Jo Boaler is the Professor of Mathematics Education at Stanford University. She has become a familiar figure in the world of maths education and has a knack for publicity, often by taking quite radical positions on the teaching of mathematics.

For instance, Boaler has advocated against asking students to memorise multiplication tables and then testing students on this. Multiplication facts such as 6 x 6 = 36 are part of a larger set of ‘maths facts’ that teachers have traditionally asked students to learn by heart.

Part of Boaler’s argument is that maths facts are not as essential as this practice implies. She has claimed, for instance, that, “I have never memorised my times tables. I still have not memorised my times tables. It has never held me back, even though I work with maths every day.”

The other part of her argument is that timed tests of maths facts induces ‘maths anxiety’ – a debilitating condition that turns kids off maths and harms their performance. This claim has been the subject of quite a lot of interest because, initially, the citation trail ran cold. However, it now seems as if the basis for this claim is an article that Boaler authored in 2014. I have blogged about how I dispute that this article demonstrates that timed tests cause maths anxiety.

Indeed, my view is that maths facts are essential. By removing the need for a student to work out these simple calculations, the student is able to focus on other aspects of a problem. Given that total cognitive load is limited, this frees up resources, giving students a better chance of success and this should pay forward into a greater sense of self-efficacy in maths – one component of future motivation. If we wish to intervene with a group of students who are struggling with basic maths then a focus on memorising maths facts may be part of the solution.

Now, Boaler has co-authored an article for *Time* magazine with Tanya Lamar, a PhD student at Stanford. Paul Morgan, an education professor from Penn State College of Education, has pointed out on Twitter that there is a curious aspect to this piece.

The article is about the critical issue of teaching students who have a learning disability. After discussing a literacy intervention – which is interesting in its own right but I want to stick to the point – Boaler and Lamar state that, “A similarly promising study found that eight weeks of 40- to 50-minute sessions per day made children who had been diagnosed as having mathematics learning disabilities achieve at the same levels as a group of regular performers.”

There is no link to the study in the passage. However, later in the article, Boaler and Lamar mention students who are, “not good memorizers,” and that, “These students do not have less mathematics potential.” This time, they link to this paper which indeed involves an eight week intervention consisting of 40- to 50- minute sessions.

The paper, by Supekar and colleagues, is a foray into neuroscience and the participants underwent functional MRI brains scans prior to tutoring. I am yet to be convinced that neuroscience has much to offer that is of practical value to educators. The main effect seems to be lowering the power of studies by limiting the number of participants to a figure low enough to pass them all through an fMRI machine. Boaler and Lamar seem more impressed with neuroscience than I am, insisting that we need to understand that brain pathways can strengthen and form new connections. This seems to be just a fancy way of saying that people can learn things and I am pretty sure we knew that already.

The really interesting part of the paper is the description of the maths tutoring intervention. After brain scanning, participants:

“…subsequently went through an intensive 8-wk one-to-one tutoring program focused on number knowledge tutoring with speeded practice on efficient counting strategies. Tutoring was designed to facilitate fluency in arithmetic problem solving. Before and after tutoring, we recorded arithmetic strategy use as well as speed, accuracy, and performance efficiency of arithmetic problem solving in each child using standardized procedures.” [References removed]

Say what? Did the tutoring process really rely on ‘speeded practice’? Let’s read further. “The tutoring program combined conceptual instruction with speeded retrieval of math facts,” and, “Arithmetic verification tasks involving single-digit addition problems were performed during fMRI scanning and emphasized speeded performance.” Did any of this matter? Yes, because as the abstract states, “A significant shift in arithmetic problem-solving strategies from counting to fact retrieval was observed with tutoring. Notably, the speed and accuracy of arithmetic problem solving increased with tutoring, with some children improving significantly more than others.”

In summary, this study provides evidence to support the idea that, for students with learning disabilities at least, learning maths facts is important and speeded practice is a key component of this process.

Perhaps Boaler will now rethink her earlier views on this.

Hi. I’ve always been a firm believer in memorizing multiplication tables. However, after having two kids with learning differences, I have struggled with what to do. If child’s memory is do weak that it took 2 years to learn the alphabet with everyone helping (including an Orton-Gillingham tutor), I have come to accept that memory work can be insurmountable for some. That being said, the above-mentioned kid is now somehow passing grade 9 math. The other kid, with less learning differences, has not memorized his times tables, but has an amazing ability to think through concepts. He uses finger tricks and adding tricks (7×6 is 7+7 three times added up). This has not slowed him down much and he is in grade 8 math even though he is in grade 6! The lack of memory work on multiplication weakened the one kid’s understanding of math a ton, but one can only haggle through memory work for so long. The other kid has strong concepts of math and has overcome not memorizing the times tables.

“Tmed math tests” and “combining number knowledge tutoring with speeded practice” are 2 very different things. Timed math tests have students demonstrate, typically in one minute, how many math facts they have memorized- they are, as the name suggests, “tests” of what has already been memorized. Speeded practice, on the other hand, suggests that students have access to the math facts as they “practice” them quickly as a way of coming to memorize them.

The results of this study in no way suggest that timed tests help students memorize math facts; it suggests that speeded practice may actually help the memorization process.

I think we are splitting hairs here. The most common way I have seen of students memorising multiplication facts is to complete a multiplication grid under timed conditions. I think that’s a great idea. When Boaler criticises a focus on children ‘rote’ memorising multiplication facts and the use of timed tests then I assume she is against this practice. If she is not then I am not sure exactly what it is that she is against. Are we fine as long as we don’t call it a ‘test’? What does a timed ‘test’ of maths facts look like that makes it so different to this kind of practice?

All I know is that you can work on memorizing times table until the cows come home, but if a learning disabled kid has a low working memory, there’s only so long that you can keep doing this. At some point, an alternative may have to be reached (like with my son) or an acceptance of ability may also have to be the result (like with my daughter). I can assure you that I tried every technique possible and worked on multiplication tables for as long as possible with my kids. It wasn’t for lack of trying as I fully believe that knowing this is very useful in making math easier to learn when much higher concepts have to be thought out later on.

Not splitting hairs at all. Practice is about acquiring knowledge; tests are about showing what you know- 2 VERY different things.

This is incorrect. Testing is well known to educational psychologists as a way of reinforcing learning. There’s even a Wikipedia entry on it. https://en.wikipedia.org/wiki/Testing_effect

Trudy, what you say about low working memory and memorization strikes me as the opposite of the truth. The very point of memorization is to load long-term memory to compensate for limitations in working memory. Those who have impaired or weak capacity in their working memory would therefore benefit even more, potentially, from memorized content than those of normal working memory. But in point of fact EVERYONE’s working memory is severely limited, which is why it is essential that we work to move primary domain content to long-term memory (i.e. memorize) as much as possible and as soon as possible because it frees up what limited working memory resources are available while performing tasks and it has the effect of making subsequent learning easier. It may well be more strenuous for those with weak working memory to engage in memorization exercises, but I’m quite certain that any well-designed study will bear out that the compensating effect of memorization on affects that particular demographic disproportionately positively.

You mention the limited cognitive load of learners as a justification for memorizing math facts. There is no doubt that this is true. I have seen in my classroom that when more complicated and abstract concepts are placed on the table, the students who have a firm grounding in number sense and who do not struggle to recall basic addition and multiplication facts have more cognitive energy to dedicate to the problem at hand. But if we are simply trying to address the cognitive load limit, why the focus on memorization instead of available tools to facilitate the computation. Mathematicians have forever been searching for ways to simplify the process for calculations: John Napier is famous for his tool which made multiplication of large numbers simpler and more efficient so that practitioners (from factory workers to physicists) could get down to the business of using the math, not just doing it. I think that Jo Boaler may be on to something in that when we place an over emphasis on the rote learning of math facts, as often takes precedence in grades 2-8 classrooms in the US, we sacrifice, sometimes significantly, the opportunity to create young learners who are able to see themselves as problem solvers, recognizers of patterns, and creative mathematicians even though they may not have the ability or the sheer will to commit the facts to memory. If calculators can do it, let calculators do it and let’s get on to the interesting stuff of math.

Because calculator input errors are common and the best remedy is good estimation skills. Developing estimation requires good arithmetic, especially place value skills. In short in order to be able to accurately avoid mental arithmetic you need to be able to mental arithmetic. I have taught students with dyscalculia (which seemed to just be bad arithmetic) and they found there repeated errors frustrating and distracting. Even repeating calculations on the calculator barely mitigated the effect.

I agree with you about estimation skills. These are, without a doubt, the best skills any student can have when attacking problems and developing a sense of reasonableness. The fact that calculators won’t help in such situations, I believe, actually helps the argument I favor of them. If students don’t have number sense, the use of a calculator will not help them solve problems. Furthermore, at least my own experience has shown me that timed math tests or drilling basic facts do not move the needle on that developing number sense in students either. The point being, teach number sense, not calculation.