" Based on what I’ve seen at primary level, I’m optimistic that teaching for mastery can have a major influence on our secondary maths teaching and learning. "
Thoughts on topical issues of mathematics education from the NCETM’s Director, Charlie Stripp
Teaching mathematics for mastery at secondary school
So far, the NCETM’s work relating to the teaching for mastery philosophy that my colleagues and I have been championing has focused mainly on primary school maths. It is being positively received and primary schools that are committing to the approach are reporting that, even at this early stage, the changes they have made to their practice are clearly having a beneficial impact.
My background is in secondary mathematics teaching and I believe strongly that a similar approach, suitably adapted, has the potential to support profound improvements in pupils’ learning in secondary maths. Over the coming months at the NCETM we will be focusing more of our work on secondary mathematics, whilst continuing to build on our primary maths work, and we will develop our understanding of how a teaching for mastery approach might be applied to help raise pupils’ achievement in secondary mathematics. The aims of the new curriculum and Maths GCSE are challenging and I believe teaching for mastery could be an effective way to achieve them.
This blog post gives my initial thoughts around mastery in the context of secondary maths teaching. As discussed in previous blog posts, a teaching for mastery approach in mathematics is underpinned by the belief that all pupils are capable of understanding and succeeding at school level mathematics. The approach involves the whole class accessing the curriculum together, with additional challenge provided through pupils accessing material at greater depth, rather than through acceleration to new material from later in the programmes of study: high attaining year 8 pupils shouldn’t be rushing onto to GCSE material, but instead should be going deeper into KS3 concepts. Carefully crafted lesson design and skilled questioning encourage deep mathematical thinking in all pupils, helping them to identify mathematical connections and steering them to develop mathematical reasoning and problem solving skills. Exercises and learning activities designed to provide intelligent practice enable pupils to develop conceptual understanding, at the same time as reinforcing their factual knowledge and procedural fluency. Damaging gaps in knowledge and understanding that impede progress are avoided by rapid identification and teacher intervention.
The American cognitive scientist, Daniel Willingham, has published an interesting article, Is It True That Some People Just Can’t Do Math? on teaching and learning mathematics. His work attempts to apply what we know about how our minds work to inform how we should teach, and his ideas are consistent with the teaching for mastery approach to mathematics outlined above: ‘Virtually everyone is capable of learning the numeracy content and skills required for good citizenship: an understanding of arithmetic procedures, algebra, geometry, and probability deep enough to allow application to problems in our daily lives’ and ‘For most topics, it does not make sense to teach concepts first or to teach procedures first; both should be taught in concert. Gaining knowledge and understanding of one supports comprehension of the other.’
In my previous blog post I highlighted how shallow, superficial learning can hold back even the highest attaining pupils when they make the transition from GCSE to A level in mathematics, and suggested that teaching for mastery could help prevent this, by ensuring pupils develop solid mathematics foundations, make connections and build a strong conceptual framework alongside procedural fluency. The damaging consequences of a superficial, exam-focused, rote learning approach to mathematics are highlighted in Jo Boaler’s book Experiencing School Mathematics, which examines in detail the very contrasting approaches to teaching mathematics in two English secondary schools. The book was published in the late 1990s, but it is still relevant to today’s secondary mathematics teaching, presenting interesting evidence about the effectiveness of ‘traditional’ and ‘progressive’ maths teaching, and also discussing gender issues, setting and ’mixed ability’ maths teaching – it’s definitely worth a look if you are not familiar with it. I think that a teaching for mastery approach is likely to prove an effective way to address many of the issues raised in this book because it requires that pupils engage actively with mathematical ideas and believes that all pupils can and should have access to deep learning in mathematics.
Another thought-provoking book relevant to secondary maths teaching is Malcolm Swan’s Collaborative Learning in Mathematics. Collaborative learning involves encouraging pupils to discuss carefully designed mathematical activities. These activities aim to expose misconceptions and encourage pupils to explain their thinking, challenging one another’s ideas and understanding – this paper provides a good summary. There is strong evidence that collaborative learning, focused on well-constructed tasks and activities, is very effective at engaging all pupils with mathematical thinking.
Andrew Blair, a practising mathematics teacher and regular contributor to online debate about maths education research, recently blogged a critique of the NCETM’s support for teaching maths for mastery. Based on my previous blog post, he asserts that the second aim (second in order in the curriculum document – the three aims of fluency, reasoning and problem solving are all equally important!) of the National Curriculum, namely to ‘Ensure all pupils reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language’ is not compatible with a mastery approach to mathematics teaching, and that teaching mathematics for mastery treats pupils as passive learners. I certainly did not intend to give this impression. As anyone who had the exciting opportunity to observe one of the Shanghai primary mathematics teachers’ lessons during the recent exchange visits will know, pupils are certainly not treated as passive learners. Instead, they are challenged to explain their answers and to make up their own examples of how to apply mathematical concepts. Exercises and tasks are designed so that pupils can engage with them through thinking critically about mathematical situations, actively developing their own understanding. I think many of the tasks on Andrew’s website Inquiry Maths, and the example he offered in his blog post, are excellent examples of tasks that engage pupils in mathematical reasoning and in developing their own conceptual understanding, whilst also reinforcing their factual knowledge and procedural fluency. These tasks could easily and successfully be used by a teacher committed to teaching for mastery.
I believe that carefully designed collaborative learning activities and inquiry tasks can be used as intelligent practice within the well-designed lessons that are central to a teaching for mastery approach to mathematics. However, I think a reflection from one of the Chinese maths teachers involved in the primary teacher exchange is especially relevant to their use and explains a key aspect of teaching for mastery (I paraphrase!):
The teacher holds the string of a kite and lets it unravel so that the pupils can go their own way to explore and make their own sense of the mathematics, but the teacher must also reel the kite back in on a regular basis, to ensure that all pupils have made the most valuable connections and developed correct understanding of the mathematical ideas.
The Chinese teacher then observed that teachers sometimes forget to reel the kite back in! For all pupils to develop the conceptual understanding of mathematical ideas required for sustainable learning, regular ‘reeling in’ is vital. This is a key feature of teaching for mastery and without it pupils may fail to make the key connections that are important for successful progress.
The NCETM will be working with the Maths Hubs on a secondary maths teacher exchange programme with Shanghai in the coming academic year, focusing on maths teaching at the KS2/3 transition. This should provide powerful evidence about how we can develop teaching for mastery at secondary level (I’ll have more to say about this in future blog posts, as the exchange progresses). Based on what I’ve seen at primary level, I’m optimistic that teaching for mastery can have a major influence on our secondary maths teaching and learning, enabling many more pupils to engage positively with maths, enjoy learning it and develop a sustainable understanding of mathematical ideas. This will prepare them well for using maths with confidence in their future life and work, as well as helping them to succeed in their Maths GCSE.
‘Is it True That Some People Just Can’t do Math?’ Daniel Willingham, American Educator, winter 2009-10
‘Experiencing School Mathematics’, Jo Boaler, 1997. ISBN 0335199623
‘Collaborative Learning in Mathematics,’ Malcolm Swan, 2006 ISBN 186201311X
If you find Daniel Willingham’s article interesting, you might also find his book ‘Why Don’t Students Like School’, 2009. ISBN 9780470591963 – a worthwhile read, though it is general, rather than maths specific.
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