Path smoothing or challenging maths?

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In the September issue of Mathematics Teaching, the journal of the UK’s Association of Teachers of Mathematics, there are a couple of articles that are available to non-subscribers. One of these is Models for teaching mathematics revisited, by Andrew Blair and Helen Hindle. It reflects on a dichotomy created by Alan Wigley in 1992 between two different ways of teaching maths – a dichotomy that I had not heard of before. Briefly, Wigley proposed that there are two broad models: path smoothing versus challenging. Interestingly, Blair and Hindle do not suggest the best approach is a bit of both, but come out firmly in favour of the challenging method. This apparently holds the twin needs of instruction and exploration together in a ‘creative tension’. The challenging model contains elements of ‘telling or explaining’, but the aim is student agency, with agency defined as ‘the actual ways situated persons wilfully master their own life’. That’s some profound magic.

So what are these two approaches? Path smoothing involves small steps where the teacher decides on the type of problem and teaches students a procedure to solve it. This is followed by ‘repetitive exercises’ which don’t sound great when you describe them like that. So path smoothing appears to be a form of explicit teaching, or at least what explicit teaching looks like when students are first introduced to a topic. By contrast, the challenging model involves a problem that can be tackled a number of different ways with the students being given time to explore it, come up with their own conjectures and so on. This sounds to me a lot like problem based learning.

Why do Blair and Hindle object to path smoothing? Firstly, it apparently denies the creative tension described above. Secondly, it produces ‘limited and negative results’ and they wonder why it persists ‘in the face of evidence of its ineffectiveness’. They also seem to conflate the pedagogy of maths with the epistemology of professional mathematicians.

The only research evidence Blair and Hindle produce to support these strong claims is the final evaluation of a project conducted by the UK government where schools were encouraged to adopt a mastery teaching methods inspired by maths teaching in Shanghai. This found little evidence for the impact of Shanghai mastery on primary school students’ standardised test results, although there were a number of other findings and interim reports.

The Shanghai mastery study was not a randomised controlled trial. Students in the mastery schools were compared with students matched schools but neither they nor the schools were randomly assigned to mastery. Instead, assignment seems to have been quite ad hoc. As a result of this design the researchers note the, ‘need to remember that findings from quasi-experiments are better interpreted as correlation rather than causation.’

Even if we set this caution aside, it is not at all clear, or even perhaps likely, that the comparison schools used the alternative challenging teaching model. If so, it would be hard to conclude anything from it about the relative benefits of path smoothing versus mastery. The intervention itself seemed to be implemented using a variety of resources, including those produced by NCTEM and White Rose maths. Finally, there seems to have been an odd preoccupation with mixed-ability teaching or ‘all-attainment’ teaching as the researchers described it. This may well be a feature of maths teaching in Shanghai, but it seems to bear little relationship to the path smoothing model. In fact, mixed ability teaching is described as a key feature of Hindle’s own challenging approach.

Rather than hang so much on one, admittedly large, quasi-experimental trial that didn’t test the factor in question, it may be better to look at the broader literature on explicit, interactive forms of teaching when compared with more implicit approaches. There is a wealth of evidence from process-product research in the 1960s through to more recent randomised controlled trials that demonstrates the superiority of an explicit approach. We also have correlational data from PISA which addresses the two different styles.

We even have cognitive load theory to explain why explicit teaching is superior.

Strangely, Blair and Hindle do not refer to cognitive load theory at all. When speculating on the puzzling continued existence of the path smoothing model, they suggest it may be due to teaching to exams, the influence of parents or even a response to tight school budgets in the UK. I mean, teachers surely couldn’t have reasoned arguments, supported by evidence, for maintaining such an approach, could they?



What is the cause of the Australian school behaviour crisis?

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Who should we believe? Should we believe the education academic who thinks talk of a behaviour crisis in Australian schools has been manufactured?

“Despite claims of a crisis in student behaviour – particularly in disadvantaged communities – my research team saw little evidence of student-driven disruption.”

Or should be believe the research being reported today in the Sunday Herald Sun:

“A LaTrobe University study into teacher-targeted bullying found 80 per cent of teachers were victims of harassment in the past year… 10 per cent of teachers reported being hit or punched by a student… Kids yelled and swore at teachers, assaulted them, made disparaging remarks both in person and on social media and damaged their personal property.”

How can two such contradictory views of Australian schools coexist and how can they be reconciled with international evidence that suggests Australian students suffer from a relatively large amount of lesson disruption?

Someone is spinning the facts to suit an ideological agenda, but who is it?

One answer is that people like The Herald Sun journalists and, er, me are trying to whip up a moral panic in order to impose neoliberal policies on schools. It is certainly the case that many Charter Schools in the US, and Free Schools and Academies in the UK, have used their autonomy to develop the kind of transparent behaviour policies that committees of education bureaucrats would likely never support.

And yet it seems odd to assume that people like me have invented all of this. How could we influence surveys of teachers and students?

One clue as to what is happening can be found in the Herald Sun when it discusses the, “Spiralling rates of mental illness among students.”

One interpretation is that we are clearly in the midst of a mental health crisis. Another is that behaviours that would previously have been seen as simply bad, and that would have resulted in a punishment and a moral lecture, are now being used as diagnostic criteria. There are hints of evidence of this when you look at the criteria to diagnose Oppositional Defiant Disorder. I would not advocate returning to past approaches to discipline but I do wonder if we have experienced an over correction.

Does it matter whether mental health issues have increased or been renamed? Are the effects not the same? No. If you construct misbehaviour as being part of a disorder then sanctioning that behaviour or suspending a student from school begins to look like discrimination.

There is an entire arm of education research that looks at the correlation between school suspensions and other negative outcomes and then heavily implies that the former cause the latter (e.g. here). The alternative view, that behaviours that are likely to lead to suspension are also likely to cause the other negative outcomes, is swept aside. It is a case of using hard data to tell something far from an established truth.

And when policymakers listen to these researchers, they make it harder for schools to act to manage problem behaviour which, in turn, increases the pressures on teachers and leads to the kinds of survey results reported in the Herald Sun.

Lewin’s equation

I don’t know if this ever happens to you, but sometimes I find myself wandering the corridors of the internet and arriving in a room where I never realised I wanted to be. That is a joy of our age that arises out of the ease with which we can no follow interesting people and interesting conversations.

In this case, I started with a Quillette article about gender which I would probably never have clicked on if it had not been for my previous post on boys’ educational underachievement. The Quillette piece drew heavily on a 2018 academic essay by Professor Alice Eagly. The essay makes the case that researchers should at least be open to the idea that differences between men and women may have both biological and environmental components.

To support her point, Eagly draws on an equation set out by Kurt Lewin in 1936 in his Principles of Topological Psychology:


This represents the almost trivially simple idea that behaviour is a function of both the person and the environment. Or, as Lewin puts it, “Every psychological event depends upon the state of the person and at the same time on the environment, although their relative importance is different in different cases.”

“Tell me something new!” you might well exclaim. And yet bias, whether it is of the common kind or of the ideological kind that concerns Eagly, can cause us to focus on one to the exclusion of the other.

In Eagly’s case, her concern is that ideology causes feminist psychologists to neglect the role of the person. However, I would suggest that in education, we are equally inclined to reject the role of environment. We may, at one extreme, suggest a child has a learning disability or a behavioural disorder, or at the other extreme suggest a child is inherently lazy or malevolent, when in reality, the single most effective thing we could do is modify the environment that child is in. I would argue that an environment involving structure, routine, a clear and consistent approach to poor behaviour and explicit teaching can mitigate many of the issues that we may otherwise ascribe to the person. You may have other ideas.

Nevertheless, Lewin’s equation represents a useful heuristic we may deploy when we catch ourselves going to far in one direction when attributing cause.


The problem with boys

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Falsifiability is a key aspect of any scientific theory. If you are going to put forward a testable proposition then you have to be able to answer the question: What evidence would prove this proposition wrong? If you cannot think of anything, then you have a problem.

Sociological theories are often unfalsifiable, particularly when interpreted by amateur sociologists drawing on these theories in the pursuit of other goals. For instance, the concept of ‘patriarchy’ can be used to explain any and all evidence about gender disparities. When women are disadvantaged then this can be explained by the patriarchy working to oppress women, but when men are disadvantaged, this can also be explained by the patriarchy and, specifically, the role of toxic masculinity and its deleterious effect on men.

I was struck by this when listening to a podcast from the BBC about the enduring problem of boys’ educational underachievement.

I have been interested in this issue for my whole career. As a trainee teacher, I was asked to write a couple of extended literature reviews. One was to be specific to the subject I taught and the other was to be more general. So I wrote about girls’ under-representation in the sciences, particular physical sciences and maths, as well as boys’ underachievement across the board. Writing about both topics enabled me to draw an interesting contrast. Many were quick to blame societal factors and discrimination for girls’ under-representation in science, even though a closer reading suggested it was more complex, and yet they did not do this when considering boys’ achievement.

The BBC podcast is interesting because David Grossman, the journalist, takes a refreshingly naive and open approach to the issues. His stance is that he is a father of boys and he just wants to figure out what is going on. He quickly determines that key differences between boys and girls emerge early, prior to school, and so schools are unlikely to cause these differences.

However, after exploring arguments about the relative roles that biology and the environment play in early boy/girl differences – for instance, does testosterone in the womb have an effect on boys? – he expresses surprise that so many of the academics he contacted for the programme did not want to discuss this aspect.

This is not surprising to those who are more accustomed to this debate. The average achievement of boys compared to girls is likely to be related to a complex mix of biological and environmental factors which, as Grossman eventually establishes, is ultimately irrelevant to the question of what works best to address this difference. However, anyone with an understanding of the academy in 2019 will be aware of the consensus that gendered behaviour is entirely socially constructed as a response to the environment (e.g. the patriarchy causes toxic masculinity). One academic that Grossman interviews suggests this may be an overcorrection of past errors made by ‘biological determinists’ who sought to simplistically attribute behavioural traits to biological differences in a manner that reinforced negative stereotypes and prejudices. This is plausible.

I still think these are interesting issues. However, I have noticed through the course of my career that the conversation around boys’ underachievement has become more muted. Even if it can be explained by toxic masculinity and therefore does not threaten sociological theories, it perhaps focuses on the wrong subjects. Maybe boys are not a fashionable group to be concerned about. If I am right, this represents a major problem, because boys’ underachievement is not going away and its consequences for all members of society are potentially dire.

How to fix peer review

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Twitter has been buzzing with discussions about the merits of the process of peer review. This was prompted by a blog post by Dr Pam Jarvis of Leeds Trinity University on the British Educational Research Association (BERA) website. Jarvis draws a distinction between peer-reviewed research and blogging:

“Blogs based purely on the personal experiences and opinions of one person may be presented by some writers (and consumed by some readers) on the same basis as more in-depth, academic peer reviewed publications… Blogging is certainly an excellent tool for both personal and collective reflection. However, all must be fully aware that the meticulous research and peer review processes that are undertaken within academic publishing means that self-published polemic can never be constructed as its equal in a policy advisory role.”

This is an interesting opinion to express in a blog post, not least because there is a slew of academic peer-reviewed research in education that is based largely, if not purely, on the personal experiences and opinions of one person. Such papers arise out of the postmodern shift towards subjectivity and academics have made-up a variety of terms to describe this kind of research. Andrew Old posted an amusing thread of such papers on Twitter and just today a research article came through my RSS feeds in which, “the participants explore the possibilities opened up in a diffractive, post-human space-time-mattering, where memory stories are not taken to be signs of individualized essential selves, but a means of tracing the space-time-mattering of the worlds they live in.” I am no expert in this kind of research, but it seemed to involve turning an anecdote about the classroom into a play. The researchers even made a diorama with the teacher represented by a plastic figure of a police officer.

Perhaps blogs are not the worst thing that can happen. Perhaps peer review is no guarantee that a piece of research has value, particularly in a policy advisory role. Perhaps this is down to the nature of peer review.

Technical review

I have been through a number of peer-review processes and it does not seem to be an optimal system. The fact that established researchers have a role as gatekeepers means that research that challenges established views and theories is always going to have a hard time. I am also unsure why reviewers should remain anonymous. Those who hold a professional opinion about a piece of research should stand behind that professional opinion. If it is a personal opinion then what value does it hold?

Part of the problem is that peer review aims to do a number of quite separate things from basic fact-checking to evaluation of the worth of a piece of research. Some reviewers have passed back to me useful technical corrections and even commentary on grammar. Others have made suggestions about how to present data and aspects of analysis. Comments on the validity of a particular experimental method and what it is capable of demonstrating are also largely technical judgments.

Although I have received a few comments on references, this strikes me as an area of priority. Researchers are evaluated on the number of citations they receive and this can create an incentive for researchers to cite their own papers or those of close colleagues, perhaps with the understanding that they may be cited in return. All papers should therefore have their references reviewed with this in mind. Computer programmes could do some of the leg work and then a researcher could take a view, perhaps deciding whether references are superfluous or redundant and being pretty ruthless in asking for excisions.

In addition, particular journals could publicly lay out their own technical requirements according to editorial decisions. For instance, they could specify whether certain data should be available for double-checking or whether they view it as appropriate to publish p-values. All of these technical aspects could be performed before publication.

Ideological review

However, alongside these technical review functions, peer review also aims to make more subjective evaluations of the value of research and whether it adds anything significant to the field. Reviewers are even free to take issue with the tone of an article. This is where bias has the greatest potential to affect the corpus of work that is published. If you have built your career around a particular model or theory then research that appears to falsify this theory will be unwelcome. Confirmation bias may cause you to dismiss this research by questioning its value or generalisability – which you can do with any research – and you are likely to perceive the authors as impolite, lacking respect for the foundations of your particular field.

Open-access journals

We can disrupt this system a number of ways. Firstly, we need to separate technical and ideological review. It would also seem wise to take technical review out of the direct control of the luminaries in any particular field so that this cannot be easily used as a Trojan horse for ideological review.

In my proposed system, journal publishing would cease to exist as it does at present. Instead, universities and learned societies would set up open-access journals committed to posting as much research as possible online. There would then be a two stage process.

Funds previously spent by university libraries on subscriptions would be diverted into the process of technical review, which would be performed by a mix of technical measures – checking grammar, checking the pattern of citations – and human judgement. The latter could be performed by graduate students as a modestly remunerated role for which they would also gain academic credit. They could even re-run the data, checking statistical models and perhaps learning about these models in the process. A graduate student who wants to check the validity of a non-parametric test that they are unfamiliar with would have a useful prompt for discussion with his or her supervisor.

The reviewers would be public and acknowledged by the journal. Given there are no space constraints involved in online publishing, a journal could commit to publishing every article that meets its publicly available technical standards. This would potentially get around the ‘file-drawer’ problem where studies with null or negative findings tend to remain unpublished and therefore we overestimate the strength of evidence in a particular area. The problem this would then cause would be one of inundation, with journals lacking the capacity to technically review all the papers they are sent. However, waiting lists could be made public and researchers could then decide whether to submit to a journal that is slightly less prestigious but that has a shorter queue. Prestigious journals would have an incentive to expand capacity so that the best research doesn’t go elsewhere.

What about value?

So, am I suggesting no role at all for discussions of the worth of particular studies? Anyone who follows my blog would find that unlikely. Instead, these comments, alongside any further technical comments, should be lodged publicly by leading researchers after publication in a process similar to the comments on a blog. However, this process would be controlled by the journal which would ensure that those making comment were who they said they were and that the paper authors had no veto over who is able to comment.

Given the new publishing landscape, the main metric for researchers would be citations and second-order citations (citations of papers in which they are cited), with publication in a prestigious journal becoming less of a factor than now.

Beware magpies

Everyone knows about Australia’s dangerous creatures: Crocodiles, snakes, spiders and even jellyfish fill the imaginations of tourists, but the fact is that you are unlikely to happen upon these critters often. I have lived here since 2010, I have personally encountered two potentially dangerous spiders, no snakes and no crocodiles. A much more significant problem are the magpies.

Magpies? To folks from the UK, the idea that magpies pose any kind of threat seems strange. But our magpies are not closely related to European magpies. They are a feistier sort of bird entirely.

Frustratingly, this is all based upon a misunderstanding. Magpies nest between August and October and the males have the job of guarding the nest. Most of them do not swoop, but the ones that do are trying to protect their young from attack. Fast moving runners and cyclists must appear particularly threatening to a magpie, even though few runners and cyclists have an interest in killing and eating magpie chicks.

I think education suffers from magpie-like behaviour. Suggest that systematic phonics is a better approach for teaching early reading than balanced literacy and the education magpies are likely to swoop you for being a neoliberal shill of Big Phonics bent on making children hate reading, when the reality is that phonics proponents just want more children to learn to read and it is balanced literacy programmes such as Fountas and Pinnell or Lucy Calkins that are the big money-spinners for publishers.

Similar swooping behaviour can be provoked by suggesting that it is important for children to learn and practice standard approaches to solving maths problems, that students need to commit facts to memory, that they should learn about powerful cultural ideas or read classic texts, or that student behaviour needs to be actively managed, school leadership need a plan for this and school exclusion is sometimes necessary. Anyone putting forward these ideas is likely to have their motives questioned even though all of these ideas can be advanced on their own terms with an appeal to evidence and rational argument. You may not agree with the validity of that evidence or those arguments, but that is an entirely different matter to assuming that your opponents are prejudiced conservatives who hate children and want to kill creativity.

Let’s anthropomorphise Australian magpies for a moment. I wonder what would happen if we could talk to them and explain that we had no dastardly designs on their babies. I wonder how they would feel about that. I suspect they would feel similar to teachers on Twitter when they learn of evidence they were never taught about at university or exposed to in training sessions run by school leaders and consultants. I suspect they would be angry at first and then relieved. I suspect they would feel unburdened. I suspect they would want to share this message.

You cannot blame the magpies. They are only trying to protect their young, no matter how misguided they are and, unfortunately, we cannot explain to them why. But educations magpies are all around us and they deserve the truth. I suggest you wear a helmet.

The false and damaging choice between facts and democracy

An article on improving education about democracy has been published in The Conversation by Edda Sant, Senior Lecturer in Education Studies at Manchester Metropolitan University. Reports of the death of democracy, whether greatly exaggerated or not, are a motif of our times. The impossible equation we are all trying to balance is the relationship between populism, the rule of law, expert opinion and truth. In a feverishly polarised atmosphere, neither side takes responsibility for its substantial part of the mess. So perhaps improving education is the way forward.

Such arguments tend to throw out paradoxes. Proponents of critical thinking may be found displaying a notable lack of it and instead encouraging something approaching indoctrination. However, this not Sant’s position. Instead, Sant’s view is possibly even more dangerous.

Firstly, what can we agree on? I am happy to support the following sentiment:

“Promoting democracy has always been one of the tasks of schools within democratic systems. But this demand is now on the rise. Indeed, across the globe teachers in schools are expected to engage students as democratic citizens. It’s hoped such lessons about democracy and what it is to be a good citizen will help to combat growing support for totalitarian and radical views.”

This seems uncontroversial. Yes, there are education systems that would not seek to promote democracy but they exist in undemocratic countries. Some people may differ, but I suggest it is a marginal and eccentric view to propose that education systems in democratic countries should be neutral on democracy. Let’s promote the idea.

Yet, what’s this? According to Sant, the promotion of democracy is in opposition to an alternative focus of schools in democratic systems:

“…when topics such as patriotism or historical conflicts are presented as ideas to be debated rather than facts to be learnt, students have time to form opinions and democracy benefits… Part of the problem is that in recent decades, there has been an increasing insistence on standardised tests… As a consequence, students learn there is a single correct answer for everything, including politics and democracy.”

Instead, we should apparently follow the example of, say, Brazil, where, “members of the school community democratically agree on school rules.”

I am not convinced that school rules represent the greatest democratic fulcrum of our times. What they do represent is the kind of topic that school students can construct a coherent argument about without much support because they know about the issues involved. This is why, in the absence of a properly sequenced, knowledge-rich curriculum, standardised writing assessments tend to ask students about whether schools should have uniforms or whether dogs make better pets than cats.

However, if we want to learn, as Sant suggests, from historical conflicts, then we hit a snag. Take the War of 1812, for instance. What would it take for students to have an opinion on something like that? First of all, you would need to know that it was a war between Great Britain and the United States. You would need to know about the Chesapeake-Leopard affair and the Little Belt affair, the British naval blockade of Napoleonic France (and hence some conception of what Napoleonic France was, how it arose out of the French revolution and why it was an antagonist of Britian). You would need to understand the context of the westward expansion of the U.S. and how this led to Britain forging alliances with first nations peoples. And you may want to be able to illustrate your argument with some vivid details such as the British attack on the White House.

But all these things you would need to know could be described as ‘facts’ – the kind of facts you might be expected to know for an exam. Yes, constructing an argument relies on more than just knowing facts, but you still need to know those facts otherwise what kind of argument can you construct? It has always been this way. Classic essay questions have involved students of history in evaluating the causes of the first world war or discussing whether the bombing of Hiroshima was justifiable. This is how history has traditionally been taught and Sant’s idea that exams cause students to learn a single correct answer to such questions is faintly absurd. We might prompt our notional history students with, “The War of 1812 was the result of British naval arrogance. Discuss.” We would not expect much from any responses in the absence of sufficient facts.

And it is odd to cite knowledge of facts as part of the problem in an argument about democratic malaise. One of the criticisms of populists such as Trump is their apparent disdain for facts or their preference for manufacturing their own alternative facts. If there is one thing that we may call upon to see us through the current mess it may indeed be facts. A populace educated to the point of knowing lots of facts and being able to deploy those facts in sophisticated arguments may take a dimmer view of demagoguery. I hope so.