The subtext of the new maths curriculum may be a return to drill and practice, but only if we choose to interpret it that way, says Mike Askew...
“A high-quality mathematics education therefore provides a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject.” (NC p.99)
Thus England’s new national curriculum sets out its overarching aims, aims dear to the heart of anyone who believes maths can enrich learners’ lives and is about more than international test scores.
The introduction to the new curriculum emphasises the need for striking a balance between fluency, reasoning mathematically and solving problems – a sentiment readers of this column will know I support.
Qualities of disposition and engagement are also emphasised in the introductory remarks. These include encouraging learners to persevere in seeking solutions, move fluently between different representations of mathematical concepts, and express their thinking to themselves and others. All well and good and encouraging.
But of course it is the more frequently referenced programme of study, not its introduction, that shapes what happens in the classroom. So does this follow through on these laudable aims and directions? A quick read through would indicate it does not – there’s much talk of ‘practice’, ‘memory’ and ‘columnar’ methods (what was wrong with ‘algorithm’?). For example, in KS1 the point is made that ‘An emphasis on practice at this early stage will aid fluency’ (NC p.101), while by the end of year 4 pupils ‘should have memorised their multiplication tables up to and including the 12 multiplication table’ (NC p.113) and upper key stage 2 pupils ‘practice…formal methods of long and short division’ (NC p. 136).
It is easy to read such devilish details as a ‘back to basics’ curriculum – a return to learning mathematics through drill and practice (death by a thousand worksheets) with memory being the overarching means of success. But there are other ways to read the curriculum.
The writer Roland Barthes suggests we can approach texts such as the National Curriculum from a ‘readerly’ perspective, which means trying to extract from the text the writer’s original meaning. From a readerly perspective, the curriculum might be seen as revealing the government’s intention to promote an old-fashioned style of teaching based on ‘training’ learners to carry out procedures and commit these to memory through extensive practice.
Barthes also talks about a ‘writerly’ approach to engaging with texts. Writers generally accept that, once something is committed to paper and sent out into the world, they have little control over how the reader makes sense of the text. Texts, from a writerly perspective, are open to a variety of interpretations. Even if, in the mind of the writer, ‘practice’ means working through pages and pages of examples, the reader can still interpret the word in a variety of ways. Learners can practice through games; through brief, rapid, whole-class questions and answers; through playing with mathematical ideas and constructing their own examples; or through working on rich problems and enquiries that embed practice – the list could go on.
Taking a writerly stance towards the National Curriculum means we do not have to assume the most immediate and obvious meaning of what is written. Whatever the intent, there is room for a variety of interpretations.
Of course, different interpretations could give rise to flashpoints – areas of disagreement between folks who read the curriculum in different ways (based on different values). Take, for example, ‘Solve one-step problems involving multiplication and division’. Do we take this to mean an emphasis on word problems, which are basically calculations dressed up in a context? Or could it mean the type of problem that gives rise to a rich conversation about where fractions come from, as opposed to simply getting the correct answer, e.g. Six children equally shared four bars of chocolate. How much chocolate did they each get?
Educational theories are like buses. Don’t worry if you miss one, another will be along soon.
Theories have come into fashion and gone out again – discovery learning and behaviourist training to name but two. Other theories hang around like guests who don’t know the party is over – it’s generally accepted there is not a shred of evidence for different ‘learning styles’, yet talk of these persists (This website has links to many studies about the myths around learning styles theories: http://goo.gl/s5QVt).
Some theories are helpful in challenging how we think, but are less useful in directing practice. Gardner’s theory of multiple intelligences (how many are we now up to 7? 8? 9?) successfully replaced views of a central, overarching, general intelligence, but the teaching implications arising from the theory are not clear.
Don’t get me wrong, in my role as an academic I like a good theory (I’m most influenced by the work of Vygotsky) and sometimes I stumble across one that looks like it might have some use (theories are probably more like iPads than buses – great to play with for the moment, until the next, more sophisticated version comes along).
One such theory is ‘successful intelligence’, which Sternberg, its originator, argues rests on three core intelligences: creative, practical, and analytical. We all, according to Sternberg, have these three intelligences, but the balance is different across the population.
What, you may be asking, has this got to do with the new national curriculum for mathematics? Well, Sternberg’s empirical studies have looked at how learners in parallel, matched classes respond to different types of teaching. He set up an experiment where, in one class, teaching focused on learners committing knowledge to memory through the traditional approach of worksheets and practice. By contrast, pupils in another class were taught through learning activities balanced across creative, practical and analytical approaches (successful intelligence).
When both groups of learners were given a traditional, memory-based test, those learners in the successful intelligence classes actually did better on the test of knowledge recall than the learners who had been explicitly taught to that end. So it seems you could have your mathematical memory cake (if that’s what you value) and bake it in ways that encourage the creative, practical and analytical.
In this series of articles, I’ll be looking at how we might read the curriculum and the ways in which we can teach potential flashpoints for successful mathematical intelligence. The intention is to provoke staffroom conversations about the curriculum and how we teach it. Sparks may fly, but I hope everyone agrees children deserve to learn mathematics by engaging with rich activities, solving challenging problems, reasoning about the whys and hows of maths, engaging in an appropriate amount of practice and, most of all, talking about mathematical ideas.
1. ‘Pupils should make rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems.’ (NC p. 99)
What might be the balance across fluency, reasoning and problem solving? Does fluency have to be in place before pupils can engage in problem solving or reasoning?
2. ‘Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on.’ (NC, p.99)
Is there a contradiction here? Is the expectation that some pupils get enrichment activities while others are given yet more practice to do? How do we strike a balance between consolidation and enrichment?
3. ‘Calculators should not be used as a substitute for good written and mental arithmetic’. (NC, p. 100)
ICT seems to be equated with calculators – there is no explicit mention of computer technologies. What is the place of ICT in the new maths curriculum?
4. Discussion is to be used to ‘probe and remedy’ misconceptions (NC, p.100).
How exactly do we remedy misconceptions? Japanese guides to teaching advise teachers that some misconceptions may take several weeks for learners to sort out. What does that mean for classroom teaching?
5. Year 2 pupils ‘connect unit fractions to equal sharing and grouping’ and recognise simple equivalences (NC, p.109)
Can young learners deal with a topic that traditionally is thought to be better taught later? The evidence is yes, they can, but it requires some changes to the way fractions are usually taught.
6. Year 3 pupils solve multiplication problems that include ‘positive integer scaling problems and correspondence problems’ (NC, p.115)
The curriculum encourages a breadth of models of multiplication and division that go beyond the usual view of multiplication as repeated addition. What might these look like?
7. Year 4 pupils write ‘statements about the equality of expression’ using the distributive and associative laws (NC p.122)
What sort of teaching can encourage this?
8. Year 5 pupils solve problems involving ‘scaling by simple fractions’ and ‘simple rates’ (NC, p.129)
This can be read as introducing ideas around ratio earlier than is often the case. What sorts of ratio problems can learners solve and how do we support their understanding?
9. Year 6 pupils ‘encounter and draw graphs relating two variables’ (NC, p.141)
What sort of variables can Year 6 pupils work with? What challenges does this present?
I’ll address each of these questions in detail over the course of the year in the magazine, with older articles archived on teachprimary.com so they can be back referenced.
All National Curriculum references are taken from: Department for Education (September 2013) The national curriculum in England. Key stages 1 and 2 framework document.
Based on his research, Sternberg argues that learners taught for creative, practical and analytical intelligences do better on the assessment of memory items because teaching for successful intelligence ‘enables children to capitalize on their strengths and to correct or compensate for their weaknesses, and it allows children to encode material in a variety of ways’ (Sternberg, 2008, p. 155). In other words, a curriculum that appears to be rooted in the rote memorisation of facts and recall of procedures still benefits from what we know about good teaching: that learning based in tasks that engage learners in finding creative solutions, reasoning about methods and applying knowledge not only leads to deeper understanding, but also higher standards.
Sternberg, R. (2008). Applying Psychological Theories to Educational Practice. American Educational Research Journal, 45(1), 150–165.
Mike Askew is adjunct professor at Monash University. Now working freelance, Mike teaches, researches and writes about primary mathematics, working internationally with schools and universities. He can be contacted through mikeaskew.net
Top tips for differentiation
5 friendship and emotions intervention ideas