The TES has quoted maths education professor Jo Boaler as stating that the increased focus on memorising times-tables in England is “terrible”:

“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.

“It is not terrible to remember maths facts; what is terrible is sending kids away to memorise them and giving them tests on them which will set up this maths anxiety.”

Boaler is obviously alluding to some research here although it’s not clear what this is. What is clear is that she is wrong.

Tables help

Knowing maths facts such as times tables is incredibly useful in mathematics. When we solve problems, we have to use our working memory which is extremely limited and can only cope with processing a few items at a time.

If we know our tables then when can simply draw on these answers from our long term memory when required. If we do not then we have to use our limited working memory to figure them out when required, leaving less processing power for the rest of the problem and causing ‘cognitive overload’; an unpleasant feeling of frustration that is far from motivating.

An example would be trying to factorise a quadratic expression; tables knowledge makes the process much easier.

The fact that Boaler never uses times tables as a maths education professor tells us something but I’m not sure it tells us much about the value of tables in solving maths problems.

I am sure that testing can induce anxiety but it certainly does not have to. Skilful maths teachers will communicate with their students and let them know that the tests are a low stakes part of the learning process.

Tests are an extremely effective way of helping students learn, particularly for relatively straightforward items such as multiplication tables and so, appropriately used, they should be encouraged.

We also know that how students feel about their ability – their self-concept – is related to proficiency and that it is likely that proficiency comes first ie proficiency causes increased self-concept.

With this in mind, if we want students to feel good about maths and reduce maths anxiety in the medium to long term then we need to adopt strategies that improve their ability to solve problems.

Learning multiplication tables is exactly such a strategy.

Is you read Jo Boaler’s article carefully she states, “It is useful to hold some math facts in memory. I don’t stop and think about the answer to 8 plus 4, because I know that math fact. But I learned math facts through using them in different mathematical situations, not by practicing them and being tested on them.” She is not saying that we should not know them, but that memorization is not the method we should be using. As teachers, we should be teaching the students strategies that they can use such as the doubles +1 or make 10. It is far more important for students to know strategies than to blurt an answer in 3 seconds when they do not understand that multiplication is repeated addition. If I don’t know 6*7, then I can use 6*6 and add another 6.

I was given a class of lower math students. I follow Jo Boaler’s principles. These students started the year hating math and thinking they were horrible at it. I had grade 4 students that were doing their addition tables starting from 1! Through the teaching of strategies and teaching students to visualize what they were doing with their hands, they grew more confident and started feeling more and more successful. We practice our math facts, but through specially designed activities to target the ones they do not know. To this day, I do not have all of my multiplication facts memorized, but I have many strategies that I use to figure them out and this has never hindered me in my teaching or life.

When I was at school my maths teacher (she completed 50 years of teaching in 2011!) used to give us a 5 minute test at the start of each lesson. It never stressed us out.

I wish Boaler and her ilk would stop referring to retrieval practice as “testing”. Apparently that’s what she’s referring to, as the standard way to memorize times tables is to set up a frequent feedback cycle to give the student the rewarding experience of watching their own progress in a low-pressure environment. I fail to see how that can cause math anxiety. Seeing one’s own progress, in fact, is a pretty good way to improve self-esteem when faced with the daunting process of mastering a complex and abstract discipline covering a huge body of knowledges. One step at a time … Further, when that retrieval practice is very similar to the eventual, far less frequent achievement testing, there is likely to be far less anxiety about that, as the most common cause of testing anxiety is unfamiliarity with the process of testing and insecurity about one’s own performance level. I like @governingmatters’ comment here: When you practice for 5 minutes a day, something you are given effective ways to master, and this is entirely routine … what stress?

I simply can’t get to grips with this argument. I think it is fairly straightforward. If you know your times tables by heart you wont be taxing your working memory as much and you will have a greater capacity to work out more complex problems, equations etc.

As for anxiety, I think there is a bit of “bubble-wrapping” here. We must be careful to try not to eliminate all anxiety from our childrens’ lives. Anxiety should not be viewed as a bad thing (unless of course it becomes pathological and effects a persons life) it should be viewed as an inevitable consequence of life and a lesson on building resilience. Also,sometimes, a little anxiety has positive repercussion, such as using the adrenaline to spur us on and heightening our senses.

I feel very cross that so many teachers are opposed to any form of memorisation (drill & kill, as they call it). To my mind instant retrieval of your times tables is going to help you succeed in maths, therefore the should be practiced in schools and not outsourced to parents. By out-sourcing many children will never learn them, particularly those that are already disadvantaged. I’ve put this to some teachers and they’ve told me that if the parent wont get involved then the child may be offered “invention” at school. Why not do 5-10 mins a day practise in the first place to avoid this? I’m actually stunned that the Ed. dept. haven’t jumped on this and rectified the situation, specifying that it is the responsibility of the teachers to teach the times tables, but I guess many teachers would view that as messing with their autonomy.

When you see how teachers reacted to being told to teach systematic synthetic phonics – total outrage from many quarters at the attack on autonomy – you can guess fairly accurately their reaction to being made to actually teach times tables (or anything) , with backing from unions etc.

When I taught senior high school math I found that about 80% of the errors that senior students made on exams were related to not knowing their basic facts. I also found that very few of the students, who had weak basic skills, ever spent the time to rectify this problem, despite me giving them a plan for addressing their shortcomings. It seems that most students, who failed to learn these facts at the age when they were first introduced, never take the time to learn them in later life. Hence, their math issues simply compound until the student eventually gives up on taking any further math courses.

Oh I have heard this whole ‘They don’t need to know anything off by heart’ many times. It’s not just that the sentiment is wrong, it’s the way it is spoken with such confidence and belief that irks so much and the fact that these people never seem to have said subject qualifications beyond GCSE either. Said teachers also tend to fall into the anti-practise ‘Children should never do more than one question from the textbook’ category too.

I understand the cognitive load argument well enough, but I think people are oversimplifying – nobody is really saying that long-term memory is unimportant. I wonder how many of you know your times tables up to 12? to 20? to 100? The reality is that you don’t need to, because you have automated the key calculations to memory, and can easily work out 35×17 or whatever. Why, then, not say the same for the other times tables?

You might think that you can only do the above example because you learned your times tables, but is there really any evidence to support that assumption? Can we be sure of cause and effect? One piece of counter-evidence is very powerful (see ‘black swan theory’!). If people like Boaler can reach a high level in maths without being to recite tables (and I have heard of other examples than this, e.g. dyslexic mathematicians), this suggests that recitation of tables is not necessary. Likewise, most kids that are great tables-reciters at primary do not go on to become great mathematicians (many hate it by high school), so it is not sufficient either.

Yes, automating calculation to long-term memory is good. You can probably kind of ‘see’ in your head, without having to give it much attention, that 5×7 is 35. Did you have to mentally reel though 1×7, 2×7 etc to get there? No! Actually, I’d suggest that reciting the tables in order is considerably less helpful than doing random arithmetic problems. See the (extensive) research on interleaved practice.

Test anxiety is a separate issue – bad tests lead to anxiety. For some kids, anxiety over recitation of times tables is almost unavoidable. However, even good testing of times tables is apparently neither necessary nor sufficient for maths competence.

By analogy, spelling tests are neither necessary nor sufficient for kids to learn to spell, but of course there is no argument that we ideally want learners to memorise vocabulary including spellings of words.

I don’t think that Jo Boaler *has* reached a high level in maths. She is a maths education professor and not a mathematician. As I understand it, her degree is in psychology.

Even so, I am prepared to accept that you can get by without times tables. If you made the claim that it is possible to cut lawns without a lawnmower then I’d also be prepared to accept that. Perhaps you could use shears or a scythe? My question would be; why would you want to? Why make life hard? I would certainly use times tables plus the standard algorithm to solve 35 x 17. It’s the simplest approach.

The maths anxiety argument doesn’t stack up. I’ve written about it here:

Surely the simplest approach to solve 35×17 is to use a calculator, and have a rough idea what the correct answer should be. I finding it interesting now reading much more research as a maths adviser how polarised the arguments are becoming.

I agree that it is the recitation that can be problematic. I came top in maths tests at school, went on to read mathematics at university (Oxford) and was not stressed by the subject at secondary school. What stressed me in primary school and made me feel that I was not good at mathematics was an inability to recite a whole times table to the rest of the class to get a star. People were praised at getting another star – my row remained blank. However, ask me any calculation like 7 x 8 and 56 or whatever the answer was would be said immediately. It was the sequencing orally that I simply could not manage and still can’t fluently. What we do want children to know is the answer to a calculation and this gets morphed into reciting a times table. There will be some children for whom reciting the times table is fun and helps them learn. That is good. Important that they are still able to have random access to one specific calculation and they do not have to recite the whole table to get there:)

I think the fact that I know automatically that 5×7 is 35 is very important to my ability to solve maths problems. Not knowing that fact means I have to do the calculation which takes time and effort. Very rarely does an adult go through a multiplication table to get to an answer, because they know all the bits off by heart. Your argument that because they don’t recite tables means knowing them is irrelevant is by its own analogy incorrect.

Math anxiety is alleviated, in no small measure, by providing students with the tools they need to solve problems, whether it be times tables, or procedures for solving types of word problems.

In her paper, “Fluency Without Fear,” Jo Boaler agrees that math facts are incredible important and useful; her research is about how those math facts are learned and internalized.

“Teachers should help students develop math facts, not by emphasizing facts for the sake of facts or using ‘timed tests’ but by encouraging students to use, work with and explore numbers. As students work on meaningful number activities they will commit math facts to heart at the same time as understanding numbers and math.”

It’s a matter of supporting students to “commit math facts to heart” as well as develop a deep understanding of and connection between those math facts.

Daniel,
When someone supports a very specific claim by referencing an entire book should we:
A. Read the book to see whether this research is solid.
B. Accept the claims without further question because a book.
C. Realize we are wasting our time with the person making the claim as they are too either too lazy or not interested in giving references to the specific research that supports their claim.

In particular you will find the very first reference in the paper you link to (Beilock, 2011) is a book. Yet this is given as support for the claim that math facts are held in working memory and that this becomes blocked under stress.

First just try and untangle the ambiguous statements in Boaler’s paper. Math facts are held in working memory – what does that mean. Probably not that they are stored there but if we are just talking about a math fact that are currently held in working memory this is just a math fact we are currently thinking about. Nothing to do with memorization there. Once you have a worthwhile guess at what she means go and read the whole book she references to see if it supports that or choose plan C above.

Further, it is not going to surprise anyone that too much stress is bad for math performance. Extrapolating from a single point, here high stress, and assuming a linear relationship is just bad science. Anytime someone does that we should point that out and then ignore them until they stop making worthless arguments.

Greg has already pointed out that conflating the creation of stress and activities aimed at memorization is just another way of not helping make things clear.

What cognitive experts all agree upon is that FIRST a fundamental fact or procedure must be memorized (moved into long-term memory), a process which nearly always requires effort. Only then can its uses and meaning become more understood, by encountering the fact or procedure in a variety of contexts.

I have slides that document why cognitive scientists say that memorization must precede conceptual understanding in math and the sciences. They are posted at http://www.chemreview.net/ChemEd2015Post.pdf

What is terrible is giving so much air time to people who claim to know how children should learn maths when they have little evidence of their own success in the subject, let alone teaching it. Stanford University should be concerned when their academic staff come out with loaded statements like “sending kids away to memorise [their times tables] AND giving them tests on them which WILL set up this maths anxiety” – a statement that says more about their own personal negative experiences than anything else.

It is worse than talking about their own personal negative experiences. Boaler claims she grew up when rote was not popular. She is referring to imagined negative experiences. Once you allow imaginary experiences as evidence you can build up a lot of thoughtful evidence for your views. Unfortunately this is an entirely different use of the word thoughtful than most people are used to.

Memorised ‘times tables’ are (extremely) useful if you’re the sort of person who a) can memorise things, b) can use this information in working memory, c) has intuitive grasp of number/quantity (ie not dyscalculic – here memorization is useful but has no corrective mechanisms in case of poor recall). But certainly mathematics (even at high level) and even to some extent arithmetic (albeit at slower speed) can be done with no memory of times tables (even without aids; e.g. with consecutive addition). Personal experiences (either for or against) are only helpful as illustrations of diversity but on their own do not represent a good foundation as policy for all. People vary in how well they remember all of their times tables – many people’s memory of the times tables is very patchy and only goes up to a certain level (with gaps in the middle) – so it is certainly possible to go through life with little to no times tables knowledge. Our ability to navigate the world is social and many people with huge gaps in basic numeracy and literacy make do relying on support of people around them – it certainly limits what they can do but often less than is assumed (often higher leadership roles require less of these skills than support ones and many successful business founders struggle with either).

Amazing how people can consciously or unconsciously decide to not read what someone says when they deeply need to have it be something vaguely similar, but fundamentally different.

Here is what you quoted from Jo Boaler: “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.

“It is not terrible to remember maths facts; what is terrible is sending kids away to memorise them and giving them tests on them which will set up this maths anxiety.”

It takes a particular sort of bias to turn the above into: “CHILDREN DO NOT NEED TO LEARN MULTIPLICATION FACTS,” or “CHILDREN SHOULD NOT LEARN MULTIPLICATION FACTS” or any number of similar absolute claims that Boaler never makes.

What I see on the page is, first: some claims about her personal experience and work, neither of which is particularly relevant to the second remark, which is really the crux of what is getting most commenters here into a snit. And to ensure that y’all CRUSH Boaler, we get comments about whether she’s a “real” mathematician (which, of course, she isn’t claiming or implying in that first paragraph, nor has she or anyone who knows her work claimed about her). She’s a mathematics educator, and much as that upsets the apple carts of any number of people who are (or see themselves as well-positioned to look down their noses at the work of anyone who isn’t), it’s quite possible for someone without a doctorate in mathematics to know a great deal more about teaching and learning mathematics than someone with such a degree. There are mathematics educators who are Ph.Ds in mathematics and mathematics educators with degrees in other fields. You don’t find the latter sort teach graduate mathematics courses in mathematics to would-be mathematicians, physicists, etc. So what, exactly, is the gripe?

Lots of K-12 mathematics teachers lack a PhD. in mathematics. Lots lack a master’s degree in mathematics. I don’t hear the traditionalists whining much on that score (as long as a given teacher adheres to traditionalist practices and views of K-12 math teaching, of course).

Now to the second quoted paragraph. It seems to suggest that some kids who haven’t yet mastered the multiplication or addition tables are negatively affected by being “sent away to memorize them” and by “giving them tests on them.” Doesn’t strike me as controversial. Since I work in K12, have raised a son who went through K12, was around when his sister was in grades 2-7, etc., I’ve seen how different children from various backgrounds respond to what Boaler describes. My son dealt with it all fairly easily. His sister was traumatized by it for a couple of years. The experience left a very bad taste in her mouth about mathematics, and based on what I saw going on at her schools, I”m not surprised.

I’ve seen teachers use the timed facts, minimal errors, testing in primary grades in ways that could and seemed to in fact upset kids. Not all kids, but I’d venture that one is too many, and it’s a lot more than one in a given class, let alone school, district, state, or this country.

The claim from most weighing in HERE, though, is that this simply can’t be and isn’t the case. Well, pardon me for suggesting that there just might be some selective blindness, coupled with lack of frequent contact with “other people’s children,” necessary to see what is obvious to those who don’t desperately need to deny that the “tried and true” methods they were raised with and believe in so deeply could possibly be bad for anyone, let alone for a lot of someones.

But all of that aside, let’s ask what Boaler is actually claiming and calling for. Does she, in fact, EVER state that students shouldn’t learn arithmetic facts? If so, I don’t see it. What I read is someone saying that there’s no need to spend time dedicated to ROTE memorization of those facts. That’s not tantamount to the distortion preferred by the traditionalists that she and anyone who isn’t a back-to-basics and/or direct instruction, rote memorizing fan oppose kids doing anything that could reasonably be expected to result in students learning their arithmetic facts. And that’s simply not the case. Like most knowledgeable and experienced mathematics educators, Boaler knows full-well that there are other ways that kids learn those facts besides reciting them in groups, using flash cards, or doing some sort of computerized equivalent of drilling. For decades, American children have learned addition facts from playing board games that entail adding numbers up to 6 on two dices. (With other sorts of dice besides cubes, the numbers can be greater than 6, of course). Curricular materials usually despised by traditionalists include many games that require kids to practice arithmetic facts, but not via sitting through the sorts of rote practice that are so honored and beloved by the narrow-minded.

I could continue, but either the point is made and interested parties will look into what sorts of alternatives to drill are out there, or it can never be made to a given person (indeed, such folks are likely to have stopped reading in any meaningful way, if not literally so, many paragraphs above). Those who deeply need to dismiss any challenge to the pedagogy of Prussian military education that has informed so much American schooling for over a century, aren’t going to listen to anything else. Whether the dismissal comes from simple ridicule or some fancy-sounding “scientific” theory like “cognitive load,” the notion that kids pick up enormous amounts of information, including mastery of arithmetic facts, via non-rote, non-drill activities (indeed, often non-school activities) is simply unacceptable. It always has been and always will be. Luckily, not ALL kids are doomed to be controlled by those with such limited understanding of how children learn.

Jerry Becker shared your response via email. I was curious to see if your post remained here… Thanks for taking the time to post [this] reply, I agree with you. Jennifer V. Jones, Ed.D. Rutgers, mathematics education, math teacher PD

Well stated Michael. I am currently reading her current book on MIndsets and do not agree with all this negativity at all. As a high school math teacher and a father of elementary aged children, I really see the need for change. They seem to be taking what is said out of context.
Seems the old guard is protecting the old school.

Great response, Michael. I was also amused to see a “scientific” diagram accompanying the post – the sort that Greg tends to diss so easily when the argument doesn’t suit him!

I think many see that children pick up info like times tables outside school. The problem is that such children are likely to come from a more privileged background. Not teaching e.g. times tables methodically and using retrieval practice to embed them, disadvantages children who don’t get support at home or do not have a middle class culture to help them. Not teaching knowledge methodically and making sure it is embedded disadvantages the already disadvantaged.

Yes, we must make sure disadvantaged kid, particularly inner city kids of color, get the worst of 19th century teaching regardless of anything we may have learned about kids, teaching, learning, etc., in the last 150 years, because, um, why? I guess those kids don’t play games or in any other ways encounter multiplying, and if they did, their little deprived ghetto brains couldn’t possibly learn without drill.

Clearly this author is missing the point. Boaler is not stating that memorising is bad. All she is trying to say is that if mathematics learning is based on merely memorising facts then the learner will not develop an understanding of mathematics. They might know the answer but have no understanding of how they got to that answer. This can be problematic in the long run when they have to problem solve. Memorising should be used in conjunction with number sense. At a young age children should not just memorise mathematics facts. Number sense at a young age sets the foundation for mathematics learning.

I am in complete agreement with everything you said, based on my experience teaching students math. Thank you for pointing out the challenges to working memory if nothing is stored in long-term memory.

I have never seen such low students in my 5th grade class!!! Somehow Jo Boaler’s article has given teachers permission not to give time tests at all? How do you assess growth without timings? I had only 4 students come from fourth grade knowing their multiplication facts fluently and 2 knowing division facts fluently. 14 addition, 8 subtraction. They figure them out eventually but they take twice or three times the time as the other students.
Now I am trying to teach 2 digit by 3 digit multiplication!!

Reblogged this on The Echo Chamber.

Is you read Jo Boaler’s article carefully she states, “It is useful to hold some math facts in memory. I don’t stop and think about the answer to 8 plus 4, because I know that math fact. But I learned math facts through using them in different mathematical situations, not by practicing them and being tested on them.” She is not saying that we should not know them, but that memorization is not the method we should be using. As teachers, we should be teaching the students strategies that they can use such as the doubles +1 or make 10. It is far more important for students to know strategies than to blurt an answer in 3 seconds when they do not understand that multiplication is repeated addition. If I don’t know 6*7, then I can use 6*6 and add another 6.

I was given a class of lower math students. I follow Jo Boaler’s principles. These students started the year hating math and thinking they were horrible at it. I had grade 4 students that were doing their addition tables starting from 1! Through the teaching of strategies and teaching students to visualize what they were doing with their hands, they grew more confident and started feeling more and more successful. We practice our math facts, but through specially designed activities to target the ones they do not know. To this day, I do not have all of my multiplication facts memorized, but I have many strategies that I use to figure them out and this has never hindered me in my teaching or life.

When I was at school my maths teacher (she completed 50 years of teaching in 2011!) used to give us a 5 minute test at the start of each lesson. It never stressed us out.

I wish Boaler and her ilk would stop referring to retrieval practice as “testing”. Apparently that’s what she’s referring to, as the standard way to memorize times tables is to set up a frequent feedback cycle to give the student the rewarding experience of watching their own progress in a low-pressure environment. I fail to see how that can cause math anxiety. Seeing one’s own progress, in fact, is a pretty good way to improve self-esteem when faced with the daunting process of mastering a complex and abstract discipline covering a huge body of knowledges. One step at a time … Further, when that retrieval practice is very similar to the eventual, far less frequent achievement testing, there is likely to be far less anxiety about that, as the most common cause of testing anxiety is unfamiliarity with the process of testing and insecurity about one’s own performance level. I like @governingmatters’ comment here: When you practice for 5 minutes a day, something you are given effective ways to master, and this is entirely routine … what stress?

I simply can’t get to grips with this argument. I think it is fairly straightforward. If you know your times tables by heart you wont be taxing your working memory as much and you will have a greater capacity to work out more complex problems, equations etc.

As for anxiety, I think there is a bit of “bubble-wrapping” here. We must be careful to try not to eliminate all anxiety from our childrens’ lives. Anxiety should not be viewed as a bad thing (unless of course it becomes pathological and effects a persons life) it should be viewed as an inevitable consequence of life and a lesson on building resilience. Also,sometimes, a little anxiety has positive repercussion, such as using the adrenaline to spur us on and heightening our senses.

I feel very cross that so many teachers are opposed to any form of memorisation (drill & kill, as they call it). To my mind instant retrieval of your times tables is going to help you succeed in maths, therefore the should be practiced in schools and not outsourced to parents. By out-sourcing many children will never learn them, particularly those that are already disadvantaged. I’ve put this to some teachers and they’ve told me that if the parent wont get involved then the child may be offered “invention” at school. Why not do 5-10 mins a day practise in the first place to avoid this? I’m actually stunned that the Ed. dept. haven’t jumped on this and rectified the situation, specifying that it is the responsibility of the teachers to teach the times tables, but I guess many teachers would view that as messing with their autonomy.

When you see how teachers reacted to being told to teach systematic synthetic phonics – total outrage from many quarters at the attack on autonomy – you can guess fairly accurately their reaction to being made to actually teach times tables (or anything) , with backing from unions etc.

When I taught senior high school math I found that about 80% of the errors that senior students made on exams were related to not knowing their basic facts. I also found that very few of the students, who had weak basic skills, ever spent the time to rectify this problem, despite me giving them a plan for addressing their shortcomings. It seems that most students, who failed to learn these facts at the age when they were first introduced, never take the time to learn them in later life. Hence, their math issues simply compound until the student eventually gives up on taking any further math courses.

Children need to know their Times Tables to instant recall; most adults don’t need to: Time for Tables? http://wp.me/p3watv-1U via @chemistrypoet

Oh I have heard this whole ‘They don’t need to know anything off by heart’ many times. It’s not just that the sentiment is wrong, it’s the way it is spoken with such confidence and belief that irks so much and the fact that these people never seem to have said subject qualifications beyond GCSE either. Said teachers also tend to fall into the anti-practise ‘Children should never do more than one question from the textbook’ category too.

I understand the cognitive load argument well enough, but I think people are oversimplifying – nobody is really saying that long-term memory is unimportant. I wonder how many of you know your times tables up to 12? to 20? to 100? The reality is that you don’t need to, because you have automated the key calculations to memory, and can easily work out 35×17 or whatever. Why, then, not say the same for the other times tables?

You might think that you can only do the above example because you learned your times tables, but is there really any evidence to support that assumption? Can we be sure of cause and effect? One piece of counter-evidence is very powerful (see ‘black swan theory’!). If people like Boaler can reach a high level in maths without being to recite tables (and I have heard of other examples than this, e.g. dyslexic mathematicians), this suggests that recitation of tables is not necessary. Likewise, most kids that are great tables-reciters at primary do not go on to become great mathematicians (many hate it by high school), so it is not sufficient either.

Yes, automating calculation to long-term memory is good. You can probably kind of ‘see’ in your head, without having to give it much attention, that 5×7 is 35. Did you have to mentally reel though 1×7, 2×7 etc to get there? No! Actually, I’d suggest that reciting the tables in order is considerably less helpful than doing random arithmetic problems. See the (extensive) research on interleaved practice.

Test anxiety is a separate issue – bad tests lead to anxiety. For some kids, anxiety over recitation of times tables is almost unavoidable. However, even good testing of times tables is apparently neither necessary nor sufficient for maths competence.

By analogy, spelling tests are neither necessary nor sufficient for kids to learn to spell, but of course there is no argument that we ideally want learners to memorise vocabulary including spellings of words.

I don’t think that Jo Boaler *has* reached a high level in maths. She is a maths education professor and not a mathematician. As I understand it, her degree is in psychology.

Even so, I am prepared to accept that you can get by without times tables. If you made the claim that it is possible to cut lawns without a lawnmower then I’d also be prepared to accept that. Perhaps you could use shears or a scythe? My question would be; why would you want to? Why make life hard? I would certainly use times tables plus the standard algorithm to solve 35 x 17. It’s the simplest approach.

The maths anxiety argument doesn’t stack up. I’ve written about it here:

https://gregashman.wordpress.com/2015/09/19/tips-for-reducing-maths-anxiety/

Surely the simplest approach to solve 35×17 is to use a calculator, and have a rough idea what the correct answer should be. I finding it interesting now reading much more research as a maths adviser how polarised the arguments are becoming.

I agree that it is the recitation that can be problematic. I came top in maths tests at school, went on to read mathematics at university (Oxford) and was not stressed by the subject at secondary school. What stressed me in primary school and made me feel that I was not good at mathematics was an inability to recite a whole times table to the rest of the class to get a star. People were praised at getting another star – my row remained blank. However, ask me any calculation like 7 x 8 and 56 or whatever the answer was would be said immediately. It was the sequencing orally that I simply could not manage and still can’t fluently. What we do want children to know is the answer to a calculation and this gets morphed into reciting a times table. There will be some children for whom reciting the times table is fun and helps them learn. That is good. Important that they are still able to have random access to one specific calculation and they do not have to recite the whole table to get there:)

I think the fact that I know automatically that 5×7 is 35 is very important to my ability to solve maths problems. Not knowing that fact means I have to do the calculation which takes time and effort. Very rarely does an adult go through a multiplication table to get to an answer, because they know all the bits off by heart. Your argument that because they don’t recite tables means knowing them is irrelevant is by its own analogy incorrect.

Math anxiety is alleviated, in no small measure, by providing students with the tools they need to solve problems, whether it be times tables, or procedures for solving types of word problems.

In her paper, “Fluency Without Fear,” Jo Boaler agrees that math facts are incredible important and useful; her research is about how those math facts are learned and internalized.

“Teachers should help students develop math facts, not by emphasizing facts for the sake of facts or using ‘timed tests’ but by encouraging students to use, work with and explore numbers. As students work on meaningful number activities they will commit math facts to heart at the same time as understanding numbers and math.”

Full text here: http://www.youcubed.org/wp-content/uploads/2015/03/FluencyWithoutFear-2015.pdf

It’s a matter of supporting students to “commit math facts to heart” as well as develop a deep understanding of and connection between those math facts.

Daniel,

When someone supports a very specific claim by referencing an entire book should we:

A. Read the book to see whether this research is solid.

B. Accept the claims without further question because a book.

C. Realize we are wasting our time with the person making the claim as they are too either too lazy or not interested in giving references to the specific research that supports their claim.

In particular you will find the very first reference in the paper you link to (Beilock, 2011) is a book. Yet this is given as support for the claim that math facts are held in working memory and that this becomes blocked under stress.

First just try and untangle the ambiguous statements in Boaler’s paper. Math facts are held in working memory – what does that mean. Probably not that they are stored there but if we are just talking about a math fact that are currently held in working memory this is just a math fact we are currently thinking about. Nothing to do with memorization there. Once you have a worthwhile guess at what she means go and read the whole book she references to see if it supports that or choose plan C above.

Further, it is not going to surprise anyone that too much stress is bad for math performance. Extrapolating from a single point, here high stress, and assuming a linear relationship is just bad science. Anytime someone does that we should point that out and then ignore them until they stop making worthless arguments.

Greg has already pointed out that conflating the creation of stress and activities aimed at memorization is just another way of not helping make things clear.

What cognitive experts all agree upon is that FIRST a fundamental fact or procedure must be memorized (moved into long-term memory), a process which nearly always requires effort. Only then can its uses and meaning become more understood, by encountering the fact or procedure in a variety of contexts.

I have slides that document why cognitive scientists say that memorization must precede conceptual understanding in math and the sciences. They are posted at

http://www.chemreview.net/ChemEd2015Post.pdf

— rick nelson

What is terrible is giving so much air time to people who claim to know how children should learn maths when they have little evidence of their own success in the subject, let alone teaching it. Stanford University should be concerned when their academic staff come out with loaded statements like “sending kids away to memorise [their times tables] AND giving them tests on them which WILL set up this maths anxiety” – a statement that says more about their own personal negative experiences than anything else.

http://mathmo.co.nz/2014/05/05/automaticity-times-tables/

It is worse than talking about their own personal negative experiences. Boaler claims she grew up when rote was not popular. She is referring to imagined negative experiences. Once you allow imaginary experiences as evidence you can build up a lot of thoughtful evidence for your views. Unfortunately this is an entirely different use of the word thoughtful than most people are used to.

Memorised ‘times tables’ are (extremely) useful if you’re the sort of person who a) can memorise things, b) can use this information in working memory, c) has intuitive grasp of number/quantity (ie not dyscalculic – here memorization is useful but has no corrective mechanisms in case of poor recall). But certainly mathematics (even at high level) and even to some extent arithmetic (albeit at slower speed) can be done with no memory of times tables (even without aids; e.g. with consecutive addition). Personal experiences (either for or against) are only helpful as illustrations of diversity but on their own do not represent a good foundation as policy for all. People vary in how well they remember all of their times tables – many people’s memory of the times tables is very patchy and only goes up to a certain level (with gaps in the middle) – so it is certainly possible to go through life with little to no times tables knowledge. Our ability to navigate the world is social and many people with huge gaps in basic numeracy and literacy make do relying on support of people around them – it certainly limits what they can do but often less than is assumed (often higher leadership roles require less of these skills than support ones and many successful business founders struggle with either).

Amazing how people can consciously or unconsciously decide to not read what someone says when they deeply need to have it be something vaguely similar, but fundamentally different.

Here is what you quoted from Jo Boaler: “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.

“It is not terrible to remember maths facts; what is terrible is sending kids away to memorise them and giving them tests on them which will set up this maths anxiety.”

It takes a particular sort of bias to turn the above into: “CHILDREN DO NOT NEED TO LEARN MULTIPLICATION FACTS,” or “CHILDREN SHOULD NOT LEARN MULTIPLICATION FACTS” or any number of similar absolute claims that Boaler never makes.

What I see on the page is, first: some claims about her personal experience and work, neither of which is particularly relevant to the second remark, which is really the crux of what is getting most commenters here into a snit. And to ensure that y’all CRUSH Boaler, we get comments about whether she’s a “real” mathematician (which, of course, she isn’t claiming or implying in that first paragraph, nor has she or anyone who knows her work claimed about her). She’s a mathematics educator, and much as that upsets the apple carts of any number of people who are (or see themselves as well-positioned to look down their noses at the work of anyone who isn’t), it’s quite possible for someone without a doctorate in mathematics to know a great deal more about teaching and learning mathematics than someone with such a degree. There are mathematics educators who are Ph.Ds in mathematics and mathematics educators with degrees in other fields. You don’t find the latter sort teach graduate mathematics courses in mathematics to would-be mathematicians, physicists, etc. So what, exactly, is the gripe?

Lots of K-12 mathematics teachers lack a PhD. in mathematics. Lots lack a master’s degree in mathematics. I don’t hear the traditionalists whining much on that score (as long as a given teacher adheres to traditionalist practices and views of K-12 math teaching, of course).

Now to the second quoted paragraph. It seems to suggest that some kids who haven’t yet mastered the multiplication or addition tables are negatively affected by being “sent away to memorize them” and by “giving them tests on them.” Doesn’t strike me as controversial. Since I work in K12, have raised a son who went through K12, was around when his sister was in grades 2-7, etc., I’ve seen how different children from various backgrounds respond to what Boaler describes. My son dealt with it all fairly easily. His sister was traumatized by it for a couple of years. The experience left a very bad taste in her mouth about mathematics, and based on what I saw going on at her schools, I”m not surprised.

I’ve seen teachers use the timed facts, minimal errors, testing in primary grades in ways that could and seemed to in fact upset kids. Not all kids, but I’d venture that one is too many, and it’s a lot more than one in a given class, let alone school, district, state, or this country.

The claim from most weighing in HERE, though, is that this simply can’t be and isn’t the case. Well, pardon me for suggesting that there just might be some selective blindness, coupled with lack of frequent contact with “other people’s children,” necessary to see what is obvious to those who don’t desperately need to deny that the “tried and true” methods they were raised with and believe in so deeply could possibly be bad for anyone, let alone for a lot of someones.

But all of that aside, let’s ask what Boaler is actually claiming and calling for. Does she, in fact, EVER state that students shouldn’t learn arithmetic facts? If so, I don’t see it. What I read is someone saying that there’s no need to spend time dedicated to ROTE memorization of those facts. That’s not tantamount to the distortion preferred by the traditionalists that she and anyone who isn’t a back-to-basics and/or direct instruction, rote memorizing fan oppose kids doing anything that could reasonably be expected to result in students learning their arithmetic facts. And that’s simply not the case. Like most knowledgeable and experienced mathematics educators, Boaler knows full-well that there are other ways that kids learn those facts besides reciting them in groups, using flash cards, or doing some sort of computerized equivalent of drilling. For decades, American children have learned addition facts from playing board games that entail adding numbers up to 6 on two dices. (With other sorts of dice besides cubes, the numbers can be greater than 6, of course). Curricular materials usually despised by traditionalists include many games that require kids to practice arithmetic facts, but not via sitting through the sorts of rote practice that are so honored and beloved by the narrow-minded.

I could continue, but either the point is made and interested parties will look into what sorts of alternatives to drill are out there, or it can never be made to a given person (indeed, such folks are likely to have stopped reading in any meaningful way, if not literally so, many paragraphs above). Those who deeply need to dismiss any challenge to the pedagogy of Prussian military education that has informed so much American schooling for over a century, aren’t going to listen to anything else. Whether the dismissal comes from simple ridicule or some fancy-sounding “scientific” theory like “cognitive load,” the notion that kids pick up enormous amounts of information, including mastery of arithmetic facts, via non-rote, non-drill activities (indeed, often non-school activities) is simply unacceptable. It always has been and always will be. Luckily, not ALL kids are doomed to be controlled by those with such limited understanding of how children learn.

Jerry Becker shared your response via email. I was curious to see if your post remained here… Thanks for taking the time to post [this] reply, I agree with you. Jennifer V. Jones, Ed.D. Rutgers, mathematics education, math teacher PD

Well stated Michael. I am currently reading her current book on MIndsets and do not agree with all this negativity at all. As a high school math teacher and a father of elementary aged children, I really see the need for change. They seem to be taking what is said out of context.

Seems the old guard is protecting the old school.

Great response, Michael. I was also amused to see a “scientific” diagram accompanying the post – the sort that Greg tends to diss so easily when the argument doesn’t suit him!

I think many see that children pick up info like times tables outside school. The problem is that such children are likely to come from a more privileged background. Not teaching e.g. times tables methodically and using retrieval practice to embed them, disadvantages children who don’t get support at home or do not have a middle class culture to help them. Not teaching knowledge methodically and making sure it is embedded disadvantages the already disadvantaged.

Yes, we must make sure disadvantaged kid, particularly inner city kids of color, get the worst of 19th century teaching regardless of anything we may have learned about kids, teaching, learning, etc., in the last 150 years, because, um, why? I guess those kids don’t play games or in any other ways encounter multiplying, and if they did, their little deprived ghetto brains couldn’t possibly learn without drill.

Got it.

Clearly this author is missing the point. Boaler is not stating that memorising is bad. All she is trying to say is that if mathematics learning is based on merely memorising facts then the learner will not develop an understanding of mathematics. They might know the answer but have no understanding of how they got to that answer. This can be problematic in the long run when they have to problem solve. Memorising should be used in conjunction with number sense. At a young age children should not just memorise mathematics facts. Number sense at a young age sets the foundation for mathematics learning.

I am in complete agreement with everything you said, based on my experience teaching students math. Thank you for pointing out the challenges to working memory if nothing is stored in long-term memory.

I have never seen such low students in my 5th grade class!!! Somehow Jo Boaler’s article has given teachers permission not to give time tests at all? How do you assess growth without timings? I had only 4 students come from fourth grade knowing their multiplication facts fluently and 2 knowing division facts fluently. 14 addition, 8 subtraction. They figure them out eventually but they take twice or three times the time as the other students.

Now I am trying to teach 2 digit by 3 digit multiplication!!