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.