Mathematics Resilience - The Shard Conference

Roger Titcombe's picture
On 16 January I attended the Shard Conference of the Mathematics Resilience Group led by Sue Johnston-Wilder (Warwick University) and Clare Lee (Open University).

It was held in the Warwick University Business Centre, which occupies most of the 17th floor. The view is superb. The main Conference Area looks across directly at all the iconic totem poles of capitalism thrusting skywards in the City. Just below is the massive building site of the London Bridge Station development.

Mathematics Resilience is a hard sell because it is profoundly counter-intuitive. It argues that the adults and children of the western world, and especially the UK, are gripped by a destructive, deep rooted anxiety about maths. And not just hard maths, but also numeracy at the most basic level.

Fine so far, who would argue with that?

But Mathematics Resilience argues that the solution is not to break down the teaching of maths into a progression of even more small, easily accessible, shallow steps, for which pupils can be rewarded with success as they mount each stage, but to deliberately and methodically confront them with stuff that makes their brains hurt because they can't readily understand it.

The argument is that maths anxiety is the inevitable consequence of being forced to participate in a world of numbers, in maths classrooms and elsewhere, that is a complete mystery in terms of understanding, and that being trained through repetition and memory exercises to 'get sums right' just increases the anxiety because, just as you thought you could it do it, along comes a vaguely similar but slightly different problem that you haven't a clue how to tackle. And often you don't even understand the question.

So the problem is not primarily a lack of knowledge but that the knowledge, and even successfully taught calculation skills, are floating about like detritus in a cognitive ocean of disconnected misunderstandings.

Now comes the anti common sense part. The solution is to give pupils harder, not easier problems. And to especially praise them when they get the wrong answers!

Followers of Vygotsky (who else) will see where this is heading: yes, to the pedagogic principle of the Zone of Proximal Development (ZPD) and the necessity of cognitive struggle for the process of gaining understanding.

We mustn't forget Piaget either, because the understandings gained are never specific to the problem that was the context for the particular cognitive struggle. The resulting cognitive gains are general. The pupil can then not only understand different sorts of maths problems for the first time, but can now tackle harder stuff in all subjects and in real word situations. A developmental, stage-related progression in personal cognitive function has resulted.

Of course none of this will work if the learner is left to drown without support in an ocean of misunderstanding. Such support involves the following features that have all been proven to facilitate successful developmental learning.

1. A teacher that understands the pedagogy and is sufficiently skilled to apply it. There is therefore a training need.

2. Pupils that are familiar with the personal cognitive processes of metacognition, and who readily practise it.

3. Unselfconscious and uninhibited peer to peer discussion about the problem in question and how it might be solved. This is Vygotsky's 'social plane learning'.

4. The cultural expectation of mental struggle. Guy Claxton refers to this as developing 'a positive learning disposition' that expects dead ends and wrong answers. This might be called having a 'good mental character', almost in the VIctorian sense.

5. The need to consciously engage Daniel Kahneman's 'System 2 thinking' as a mental habit.

6. The explicit absence of external incentives and competition.

So how did the day go?

Pretty well. I am sure all the delegates left feeling much the wiser and more energised. Here are some of my personal highlights.

The contribution of Mike Ellicock of 'National Numeracy'

"Being numerate goes beyond simply ‘doing sums’. It means having the confidence and competence to use numbers and think mathematically in everyday life, for example being able to make estimates, identify possibilities, weigh up different options, decide which is most appropriate and choose the correct skills to tackle and solve the problem or situation."

A debate about why maths anxiety is largely absent in China, Singapore and South Korea

As might be expected this was inconclusive, but I was very interested in Sue Johnston-Wilder's analysis of the teaching of one of China's 'superstar status', high earning, private maths coaches. She said that she was particularly impressed by the amount of time spent developing conceptual understanding. The example she gave was in relation to 'fractions'. The coach clearly recognised the cognitive struggle involved in the understanding of the concept of 'parts of things' and spent a great deal of time on it prior to the teaching of how to 'do sums with fractions'. She also talked about the growing interest from the high achieving far east education systems in the 'mathematical resilience' movement here.

A debate about the importance of parental support

I was a bit worried about this. The speaker talked about 'parental engagement' and support being more important as a learning outcome indicator than 'FSM' and 'social class'.

These latter are 'proxy indicators'. It is 'cognitive ability that counts', as I am always banging on about.

I wondered how parents might react to a teacher telling them that their children 'didn't make enough mistakes'. If the principles of learning resilience are a hard sell to teachers, then how can we get parents on board?

I reflected on being told by a Hackney insider that Mossbourne Academy's SEN provision is superb, largely because of the time and money spent on doing all the things in school that are necessary to ensure that all pupils get what they need, regardless of parental support. Isn't this what the pupil premium is for?

The relevance of my book, 'Learning Matters'

Part 5 is a summary of the important work of many of those that share the basic principles of 'Mathematics Resilience'. These include the following.

5.2 Shayer and Adey - The 'Cognitive Acceleration' programmes
5.3 Mortimer and Scott - The importance of pupil talk - CLIS at Leeds
5.4 Johnston-Wilder and Lee - Mathematical Resilience
5.5 Claxton - Developing Learning Capacity and Resilience
5.6 Kahneman - Thinking Fast and Slow
5.8 The Rev Dawes - Kings Somborne School 1837-1860

There are others not included, especially Mike Grenier and the 'Slow Education' movement.

This latter is a serious omission

There is clearly a lot of excellent work going on.

I would like to see more of a common front. This is mainstream 21st century pedagogy. It needs to be promoted as such by all involved pushing forward together on as many fronts as possible.
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Neil Moffatt's picture
Fri, 30/01/2015 - 13:30

Thanks Roger. This is something that I feel intensely strongly about.

I inherited a very analytical and creative mind and was able to flourish at maths at school and University. But in spite of my undoubted passion for this subject, I felt that much of what we were being taught was both 'floating like detritus', as you describe, and therefore meaningless as a result of this disconnect with the real world, but also that a lot of what I was taught would have found only esoteric connections with the real world. So there is a double sense of alienation for many. No wonder that fear and the like were fostered.

Where my sister teaches GCSE maths, three numeracy experts have been recruited to try to 'fix' the basic numeracy shortcomings of many children. Her son is studying Pharmacy at University but is still not sure how to do long division.

This loss of basic numeracy skills is another consequence of the disconnect - the many rules of more advanced maths relegates more basic familiarities into a state of unfamiliarity through underuse.

The initial problem may be at primary school - where many teachers are excellent at empathy, but are frequently not themselves comfortable with Maths. They can teach the rules, but often fail to teach feeling. So they tend to engender a narrow way of thinking that can become claustrophobic in later years.

Every mathematical operation a child carries out must be equatable to the real world. Frequently, this is not the case, and any problem that requires a deviation from the rules and methods taught will find many flounder as they are neither taught to fully understand what they are doing above how to do it, but do not have the flexibility to attack problems from multiple directions encouraged and fostered.

I am writing a set of animated web pages that hope to teach this feeling, so strong is my concern about the omission of such teaching.

Roger Titcombe's picture
Sat, 31/01/2015 - 09:33

Neil - It is not just maths, which is the last and probably the most important point I am trying to make.

"the problem is not primarily a lack of knowledge but that the knowledge, and even successfully taught calculation skills, are floating about like detritus in a cognitive ocean of disconnected misunderstandings." As for maths, so for science.

The front page of today's Guardian contains this sentence by Vikram Dodd.

"Tasers use an electric current of up to 50,000 volts to incapacitate people and critics say that the weapon is often too lethal."

Vikram, who I guess is very highly educated, reveals that he has no understanding whatever of the physics of electricity. Electric current is not measured in volts. It is like saying that somebody is obese if they weight 20 litres.

That the fact that Vikram's ignorance is shared by millions of people, including our top Oxbridge graduates, does not lessen the seriousness of the mistake.

As a physics teacher I regularly used to charge my pupils up to 50,000 volts or more. It made their hair stand on end and they begged in excitement for me to do it to them. It did them no harm at all. At the same time we all know how easy it is to be electrocuted by a mains voltage of 230V. Much less will do the job if you are sitting in a bath of water.

I am not even going to attempt to explain this here. Does such ignorance matter? Of course it does. The entire modern world works by electricity. What must be the judgement on a pedagogy that unleashes into the modern world top graduates who, after 17 years of full-time education, do not have the faintest clue about the energy form that makes it possible for me to make myself a cup of tea to drink as I am typing this on my computer?

Whereas a lack of basic maths understanding causes serious anxiety, a lack of basic science understanding can be happily ignored. The important point is that both problems can be overcome with the right kind of pedagogy.

But, like teaching for 'mathematics resilience' it is counter intuitive and unlikely to be adopted in KS4 by schools threatened by OfSTED and league tables for failing to meet floor targets.

Neil Moffatt's picture
Sat, 31/01/2015 - 12:51

Thanks Roger. Sorry to be myopic, but I of course understand and share your concern about science. I loathed Chemistry because it was presented to me as a countless series of abstractions. No feel for the subject was ever engendered in me. Encountering it presented in a meaningful sense in subsequent years served to illuminate the degree of failure to contextualise and humanise that teaching.

Likewise, for me, subjects such as History, where the overt focus on discrete facts swamped an appreciation of the lessons to be learned from past behaviours.

Andy V's picture
Sun, 01/02/2015 - 15:19

This blog may have some merit. It too reaffirms the Ofsted approach that it is not a matter of zipping through the curriculum content (e.g. tick box style) but rather one of ensuring understanding and enriching that through differentiation via more sophisticated work for those pupils who appear to grasp a concept sooner than others.

Treat it as one for the melting pot:

Roger Titcombe's picture
Sun, 01/02/2015 - 15:39

Andy - My virus checker gives me a warning notice on this link.

Andy V's picture
Sun, 01/02/2015 - 21:39

Interesting. My laptop has Norton 360 virus protection and didn't flag up anything. Ah well, the vagaries and treachery of the internet.

Shame you can't access the blog article. It made for interesting reading.

Roger Titcombe's picture
Mon, 02/02/2015 - 15:59

Nicky Morgan has announced that all pupils must know their tables up to 12x by heart otherwise heads will be sacked and their schools will be forced to become Academies.

Once again we have the behaviourist assumption that all pupils can be trained to pass any specific threshold test by means of enough rote learning combined with high stakes pressure on heads and maths teachers.

How are the children to be tested that they all know their times tables by heart? Will it be quick fire whole class questioning by an OfSTED inspector? Or perhaps the same quick fire questions aimed at pupils chosen at random? But what is a standard 'quick fire question? How long does the child get to answer? Is is 2s, 3s, 5s? This is vital. The number of children that fail the test will depend on the answer time allowed. Furthermore, the shorter the time allowed, the greater will be the number of children that fail the test and the closer the test result pattern will be to a Bell Curve distribution. Children are all different and no amount of rote learning will make them the same.

This post is about Mathematics Anxiety. This is what the Shard Course Leader Sue Johnston-Wilder had to say in her 2010 BERA address.

"The more that we studied stories from people who exhibit mathematics phobia, and read the related literature, the more that it appeared to us that the way that mathematics is often taught in English mathematics classrooms is an unwitting form of cognitive abuse. Instances of ways of working that seem calculated to cause anxiety are asking learners to perform tasks that require feats of memory at a rapid rate or to memorise formulae without understanding in classrooms where the mathematics is divorced from the reality that it models so powerfully. These ways of working have been shown by many researchers (e.g.Boaler 2009, Jain & Dowson, 2009 and Baloglu & Koçak, 2006 ) to cause anxiety. Acting in such a way that many people are made to feel anxious, concerned or fearful seems to us to be acting in an abusive way."

What Morgan is going to force onto schools will result in even more high stakes pressure on school heads (zero tolerance of failure - oh goody). These heads will pass on the pressure to the Head of Maths (a job second in insecurity only to football managers in today's 'improve at all costs' schools). The HoD will pass on the pressure to the maths teachers and guess who will be the final recipient of the pressure? Pity the poor kids - especially those that already have maths anxiety, hate the subject and freeze in terror when the teacher tells them to stand up and then barks at them, "9 x 8 ?"

All this is described with the consequences explained in Part 2 of 'Learning Matters' - 'The Consequences of Bad Education'

But surely it is a good thing for children to 'know their tables'. Of course, and there is always a place for a bit of rote learning. That is not the problem. The problem is the red in tooth and claw behaviourism with which it is applied. When it comes to developmental learning, the best outcomes are unlikely to result from high stakes threats and pressure applied to heads, teachers and the poor kids that are ultimately on the boot end of such approaches - especially the poor terrified little sods unable to bark back the right answer in time.

Which branch of researched academic pedagogy and learning theory has this latest back of a fag packet stuff come from? Who needs any of that high faluting theory stuff when teaching kids maths is such obvious common sense that the likes of Nicky Morgan and David Cameron feel confident enough to force teachers to do it 'their way'?

Roger Titcombe's picture
Tue, 03/02/2015 - 11:28

I have come across this book.

This is the Amazon 'blurb'

"Whatever it is that makes you you, can you affect it? Your answer to that question gives us a glimpse into your 'mindset'. People with fixed mindsets believe that fundamental qualities like intelligence are essentially stable; people with growth mindsets believe that such qualities can be developed and nurtured. As teachers, if we can foster growth mindsets in our students the results will be transformative. Barry Hymer and Mike Gershon begin by explaining how learners with growth mindsets are: more open to challenges and constructively critical feedback; resilient in the face of obstacles and initial failure; convinced that effort makes a difference; able to learn well with and from others; likely to rise to the top - and stay there. Practical strategies for developing this kind of learner is the focus of the rest of the book throughout which cartoons, diagrams and visual prompts support the text. The chapter Trial and Error is about high challenge tasks and the value of errors, mistakes and initial failure. Later chapters cover what exactly is meant by effort and how to make it appeal to students; the theory and practice of feedback (as opposed to praise and prizes); acquiring meta-cognitive tools for 'thinking about thinking' and 'learning about learning'; the power of language and drama to raise awareness of growth mindsets; and growing group growth mindsets. Former teacher Barry Hymer is now Prof. of Psychology in Education at the University of Cumbria. He wrote the Gifted & Talented and P4C Pocketbooks. Education consultant Mike Gershon's online teaching tools have been viewed/downloaded more than 2 million times by teachers in over 180 countries."

There are very strong echoes of, 'Learning Matters' here.

I know Barry Hymer from my Cumbria headship years. He was behind the Cumbria LA - wide use of 'Reading Recovery', which was enormously successful including with pupils in my school.

I have just had some very positive feedback on 'Learning Matters' from Mike Ellicock of 'National Numeracy'. He talks about 'a growing movement' that is gaining momentum and recommends, 'The Expert Learner' by Gordon Stobart.

And of course we now have Melissa and Janet's brilliant new book, 'School Myths'.

All this has great relevance to Trevor Fisher's recent contributions to LSN on the question of how to turn research and argument into a 'Kuhnian' tipping point. The answer has to be to put all our efforts into pushing forward the 'broad front' that crucially includes Henry's revelations. The myths about Academies were still being peddled strongly on Ch4 News last night by both Nick Gibb and Tristram Hunt with the help of Jon Snow's hopelessly ill-informed leading questions that assumed that Academies were and are great.

We need to morph ourselves into a kind of 'collective Galileo'. The danger of sectarianism is always there as people argue about their pet 'individual trees' losing site of the forest.

We don't all have to agree in detail about all the strands in the broad front, but we do have to promote all the strands.

Janet Downs's picture
Tue, 03/02/2015 - 15:42

Roger - Nous sommes Galileo.

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