally researched learning trajectories for mathematics help us become aware of the possible directions our teaching can take. Two good examples are those outlined in Birth to Five Matters: https://birthto5matters.org.uk, and those developed comprehensively by Julie Sarama and Doug Clements here: https://www.learningtrajectories.org. Rather than describing a series of small steps for children to be walked through, these learning trajectories are written for adults, to become knowledgeable about how different areas of mathematics develop. This makes us better able to respond effectively in the moment.
Along with every other adult working mathematically with children, I realise it is essential that we build solid foundations on which to build the tall structure of mathematics. I am differing over how we might go about this. Presenting mathematics as it is, in all its rich complexity, even for our youngest children, makes my teaching focus on what the children know, on their natural powers, on the connections between different mathematical ideas, and on supporting children in the key mathematical process of inquiry. Children get to experience the breadth and depth of mathematics.
For ideas of how to flip this narrative, and for your young children to experience the breadth, depth and richness of mathematics, please take a look at the tasks described in my blogs, and at the NRich website https://nrich.maths.org/early-years . It is perfectly possible to sandwich teaching with exploration.
Enjoy your mathematical moments with your children.
Footnote
1. I describe some of these collaborative whole-group ideas in my book ‘Playful Mathematics for Children 3 to 7’. London: Sage https://blackwells.co.uk/bookshop/product/9781529755152?gC=5a105e8b&gclid=Cj0KCQiA14WdBhD8ARIsANao07jstSqk8SBULtdzeiKpiHRhIfaCHUL59bS6Ojj-7FwrseuF3Tr6TlcaAvQAEALw_wcB
References
Coles, A. and Sinclair, N. (2022). I Can’t Do Maths! Why children say it and how to make a difference. London: Bloomsbury
Gíslason, I. (2022). Book Review of ‘Classroom research on mathematics and language: seeing learners and teachers differently.’ Edited by Núria Planas, Candia Morgan and Marcus Schütte, Abingdon, Routledge (2021). https://www.tandfonline.com/doi/abs/10.1080/14794802.2021.1993981?journalCode=rrme20
Kuchmann, D. (2017). ‘Small steps on the long road to mastery.’ Mathematics Teaching 257. Derby: ATM
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