Archive for October, 2014


Fluency in Mathematics: Part 2

October 6, 2014

I gave a talk on fluency in mathematics in March at Pedagoo London (my first public appearance) and again the weekend before last at the La Salle Education maths conference. This is based on those talks and so, inevitably, it is one in a series of posts. Part one can be found here.


“And this was the greatest number of words I could fit on a single Powerpoint slide”.

At both talks I asked the audience to discuss reasons they may have been wary about giving explanations, using prolonged practice or focusing on fluency in maths. I also asked about reasons others had used to tell them not to teach that way. I then asked whether they came up with the same answers I did. Most of the following featured to some degree:

1) Understanding

The issue of whether students ability to answer questions should take second stage to understanding of some sort is not a new one in mathematics. Tom Lehrer satirised it in the 70s.

The real problem is that understanding is not well-defined. I found four definitions for it and wrote about them here, and for these talks I added another possible definition that it could mean “knowing the connections between topics”. It is only with these multiple definitions of “understanding” to confuse matters that explanation and practice can be sidelined. When “understanding” is tied down to a specific meaning we invariably find that a clear explanation and the building up of fluent knowledge, is the best way to promote understanding.

2)  Problem-Solving

While problems might well have a part to play later on in practising the application of knowledge, it is unclear why anyone would imagine problem-solving is useful for the acquisition of new knowledge. It causes a distraction from what we should be thinking about (the thing we need to know) and is likely to tax working memory unnecessary making retention more difficult. While problem-solving approaches to maths have been continually fashionable, their effectiveness is far from proven. Hattie (2009), who summarised research using effect sizes, where 0.4 showed average effectiveness, found problem-solving learning to have a very poor effect size of 0.15:

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3) Discovery/Inquiry

This is a very similar point, in that in practice it also involves engaging students in activities where new knowledge is meant to result. Again it is very fashionable. And again, the same objections apply: it is likely to distract from what is to be learnt and to limit retention. Again, Hattie found an unimpressive effect size:

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4) Engagement/Enjoyment

It is not always clear what “engagement” means as I discussed here. If it is used to mean enjoyment, then there are serious questions to be asked about the assumption that students learn best when enjoying themselves. There is too much research on that question to reference here, although hopefully I will get round to blogging about it soon, but while anxiety or hysteria won’t be great for learning, there’s strong evidence against the general principle that we learn more effectively the happier we are.

5) Groupwork/Discussion

If I’ve described anything else as “fashionable” then this tops that tree. There’s no clear evidence that groupwork is as ineffective as problem-solving or discovery, but Hattie’s effect size of 0.41 for cooperative learning would indicate it is unexceptional as a method. I would not seek to prevent groupwork, but find its aggressive promotion unwarranted.

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There is a lot of psychology literature on how people work in groups. Different conditions make it more or less effective. Its potential negative effect on motivation, known as “the Ringelmann Effect” is long-established and there are certain types of thinking that are thought to be less effective in groups. The widespread belief that groupwork is an essential part of good teaching seems implausible, and provides little reason to crowd out other methods of teaching.

Update 7/9/2014: It was pointed out to me on Twitter that Hattie also quotes one meta-analysis that takes subject into account and found an effect size of 0.01 for cooperative learning in maths.


The first time I gave this talk I could only confirm that OFSTED did seem opposed to developing fluency and this was the one point on which I could not deny the objection. The details are described here and I showed some of this video to underline the point:

When I gave the talk last week, I could be more optimistic. Inspectors should not be requiring a particular way to teach and shouldn’t be grading you individually anyway. The old subject survey guidance that laid out a depressingly trendy vision of how to grade maths lessons is now gone. Hopefully, this demon has been slain.

7) Technology

The idea here is that technological change has changed the nature of schooling or society. I’ve used these before, but in a week where the TES published this, it is worth remember just how long the claims that traditional teaching is doomed have been going on. Here is some of the usual rhetoric:

The idea that our schools should remain content with equipping children with a body of knowledge is absurd and frightening. Tomorrow’s adults will be faced with problems about the nature of which we can today have no conception. They will have to cope with the jobs not yet invented.


we find ourselves in a rapidly changing and unpredictable culture. It seems almost impossible to foresee the particular ways in which it will change in the near future or the particular problems which will be paramount in five or ten years.


Books will soon be obsolete in the public schools…Our school system will be completely changed inside of ten years.


These quotations are actually from 1966, 1956, and 1913, respectively. (Sources for the latter 2 can be found by searching my blog, and for the first from this book).

This does not disprove the argument about technology, but it does seem to shift the burden of proof. A more developed case can be found here,  here and here.

8) Independence

The philosophical arguments about independence and autonomy can be found here. And I would add to it now, that if independence from the teacher was so important, then it is hard to explain the effectiveness of Direct Instruction, a method of teaching very much focused on the role of the teacher:

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9) Research and Training

Probably the biggest reason for the poor view of fluency many maths teachers have is that we’ve often been trained by those who would deny its importance. We’ve also often been told that this is justified by the research. A proper take down of the shambles that is maths education research here and in the US would take a whole blogpost, but you can get a flavour of it by looking at the work of Jo Boaler, the criticisms of it, and the methods she uses to silence critics. This does not resemble scientific research or academic debate in anyway. The field is mainly propaganda for groupwork, mixed ability teaching and the methods of “fuzzy maths”. There does now seem to be the first signs of debate, and I was able to mention a few papers that challenge improve on the research methods and challenge the orthodoxy, but it seems early days and too much of what I have found is by economists and not published in maths education journals despite seeming to be of much higher quality than the studies that do get published:

Continued in part 3


Teachers need the time to get out more: Academies Week

October 5, 2014

Education is enough of a national concern that there is no shortage of organisations arranging meetings and conferences about different aspects of the subject. Being addicted to Twitter has allowed me to keep up with much of this. Being a blogger has also meant I receive invitations to speak at events, or am placed on the guest list for ones not open to everybody. However, such events are not evenly distributed across the year, and the last 11 days have seen my diary overloaded.

Despite living in West Midlands, the Saturday before last I was in the centre of London attending the launch of a book,
Don’t Change the Light Bulbs (edited by Rachel Jones). The following Tuesday, I heard Teach Like a Champion author Doug Lemov in Walworth, giving a lecture on leadership…

Continued in Teachers need the time to get out more : Academies Week


Fluency in Mathematics: Part 1

October 4, 2014

I gave a talk on fluency in mathematics in March at Pedagoo London (my first public appearance) and again last weekend at the La Salle Education maths conference. This post is based on those talks and so, inevitably it is long enough to take several posts and revisits some old ground.


My talk at Pedagoo London

I started by asking the audience the following questions (yes I know, exciting start), and giving them a couple of minutes to work them out:

  1. 7×8
  2. Simplify 49/84
  3. Find √729
  4. Solve 3x+40=19
  5. Write √7/√175 as a decimal

At both talks there was somebody who took not much more than a minute and others who struggled. It is possible to answer every one of those questions in a few seconds if you have memorised the correct basic facts, such as times tables, and how negative numbers and surds work, and can recall them fluently. Both times though, I appeared to be the only person in the room to have memorised the first 27 square numbers.

As well as being useful for solving problems, fluency is now one of the major aims of the new maths National Curriculum, quoted below:


The national curriculum for mathematics aims to ensure that all pupils:

  • become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.
  • reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
  • can solve problems by applying their mathematics to a variety of routine and non routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.

To have fluency as the first of three aims is a big change compared with the old national curriculum, the aims of which seemed to include everything else but fluency:

Curriculum aims

Learning and undertaking activities in mathematics contribute to achievement of the curriculum aims for all young people to become:

  • successful learners who enjoy learning, make progress and achieve
  • confident individuals who are able to live safe, healthy and fulfilling lives
  • responsible citizens who make a positive contribution to society.

The importance of mathematics

Mathematical thinking is important for all members of a modern society as a habit of mind for its use in the workplace, business and finance; and for personal decision-making. Mathematics is fundamental to national prosperity in providing tools for understanding science, engineering, technology and economics. It is essential in public decision-making and for participation in the knowledge economy. Mathematics equips pupils with uniquely powerful ways to describe, analyse and change the world. It can stimulate moments of pleasure and wonder for all pupils when they solve a problem for the first time, discover a more elegant solution, or notice hidden connections. Pupils who are functional in mathematics and financially capable are able to think independently in applied and abstract ways, and can reason, solve problems and assess risk. Mathematics is a creative discipline. The language of mathematics is international. The subject transcends cultural boundaries and its importance is universally recognised. Mathematics has developed over time as a means of solving problems and also for its own sake.

Which to me, now, really sums up an era where you could have every variation of progressive education together in one document, while missing any mention of what is worthwhile knowledge.

One of the main reasons I think this has now changed, is a change in the understanding of how we think. One of the more popular diagrams in education today is this one from Dan Willingham’s book:

It’s a very simplified, but uncontroversial, model of how we think. We have a limited working memory, where conscious thought takes place, and a potentially unlimited long-term memory.  To use working memory effectively, we draw on information that is already in long-term memory. To get things into long term memory we have to overcome the limitations of working memory. Having useful information in long-term memory, and being able to recall it without difficulty, makes thinking easier. In maths it is useful to be able to fluently recall a lot of knowledge, particularly basic number facts, rather than work everything out from first principles. The question at the start about simplifying 49/84 was an example of this, as fluency with the 7 times table makes the question trivial.

In order to build fluency we need to acquire knowledge and to learn to remember it without effort (automaticity).


See? I did say this

And if that is our aim then, in maths, the best method is to tell kids what they need to know and set them lots of questions where they practise recalling it.

Continued in Part 2


Spot The Difference: The ATL and Behaviour.

October 1, 2014

You may be aware that OFSTED recently produced a report that should actually be welcomed by teachers. “Below the Radar” raised the issue of poor behaviour in schools, and argued that schools leaders should take more responsibility for dealing with it and identified a lack of effective training as a problem. I’m happy to praise OFSTED for siding with the interests of teachers, and am even willing to believe that (finally) Sir Michael Wilshaw is succeeding in getting the organisation to respect his priorities. But I was a little surprised at the response from the unions. As ever, the unions were torn between representing the interests of classroom teachers (who suffer where discipline is weak) and simultaneously representing their bosses in school management (who are often the cause of poor discipline). The union I noticed (but this may just be a fluke) get into the biggest mess over this was the ATL.

At the start of the month, their leader Mary Bousted was on TV claiming that schools must do more about behaviour:

This followed on from what the ATL were saying at their 2013 conference (this is from a press release):

Nearly 90% of support staff, teachers, lecturers, school heads and college leaders said they have dealt with a challenging or disruptive student during this school year. The main targets of challenging behaviour were other students (cited by 72%), followed by teaching staff (46%), and then support staff (43%). Between students the most prevalent challenging behaviour was verbal aggression (cited by 77%), followed by physical aggression (57%), bullying in person (41%), and breaking or ruining other students’ belongings (23%).

Thankfully, most of the disruptive and challenging behaviour facing education staff was fairly low level with 79% of staff complaining that students talked in class, did not pay attention and mucked around. Sixty-eight per cent said students were disrespectful and ignored their instructions, 55% said they had had to deal with verbally aggressive students, and a fifth (21%) had had to deal with a physically aggressive student…

…Dr Mary Bousted, general secretary of the Association of Teachers and Lecturers (ATL), said: “Regrettably teachers and support staff are suffering the backlash from deteriorating standards of behaviour. They are frequently on the receiving end of children’s frustration and unhappiness, and have to deal with the fall-out from parents failing to set boundaries and family breakdowns. And the huge funding cuts to local services mean that schools often have to deal with children’s problems without any help.

“Schools with firm, clear and consistently enforced behaviour policies create safe learning environments for children and staff, but problems occur when schools fail to enforce good discipline policies and when children know there are weak or non-existent sanctions.

”Schools need to give their staff good and regular training so that they know how to work with students with behavioural or mental health problems and have confidence in handling pupils with challenging behaviour. Behaviour training also needs to be an integral part of teacher training.”

Now let us see what happens when OFSTED agree with the ATL about behaviour, and how graciously they accept inspectors agreeing with them about its importance, the importance of effective discipline policy, and good training. Here is the ATL’s response to the OFSTED report:

Dr Mary Bousted, general secretary of the Association of Teachers and Lecturers (ATL), said: “Once again Ofsted has revealed its deeply narrow-minded nature, attacking schools and leaders regarding pupil behaviour. Its failure to identify systemic issues weakens a system which is already creaking under huge cuts to local support services for schools, particularly for the most vulnerable and often challenging students.

“Instead, Sir Michael Wilshaw, in his Clint Eastwood mode, fires indiscriminately at teachers and leaders, wounding further the morale of staff. At a time when recruitment and retention in education are approaching crisis levels, this is a particularly short-sighted and destructive approach. Indeed, Ofsted’s report mentions that high staff turnover and insufficiencies in training have an impact on schools’ ability to consistently tackle challenging behaviour yet they have chosen to ignore the implications for Government policy around teacher training, supply and professional development.

“We know that consistency of approach and support is key to achieving high-quality pupil learning and behaviour in schools, but Ofsted’s rhetoric rings hollow based on the inconsistency of its own practices. Calling for zero tolerance and stricter approaches doesn’t reflect the evidence of what actually works in excellent classrooms.

“Yes, schools need clear behaviour policies, applied consistently by all staff. Yes, staff need to be supported by leaders when using those agreed policies. But no, Sir Michael, neither teachers or pupils do particularly well when constantly belittled nor when they have decreasing access to much-needed resources. Ofsted needs to review its behaviour policies and this needs to start from the top.”

I’m the last person to start defending OFSTED, but how can teachers ever hope to hold inspectors to account if this is how a teacher union reacts like this to OFSTED agreeing with them and arguing for something that is the interests of their members?

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