Following my piece on the reading wars, I thought it would be worth writing a brief for parents on the maths wars. These are not as high profile as the reading wars but they have had a similar impact, especially in the area of early numeracy.
1. What is ‘constructivism’ and why does it matter?
Constructivism is a theory about how we learn. It quite reasonably claims that children are not blank slates. Instead, they relate new knowledge to things that they already know. These are organised as ‘schema’ in the mind. It is the process of ‘constructing’ these schema that gives ‘constructivism’ its name.
As far as this goes, there is little to disagree with. Effective teachers will always try to tease out what students already know and give examples and analogies that they can relate to. Learning how to do this is part of learning the craft of teaching and it is probably one of the factors that differentiates more effective teachers from less effective ones. However, some educationalists take this further. They equate the active construction of mental schema, something that can occur while listening to a teacher, with the need for children to be physically doing something. They also assume that it is preferable for students to figure out their own strategies for solving problems rather than using strategies given to them by the teacher. Some assume that children may only change their ideas through the process of cognitive conflict and that this involves sustained struggle.
The result of these fashionable beliefs is to favour a style of maths teaching where students are expected to learn key principles through engagement in various activities rather than being taught them directly by the teacher. The teacher acts as a kind of guide or facilitator. Unfortunately, we now have plenty of evidence that this is a less effective approach than explicit teaching, especially for students who struggle with maths.
2. There is nothing wrong with timed testing.
Jo Boaler, Professor of Mathematics Education at Stanford University, has a rock star status among many maths teachers. She can attract large crowds to conferences and has set-up a popular maths teaching site, YouCubed. One of Boaler’s contentions is that timed maths tests are harmful because they trigger maths anxiety, a phenomenon where children start to feel worried about maths lessons and assessments.
This is a plausible idea. We can all imagine that timed tests could be stressful for children. However, I think it is also important to point out that a skilled teacher should be able to use these kinds of teaching methods in a healthy and supportive way. In fact, it is not quite clear which research Boaler is citing when making this claim. In a review of Boaler’s “Mathematical Mindsets” book, Victoria Simms, lecturer in psychology at Ulster University, chased the contention about maths anxiety back through the references to the YouCubed website where the trail went cold.
On the other hand, the potential benefits of timed testing are clear. Essentially, we would like children to simply know plenty of basic maths facts e.g. that 7 x 7 = 49 without having to work them out. This is because they can then devote mental effort, which is relatively limited, to other aspects of solving a problem. By using timed tests of maths facts such as these, we can determine that students have learnt them to the point of automatic recall rather than that they are working them out using some kind of strategy. This is so important that I would encourage parents to supplement school maths with these kinds of timed tests at home, especially if they do not form part of the school programme.
3. There is nothing wrong with learning standard algorithms.
Another aspect of mathematics that has gone out of fashion is teaching children the standard algorithms for addition, subtraction, multiplication and division. The first three of these involve arranging the numbers above each other in columns and working from the smallest place value, such as the units, up to the largest place value which might be tens, hundreds, thousands or more. The process involves what many people refer to as ‘borrowing’ or ‘carrying’ but that experts tend to call ‘regrouping’.
Instead of teaching the standard algorithms, many educators prefer students to invent their own strategies for performing these operations. The extent to which children actually invent them is questionable because they all follow a similar pattern – they are basically variations on ways of doing mental maths. For instance, if you want to add 25 to 47 you might take 3 from the 25 and add it to the 47 first in order to make the 47 up to 50. You can then perform the relatively simply calculation of 50 + 22 = 72.
Such strategies are great and should be in the repertoire of students but not at the expense of the standard algorithms. The standard approaches are far more powerful because of the way that they work from small to large. They also require a strong understanding of place value and develop understandings that are needed for higher level maths. For instance, students of Mathematical Methods, a higher level maths course for Year 11 students in my state of Victoria, are required to do polynomial long division, something that is tricky if they have never learnt ordinary long division.
The case against the standard algorithms seems to be that they can be learnt without understanding, something we will return to below. A paper by Kamii and Dominick is often cited as providing evidence that invented strategies are superior to standard algorithms. However, in my view it does not use the strongest experimental design. Moreover, when Australian researcher Stephen Norton completed similar research using more complex calculations, he found the reverse effect; an advantage for standard algorithms.
So children should be exposed both to mental arithmetic strategies and the standard algorithms. If they don’t get the latter at school then you should consider ways of providing this at home.
4. Understanding can develop alongside procedural knowledge.
The idea of developing an understanding of mathematics is often set in opposition to students learning procedures for solving problems. Actually, it is teachers in Western countries that tend to do this. East Asian maths teachers also want students to understand but seem more relaxed about whether this comes before or after learning a procedure.
The reality is the common sense idea that procedural fluency and understanding develop in parallel with one feeding into the other. The idea that procedural fluency can somehow be harmful to understanding is probably as absurd as it sounds and I am aware of little evidence to support it.
5. Motivation comes from a developing sense of competence
Maths is often seen as boring or hard work. Many students are turned off by the subject and there is constant commentary in the Western media about the need for more maths graduates or graduates from the numerate science, technology and engineering disciplines.
This leads to a lot of woolly thinking: We need to make maths fun! We need to make it more engaging! We need more games! We need more projects! We need more visits from real mathematicians! We need to make maths more real life! We need to make school maths more like what professional mathematicians do!
All of these approaches may lead to a passing or ‘situational’ interest. However, if we wish to build students’ long-term motivation for a subject then a better strategy might be to teach them well so that they become more competent. It seems likely that students will be turned off a subject they find frustrating and in which they have little success. On the contrary, gradually gaining mastery will make it more appealing. This is why teacher effectiveness research suggests that we ensure students obtain a high success rate and why evidence from long term studies suggests that achievement leads to later motivation.
End the war and start rebuilding
It’s time to walk away from grandiose ideologies and focus on practical strategies, based in the science of learning.
A good start for parents would be to ensure that children know their maths facts, whether they learn these at school or not.
[This article has been removed because it was a comment made in good faith on a piece that has been retracted by the publication in which it was published. You can read about this below:
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How can a maths teacher use the power of storytelling? I used to think this would involve narratives about great mathematicians and their discoveries. Yet, ultimately, such tales are not the most important thing for students to remember.
I started to think about stories after reading Dan Willingham’s, “Why don’t students like school?” but I couldn’t really settle on what I needed to do and threw narrative out as unsuitable for maths. Then, as I became more involved in the development of writing at my school, I started to understand more about the narrative arc that most stories follow; exposition, complication, crisis and resolution. There seems to be something primal about this pattern because it is replicated across cultures.
So I started to think about worked examples. A worked example is, of course, a story. It starts by defining the characters in the story, perhaps by picking them out from a background scenario. Then some action happens before the problem is resolved. Yet if we think of the performance of a worked example as the telling of a story then we can emphasise the stages of the narrative arc.
What does this look like? Well I have started to sometimes inject a little jeopardy; will I or won’t I solve the problem? So I become a character too. And I ham that up a little. Just as with bedtime stories, the students know – I hope – that it will all work out in the end. I would certainly be very careful with this approach with a new class where the students might be sceptical of my abilities.
There are some hints from the research that focus on similar ideas. For instance, the embodiment principle from Mayer’s cognitive theory of multimedia learning suggests that students learn more from a human like on screen agent than they do if that agent is not present. This may be because the human element triggers empathy and so makes the story more compelling.
When I perform examples as stories, I am trying to gain and focus my students’ attention. I am trying to make the examples more compelling so that they will stay in the memory. And yes, I am trying to make the lesson interesting.
A couple of years ago, I reported on a large randomised controlled trial in the U.S. of Reading Recovery. I pointed out that, as with other studies of Reading Recovery, it was impossible to tell whether the instructional procedures used were responsible for any effect. Instead, any gains may have been due to the one-to-one tuition format of the intervention. After all, one-to-one tuition has been held up as a ideal form of instruction by none other than Benjamin Bloom.
Since my original post, I have also pointed out that, when compared on similar outcome measures, Reading Recovery tends to generate smaller effects than programmes based on systematic synthetic phonics (SSP). I am cautious about comparing effect sizes but such an approach has the greatest validity when comparing children of the same age learning the same content, as in this case. The greater effectiveness of SSP hardly surprising given the probable status of SSP as the educational intervention with the greatest amount of empirical evidence to support it.
Reading Recovery, on the other hand, seems to have evolved from a whole language approach to one that now incorporates phonics, although not in the same systematic way as SSP. It also seems to influence ‘balanced literacy’ teaching methods and so its impact stretches much further than the realm of intervention. I think that people are drawn to the narrative at the heart of Reading Recovery and start seeing early reading from this perspective.
So it was with interest that I read a new peer-reviewed paper published in the official journal of the Learning Disabilities Association of America. It returns to the trial that I commented upon in 2015. The authors note that the long term effect of the intervention was ‘not significant’ and that there was evidence that some of the lowest achieving groups of students were systematically excluded from Reading Recovery condition.
This is very worrying if policymakers, swayed by the original study, have decided to invest money in Reading Recovery as a strategy to tackle reading difficulties.
Perhaps, as this new evidence comes to light, the U.S. will make the move away from Reading Recovery that has already been initiated in Australia following reviews of the evidence here.
“If you feel permanently scarred from your experience in mathematics, you’re not to blame; it’s your teacher who is to blame.”
Dr. Daniel Mansfield speaking on ABC radio, 15th August 2017
Dr. Daniel Mansfield has been working with an ancient Babylonian clay tablet. What he has discovered about it is fascinating. A long time before Pythagoras, the Babylonians knew of Pythagoras’s theorem. Moreover, they had a version of trigonometry; the maths of triangles. However, instead of using angles to describe their triangles, the Babylonians used exact ratios of side lengths. This is similar to the way road signs display the steepness of a road as a ratio of rise over run.
The Babylonians had another trick to make things work better. Instead of basing their number system on ten, a hundred and so on – the decimal system – they based it on sixty. Sixty has the advantage of having more factors. You can divide 60 exactly in half and exactly into thirds but you can’t do the latter with a hundred. This is useful for expressing ratios and it is the reason why we have developed time measurement systems based on sixty. It is also the reason behind non-decimal weights and measures systems and, funnily enough, the way we measure angles in degrees.
It is odd to consider the idea that this ancient tablet might tell us something about a better way to teach maths, the subject of the ABC news report. Mansfield seems to think that using exact ratios instead of angles would transform the teaching of trigonometry, making it accessible and engaging.
This seems far-fetched, not least because the concept of ratio that this system relies upon, and related concepts such as fractions, are extremely difficult to learn. Learning these concepts is challenging for most learners. And yet we persist because we already know how important they are for mathematical understanding, particularly as students get older and move on from basic numeracy. And is Mansfield suggesting that this should go hand-in-hand with a base-sixty number system in order to rid us of all those decimals? As a maths teacher who has experience of teaching children to solve time problems, this sounds like a poor plan.
On the other hand, I suppose a bit of teacher-bashing makes for a good story.
This last week saw an explosion of pent-up frustration. It started with an argument sparked by a prominent edu-tweeter and then rapidly moved on to a critique of researchED.
You could almost sense the release of pressure. All of a sudden, armchair pundits everywhere were having their say on what researchED should be and how it should be run.
Since the initial diffuse criticisms, it has emerged that there are those out there who value different things. To them, researchED is ‘positivistic’, too critical of teacher education and too concerned with a narrow view of ‘what works’. They would prefer it to pursue and promote their own priorities rather than those of the people who run it and organise it all. And you can understand why. After all, researchED is pretty successful.
As a grassroots movement, researchED is capable of mobilising teachers on their precious weekends to come and listen to researchers and fellow teachers discuss the practical application of empirical education research. It has now grown into an international movement, running conferences in different continents. And it has achieved this without charging $600 a ticket thanks to partnering with a range of people and organisations who see its value.
Who wouldn’t want a piece of that?
Well, I have an idea. Don’t be disheartened if researchED doesn’t promote your agenda. Instead, why not consider starting a movement of your own? For instance, there could be a pomoED that is largely uninterested in practical applications and ‘what works’. Instead, it could focus on dialectic, critical pedagogy and all that kind of caper. Organisers would be free to monitor presenters’ ethnicities and genders in whatever way they saw fit.
All you would need to do is find the right partners, put it out there and see what happens. I’m sure that there are plenty of teachers literally dying at the thought of an opportunity like this.
Back in the mid 1990s, the U.K. was in the grip of a crisis caused by the emergence of Bovine Spongiform Encephalopathy (BSE). This was a fatal brain condition affecting cattle that became colloquially known as ‘mad cow disease’. People were concerned because it seemed as if cows had acquired the illness by eating sources of protein containing the remains of sheep that suffered from a similar disease, scrapie. If cows eating the remains of sheep could catch it then then could humans eating beef also be at risk?
I remember the reassurances at the time. We were repeatedly told that there was no evidence of a risk to humans. That was true. At that time there really was no evidence. That evidence came later with the outbreak of a related condition in people.
BSE is the best example that comes to mind to demonstrate that absence of evidence is not evidence of absence. This is important when considering the matter of the overreach of science.
So what has this to do with school uniform? If I set up a school in Britain or Australia then I would probably wish to have a uniform. I think it acts as a social leveler; children are less concerned about their clothes. It can also be a blessing to parents who only have to buy two or three versions of something and schools can choose to make these relatively cheap. Parents can also avoid a daily confrontation about what their children wear.
However, I recognise that this is no panacea. Some kids will still buy expensive shoes or push the boundaries of what is acceptable. Yet I’d rather a ritualistic confrontation centred on uniform than this same rebellious need breaking out in many and various other forms. It’s almost like the agreed theatre in which to do battle, like a football pitch or a chess board.
Yet I would never claim that there is evidence to support school uniforms. The Education Endowment Foundation have this about right in their summary. There is no evidence to say uniforms work and there really couldn’t be. Short of running a massive, unfeasible experiment, all we can do is look for correlations and schools rarely move from having no uniform one day to having a uniform the next day while changing nothing else.
And uniform is a few steps removed from achievement or behaviour. In order to affect either, a lot of other culturally specific chains of cause and effect would have to kick in. I once visited a grammar school in Germany and was shocked to see tables totally covered in graffiti. My hosts were unfazed: Teachers taught the lesson and it was entirely up to the students whether they paid any attention or not. The idea of adding a school uniform to this context and expecting it to have any effect is clearly absurd.
There is no evidence that school uniforms affect behaviour or achievement but that does not mean that there is evidence that they do not. That would be overreach. If someone suggested that a lack of scientific evidence disproves the case for uniforms then this would be an accurate example of a much misused word; ‘scientism’. Absence of evidence is not evidence of absence. It is not scientism to draw on scientific evidence where such evidence exists. It is scientism to imply a scientific basis for a conclusion that cannot be supported by science.
Differentiation is a similar issue to school uniforms in that it is big, amorphous and difficult to test. And yet differentiation is often promoted in a way that resembles scientism. It is highlighted as best practice and written into teaching standards rather than being something left to the professional judgement of teachers and schools.
I choose my positions with care. I argue that there is plenty of evidence for explicit instruction because there is. I argue for cognitive load theory because it has a pretty good set of studies behind it, even as I highlight the contentious parts. I point to the evidence amassed by three government reports when I promote systematic synthetic phonics.
Yes, I often object to ideas on the basis of a lack of evidence but that is when they are being promoted as if such evidence exists or is beyond question, or if they are being imposed on teachers by a power structure. In some cases, such as implicit forms of teaching, there really should be evidence by now.
I have an agenda and it’s pretty straightforward. But I’m not silly enough to make claims I can’t support.