The trouble with men

I’ve never really liked alpha males. I remember being in my late teens. My group of friends would always go to the same pub, sit around the same table and have a conversation dominated by our alpha male. It was mostly about mocking various members of the group or quoting bits from “The Smell of Reeves and Mortimer”. If someone came up with an original joke then it was promptly forgotten. Unless it was the alpha male, in which case it would be repeated until the end of days.

I hate that thing where alpha males hold court and you have to listen to their ill-informed opinions, waiting to get a word in edgeways. At that time, I had a reputation for grumpiness which was partly well-deserved as an angsty, guitar-playing teen but partly because I sometimes urged us to go to other places and do other things. I remember the exhilarating feeling of arriving at university and suddenly being able to stay up until 3.00 am and talk about politics without anyone laying into me for being pretentious. I went to parties. I talked to girls. I danced. I didn’t look back.

There are men and there are men. We are a diverse bunch. Yes, we are overrepresented in the upper echelons of business and politics but we are also overrepresented in the suicide statistics. For every mediocre man lifted into a position of wealth and power on a tide of privilege, there is a tortured soul unsure how to continue.

Boys also seem to fare badly in our education systems, particularly the poorest boys. And men’s health is generally worse. We die younger and more money is spent on women’s medical issues than men’s. And yet men retire later.

It is worth noting that this is largely the fault of men. The difference in suicide rates between the sexes has stayed remarkably constant over the years and extends well back into a time before feminism was a societal force. So it’s not about that. And it’s not as if greedy women are stealing men’s health money; it’s because men simply don’t talk about their health very much; they don’t seek help or campaign. It is men who keep it low profile.

Even the problems in boys’ education are our own fault. We have maladapted views of manhood that don’t fit well with hard, academic work and gaining the best qualifications.

So that’s alright then. Some men win big in life so there’s a kind of justice in the fact that others are unhealthy, unhappy and uneducated.

I’m not sure.

An open, generous kind of feminism would look at ‘men’s problems’ and see them as a manifestation of the same gender stereotypes that harm women. An open, generous kind of feminism would be progressive and seek to improve on the past rather than accept what has always been. It might be sceptical, but it would hope that “International Men’s Day” might raise the profile of these issues and start men talking about them more. It would accept that problems such as male-rape and violence against men are worth addressing and are more likely to be addressed if there are recognised forums in which to talk about them.

And an embracing attitude that recognises the legitimate components of the men’s movement offers the prospect of moving us forward in a broad, humanist endeavour. It marginalises the cranky, reactionary, anti-feminist elements.

On the other hand, an inward-looking feminism would instead see the position of the sexes as a zero-sum game. An inward-looking feminism would imagine that nothing can be improved for men without incurring a cost to women. Some feminists and their virtue-signalling male supporters might take the position that, in a culture of victimhood, women are the greater victims – which they are – and must therefore have only their issues addressed.

I have no idea whether the organisers of International Men’s Day can help deal with the issues facing today’s men. But we should at least have the generosity of spirit to let them try. Remember, we’re not all alpha males.

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Hattie on inquiry learning

Penny Bentley (@penpln) recently tweeted a link to the following clip where John Hattie discusses inquiry learning:

I have a few comments that I would like to make.

Hattie’s view is consistent with cognitive load theory

It is reasonable to propose the use of inquiry learning once sufficient domain knowledge has been developed (even if Hattie’s choice of the pejorative term ‘shallow’ is unfortunate). In fact, this is a structural feature of our education system where the culmination of study in a particular area might be a research-based masters degree or a doctorate. However, it is also possible to conceive of ways of employing inquiry more immediately.

Imagine a science unit of work where, instead of a test at the end of the unit, there is a test two-thirds of the way through. Students may then move on to conducting some form of inquiry within this domain. Perhaps it might be contingent upon them demonstrating sufficient knowledge and understanding in the test and this would therefore act as a hinge-point, with some students being retaught the concepts rather than continuing to the inquiry.

We then might derive some significant gains from the inquiry component. Students will maybe deepen their understanding through application. For instance, if students had sufficient grounding in projectiles then a projectiles-based inquiry would further develop ideas around experimental methods and uncertainties, the effect of air resistance and so on.

It ain’t going to happen

The problem with such a model is that the reason teachers choose inquiry is linked to students having little domain knowledge of what they are inquiring into. Teachers think that students finding things out for themselves is motivating. They confuse the way that experts gain knew knowledge in a domain – epistemology – with the best way of teaching well-established knowledge to novices – pedagogy. This fundamental confusion between experts and novices is both widespread and misguided.

A student who already knows the theory behind projectiles is going to have a good idea about what is likely to happen and is not going to develop their own idiosyncratic hypotheses. Yet proponents of inquiry tend to value the ‘skill’ of inventing such hypotheses.

At best, inquiry enthusiasts will promote ‘just in time‘ teaching of concepts i.e. the focus is on the inquiry element, with teaching of domain knowledge minimised to as great an extent as possible.

The precise opposite has been proposed

The book “How People Learn” by Bransford et. al. for the National Academies Press in the US is highly influential and certainly good in parts. If anything, it can be be given credit for being freely available and for helping raise the profile of cognitive science in education, but I have two main problems with it. The exemplars of good teaching do not reflect what we know from research, and the following passage from Chapter 1 seems to conflate explicit instruction with ‘simply providing lectures’ and is in direct opposition to the position described by Hattie [my emphasis]:

“Fish Is Fish (Lionni, 1970) and attempts to teach children that the earth is round (Vosniadou and Brewer, 1989) show why simply providing lectures frequently does not work. Nevertheless, there are times, usually after people have first grappled with issues on their own, that “teaching by telling” can work extremely well (e.g., Schwartz and Bransford, 1998).”

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What it means if 20% of students have special educational needs

This is not a new fact, but I was recently reminded that 20% of students in English schools are classified as have a special educational need. I am sure that I heard this when I lived in the U.K. but I didn’t think about it much at the time. Reflecting on it now, it seems extraordinary.

Imagine we used some kind of test to identify students with special educational needs. Now, I know that we would not actually do this but humour me for a minute as I explain. We would expect the results to follow a normal distribution around a mean value. Normal distributions are mathematically well-described :

normal distribution

If we selected the lowest 20% of scores then some of these students would actually be within one standard deviation of the mean.

For comparison, the mean adult male height in Australia is about 176 cm (5’9″) and we could expect, based on other data, that the standard deviation is around 7 cm (3″). So a man who is 169 cm (5’6″) would be about one standard deviation below the mean. Would you classify such a man as being especially short? Indeed, is a guy who is 183 cm (6′) especially tall?

For interest, if we decided to only classify those students who are two standard deviations below the mean as having a special educational need (equivalent to an adult male height of 162 cm or 5’4″) then this would represent just over 2% of the population.

There is a problem with this comparison. Students are identified as having a special educational need in a number of different categories. So the 20% could be composed of students who are all in the bottom 16% in that particular category. I wonder how much of a factor this is. It seems likely that one special educational need will correlate to another.

One of the invidious aspects of seemingly benign education theories like learning styles is that they imply labelling students. You have to wonder how a teacher’s expectations change when a child is labelled as a ‘kinaesthetic learner’. Clearly, a label of ‘special educational needs’ in some way has to alter teachers’ perceptions of a student. What will this mean for the student?

Well, there is evidence that it could affect teacher assessments.

And what about early reading? The following is quite possible: Students are taught with less effective methods, a significant minority of these children therefore do not learn to read, they then gain the label of having a special educational need which explains this and so nobody seeks to scrutinise the teaching approach. I am sure that this doesn’t happen but it seems to me that it could.

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Can Jo Boaler grow your brain?

In September last year, UK broadcaster, Radio 4, transmitted an interview with Jo Boaler, a professor of maths education. I remarked on this at the time on my old blog. There were a number of claims that I disagreed with but one claim stood out as simply very strange.

Boaler stated that, “One of the recent studies showed us that when you make a mistake, your brain grows.” She then equated the firing of synapses with the brain ‘growing’ and went on to explain that, in the study, there were two bursts of synapse firing, with the first occurring before the participants knew that they had made a mistake.

Sarah Montague, the interviewer, was surprised by this. “How on earth can that be?” she asked. “If you don’t know you’ve made a mistake, why should a synapse fire…?”

It’s a good question and, at the time, I wrote that I would love to read the study in question.

I was reminded of this when I saw a new video that Jo Boaler has produced to explain the idea of a growth mindset to students (a video that coincides with the release of her new book; a book that I’m happy to review if anyone wants to send me a copy). She makes a very similar claim. “A research study found that when people make mistakes their brains grew more than when they got work right.” Again, referring to the firing of synapses, she states, “The first comes when you make a mistake and the second comes if and when you’re aware that you’ve made a mistake.”

This time, one of her students has the role of asking the obvious question, “But how can your brain grow if you don’t know you’ve made a mistake?”

Boaler replies that, “Your brain grows when it makes a mistake because it’s a time when your brain is challenged and struggling.”

I went back to my old post and, in particular, a comment from “Luke” who located the study in question. Boaler had mentioned the paper on Twitter when asked about it by Daniel Ansari. It is a 2011 article by Moser et. al. and is freely available on-line so you can read it for yourself.

In this experiment, 25 participants have to pick out whether the central letter in a string of letters is congruent with the surrounding or ‘flanking’ ones. It is an example of a flanker task. Two letter are used for each trial. For example, in the first trial, “M” and “N” are used. The string “MMMMM” is congruent and “MMNMM” is incongruent. It took me a while to get my head around this. Surely, this is the easiest task ever? Then I realised that the participants are under severe time pressure.

I am not a neuroscientist so forgive me if I misrepresent some of the subtleties in what I am about to explain. The participants had electrodes attached to their heads to measure electrical activity known to occur in the processing of mistakes. Note that they do not measure other brain activity. This means that the fact that they found more activity when participants made mistakes is both unsurprising and a finding that does not rule out the possibility that there was even more activity of other kinds in other areas of the brain when participants got the answers correct. Also, recording a voltage in this way is not the same as concluding that the brain has ‘grown’. Yet brain growth is the repeated claim.

The fact that one of these electrical bursts – the “Error-related negativity” (ERN) – occurs before participants are aware of their mistake now seems less mysterious. As the paper explains:

“Current conceptualizations suggest that the ERN and the Pe are dissociable neural signals involved in error processing, with the former reflecting conflict between the correct and the erroneous response and the latter reflecting awareness of and attention allocation to errors.”

Clearly, the participants have all of the information that they need in order to figure out if they are right or wrong – they don’t need any external feedback. This is one of the reasons that the researchers chose this particular task. It is quite possible that some of this is processed unconsciously before the participant becomes aware of the mistake. I sometimes ‘feel’ something when I write a typo before I realise exactly what I have done and I put this down to subconscious processing.

If you mine the references of the Moser paper, you eventually arrive at this paper by Nieuwenhuis et. al. (2003) which seems to be the source of a stronger claim. In this case, participants completed a ‘antisaccade test’ and the researchers found the ERN response even when the participants were unaware that they had made an error (although it was weaker than if they did become aware). However, it is worth looking at Wikipedia’s description of an antisaccade test:

“To perform the anti-saccade task, a patient is asked to fixate on an motionless target (such as a small dot). A stimulus is then presented to one side of the target. The patient is asked to make a saccade [eye movement] toward the opposite direction of the stimulus. For example, if a stimulus is presented to the left of the motionless target, the patient should look toward the right. Failure to inhibit a reflexive saccade is considered an error.”

So the purpose is to resist a reflex. Again, although not consciously aware, the participants do have access to the information that they need to determine whether they have made an error. So it is possible that they are processing this unconsciously.

The ERN is interesting and the literature on it contests whether it really is about error processing or noticing conflict. However, these particular experiments seem very far removed from somebody solving a typical school maths problem, making an error, not realising they have made an error and having their brain grow in response. Which was the meaning that I had taken from Boaler’s statements.

The new video also adds another layer to the brain growth argument.

“Your brain is like a muscle,” we are told, “the more you exercise it, the bigger it gets.” This is an alarming prospect given that our skulls are a fixed size.

We are then introduced to Cameron, a little girl who had the right hemisphere of her brain surgically removed in order to treat a rare disease that produced violent seizures. Boaler’s students inform us that, “The doctors expected Cameron to be paralysed for a long time, maybe forever because she lost part of her brain that controls physical movement. But she shocked doctors and scientists as, within months, she was running around again.” Apparently, “the doctors could only conclude that the missing side of her brain had, in effect, regrown.”

It is not hard to research Cameron’s story. This news report contains a quote from Cameron’s surgeon:

“We like to do children because of their ability or their plasticity — that’s the ability of the other side of the brain that we haven’t removed to take over and control the function of the diseased half we’re removing,”

So it seems that the doctors expected Cameron’s left hemisphere to take over. They weren’t shocked by this at all. Indeed, it is unclear why anyone would want to perform such surgery without expecting a positive outcome. And it doesn’t sound like the brain had ‘in effect, regrown’. Instead, the left hemisphere rewired itself to take over some of the functions of the missing right brain.

In this piece on the University of New South Wales wikispaces, there is a much more detailed discussion of the case. It seems that there have been similar cases before, enough that, “The neurosurgeon, Dr. George Jallo was confident that Cameron would make a full recovery after the hemispherectomy. This surgery can be performed successfully on children because of the ability of the remaining hemisphere to compensate for the removed, diseased hemisphere,” and that, “Children who have undergone hemispherectomies are often able to regain the ability to talk and walk, although fine motor control of the contra-lateral side remains impaired.”

In short, these are all very odd pieces of evidence to use in order to make the claims that Boaler makes. I do understand that the new video is intended for children but, as I noted at the start, similar claims have been made in other forums.

Can your brain ‘grow’ in the way Boaler describes? Only if you consider electrical activity in the brain – which every living person has – to be ‘growth’ or the ability for a child’s damaged brain to rewire itself to be ‘growth’.

grow your brain

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Some comments on explanations in maths

Barry Garelick and Katharine Beals wrote an excellent piece in The Atlantic on their scepticism about requiring students to write explanations for maths problems. Dan Meyer then wrote a response and I, along with many others, jumped in on the comments. It’s a great discussion but I wonder whether a couple of points touched on by Garelick and Beal have been missed along the way. They are:

1. Understanding is latent and cannot be measured directly

2. There is no reason to think that prose explanations are better at exposing understanding than correct or incorrect solutions

It seems that many people – let’s call them the ‘explanationists’ – think explanations are important because they give us direct access to what the student understands. Well, they might. In class, I ask my students to explain their thinking all the time. However, in the numerous examples that people tend to post online (this is my favourite), these explanations sit in assessments and the debate in the US seems to centre around Common Core ‘aligned’ tests.

The issue raised by the explanationists is that perhaps students have ‘rote’ learnt a procedure. If this is the case, they will be able to get the right answer without understanding why. Even showing the correct mathematical steps – the ‘workings’ as I would call them – is not sufficient for the explanationists because a student might have ‘rote’ learnt these too. I am quite sceptical about this elevation of ‘understanding’ as the primary aim of maths and maybe we could leave it at that, as some more traditionally-minded teachers are inclined to do.

Yet strangely, we may take the argument of the explanationists and actually use it against explanations. Who is to say that these cannot be ‘rote’ learnt either? Have these teachers never taught any other subjects? I do. I teach VCE physics and I teach lots of explanations. For instance, if asked to explain the role of a split-ring commutator in a D.C. Motor, I tell students to write:

“The split ring commutator reverses the direction of the current through the coil every half turn, thereby keeping the torque in the same direction and therefore the coil rotating in a constant direction.”

could rely on them to construct a response from their own understanding. However, this would be fraught. It’s a pretty tricky thing to explain, there is a chance that they won’t cover all of the points and they might say something like “keeping the torque the same” rather than “keeping the torque in the same direction”. This would be technically wrong for the kind of motor that students are typically asked about. And so I teach them an answer that I’ve derived from examiners’ reports.

I therefore cannot really tell whether students understand the principle of the split-ring commutator by analysing responses to this question, although I suspect it is far easier to memorise an answer if you have a good understanding of what it means.

You may disagree with the principle of me teaching this explanation in this way. You might hiss that I am “teaching to the test”. Perhaps, but that’s not what this post is about. The point is that it is quite possible to memorise an explanation. If we think students might be motivated to ‘rote’ memorise a procedure for a test then the same motivation might make them ‘rote’ memorise an explanation. Of course, a skilful questioner will vary the questions to avoid some of this. This is why externally set tests are so valuable – you don’t know what’s going to come up. But a good question will expose a misunderstanding just as easily as asking students to write an explanation.

For instance, I am going to draw heavily on Dylan Wiliam here and propose the following question:

fractions question

 

 

 

 

A student who selects ‘B’ is likely to have the misunderstanding that the larger the denominator, the smaller the fraction. This is a classic maths misconception because it results from overgeneralising something that is true. A student who picks ‘C’ is very likely to have the correct understanding. Of course, we can’t be sure. It could be chance. Or, the student might have been trained in a procedure for answering this kind of question without really understanding how it works (I’m not sure what that would look like). Even something as simple as seeing a question like this before might prompt a student to pause, remembering that it wasn’t as straightforward as they had first thought, rather than just writing down ‘B’.

Judicious use of such questions is at least as likely to expose a students’ understanding as requiring written explanations.

Of course, everyone irrationally hates multiple-choice questions. Perhaps we should be assessing students using real-world investigations that are marked against rubrics? Would such an extended response solve the problem of memorising procedures and/or explanations and ensure that we are assessing true understanding? No. Talk about this with your colleagues who teach English or history and you will realise that assessment through rubrics is no picnic. Rubrics are just as gameable as any other form of assessment:

How rubrics fail - Greg Ashman

For some students, maths is a respite from literacy-based activities; reports, investigations, researching stuff on Google. If we require written explanations in maths tests then we are making maths performance contingent on literacy ability. This will disadvantage those who can do the maths and understand the maths but have low literacy, such as those who are still learning English. It means that our test is not valid; it is not measuring the thing that it is supposed to be measuring.

Given that the language of maths – the symbols and operators – have been developed over time to precisely describe notions that are often quite difficult to put into words, it also seems a little perverse to insist on such backward translation from maths into English as evidence of understanding when there is a good chance that it is nothing of the sort.

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5 Things to consider in Primary School Maths

Following my recent posts on questions to ask your child’s primary school teacher (here and here), I had a request to expand on my comments about the teaching of mathematics. There are a few issues surrounding maths that I believe parents should know about but before I go into that, I wish to make two points. Firstly, a fundamentally misconceived maths program taught by dedicated, evaluative teachers will always be better than one that meets the highest standards of evidence but that is taught badly. In my school we have specialist mathematics teachers and I think this is far more important than the specific details of the program. Secondly, the intention of this post is to inform parents and not to have a go at primary school teachers. Some teachers were offended at my comment that time-tables songs were not the best way to memorise tables. They thought I was suggesting that this is what many primary school teachers do. No – I was just setting up two contrasting alternatives in order to explain my point.

1. Discovery learning

Discovery learning is ineffective and most people tend to recognise this. So you don’t see many schools advertising their programs as discovery learning (apart from in AITSL’s illustrations of their teaching standards, bizarrely). Yet if it looks like a duck and quacks like a duck then it probably is a duck. And there are two powerful fallacies that drive people towards discovery learning. The first is the idea that we understand something better if we discover it for ourselves. We don’t. Secondly, we tend to assume that by asking students to emulate the behaviour of experts then our students will themselves become experts. Experts in maths are research mathematicians who make new discoveries so we should get our students doing that. Yet this is also fallacious thinking.

In primary maths, discovery learning takes on the form of ‘multiple’, ‘alternative’ or ‘invented’ strategies. Students are intended to make-up their own ways of solving problems and to solve a single problem in several different ways. Explicitly teaching a standard approach, such as the standard algorithm for addition, is discouraged. Of course, many students don’t discover much and so they pick-up these strategies from others or are led toward them by the teacher.

Standard Algorithm for Addition

Standard Algorithm for Addition

2. Big to little or little to big?

The kinds of alternative strategies that the students ‘discover’ are generally variations of strategies that we might use for mental arithmetic. Imagine I wanted to add 25 and 49. I would probably first add the 20 and the 40 to make 60. Then I can add the 5 and the 9 to get 14. Sometimes, little sub-moves will be encouraged as part of this e.g. take 1 from the 5 and add it to the 9 to get another 10 so that we have 70, then add the remaining 8.

Notice how this proceeds from big to little. We add the tens first and then the units. But when we add the units we find that we have yet another ten so we have to loop back and add this to the tens that we already had. This is inefficient when we get to larger numbers and is the reason why the standard algorithms generally start with the units first, then tens and so on. Indeed, students who use the standard approach seem to have more success, particular with larger and more complex calculations.

The objection to standard algorithms seems to be that kids can learn them as a process without understanding how they work. Presumably, they have to understand procedures they’ve invented themselves? This may be true if they really have invented them but I suspect such genuine invention is rare, with most children latching on to the ideas of others. In this case, these alternative procedures could also be replicated as a process without understanding.

In my view, students should be taught the standard approach and this approach should be explained to them. This requires the teachers to also understand how these processes work.

3. Words and pictures or actual maths?

Given that alternative strategies are meant to be ad hoc and contingent, there is no formal way for expressing them. You may see it done with pictures or even in words. Contrast this with the standard algorithms – their universality means that they follow a tightly defined set of notation. One way that alternative strategies may therefore be promoted is by insisting that students ‘explain their reasoning’ or draw diagrams when answering questions on homework or assessments. This is basically a way of marginalising the standard algorithms.

For instance, imagine the following question:

“A lottery syndicate of 13 people wins a total of $3 250 000. If the money is shared equally then how much would each member receive?”

A simple use of the long division algorithm is sufficient to explain what the student is doing, why they are doing it and to determine the right answer. If the student has gone wrong then the error will be easy to find. An insistence on words or pictures would be redundant unless you wish to privilege alternative strategies.

4. Maths anxiety and motivation

Maths anxiety is real. Some people struggle with maths – perhaps because they were not taught very well – and develop a fear of maths tests and maths more generally. It is complex and the chain of cause and effect is not entirely clear. Evidence does seem to point to timed tests as being associated with anxiety but perhaps better test preparation and framing would mitigate this. However, as well as advising us against timed tests, a whole raft of things that look and smell a lot like alternative strategies and related ideas such as the use of ‘authentic’ problems are proposed as possible solutions.

Authentic, real-world problem are considered good because the idea is that they will motivate children and so the children will learn more. In fact, a lot of discussion centres around motivating and engaging students. I am sceptical that many of the activities that are suggested as motivational are actually motivating for students and evidence suggests that motivation works the other way around. Maths achievement predicts motivation but motivation does not predict achievement. In other words, teach them maths, increase their competence and then they will start to feel more motivated about maths.

5. Cognitive load

Finally, it is worth mentioning that a lot of fashionable strategies are at odds with what we know about human cognition. Children should know their maths facts because that means that they don’t have to work out 5 x 8 whilst attending to other aspects of a complex problem. Those people who dismiss times-tables as ‘rote’ learning fail to take account of this. And so do those who propose big, messy, open-ended, real-world problems. Such problems have many facets and often contain information that is irrelevant to finding a solution. All of this needlessly increases cognitive load and makes learning less efficient.

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Narrative Do-ology

Making Learners Extraordinary ™

I am in the classroom ofJulian Malvolio atBayswater Elementary in Kunnunna, UE. He is setting up for theday; about to teach his Grade 5 creative writing class. We take five to have a chat about what’s bugging him.

“I wasfinding that the kids were just replicating a procedure. They weren’t thinking about it. There were just going through it.”

I have been thinking about this problemfor some time and, after much deliberation, I have decided that I am right. The crux-pointis one of a lack ofunderstanding. Sure, kids can go through some kind of procedure to write a story, select characters and so on. But the ways this isdone can be learnt and practised rote. There is much more to writing than simply placing words together, one after the other. Often, the writing is derivative, focusing on wizards and dragons.

This is why here, at the Extraordinary Learning…

View original post 217 more words

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Motivating students about maths

One of the long running arguments that I have been involved with is an argument about motivation. When I promote explicit instruction, it is sometimes suggested that I have neglected the role of motivation altogether. People think that it is little use showing that explicit instruction maximises learning if it puts kids off; if it’s demotivating. In commenting on one of my blog posts, Dan Meyer claimed that I, “hypothesize that learning and motivation trade against each other, that we can choose one or the other but not both.”

This is not the case. I am actually pretty interested in motivation. I just happen to have a different theory of motivation to many people. I have even gone to the trouble of creating a graphic to explain this (I have also written that I reject the dubious idea that explicit instruction is inherently less interesting and therefore less motivating than the proposed alternatives, but that’s a separate issue).

The problem is that achievement and motivation are correlated with each other and so many commentators then assume that we can increase achievement by taking steps to increase motivation; motivation causes achievement. This is often the rationale for inquiry learning initiatives (e.g. here and here). Yet we can’t be sure that it works this way around. It could just as well be true that achievement causes intrinsic motivation. This is what I tend to think. Or it could be the case that the cause acts in both directions; a virtuous circle.

A new paper has been published (thanks to @MrPABruno for the link) that offers some support for my view. It looked at the interaction between achievement and intrinsic motivation in maths in Grades 1 to 4. The discussion is full of caveats, as we might expect of a good academic paper. We can’t necessarily generalise the claims beyond the scope of the study. For instance, we should not assume that the results would be replicated in Middle School science.

Yet the findings are quite clear. “Cross-lagged models showed that achievement predicted intrinsic motivation from Grades 1 to 2, and from Grades 2 to 4. However, intrinsic motivation did not predict achievement at any time.” This falsifies a key prediction of self-determination theory. A separate finding of perhaps even greater importance was that attitudes harden very early in education and so primary school maths is critical for how students will see themselves mathematically in the future.

It would be cool if I could claim credit for great foresight but my views about this are actually based on research from the early 1970s. In Project Follow Through, students who were taught using the model that did the most to improve achievement – Direct Instruction – saw greater gains in self-esteem than students taught using models that directly targeted self-esteem.

So we seem to be building quite a consistent body of evidence.

Mathematics Achievement

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‘Burden of proof’ and other poor arguments

There are lots of poor arguments used in education discussions. I have recently been at the wrong end of one when a genuine disagreement with another blogger was unpleasantly conflated with trolling and abuse. This is upsetting and yet it is sadly par for the course.

 

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I am with Paul Graham on this – if we avoid bad argument and focus instead on disagreeing with the main idea that somebody is putting forward then we would all feel a lot better. I would certainly prefer that. And it is a big improvement on endless, unresolvable discussions of ‘tone’. As Graham says, tone is, “a weak form of disagreement. It matters much more whether the author is wrong or right than what his tone is. Especially since tone is so hard to judge. Someone who has a chip on their shoulder about some topic might be offended by a tone that to other readers seemed neutral.”

I wish to particularly note a logical error that crops up every time I write about E D Hirsch and that arises for seemingly ideological reasons. His more strident critics will openly accuse him of racism but, more often, it is heavily implied. The error is the fact that these critics have never produced, as far as I am aware, any evidence of racism at all. Instead, they tend to demand that I prove that he is not a racist.

This is known as ‘shifting the burden of proof’.

In a rational argument, it is for the person making the claim to substantiate that claim. It is not for those who are sceptical to prove the claim wrong. To do so is often quite impossible. The most famous example of this kind of argument is Bertrand Russell’s teapot and it goes like this:

Simplicio: There is a teapot orbiting the Sun at some point between the Earth and Mars.

Sagredo: That seems unlikely.

Simplicio: Prove to me that it is not the case.

Sagredo: I cannot.

Simplicio: Aha! I am right!

Please try to avoid this fallacy when debating education.

And it is worth listening to Bertrand Russell himself:

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Inquiry maths is not the answer

I wrote this for an Australian newspaper but they decided not to run with it so I’m posting it on my blog:

New federal education minister Simon Birmingham recently announced nearly six and a half million dollars of government funding for the ‘Mathematics by Inquiry’ initiative for Prep to Year 10 students. Noting that, “when people think of maths they usually think textbooks,” he suggested, “These new curriculum and teaching materials will help make maths more meaningful and more attractive, to students by showing them how they use maths in their own experiences, careers and lives, in a range of everyday situations.”

What’s not to love about such a project? Participation in maths and science seems to be on the slide and teaching maths through real-world contexts sounds like an obvious way to make it more  interesting and relevant to students. The idea has the quality that the American comedian Stephen Colbert has termed ‘truthiness’. It feels right; from the gut.

Indeed it is such an obvious solution that it has been thought of before. As far back as the 18th Century, the philosopher Rousseau was expressing similar ideas about how children should be taught but perhaps the most influential exponent of ‘learning by doing’ was the American academic, John Dewey. At the start of the 20th century he did more than any other to popularise this idea and kick-start the progressive education movement.

Unfortunately, this is one of those grand ideas that sound plausible and appealing in theory but don’t work in practice. The benefit of large-scale educational research is that it can sift through the plausible to find the stuff that has real benefits. And inquiry learning does not appear to be such an approach.

For instance, Professor John Hattie of Melbourne University compared research into different teaching methods and concluded that, “the results show that active and guided instruction is much more effective than unguided, facilitative instruction,” such as inquiry learning. Moreover, a study by the American researchers, David Klahr and Milena Nigam, found that the few students who managed to discover a key scientific principle for themselves understood it no better than students who were explicitly taught the principle.

To understand why inquiry learning is not as effective as explicit instruction – where students are given carefully sequenced and structured lessons of increasing depth and complexity — apply the same idea to a trainee surgeon. Dull classes in anatomy and physiology would be cast aside. Instead, we would hand over the scalpel and see if our students could figure out what was wrong. This would never happen, of course, because people would die. Giving a novice maths student a real-world problem to solve is a similar affair. She won’t know what to focus on. Her lack of experience will mean that she cannot draw parallels with other problems. She might be lacking in the basic skills needed. Trying to execute these skills will take up most of her attention.

Cognitive science research has shown that the number of things that we can focus on at any one time is severely limited. A well thought out, sequential teaching sequence introduces new concepts drip by drip whereas an inquiry learning approach tends to overload students who are new to an area of study. We all experience being thrown in at the deep end from time-to-time and it is not pleasant.

We could counter-argue that inquiry learning should have a bit more scaffolding; a few hints and tips to send our student in the right direction. Even so, we are still not teaching skills in a systematic way that builds logically from one to the next. It’s a little like trying to learn to catch a ball by playing games of cricket. You might go through a whole match without having to make a catch.

The evidence may show that inquiry is not the best way to learn something new, but what if it turns more students on to maths? That has to be a bonus, right?

Let’s take this argument at face value and assume that a mathematics lesson that avoids the abstract in favour of the commonplace and mundane is something that kids really enjoy, even if they don’t learn a great deal from it. This still brings to mind a quote attributed to the American Entrepreneur Jim Rohn, “motivation alone is not enough. If you have an idiot and you motivate him, now you have a motivated idiot.”

The answer to declining mathematics and science performance and participation is not inquiry learning and not a dumbing-down of the curriculum. Those strategies are not going to help us outperform the Chinese. Rather, giving children sound basic mathematical skills on which to build provides them with a firm foundation and the confidence to tackle more advanced courses later in schooling.

This is a more long-term and challenging approach but it is more in keeping with what we know about learning than throwing money at a gimmicky scheme.

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