“Knowledge is produced within mathematics but policed within the mathematics curriculum,” says Dr Cathy Smith.
“How do people decide what is or isn’t maths? Largely through their school experiences.”
She is explaining Basil Bernstein’s work on how knowledge is formed, organised and reproduced; something that has particular resonance for current discussions in the edusphere about knowledge-rich curricula and teacher agency over curriculum decisions.
Is maths more like science or sociology? Smith suggests that, if we are convinced by Bernstein, it is in fact the latter.
Science is a hierarchical type of knowledge - when we learn new things, they tend to replace the old ones, overwriting existing ideas as new information comes to light.
Maths, says Bernstein, is a horizontal type of knowledge - new things add to what is already there without replacing them.
“Maths is made up of specialised languages like probability, algebra or graphs with which you can describe the world,” says Smith. “But one language doesn’t displace or disprove the other.”
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So what is the knowledge structure of mathematics, and how does that relate to teaching it?
“Maths has organising power,” she says. “It has a strong grammar, not like nouns, verbs and adjectives, but in the sense that it organises the world in relatively precise ways.”
One surprising consequence of this is that the way maths works, with its own evaluative rules contained inside it, could make it a particularly empowering subject.
“If students have access to the recognition rules (how do you know what is right and what is good?) and the realisation rules (how do you make maths for yourself?) of maths, then the teacher is not the only authority in the classroom - the students can also take some of that authority.”
Smith is senior lecturer in the School of Mathematics and Statistics at the Open University, and leads the mathematics education team there.
She’s an expert in maths teacher education - in particular, but not restricted to, post-16 teaching and learning.
“Post-16 mathematics is one of my passions,“ she says, smiling.
The bedrock of learning
In the podcast, we talk about the distinction between mathematical (subject) knowledge - what she calls the bedrock under the landscape - and knowledge of mathematical pedagogy, which is the trees, the grass, the more visible structures that are overlaid on the top.
“The maths part is knowledge of curriculum and how it fits together and the progressions; the pedagogy part is examples, useful language, misconceptions etc,” she says. “And it is so important to consider which of them you are foregrounding and how to pivot between them when helping teachers examine their own practice.”
We also talk about issues of control in the maths classroom; the arbitrary and the necessary in maths teaching; why it is important not to waffle; Stem (science, technology, engineering and maths) sentences; the reason behind Smith’s Twitter handle (@Itchymaths); implicit and explicit classroom rules; and the secret club of subject association members.
Plus, the usual jokes, suggestions for further reading, and some particularly thorny questions from Twitter…
Dr Cathy Smith mentions:
Lucy Rycroft-Smith mentions:
Lucy Rycroft-Smith produces the Mathmatips podcast for Tes
