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.

Great piece and very informative – in the USA the new approach was described as ‘fuzzy maths’ and heavily criticised as it was not based on sound research. In Australia John Sweller also argues that repetition, rote learning and being able to automatically recall basic numbers and times tables are very important.

I read Greg’s article with great interest and agree with his comments. My argument , coming from a special ed. perspective, is that there are children who will never construct an understanding of maths concepts, if left largely to their own devices and a bucket of blocks etc. Children who are well taught even those who find maths challenging have a better chance of improving their maths abilities and actually developing an interest in maths.

Knowing maths facts has a life long advantage.

Good research based interventions in maths teaching just as have been shown in reading, are worth their weight in gold!

Reminds of me of a bizarrely bad PD by Professor Peter Sullivan I once attended where he told primary school teachers to give students the most difficult question on a concept first so that all students struggle and then get easier. Apparently when you finish off with the easy ones and the struggling students get it- it builds self-esteem. I was like ‘great they can do the easy ones, now what?!?’

6 x 9 = 54, or is it 8 x 7. Anybody,s guess!!!!

3x^4 + 2.5x^3 – 14x^2 + x -21.4 divided by 4x^2 – 7x – 10

Well go on, do it then !!!!!!!!!!!!!

Thanks Greg, very good read. Your point 4 here has applications well beyond maths – it also relates to the basic verb and noun paradigms in the study of an inflected language, which are avoided like the plague by most younger teachers these days (because they are so redolent of “older” forms of language teaching, before the days when the communicative activity dogma became pervasive). And of course point 5 is especially relevant for language teachers too.

…This is so important that I would encourage parents to supplement school maths with these kinds of timed tests at home…

I am seriously considering doing this, given the weird patchwork coverage of maths at my daughter’s school.

Good post. I really don’t think many parents realise how things have changed in schooling in general and maths in particular. Memorisation of any kind seems to be discouraged. I am finding this is still a problem in high school. My daughters don’t understand that they have to commit formulas/procedures/rules to memory to succeed and I am the one that must insist that they practise this. Both my daughters were actively discouraged from using standard algorithms in early-mid primary and replace them with “supposed” invented ideas which were in fact just very convoluted ways of doing something simple and completely decimated any confidence they may have had.

Luckily I realised early in my kids schooling what was happening – ie yr 3 – so while teachers where brushing off my concerns with, “don’t worry maths isn’t their thing but they are excellent English students”, I took them to Kumon where both learnt their times tables. That was absolutely invaluable and they were both really proud of themselves for accomplishing that task.

Again, any memory work was outsourced to parents not done in schools where it should be standard. As a result my daughter’s yr 6 teacher told me a few weeks ago that less than half his class of 25 know any of their times tables and many can not add or subtract simple numbers!

IN 2004 I wrote ‘Why our schools are failing’ and argued as below – very little has improved since then.

Creativity requires structure and discipline. After outlining the debates surrounding fuzzy maths and referring to the work of experts in the area of the psychology of learning mathematics, Hirsch states:

I believe you will get strong agreement from them on the following points: that varied and repeated practice leading to rapid recall and automaticity is necessary to higher-order problem-solving skills in both mathematics and the sciences. They would probably explain to you that lack of automaticity places limits on the mind’s channel capacity for higher-order problem-solving skills. They would tell you that only intelligently directed and repeated practice, leading to fast, automatic recall of math facts, and facility in computation and algebraic manipulation can one lead to effective real-world problem solving. Anderson, Geary, and Siegler would provide you with reliable facts, figures, and documentation to support their position, and these data would come not just from isolated lab experiments, but also from large-scale classroom results.

The Australian academic, John Sweller, is also highly critical of the belief amongst progressive educators that learning somehow arises intuitively or by accident. In arguing in favour of direct instruction, as opposed to discovery learning, Sweller states:

…information should always be presented in direct rather than indirect form… This principle applies equally to all educational contexts but flies in the face of much educational theory of the last few decades. Beginning with discovery learning in the 1960s and extending to the constructivist learning techniques of the 1980s and 1990s, enquiry based instructional techniques have gained a considerable following amongst educational theorists… In all cases, learners are required to discover information that needs to be learned rather than having the same information presented to them. There is no aspect of human cognitive architecture that suggest that enquiry-based learning should be superior to direct instructional guidance and much to suggest that it is likely to be inferior.

Well said.

Well said Greg. We’re having a very heated discussion over arithmetic in Canada and I’m really looking forward to our first ever ResearchED event in Toronto this November. Hoping that might shake some things up a bit.

On another front, it seems here that the conversation is really starting to shift. Primarily I think, because parents are fed up. When we currently have upwards of 50% of school aged children now attending outside tutoring centres just to learn their timed tables or spend hours with mum and dad over the dinner table…ever single night…parents are beyond frustrated, As are teachers. Maybe I’m a bit delrious but groups such as WISE Math, do-founded by math professors, Dr. Anna Stokke and Dr. Rob Craigen, have started somewhat of a revolution (www.wisemah.org). We now have multiple advocacy groups all asking that evidenced based arithmetic be taught in our schools, and policies are starting to change…slowly.

The best advice I could offer on this topic, if you’re a parent, is to get informed and to get talking – to anyone. Your neighbour, your teacher, your local representative, the media. Don’t let the educationalists be the only voice in this discussion. Fight for your kids, because it’s their future that’s at stake.

That’s good advice.

I’d also recommend parents take note of the news and radio discussions and follow up via twitter or email to point the journalists and interviewers to Greg’s blog.

If you’d like some free timed 20 minute hierarchical tests try these from 5 years to 16

http://learninglighthouses.webs.com/

Reblogged this on The Echo Chamber.

Could I suggest one way to possibly educate some Australian parents as to why their children might be experiencing difficulties with maths. People who agree with Greg’s post – parents & teachers – post their views/ideas on a site called Essential Kids under Your Child’s Education.

Many parents visit this site to get advice but from my experience I was the only person NOT advocating for progressive methods, bar one or two others. This site has a large membership so I see some benefit in discussing topics here to get the message out.

I think it could be useful for parents to read the views of teachers who have a different perspective from the dominant orthodoxy and explain to them why they believe their kids are having so much trouble. There are often questions in relation to maths.

Here’s the site for anyone who might want to join and hopefully help parents and teachers understand what is going on.

http://www.essentialkids.com.au/forums/index.php?/forum/96-your-childs-education-year-one-and-beyond/

great points Tempe. It’s very hard for parents to know what to believe because we all think schools know best for our kids. And many serve this purpose very well. However with the onslaught view of education as a “business”, too many salespeople have come calling, duping teachers and parents into their slick way of thinking.

Parents shouldn’t “have” to become educated on this stuff, unfortunately it’s now mandatory in order to keep our kids from falling through the cracks. It would be great if you could also post this on facebook, and I’d be happy to share this far and wide. Cheers.

All of my posts are on Facebook via the Filling the Pail page

https://www.facebook.com/fillingthepail/

Yes, Tara parents simply don’t know what’s going on or many have bought into the idea that education is about 21st skilling for jobs that don’t exist yet. It’s said so often that it has become “truth”. Of course when teachers use terms like deeper understanding and critical thinking we all nod in agreement because that’s what we want for our kids but what we are failing to question is how we get that deeper understanding.

Also, parents are not confident in questioning (it’s not their area of expertise) and they trust the experts. Lastly, they don’t want to be seen to be upsetting the apple cart. But upset it we must.

It is fantastic to see so many Canadian teachers/professors/parents across Canada critiquing the 21st C push. The press also seem to pick up stories there which doesn’t happen here. It would be good if occasionally our public broadcaster did a story on education that didn’t involve the latest fad or we were to see Greg’s post used as opinion pieces in newspapers etc.

To parents reading this, and looking for an affordable resource to help educate their kids, may I suggest a book called ‘How to Tutor’ by Samuel Blumenfeld. It is available as a free PDF at his website http://blumenfeld.campconstitution.net.

He starts with a fantastic historical explanation of the development of modern arithmetic, going back to ancient Greece. The rest of the book is a practical explanation of how to teach, by memorising in a way that helps the student see the patterns that facilitate understanding.

He also wrote a phonics program called Alpha Phonics.

Well articulated Greg. You’ve got a good read on the key issues in the Math Wars.

While I absolutely agree that there is nothing wrong with learning standard algorithms, and I’m not here to advocate one method over another, it isn’t true that standard algorithms require a strong understanding of place value. You need to remember a process, but that can be done with no great depth of understanding about place value, and when this isn’t addressed (they’re getting the answers right so the teacher doesn’t rock the boat, or the teacher focuses only on the children who haven’t remembered the process yet) then each subsequent year of maths becomes more bewildering, as connections are not being made. It also isn’t true that standard algorithms all work from small to large. Long division starts with the highest place value and works towards the lowest. I know anecdotes are not data, but I’ve spoken with many adults (people do tell you about their own school maths experiences when they hear that you teach maths) who have told me that they are still bemused by the ‘bringing down’ of digits in long division – they had learned to perform long divisions without understanding the role of place value in the standard algorithm.

There is no reason that the understanding of place value can’t be taught alongside the standard algorithm. But I’d be wary of placing to much emphasis on understanding in the younger years.

I found, from a personal perspective, that my kids really needed to get the algorithm down pat and then explaining place value made more sense and they didn’t feel overwhelmed and confused. The other way round didn’t work for them ie years of understanding in early primary without actually leaning how to execute easily and efficiently. They really needed to learn how to do some basic maths. I felt so much explanation was overwhelming them. When they could actually add, subtract, multiply etc, courtesy of me and Kumon, they became more relaxed, believed they were more capable and got better marks in maths on their report cards. It made a difference very, very quickly.

Reblogged this on No Easy Answers and commented:

I agree with all of this.

Thanks for this. My kids are 8 and 10 and getting more and more confused by their maths classes. They seem to have missed the basics entirely. Now they’re being asked to do ‘maths fluency’ tasks but they can’t do the basic sum the fluency task is based on. I feel they’re not getting the help they need so I guess it’s up to me to keep paying the school fees but find another way. It’s frustrating for me and stressful for them. I’m so angry it’s come to this.