I will argue here the government has sabotaged its own policy on maths teaching through the power it has given to OFSTED and to the NCETM, but first I will review what the debate in maths education is about. As with history teaching, or phonics, the trouble with writing about maths is the long history of the debate about the subject. My usual method of providing an introductory summing up about what’s at issue is to quote the following from the American maths professor, W. Stephen Wilson description of the opposing side:
There will always be people who think that calculators work just fine and there is no need to teach much arithmetic, thus making career decisions for 4th graders that the students should make for themselves in college. Downplaying the development of pencil and paper number sense might work for future shoppers, but doesn’t work for students headed for Science, Technology, Engineering, and Mathematics (STEM) fields. There will always be the anti-memorization crowd who think that learning the multiplication facts to the point of instant recall is bad for a student, perhaps believing that it means students can no longer understand them. Of course this permanently slows students down, plus it requires students to think about 3rd-grade mathematics when they are trying to solve a college-level problem. There will always be the standard algorithm deniers, the first line of defense for those who are anti-standard algorithms being just deny they exist. Some seem to believe it is easier to teach “high-level critical thinking” than it is to teach the standard algorithms with understanding. The standard algorithms for adding, subtracting, multiplying, and dividing whole numbers are the only rich, powerful, beautiful theorems you can teach elementary school kids, and to deny kids these theorems is to leave kids unprepared. Avoiding hard mathematics with young students does not prepare them for hard mathematics when they are older. There will always be people who believe that you do not understand mathematics if you cannot write a coherent essay about how you solved a problem, thus driving future STEM students away from mathematics at an early age. A fairness doctrine would require English language arts (ELA) students to write essays about the standard algorithms, thus also driving students away from ELA at an early age. The ability to communicate is NOT essential to understanding mathematics. There will always be people who think that you must be able to solve problems in multiple ways. This is probably similar to thinking that it is important to teach creativity in mathematics in elementary school, as if such a thing were possible. Forget creativity; the truly rare student is the one who can solve straightforward problems in a straightforward way. There will always be people who think that statistics and probability are more important than arithmetic and algebra, despite the fact that you can’t do statistics and probability without arithmetic and algebra and that you will never see a question about statistics or probability on a college placement exam, thus making statistics and probability irrelevant for college preparation. There will always be people who think that teaching kids to “think like a mathematician,” whether they have met a mathematician or not, can be done independently of content. At present, it seems that the majority of people in power think the three pages of Mathematical Practices in Common Core, which they sometimes think is the “real” mathematics, are more important than the 75 pages of content standards, which they sometimes refer to as the “rote” mathematics. They are wrong. You learn Mathematical Practices just like the name implies; you practice mathematics with content. There will always be people who think that teaching kids about geometric slides, flips, and turns is just as important as teaching them arithmetic. It isn’t. Ask any college math teacher.
Roughly speaking, the tensions are between those who emphasise the procedures (both written and mental) and those who emphasise applications in context and the ability to talk about maths. In the most recent versions of the debate that I have encountered the former would claim that they are aiming for fluency and the latter would claim that they are aiming for conceptual understanding. Neither would claim not to be pursuing the other’s goal; those emphasising fluency would claim it leads to greater understanding and those emphasising conceptual understanding would claim it leads to greater fluency. Ultimately, both sides would say you have to teach both fluency and conceptual understanding, but the choice of which to emphasise will have a huge impact on teaching methods. The more you emphasise fluency, the more you will spend time on deliberate practice and relying on explanations, rather than “discovery” to result in understanding. The more you emphasise conceptual understanding, the more time you will spend using card sorts and cuisenare rods, having discussions, trying to learn maths from problem-solving and outsourcing calculations to calculators. It is best described as a spectrum rather than a dichotomy, with only the most extreme of the advocates of conceptual understanding saying kids don’t need times tables and only the most extreme (or possibly those who don’t understand the word “rote”) saying kids should learn procedures without understanding them. Emphasis also changes depending on which key stage is being discussed, with fluency being the most obvious goal for very young children learning to count or add numbers under 10 and conceptual understanding being a clear priority at A-level.
This debate maps fairly closely to wider debates about knowledge and skills. However, maths, probably more than any subject, is probably the one that most easily lends itself to the traditional emphasis on knowledge and fluency. It is very easy to find maths teachers on the less fashionable end of the spectrum, particularly among those with maths degrees or those old enough to have lived through the National Numeracy Strategy, one of the rare times that the official pedagogy in any subject was traditional. It is common to find schools where setting in maths in every year groups reflects the need to address differences in knowledge. However, the conceptual understanding side have remarkable dominance over university departments of education and, unfortunately as this is where so much power lies, OFSTED.
Politically, there have been sudden changes in which side was dominant. The emphasis on fluency resulting from the National Numeracy Strategy in the late 90s was followed by a period of weak leadership that let the conceptual understanding side gain the upper hand. Things have changed since 2010. Ministers have advocated fluency and most importantly, the new National Curriculum in maths has emphasised it. However, the big question to me has been whether ministers and documentation backing fluency would make any difference. When the ideological difference is one of emphasis people can always play lip service to one idea while focussing on another. Nobody need say they are against fluency to sabotage this policy direction; they need only say that, while of course fluency is important and they assume people will be working on it, the really important good practice to be shared is that focussing on conceptual understanding. If those with power over education take that attitude then nothing will change. Ministers will change the documentation but not the education that is being delivered.
Now, it was always going to be the case that OFSTED would sabotage the policy in this way. Their latest maths report, Made To Measure in May 2012, emphasised conceptual understanding (the word “understanding” appears 97 times in the report including 11 mentions of “conceptual understanding) and relegated fluency (“fluent” appears 3 times in the report but every time is mentioned only alongside understanding; fluency is mentioned 6 times but in 4 of those cases it is alongside “understanding”). Examples of best practice in maths given by OFSTED have been at the extreme trendy end and I summarised them (and compared them with Michael Wilshaw’s views) here. Worse, OFSTED have claimed (for instance, in these course notes released under the Freedom of Information Act) in a truly Orwellian fashion that “The definition of fluency incorporates conceptual understanding”, an interpretation which would make arguing for one side impossible. Although they accept that students should be fluent (presumably with that definition of fluency), this is OFSTED’s description of what outstanding maths teaching looks like:
Teaching is rooted in the development of all pupils’ conceptual understanding of important concepts and progression within the lesson and over time. It enables pupils to make connections between topics and see the ‘big picture’. Teaching nurtures mathematical independence, allows time for thinking, and encourages discussion. Problem solving, discussion and investigation are seen as integral to learning mathematics. Constant assessment of each pupil’s understanding through questioning, listening and observing enables fine tuning of teaching. Barriers to learning and potential misconceptions are anticipated and overcome, with errors providing fruitful points for discussion. Teachers communicate high expectations, enthusiasm and passion about their subject to pupils. They have a high level of confidence and expertise both in terms of their specialist knowledge and their understanding of effective learning in the subject. As a result, they use a very wide range of teaching strategies to stimulate all pupils’ active participation in their learning together with innovative and imaginative resources, including practical activities and, where appropriate, the outdoor environment. Teachers exploit links between mathematics and other subjects and with mathematics beyond the classroom. Marking distinguishes well between simple errors and misunderstanding and tailors insightful feedback accordingly.
Of course, as far as I am concerned any government which tolerates the existence of OFSTED is already limiting their own power to make meaningful educational change and can expect to be overruled in this way by a body that has more power over teachers’ practice than government.
The other big factor will be how the new curriculum is explained to teachers, particularly primary teachers. Will fluency or conceptual understanding be emphasised? Will teachers be given advice on how to train kids in recalling number bonds and times tables and applying traditional algorithms or will they be encouraged to get kids talking and playing with pictures? I was shocked to hear that NCETM had been funded to “roll out” the policy. Subject associations are not known for their traditionalism and NCETM has in the past produced one of the most destructive explanations of what an OFSTED lesson should look like and a “consultation” where:
Participants consistently reported that:
• too much time is spent developing “fluency in recalling facts and performing skills”to the detriment of other aspects;
• much greater emphasis should be placed on the remaining four learning outcomes, with particular emphasis being placed on “conceptual understanding and interpretations for representations” and “strategies for investigation and problem solving”.
In 2010, a submission to parliament from the NCETM claimed:
1. There is substantial evidence of what constitutes effective mathematics teaching, which includes the Cockcroft reportMathematics Counts (1982), A Study Effective Teachers of Numeracy (1997), a review led by the NCETM Mathematics Matters (2008), Ofsted’s report Understanding the Score (2008).
Together these reports have identified the following characteristics of effective mathematics teaching:
- Builds on the knowledge learners already have.
- Exposes and discusses common misconceptions and other surprising phenomena.
- Uses higher-order questions.
- Makes appropriate use of whole class interactive teaching, individual work and cooperative small group work.
- Encourages reasoning rather than “answer getting”.
- Uses rich, collaborative tasks.
- Creates connections between topics both within and beyond mathematics and with the real world, in particular drawing out connections between different representations of mathematics (eg graphical, numerical, algebraic).
- Uses resources, including technology, in creative and appropriate ways.
- Confronts difficulties rather than seeks to avoid or pre-empt them.
- Develops mathematical language through communicative activities.
- Sets high expectations for pupils in mathematical challenge, achievement and enjoyment.
2. These principles of effective teaching are widely accepted by teachers who have specialised in mathematics teaching and learning in their ITT or later in their career.
Their director has a background in Key Stage 5 teaching and advocates “a way of thinking of teaching maths which involves understanding and enjoyment” and criticises “the idea of maths as just being a set of questions where your aim is to get the answer right to the question without any kind of meaning” although he admits this “has a place” in learning. Of course, what this means in practice is what matters. So far it doesn’t look promising. A blogger who went on their training course for those implementing the primary curriculum observed:
The NCETM approach is to emphasise that fluency can only be achieved, and should only be achieved by building on a foundation of good conceptual understanding. Their training and the training that we in turn will be passing onto schools explores the key role that representation and the use of concrete apparatus has in building up this conceptual understanding.
This idea, that fluency will happen without teachers being trained to teach it well and, instead, teachers need to be trained to deliver conceptual understanding is also evident in the training videos NCETM have produced. These are the videos they have produced on times tables and advertised on their website as “Videos to support the new implementation of the new Primary Curriculum”. There are suggestions that students will also be practising recall of times tables, but look at what they think needs to be passed on to teachers:
Under the influence of OFSTED and the NCETM schools are going to continue to think “conceptual understanding” and the activities alleged to promote it, rather than deliberate practice, are the key to maths education. The government has created a curriculum which emphasises fluency in maths and then given all the power over maths education to people who have a completely different emphasis. This is not going to be half as good as the National Numeracy Strategy, this is a change in words at the top which won’t reach the ground. This is the government sabotaging its own policy.