This is a blog post about how I believe Jo Boaler is wrong when she asserts that learning maths facts off by heart and timed tests are detrimental to children’s well-being and mathematical ability. I’ve tried to take the time to read pretty much every piece of research she has linked to in her article and it’s been an interesting reading journey, not least because some of the research she cites seems to provide evidence that learning maths facts off by heart and the use of timed tests are actually beneficial to every aspect of mathematical competency (not just procedural fluency). To help me get my head around what she’s saying, I’ve summarised the entire article and analysed each part. It took me ages, but it was worth it:
- The new UK curriculum requirement for children to learn times tables off by heart will lead to children being scared of and then turn away from maths
On the ground, I have seen the opposite: children are more confident, happier and definitely better mathematicians as a result of the new curriculum bringing back all the ‘old fashioned’ requirements such as knowing maths facts off by heart. I have worked to develop a system of fun, competitive, weekly timed tests with direct feedback and co-opting the language of sport in my classes and every year I have seen the ‘orange’ children on target tracker (particularly white, working class males/PP children) make accelerated progress and across entire classes there will be an improvement in confidence, ability to concentrate and persevere with increased love of mathematics as a result. It’s too early to say whether these same children will then, according to Boaler, turn away ‘in droves’ because the children who have, in my view, fully experienced the new curriculum are still only in LKS2. I predict that her prediction is wrong, but we will need more than my evidence alone, obviously.
2. Teachers in the US, despite the Common Core curriculum [allegedly] not requiring children to learn maths facts, have misinterpreted ‘fluency’ and are forcing children to learn maths facts off by heart
At this point, I worried that I didn’t understand the word ‘fluency’ and then Boaler started talking about ‘number sense’ which confused me a bit. The two are different; here are the definitions to help us all understand:
a) ‘Fluency’ usually means ‘procedural fluency‘: the ability to apply procedures accurately, efficiently, and flexibly [maths facts and algorithms need to be at ‘instant recall’ status i.e in long term memory for this to happen]
b) ‘Number sense’ is a phrase that is used to mean ‘conceptual fluency‘: understanding place value and the relationships between operations
Boaler seems to pit the two against each other, and if I am right in my interpretation, she is saying that teachers should really concentrate on the conceptual fluency (to me, this is the same as conceptual understanding, the ‘seeing’ of the calculation within the maths problem as well as the ‘picture’ of what is happening to the numbers) and that through concentrating on children’s understanding, the procedural fluency and learning of maths facts that are integral to procedural fluency will be developed indirectly and naturally.
This is sort of a chicken and egg situation, isn’t it? In my view, it is through procedural fluency that understanding is really developed, so I’m in the opposite camp to Boaler (who seems to be in the classic progressive camp). Of course, I’m not advocating that we don’t teach for understanding, but I could teach you how an engine works and then for a fleeting moment you might understand it, but until you take it apart and put it back together over and over will you really know the parts that go together, what might be missing and how it works (I might time you to see whether you know this off by heart – because you will be quick). I find that the children who don’t understand tend not to understand because they’re stuck at using fingers, repeated addition, stringing and they cannot see the mathematical wood for the trees – they haven’t practised enough and they haven’t committed number facts (sometimes including the fact of what a number is) to long term memory.
That US teachers are choosing to help children with their procedural fluency, even though the Common Core curriculum has [allegedly] de-emphasised learning maths facts and algorithms, is a good thing in my view and it also gives me comfort that I’m not the only maths teacher who take this view.
3. Example/proof of not needing some maths facts: why bother memorising 7 x 8 when you can work it out by, say, using 7 x 10 and then subtract 2 x 7
This was the author’s example of how the better mathematician has well developed conceptual fluency rather than relying on procedural fluency. Reality: the person who knows the two maths facts of 7 x 10 and 2 x 7 as well as the number bond 14 + 56 = 70 also tends to know the maths fact 7 x 8 = 56 off by heart. So, this was actually an example of someone with better procedural fluency being better at procedural fluency. The other glaring reality is that the young child who does not know 7 x 8, even if he/she does know the maths facts 7 x 10 and 2 x 7 (which is also unlikely since they’d probably use repeated addition to get there), will not necessarily see how to use them together because he also wouldn’t know the number bond 14 + 56 = 70 or even the two basic number bond facts (6 + 4 = 10 and 5 + 1 = 6) together to make 70 to then know that subtracting one from the other will arrive at the final result of 56. Already you’re feeling exhausted for the child and this is because we all know that noodling our way to 7 x 8 is inefficient and with each layer of calculation a little child is much more likely to make a mistake, possibly arriving at 57. Which is the wrong answer. I think this example is actually proof of the importance of knowing maths facts off by heart.
4. Dr Boaler, professor of mathematics, did just fine without having to learn maths facts off by heart and naturally developed ‘number sense’ because her school developed the ‘whole child’
Unfortunately, a population study n = 1 does not qualify as statistically significant. I know many who were well and truly failed by progressive education and the maths ‘teaching’ that went with it – yet even today you will see year 6s struggling with the basics, forever stuck at repeated addition and inefficient methods like grid method, the complete lack of systemisation in their calculations and how this manifests as a scatter-gun approach to layout in their maths books, even number formation would be awry, yet they were being praised for ‘creativity’ in trying to find an answer, thus demonstrating their ‘understanding’ (the answers were wrong, by the way). All it takes is one teacher or a maths lead who has bought in to the whole ‘You don’t need to have instant recall or be quick with your algorithms because it’s all about the understanding‘ (because it justifies her own grade C maths GCSE) to let a child spend a year meandering through the leaves and branches of numbers, never to see the full mathematical forest in all it’s glory because they don’t know, off by heart, that certain leaves and branches make trees. Perhaps Boaler was lucky then? I don’t think so. Luck has nothing to do with this, so let’s just read on.
5. A study showed that low achievers have no number sense, and tend to resort to counting back in order to solve problems like 21-16, but higher achievers do have number sense and are able to do 20 – 15 + 1 instead, for example [thus showing that you don’t need to know maths facts off by heart] (research link: 404 error!)
Again, you could argue that ‘higher achievers’ have better procedural fluency because they know their number bonds to 20 off by heart as well as having done enough practice to know and apply – 1 + 1 = 0 each side of the equation. This is pretty much the same situation as point 3.
6. Problem solving is the best way to develop ‘number sense’ and indirectly learn maths facts off by heart (Feikes & Schwingendorf, 2008)
Now, this is where things got really interesting because I read the research (I’ve linked them all, this one above is on p.83) she cited as evidence for this claim. The study looked at how children begin their maths journey well before attending school by ‘compressing’ the concept of number. In lay man’s terms this means, for example, that a 4 year old initially knows there are 5 pencils in front of him because he counts them one by one, and then he eventually is able to look at the 5 pencils and instantly ‘see’ that their are 5 without even counting. It then goes on to state that through practice of addition, a child then might used his compressed concept of number (ie know what ‘5’ means) to add 4 + 5 initially by adding 5 + 5 and then -1, but then eventually he just knows that 4 + 5 = 9 off by heart as well. The premise of the paper wasn’t to imply that problem solving helps children to develop conceptual understanding and then naturally acquire key maths facts off by heart (which is what Jo inferred), but rather to make early years teachers aware of how children learn those crucial early maths facts (eg what ‘5’ is), to think about their teaching of early maths and to provide lots of opportunities for children to learn these early maths facts off by heart with manipulatives and plenty of counting practice until they are able to just glance at 5 pencils and say ‘There are 5’. If anything this paper supports the use of lots of plain arithmetic practice in order to put basic maths facts into long term memory.
7. Lack of ‘number sense’ is the reason why the Hubble Telescope once missed some stars – and number sense is inhibited by too much rote memorisation [therefore the dude in charge spent too much time learning maths facts?]
OK this is ridiculous. People make mistakes, even the good people at NASA.
8. Some people are better than others at memorising maths facts – but they’re not necessarily better at maths, nor do they have higher IQs (Supekar et al, 2013) because maths facts are only a small part of maths learning
Again, interesting to read the research here because it seems that Boaler is implying that research is tells us that learning maths facts off by heart isn’t that important and that those who take the time to memorise maths facts aren’t always going to be better mathematicians as a result. As ever, the devil is in the detail: the study looked at whether differences in morphology and connectivity in different parts of the brain affected how a child responds to individual maths tutoring. It turns out it does, but the study doesn’t imply that children shouldn’t be required to learn maths facts, since all the children in the study experienced an improvement in mathematical ability via ‘a significant shift in arithmetic problem-solving strategies from counting to fact retrieval‘, it was just that the children varied in degree of improvement. Other factors such as IQ, working memory, behavioural measures had no bearing – it was all down to variations in the regions of the brain associated with long term memory. I certainly didn’t take away the message of ‘Don’t bother getting children to learn maths facts off by heart’! What I did take away was the message that all children can become good mathematicians, it’s just that some need more time to practise and more teaching in order to make the same progress as others. But, you know what else I found in this research (and this is where my eyes popped out)? Take a look at this golden nugget:
So, the very same people who conducted this study about differences in the brain had also established that learning maths facts off by heart and doing timed/speeded practice leads to significant improvements in:
- automatic retrieval
- arithmetic fluency
- procedural fluency
- problem solving
The method of the study, understandably, used these findings as the basis of structuring the tutoring sessions as a ‘program focused on number knowledge tutoring with speeded practice on efficient counting strategies‘. At this point, I did wonder why a professor of mathematics, someone who has adopted a position against the learning of maths facts off by heart and the use of timed/speeded tests, would refer to a research paper that clearly provides evidence in favour of rote memorisation of maths facts and the use of timed/speeded tests?
9. The best way to develop fluency is to develop number sense by working with numbers in different ways (problem solving), not by learning maths facts off by heart (Parish 2014, p 159)
Boaler still maintains her position by citing another study in support of problem solving as way of learning maths facts off by heart, only it’s not a study, but a resource called ‘Number Talks’ that guides teachers in their teaching for conceptual understanding through problem solving using open ended questions for children to discuss. I did have a look at it and you know what? I quite liked it – but then I remembered that I’m pretty confident and do this sort of thing with children anyway (I’m fond of an array or a bar model), encouraging children to fully explain the reasoning behind their calculations and then demonstrate (I do this when we mark our weekly arithmetic tests) by coming to the class board. But, you know who’s sat there looking bamboozled? It’s the kid who doesn’t know any maths facts off by heart, so as soon as her friend launches into an explanation which begins with ‘Well, I know that 10 lots of 7 apples are 70 apples and 2 lots of 7 apples are 14…..’ she’s lost because she didn’t know 10 x 7 = 70 off by heart – that’s definitely not learning through problem solving. The resource itself is not evidence that children can just problem solve their way to procedural fluency.
10. Maths testing causes the life-long, debilitating condition called maths anxiety (research link yielded 404: error)
I don’t think there’s a special condition called maths anxiety, just ‘anxiety’. I used to have anxiety about performing as a musician; doing more performances made me a better musician and helped me get over said anxiety. If someone had made a big deal out of it, sent me for therapy, generally pussy-footed around me and made me feel like I had something terribly, irreversibly wrong with me, some sort of ‘condition’ like extreme asthma that I should be ever vigilant and frightened of, then I would have avoided performances and never got over that anxiety. I think it’s the same with maths testing – help the child by teaching them and letting them practise, don’t make a massive fuss about maths tests like you’re about to send the child to war (in fact, they’re fun, like a quiz!) and let the child get over their anxiety in their own time because it’s definitely not a life-long condition.
The first reference is a book that quite clearly states that with practice, and using certain mental strategies, you can overcome performance anxiety (and its tendency to befuddle the working memory) and do really well in your chosen field – nothing about abandoning maths or that performances should be avoided. The second reference, which referred to previous research that found that worrying about a maths test diminished working memory and attention available for the maths (if you’re thinking about how worried you are, you’re not thinking about the maths), was for a study that found that maths anxiety was correlated with lack of self-control of emotions and concentration as well as lack of maths facts committed to long term memory (which is where the maths facts should be anyway). Previous research had also found that maths anxiety didn’t necessarily impair performance because sometimes it leads to better concentration, and for those who had more working memory available (because they had committed facts and algorithms to long term memory) the anxiety actually had a positive effect. The study itself does go on to recommend making tests less anxiety-promoting by avoiding timed elements even though it identifies weak maths ability and low working memory (because of distraction of the anxiety itself) as being risk factors for poor performance due to maths anxiety (surely we should target the risk factors?). It certainly doesn’t say that as a result of tests, children have maths anxiety and then ‘leave’ mathematics. What I took away was that children need to make sure that they have instant recall of maths facts and also to find thoughts and methods that help with control of emotion and concentration in order to avoid the vicious circle of maths anxiety in the first place.
12. Putting pressure on children to recall maths facts at speed will not reduce maths anxiety (Silva & White, 2013; National Numeracy, 2014/404 error).
The first reference is to quite a long publication about the results of intensive courses in remedial maths for young people in the US looking to go to college. It’s quite long, but a central jist was that (and the excellent work of one of my favourite researchers, Stigler, was cited for this) they found that these young people had internalised that they weren’t good at maths and as a result, had poor work ethic and tended to give up quite easily when faced with a bit of struggle – hence doing badly in maths. A contrasting example was provided in that students in the Far East were known to persevere more because they believed that getting better at maths required practice, hard work, concentration (and being ‘good’ at maths was open to all, and they are indeed very good at maths). Part of the program was about getting student to really understand how hard work leads to success (through a bit of struggle – which they were made to push through) and the course content also attempted to get students to understand the purpose of maths in the real world as well as work with each other a bit more. There was nothing that I could see stating that getting children to learn maths facts off by heart (and therefore being speedier at recall) resulted in no change in maths anxiety or that testing was causing maths anxiety, because the ‘study’ wasn’t really focusing on that. If anything, it highlighted that maths anxiety is a problem arising from student mindset, not because of tests or ‘pressure’ to learn maths facts.
13. The best problem solvers use both numerical/symbolic and spatial/intuitive reasoning neural pathways (Park & Brannon, 2013)
This research seems to support the notion that visual aids are a great way to get children to understand a problem and then they tend to do much better. I found this paper a bit much, but I certainly didn’t infer a message that using different parts of the brain for solving problems diminishes the importance of the parts of the brain associated with long term memory and quick recall of maths facts.
14. Studies have shown that you can learn maths facts two ways – by memorisation, or by ‘strategies’, but the latter produces superior performance (Delazer et al, 2005)
The research does indeed state that drills vs. ‘strategies’ involve different parts of the brain, but then of course that makes sense really; I bet I’d use different parts of the brain to look at and listen to the sea compared to thinking about the sea as if it were written in musical notation. I had to really work hard to understand this paper, but it did eventually dawn on me that yes, while the drill and strategy people initially used different parts of the brain, the research also showed that both methods caused the ‘thinker’ to retrieve and use previously existing networks of arithmetic processing and memory – so everybody relied on their long term memory after all in order to perform the calculations. Surely this is evidence for committing maths facts to memory?
Maths is the only subject where children get upset, have to do timed tests and are made to work towards instant recall (be speedy). Why?
I think Boaler is trying to imply here that maths people need to learn from and be like teachers of other subjects? Clearly Boaler has never experienced being the fat kid in a dance class then.
15. It is a misconception that maths is about getting correct answers or about calculating, when actually it’s all about methods and reasoning (Boaler, 2013)
Her article which she references in support of this statement talks about ‘mathematical democratization’ through making maths lessons more about problem solving, reasoning, enquiry, creativity and encouraging the use of software to help with problem solving (avoiding having to rely on long term memory to do the calculations) – apparently lessons that are focused on procedural fluency are racist and sexist! I don’t think any mathematician thinks that maths is only about getting correct answers and calculating, or that this area is mutually exclusive to methods and reasoning – it’s about both sides of the coin, but actually the former is really, really important as a foundation – otherwise you wouldn’t know if your methods and reasoning were on the right track?
16. Conrad Wolfram, of Wolfram-Alpha, says we need to see the breadth of mathematics.
I looked up the website. Awesome, but I couldn’t see his quote. Do check out the website though; you won’t regret it.
17. Mathematicians, including the top mathematician Laurent Schwarz, tend to be quite slow at maths; this is because they’re taking the time to calculate in an intelligent way, so why do we try to get children to be speedy?
This statement confuses two things: problem solving, and recall of maths facts and algorithms. The latter needs to be quick because being quick is a proxy for said math facts and algorithms being tucked away in long term memory. No one is advocating rushing a student on a maths problem and yes, the slower ones tend to arrive at the correct answer whereas the quick ones are more likely to miss something crucial (like the second step – very common in UKS2). You can bet that the ‘slower’ problem solvers’ brains are working very quickly at shuffling those number facts like it’s a game of light-speed tetris. Conclusion here: the fact that mathematicians like to deliberate over a problem does not mean we need to almost encourage children to be slow at their recall of maths facts and algorithms.
18. Fluency is not based on speed of recall or memorisation of maths facts, in fact, the lowest achievers focus on memorising maths facts [and they are not fluent] (Boaler & Zoido, in press)
This is a confused and confusing statement because the article that is linked refers to children struggling with maths facts in their working memory – it’s a sort of circular reference back to the article I am writing about and she states again that conceptual understanding and the ‘joy’ of problem solving should trump procedural fluency (an emphasis on which damages mental health?). Yet, from looking at the other research above that she cites, we can quite clearly see that committing maths facts to long term memory is a great way to positively influence procedural fluency and conceptual understanding.
19. Michael Rosen is leading a cause in the UK to stop children from being tested and stressed out about tests – teachers making young children learn maths facts off by heart is contributing to this maths anxiety
This is primarily about baseline tests in EYFS reception year (children don’t even know they’re doing it) and SATs testing in schools, not about regular maths tests. In fact, we could easily draw a conclusion from this that children need more, low-stakes testing so that when it comes to the official tests, they take them in their stride.
20. Learning of maths facts should be developed through exploration with numbers
No it shouldn’t, but exploration with numbers can really help consolidate understanding. Pretty much all the research cited stated that committing maths facts to memory and doing timed tests helps with all aspects of mathematical competence, including problem solving.
21. ‘Number talks’: a package of engaging maths problems to be discussed in groups, using different strategies – helps children to learn maths facts
See point 9. Children who don’t know their maths facts end up confused.
22. When we emphasize memorization and testing in the name of fluency we are harming children, we are risking the future of our ever-quantitative society and we are threatening the discipline of mathematics
Actually, on all those counts, not we are not. In fact, it’s the complete opposite and I have to thank Boaler at this point for introducing me to so much in the way of great research that proves that learning maths facts off by heart and doing timed tests is a great way for young children to become better, and therefore more confident and happy, mathematicians.
Who’s with me?