Is it worth trying to memorise facts?

We can only think about what we know and, no mater how intelligent we might be, we cannot think about something about which we are ignorant. But how well do we need to know things? Is there any point to memorising facts?

I had an interesting discussion with some primary maths teachers recently about the benefits of memorising certain basic maths facts. While pretty much everyone agreed that if children had memorised number bonds to ten and times tables then they would have an advantage when performing calculations, there was a difference of opinion on what was reasonable to expect. Some teachers suggested that 100% fluency of times tables up to 12 by the end of Year 4 should be an appropriate minimum standard, while others were worried that it would be unfair to expect all children to learn so much in such a short time. Maybe we should have a sliding scale of expectations?

Now, I’m neither a maths teacher nor a primary specialist, but I do know that the ability to store information in long-term memory is not correlated with intelligence. There may be all sorts of physical or cognitive impairments which make it hard for children to remember things but it’s not true to say that some children are not clever enough to remember their times tables. They may not appreciate the value of knowing the times tables and they may struggle able to apply this knowledge in unfamiliar contexts, but the ability to recall information is achievable for all but the most damaged.

If we couldn’t remember we wouldn’t be able to perform anything except the most automatic process such as breathing and blinking. Even something as basic as vision requires us to learn how to make sense of visual signals; babies have to learn depth perception before they can adequately interpret their environment. The thing is, we’re unaware of many of the things we’ve learned. Very few people are ever conscious of the cognitive process of interpreting visual signals, we just see. Similarly, very few skilled readers are aware of having memorised the vast number of phoneme/grapheme relationships require to read English fluently, we just read.

The advantage of having stored such background knowledge in long-term memory is that we no longer have to think about it. Working memory – the site of conscious thought – is a small workroom in which we process information from the environment as well as retrieve information from long-term memory. Our ability to pay attention is strictly limited – there’s perhaps an average capacity to hold on to about seven items at once before we start forgetting stuff and feeling overwhelmed. Those people who have a slightly larger than average working memory have an enormous educational advantage; those with smaller than average working memories are those who most need to store the basics in long-term memory. If we have to use up some our limited processing power to think about phoneme/grapheme relationships then we’ll have less space to think about the meaning of the words we’re decoding. Likewise, if we’re able to instantly recall that the product of  7 x 8 is 56 this means we have a lot more space to solve a calculation which we have not memorised the answer to than if we have to first go through the process of working out what seven eights might be.

Everything we do depends of information stored in long-term memory and the more we know, the easier we find it to think. The good news is, storing information in long-term memory – or learning – is something we’ve evolved to find easy. Almost everyone learns to see and speak and cooperate with others with relative ease. And a majority learn to read and perform basic calculations without too much trouble. But even those children who seem to have so much difficulty remembering times tables have no trouble remembering thousands of song lyrics, extensive lists of sports trivia or whatever else they’re interested in. The general rule seems to be, no one struggles to learn what they recognise as being worth learning.

So, what’s the best process by which children might be expected to learn such information? Clearly it’s possible to memorise a relatively small domain – such as the 12 times tables – by brute force, but this kind of rote learning seems not to work for many. I remember chanting times tables as a youngster and the experience left me cold. I found that in order to retrieve 7 x 8 I’d have to work my way through the rest of the seven times table first. I never got to the point where I just knew the answer was 56. This isn’t much of an advantage and I had no more working memory capacity to solve calculations as a result.

One of the teachers I spoke to said that although some children had memorised times tables, they were unable to perform multiplications slightly outside these narrow parameters. For instance, they might be able to reel of the 9 times table up to 9 x 12 but have no idea how to work out 14 x 9.

This is the difference between flexible and inflexible knowledge. If we asked children to memorise phoneme grapheme relationships without ever getting them to think about how these re used to build words and sentences, then it’s possible that they might struggle to apply their knowledge to the task of reading a text. Similarly if all children do is rote learn number facts without thinking about what they mean then they might not be able to use their knowledge very flexibly.

When considering whether it’s worth trying to get children to memorise facts there are at least three considerations:

  1. Do the facts to be memorised have sufficient utility to make the effort worthwhile. For the basics of reading and mathematics this should be obvious, but for other, less foundational knowledge, teachers will need to consider the opportunity cost of investing curriculum time in this way.
  2. Trying to learn a fact in the absence of meaning is not only harder; it’s more likely to result in narrow, inflexible knowledge. Hopefully no teacher would insist on rote memorisation without also drawing attention to patterns. It’s not enough to simply get students to memorise basic knowledge, they also have to think about what it might means and how it could be applied.
  3. How will the process be undertaken? The word memorisation implies arduous and repetitive drill, but much of what we memorise is not learned in this way. Low stakes, distributed retrieval practice may offer a way forward, as might specific programmes like Times Tables Rockstars.

66 Responses to Is it worth trying to memorise facts?

  1. howardat58 says:

    I tried to remember poems when I was a kid. Hopeless. I tried to get the sin(A)+sin(B) formulas in maths. Hopeless. I still have to work them out. I tried to remember the lines from “We want Dames” (South Pacific) when I was 50. Hopeless.
    End result – I studied math for 7 years, nothing else !
    Finally I found ballet – no words at all.

  2. Tom Burkard says:

    Randomised presentation of number bonds is essential: the discredited practice of chanting is one reason that a lot of people have negative associations with learning their ‘tables’. This said, the profession has made a mountain out of a molehill. Once the pupil understands the commutative principle, there are only 66 facts from 2×2 up to 12×12 that must be committed to memory, and the 10x and most of the 11x are very easy to learn.

    Here I must declare an interest: I’ve published materials for teaching math facts for addition and multiplication. Sales so far have been abysmal, depsite extremely promising trials. Primary school are simply in denial: they all claim that their pupils already know their number bonds. And so they do–if they count, use repeated addition or other strategies. What they most conspicuously lack is automatic recall. Our very limited sampling suggests that very few pupils leave leave school with anything approaching automatic recall. My co-author has found that very few of his colleagues can even give the answer to 8+5 automatically.

    According to Common Core standards in the US, children should have automatic recall of number bonds for addition by the end of second grade, and for multiplication by the end of third grade. Yet here, Nick Gibb has been forced to delay implementation of online tests of multiplication facts at the end of Year 6. I fear that anything smacking of rote learning is still considered next to child abuse in many primary schools. Our experience has demonstrated that pupils do not share their aversion; even the slowest pupils looked forward to sessions.

    • David Didau says:

      Have you got a link to your books Tom? I’d be interested in taking a look

      • Tom Burkard says:

        The basic concept is that new number facts are introduced as previous ones are mastered, and that flashcards and written exercises require retrieval. I’m in the process of updating the website, especially in respect to relevant research. I’d be happy to send a pdf of the exercise book if you’re interested. We’ve found that timed exercises and progress tests provide more than enough motivation. Although I am no fan of collaborative learning, we’ve found that this is ideally suited to reciprocal peer tutoring.

        The address is

        • Michael Pye says:

          Tom I just had a peek at your website. Could I ask the rational for not teaching subtraction and division as these are traditional more difficult.
          Apologies if I missed the information, I teach SEN students in a college setting and I am very interested in your approach.

  3. Knowing and using number bonds within 20 (8+5=13 or 7+8=15) is just as important than times table knowledge but harder to learn. So many kids do not have automatic recall of these facts and this gets in the way of them learning column methods of addition and subtraction easily because so much of their attention is on the acids calculating. We have just started using a simple drill and practice computer game ‘Hit the Button’ with all classes from yr2 upwards with the expectation that with 3 minutes daily practice children will massively improve their automaticity in recalling basic number facts and then tables ( including divsion facts). We expect this to translate into much quickly and more secure learning of other maths. However, we are alert to the problems inherent in transferring learning from one context to another- pointless ifnthey can only remember in the context of a game!

  4. mikercameron says:

    I suspect the level of ‘memorisation’ depends largely on the process used to learn the times tables in the first place. The error that I think can sometimes be made is to believe that once a student can chant the tables they have learnt them.

    That’s only half the job.

    If you learnt times tables predominantly by chanting then it is likely that you do recall 8*7 by reference to either silent chanting though the list until you get to 8*7 or by reference to a commonly used value e.g. 7*7 and then moving on.

    If you learnt by completing repetitive examples it is more likely in my experience that you can instantly recall facts such as 8*7 at random. It’s the recall practise that is important here (how’s that egg by the way). Learning additional maths facts e.g. the commutative properly of multiplication enables the student, with practise, to turn half the problems into a two step process which is still faster than running through the chant.

    Over-riding all this is the need to think about efficiency. Why learn something you won’t ever use a great deal? Why take, say 100 hours to learn your times tables if you are only ever going to need them sparingly through your life? Would it be a more efficient use of time to learn something else? On this one my guess is that probably, over a persons lifetime, there is an efficiency in learning the TT (up to 10*10) given that the maths facts learn underpin most everyday numeracy a person will come across.

  5. David Moorse says:

    Lovely to hear the learning of facts held up as a helpful tool to facilitate (make easy) a plethora of other Maths processes. Freeing up working memory to focus on complex issues HAS to be worth the investment, especially as those who need it most are likely to benefit most!

  6. Paul Hartzer says:

    I don’t understand why we expanded the requisite times table to 12. When I was a child, it was 9. I personally think that 9 is BETTER than 12, because it reinforces that 10+ involves breaking it down. I suppose 10×10 is a trivial enough matter (and, honestly, it may have been 10×10 I learned, not 9×9), but 11×11 and 12×12 strikes me as not just memorizing things for the sake of roteness, but it actively conflicts with the takeaway we want in the long term, i.e., that 456 * 152 is best done as nine multiplications and a bunch of addition rather than as a table lookup from a 456 x 456 table.

    • howardat58 says:

      Or you could use a calculator.

      • Paul Hartzer says:

        Well, sure, but what if you’re lost in the middle of the woods and your cell phone dies, and you need to calculate 456 * 152?

        Or you’re taking the SAT and they tell you you can’t use a calculator. Which is more likely to happen than that first situation.

      • Michael Pye says:

        We can go to far with a reductionist argument to maths or any other subject.
        (Apologies Paul and Howard I suspect you are familiar with this, I have just had my yearly IQR and it always makes me rethink my approach)

        I teach SEN students and there is always this background argument around teaching them only what they need to know.(Usually by the non-maths teachers)

        I have a some time for it as it seems quite logical,(note: we already teach a massively reduced curriculum) but there is a clear demarcation issue about where to stop. Using estimation I can turn any money problem into a 1 digit (or two digit if I want more accuracy) question. I.e change from £5 for £3.89 becomes £5-£4=£1.

        Time is more tricky to but there are basic work around that allow most basic problems to be solved.

        The issue I find is that students struggle not with lacking a appropriate tool for any maths tasks (real world or abstract) but rather with the confidence, patience and resilience to apply them.

        I rarely teach long multiplication or division but I do a longest subtraction task early on a s a icebreaker to try and show how maths can break down seemingly impossible problems into a series of simple steps. (In this case a lot of one digit subtraction). I have always justified it as a form of over-learning and to teach a general mathematical approach.

        A more personal example:
        Before I started teaching maths my own skills had become somewhat atrophied. (I studied Physics at university but had 7-8 years of hardly using any).

        However I noticed that I had a distinctly different approach to my colleagues. (Who had studied very little maths) both in planning and dealing with any maths questions that popped up in the office.

        When I came across something I had forgot or was unsure about I immediately broke it down and started to play with it until I figured it out. I also insisted in double checking and looking for an easier approach.

        My colleagues usually had to go and look it up even though I know they had the skills to figure it out. (We are talking about basic maths skills here).

        .n summary I am not sure exactly how much help over-learning is directly but there seems to be a correlation between confidence/perseverance and maths knowledge. (Wonder if anyone ever did a study on this or if I am imaging it).
        Alternatively i may be justifying my own existence. I would appreciate your thoughts.

        • Paul Hartzer says:

          By accident or design, my math education experience was on learning how to attack unknown problems. Sure, there was lots of memorizing of formulas and such, but there seemed to be more of that in science than in math. The only memorization I remember in math was the Pythagorean theorem, standard area and volume (triangle, rectangle, circle, box, cone, cylinder, sphere), and a few identities for calculus. Even things like pyramid volume were left for us to figure out by box + cone. Distance formula? That was just the Pythagorean theorem, figure it out if you forget. The only thing I remember memorizing that, in retrospect, I wish they’d connected instead was the equation of a circle (which is PT again).

          (More later, I just glanced at the clock and am late.)

      • Paul Hartzer says:

        @chrisanicholson That’s a fair enough response, although that falls into another of my peeves: Us teaching mechanical processes without explanation. Of course I know the relevance of 12; I would also support using base twelve instead of base ten as the standard, had things evolved that way. But they didn’t, and I don’t see “memorize 12 tables” then coupled with “so you can quickly work with inches and feet” (which is a blight on THIS side of the pond, anyway… we really ought to move to metric, like the rest of the world) or (to use my example) “so you can know how many donuts to make for that order of four dozen”. If we’re not making the importance of 12 relevant and obvious to the student, then teaching the 12 tables is more harm than good: It’s fossilizing 11 and 12 as objects instead of as 10 + 1 and 10 + 2 without the ROI of the admittedly numerous but still fairly specific cases where we still use 12.

        As I’ve already indicated, though, I don’t get the point of rote memorizing a table at all… I would rather we encourage ready access of one-digit pairings so they’re available without having to call up the entire data table, and if a child really can’t memorize the things at all, here’s a calculator, we invented it for a reason. Engineers dumped the slide rule basically overnight for a reason. Let’s stop evaluating a child’s mathematical knowledge based on fact recall and start evaluating on “richer understandings.”

        • Michael pye says:

          Can we measure those richer understandings in a meaningful way?

          Measuring factual recall is easy and if correctly set up to be low stakes can have a trivial opportunity cost. Ideally the teacher would then develop those clear links to new ideas. Like you I mostly figured stuff out but we are a self-selecting cohort.

          I do know that if I had been taught clearer strategies for memorising key facts and constantly cross referencing I wouldn’t have needed to figure them out when I started teaching. It would have paid a dividend of learning beyond myself.

          • Paul Hartzer says:

            Your first question illustrates the problem with assessing mathematics mastery in an objective way. If our instruction is based on how effective and reliable our assessments are, then yes, we’re going to teach facts, not connection. But if a general art class only taught facts like history (Van Gogh lived from x to y) and science (red + white = pink), most parents would think the art teacher had failed the students. And yet, we grade students on their ability to successfully replicate a calculator.

      • David Didau says:

        Thanks! Let’s put the relevant quote up in full:

        “And finally, one objection that’s come up is the idea that tables beyond 10 are pointless. To be honest, the 11s are not terribly useful, but they are so easy to learn that the opportunity cost of learning them is insignificant. As for the 12s, I have seen it argued that this is some hold-over from pounds shillings and pence that is no longer relevant. If you think that, kick yourself now; you have just accepted uncritically one of the most ludicrous claims on the internet. The 12 times table is one of the most useful. There are 12 months in a year. There are 12 inches in a foot. The number of degrees in a half or full rotation is a multiple of 12, as is the number of seconds in a minute, minutes in an hour and hours in a day. The fact that 12 is a multiple of 1,2,3,4,6 and 12 has made 12 and its multiples extremely useful for dividing up units of measurement for thousands of years. It’s also why we often refer to “dozens” when grouping objects or indicating magnitude. And that’s without the advantage knowledge of the 12 times tables gives in the many mathematical questions that will make use of the number 12 precisely because it has so many factors. If we weren’t biased by the number of fingers on our hands, we would probably have a number system built around the number 12. Seriously, how could any numerate person have missed the importance of 12s?”

        I would remind readers that the wider point is not which tables might need remembering but the general point of memorising. I see no one has seen fit to argue against the necessity of memorising phoneme/grapheme relationships.

        • Paul Hartzer says:

          Phonics is EXTREMELY problematic. English is ridiculously non-phonetic.

          Fluent readers don’t rote memorize phoneme/grapheme relationships. Through repeated practice and exposure, they internalize what sounds letters usually make in what contexts. But the English reader who learns that “g says GUH” is in for confusion (witness the “GIF” controversy).

          • Michael Pye says:

            Paul you are misunderstanding the phonics debate. The evidence (three big studies) points to a specific intervention (systemic phonic, ideally without confusing it with mixed methods) for a very specific age group and for a limited duration. The approach tackles head on the difficulties with the language by grounding it in the phoneme/grapheme rather then the alphabet.

            I think if you read up on it and discover the details you will realise a lot of your concerns aren’t relevant as the approach is very specific in its scope. (please hunt for the studies yourself, I think its more likely to avoid the backfire effect).

            You may still have concerns but i think they will be different to the ones above.(For example the internalizing of sounds in different contexts is something I think it is trying to do better then other approaches)

            As a side note about a earlier post you are right about the difficulty of objectively measuring maths (or anything else). Performance vs learning is regularly discussed on this blog.

            From a measurement perspective (which is a maths field) the best we can do is use valid and reliable instruments to draw inferences. We can do this with knowledge.

            Extrapolating this to the point that we only teach this would be insane (which is what I think your saying). This is acknowledged by lots of people, its ignored due to a abuse and misunderstanding of data. (Really its basic maths abuse caused by a need for false accountability).

            Example: I want to test open ended problem solving but this is unreliable and often invalid if used to infer general problem solving.
            Multiple choice questions are reliable, efficient and valid but limited in what they can assess.

            it is a reasonable option to prepare students with a rich curriculum rooted on a foundation of shared knowledge (which we can test) while pushing their understanding in ways that are impractical to grade. (But nevertheless doing it)

            Please consider if this misunderstanding could lead to a flawed overvaluing of higher order skills and inquiry learning. (or putting the cart before the horse).

            In a feeble attempt to make this post relevant to Davids article I do believe memorization (and tests to assess it) are worthwhile. We can use more interesting methods then the Victorians while keeping the opportunity cost quite low allowing us to do more interesting things. For context i teach SEN students who need an abnormally high level of repetition. Around a third of my lesson time (aprox an hour) is sufficient in my opinion.

          • Michael Pye says:

            In the post below read the “teach this would be insane” part as we are agreeing with each other.

          • Paul Hartzer says:

            Were we talking about memorizing multiplication tables for a limited duration for a specific age group?

            I see David comparing learning times tables (as a needed student skill through mathematics education) to phoneme-grapheme memorization (hence, by implication, as a needed student skill throughout ELA education).

            Also, my child (7) is a very literate reader and did not go through phonics. I learned to read by 5, again without phonics. David used the word “necessity”.

            What bothers me about discussions like these is the implication that people who don’t memorize their times tables, or people who don’t learn that “g” says “guh” except when it says “juh”, are somehow deficient.

            If there are studies that somehow try to salvage phonics by narrowing the focus to the point where, hey, finally, our data supports our approach… well, that’s not convincing to me.

          • David Didau says:

            That is just not true. English is 100% phonetic, it just has a very complex orthography with 44 phonemes and over 170 different graphemes. Fluent readers may not *rote* memorise phoneme/grapheme relationships but they absolutely DO memorise them.You just have no memory of the process and little ability to introspect about this knowledge as much of it is tacit.

            i think you’re confusing rote with memorisation – I make clear in the post that I’m not advocating rote learning.

            When you say you (or your child) learned to read with phonics, I imagine you mean ‘without exposure to systematic synthetic phonics’. This may be true. It is also totally unconnected with any point I’m making. *How* you (or anyone else learned to read is irrelevant. Regardless of the process you went through, *what* you learnt (among many other things) were the phoneme/grapheme relationships which allow you to decode text in English. If you didn’t learn this then you wouldn’t be able to do it!

          • Paul Hartzer says:

            English is not 100% phonetic. (Actually, NO writing system is 100% phonetic, but I made that error in my comment as well. I meant to write “phonemic”.)

            Through tough thought, though…

          • David Didau says:

            Through tough thought, though… these words are all decodable and thus phonetic. The fact that different graphemes relate to a range of phonemes is precisely why English is an complex (or opaque) orthography.

          • Paul Hartzer says:

            Regardless of whether English writing is 100% phonemic or 95% phonemic, an argument that English is even LARGELY phonemic relies on probabilistic rules and specific lexical entries. Take the letter . A fluent reader might have something like this as an internalized rule:

            By default:
            1. is silent in the frame , where is not separated by a syllable break.
            2. is an affricate (an in “jay”) when it precedes a high or mid front vowel (“giant”, “gel”) in words that “sound French” or in the frame .
            3. Except as in the previous rule, is the nasal velar, and can only appear in the frame .
            4. Otherwise, is a stop (“get”).

            “GIF” is ambiguous because its creator wanted it to follow rule (2), but most people assume it’s rule (4) on the model of “gift” (a word of Germanic origin).

            Even with this rule, readers have to tie specific pronunciations to some words (e.g., “apply rule (2) to ‘giraffe’ but rule (4) to ‘gibbon’). And there are still a handful of spellings so odd it doesn’t make sense to have a phonemic rule. Why have ” can be /Up/” when it only occurs in a single word’s archaic spelling (“hiccough”)? It seems more plausible to me that readers simply store that as an exception.

            Comparing this to memorizing multiplication tables, though, is highly misleading because multiplication rules are probabilistic or contextual: We don’t have a rule like “3 x 6 = 18 unless it follows 4 +, in which case it’s 16”.

            Also, when people speak of memorizing TABLES, I interpret that to mean that there’s a mental, well, TABLE of information that is used to look up information. Multiplication can be done by memorizing a handful of interrelated facts. Do I have an internal memory structure that says “2, 4, 6, 8, 10, 12, 14, 16, 18” — multiply 2 x 7 by finding the seventh item in this row? Or do I have 2 x 7 = 14 stored as a fact independent of the rest? I don’t know. I DO know that I have that stored in a way that I can get it “instantaneously”, and I sense that I’ve stored 2 x 7 and 7 x 2 in the same realm of “multiples of 2” (rather than as “multiples of 7”). But I can’t tell you (a) how precisely it’s stored or (b) whether I stored it for “instant” recall because of the repeated table drills, the repeated random drills, or a lifetime of contextualized repetition.

            If the question is: Does it help to have internalized multiplication pairings for instant access? Sure. If that’s what we’re calling “memorization”, fine. If the question is: Is explicit focus on memorizing tables a more productive use of our educational time than practice and context? Well, I’m not so sure.

            If you’re making a comparison to language rule acquisition, though, the answer would be no, not in general. For students who struggle with exposure? Yes. For most children, though, explicit instruction in language acquisition is often counterproductive, when time is better spent in exposure and allowing children to create and test their own conjectures.

            HOWEVER, I’m uncomfortable with that comparison because children are surrounded by people using language nearly every waking moment, not so with math. Also, language is intricate, full of exceptions and idiosyncratic rules, as well as with dialectal variation. Brits use “u” in words Americans don’t as a matter of cultural pride for both groups; there’s no equivalent behavior in mathematics.

            Mathematics is a form of communication, but it’s dangerous to draw analogies too tightly with natural language.

          • David Didau says:

            Written language is not ‘natural’. That the point.

          • Paul Hartzer says:

            Reposting my first few paragraphs because your blog interpreted my less-than-greater-than grapheme tags as HTML codes and edited them out:

            Regardless of whether English writing is 100% phonemic or 95% phonemic, an argument that English is even LARGELY phonemic relies on probabilistic rules and specific lexical entries. Take the letter “g”. A fluent reader might have something like this as an internalized rule:

            By default:
            1. “g” is silent in the frame “Vgh”, where is not separated by a syllable break.
            2. “g” is an affricate (an in “jay”) when it precedes a high or mid front vowel (“giant”, “gel”) in words that “sound French” or in the frame “Vnge”.
            3. Except as in the previous rule, “ng” is the nasal velar, and can only appear in the frame “Vng”.
            4. Otherwise, “g” is a stop (“get”).

            “GIF” is ambiguous because its creator wanted it to follow rule (2), but most people assume it’s rule (4) on the model of “gift” (a word of Germanic origin).

            Even with this rule, readers have to tie specific pronunciations to some words (e.g., “apply rule (2) to ‘giraffe’ but rule (4) to ‘gibbon’). And there are still a handful of spellings so odd it doesn’t make sense to have a phonemic rule. Why have ”’ough’ can be /Up/” when it only occurs in a single word’s archaic spelling (“hiccough”)? It seems more plausible to me that readers simply store that as an exception.

          • Paul Hartzer says:

            “these words are all decodable and thus phonetic.”

            (1) “Hiccough” is an acceptable spelling for /hIkUp/. Since that’s the only word I’m aware of where “gh” represents /p/, that seems like a lexical entry to me, not a grapheme-phoneme link. There are a handful of other words like that.

            (2) “Decodable”, but not necessarily successfully. Words like “live”, “read”, and “bass” require context to know how to pronounce them. “Loose” is not a proper spelling for /lUz/ even though “choose” is a proper spelling of /cUz/. In languages like German or Spanish, someone who is not fluent in the language but knows the pronunciation can be reasonably confident if they meet a new word that they’re saying it correctly (or at least in the ballpark). English Language Learners have to be much more fluent in English to have a level of confidence approaching that.

            But regardless, I still don’t get the point of your analogy. Math facts aren’t contextual, grapheme relationships (especially in English and French) are. We don’t look at “3 + 4” and think “they PROBABLY mean 7” (unless we have some reason to believe the article is about something other than traditional base ten addition), but we do look at “I read for fun” and have to wonder if they mean present or past tense.

            In my opinion, the mathematical analogy to grapheme behavior is when people write their numerals in sloppy or idiosyncratic ways and I’m left to decide whether a student means 4 or 9.

          • David Didau says:

            OK – I’m moving on from this discussion. If you would like to continue it elsewhere please feel free to write your own blog on the subject and invite me to comment 🙂

  7. So how is meaning stored in memory? Is “meaning” just more knowledge or is it something else?

    • howardat58 says:

      A meaning is a fact, but it is a linking fact, joining two or more facts, in one direction or in both directions. Example: 5 and 8 makes 13 can be a basic fact, we learned it. Or it can be a derived fact – we know that 5 and 10 makes 15, and so we can get a result 2 smaller to get 13. This will be an example of a derived fact, a number 2 smaller is a count of a number which is two steps below a starting number.
      Language is more difficult than observation!!!!!

    • howardat58 says:

      “This will be an example of a derived fact, ….”
      should read
      “This more general process is an example of a linking fact, …..”

    • David Didau says:

      Yes, ‘meaning’ is the result of aggregating sufficient examples and items within a schema.

      • This doesn’t really follow from your replies, but I am very curious…

        If I tell you that:
        -troggs belong in buttles
        -every buttle holds five troggs
        -troggs do a thing called pinding
        -troggs pind spoods
        -When a trogg pinds a spood, the spood becomes three flargs.

        Does this *mean* anything? Can you memorise this?

        If I ask you how many flargs you end up with if a buttle full of troggs pind one spood each, can you answer?

        Do you know anything as a result of this activity? Do you understand anything as a result of this activity? Does it matter that you don’t actually know what any of these things really are? If we proceeded in this fashion would the fact that this is all built out of things with no meaning begin to impede your capacity to memorise the growing structure?

        • David Didau says:

          Of course I can memorise this nonsense but why would I bother? Meaning is not inherent, it’s ascribed: things become increasingly meaningful as they build up connections in relevant schema. You’ve tried to create a trogg schema which has no connections with anything else in my long term memory so it will difficult to remember.

          • Paul Hartzer says:

            In my experience, though, this is the way a lot of students are approaching mathematics: As a bunch of disconnected, meaningless facts that need to be memorized long enough for the test.

            Examples like “Solve for x when 2(x + 1) = 12” come to mind. My students have overwhelmingly memorized an algorithm that tells them to “distribute to get rid of parentheses first”, so it doesn’t matter if x + 1 = 6 is a more efficient “first step” in this particular case. They’ll start with 2x + 2 = 12, and continue from there.

            That tells me (and yes, I know, a lot of other teachers) that they don’t really know why they’re doing what they’re doing. They’re counting buttles and troggs. And, like you, they’re wondering “why would I bother?”

          • howardat58 says:

            “…an algorithm that tells them to “distribute to get rid of parentheses first….”

            Who tells the students to do this?
            Clearly someone who doesn’t have a grip on algebra.

          • Paul Hartzer says:

            My guess is that it’s an abuse of PEMDAS/BODMAS. Eek! Parentheses! Everybody run!

            Fractions create the same sense of fear and panic in the Youth of America.

            Order of Operations is a cult.

          • howardat58 says:

            2/5 is a fraction.
            No it isn’t.
            two fifths is better, and more meaningful.
            And I could go on !!!!!
            I was thinking about the written language.
            Kids need to speak before reading, particularly early on, and using the spoken word to guide the reading. Reading a bunch of words where you don’t have a grip on the spoken words is stupid. One new word per read sentence is a start, and only when the context is a clue.
            Kids don’t get read aloud to and don’t get to speak much at all.

          • David Didau says:

            If students are approaching maths as “As a bunch of disconnected, meaningless facts that need to be memorized long enough for the test.” then they’re doing it badly. Much better to memorise a lot of connected, meaningful facts to improve one’s ability to think.

          • Paul Hartzer says:

            “two fifths is better, and more meaningful.”

            That depends on the intended meaning. I know CCSS-Math is infatuated with “2/5 = two one-fifths”, and I get the relevance of that. At the same time, “2/5 = 0.8 = a point 80% of the way between 0 and 1” could have more meaning in some contexts, in which case 2/5 is division we didn’t bother to do.

            A few years ago, I asked my child (then 5) how he would distribute 4 cookies to 3 people (me, him, and his mother). He told me he would give us each a cookie, then break the last one into four pieces and give us each a piece, and then… he wasn’t quite sure. But when I asked if he’d break each cookie into three pieces (isn’t that, after all, “four one-thirds”?), he looked at me as if I’d grown another head. 4/3, for THAT context, makes most sense as 1 and then “figure it out for the last cookie”.

            Adept mathematicians have the flexibility of seeing 2/5 the most meaningful way in context, but that’s difficult to teach.

          • howardat58 says:

            Adept …. if only.

          • Paul Hartzer says:

            As far as the point about language, I agree with you. One reason why reading often needs to be explicitly taught is because there’s not enough exposure at an early enough age to the idea of written language that encourages children to make conjectures about what those weird symbols mean. I understand that spoken language is instinctive and written language isn’t, but one route to reading fluency is through “whenever Mom points to ‘cat’ in the book, she says /kaet/… I wonder what the relationship is?”. (And here, I seem to be contradicting my comment elsewhere, but I do acknowledge that MOST English words have clear phoneme-grapheme relationships, although it’s probabilistic; we tend to fill our early readers with words that follow the rules, like “the cat in the hat is fat” — with exceptions like “the” and “is” being called “sight words”.)

            The more we read to children where we hold the book and even point our fingers, the more data they have for making their own conjectures about typical, probable, and exceptional pronunciations of those glyphs.

  8. jemmaths says:

    As a child I learnt my number bonds to 20 through playing a pretty rubbish Atari game. At the end of each level you had a minute to answer as many sums as you could to get extra points. I very quickly became fluent without realising it. Learning facts can be achieved in all sorts of ways.

  9. eanelson2014 says:

    Read what cognitive science has to say about memorizing math facts at:

  10. Paul Gautreau says:

    I truly appreciate the topic of this blog post, especially considering that memorisation and practise are not a ‘hot topics’ within my professional circles. So often it seems that memorization and practice are seen as the antithesis of deep understanding, rather than as an essential complement to understanding. Done poorly, memorization and practice can impede rather than enhance learning. Done well, memorization and practice are fuel for learning.

    My only beef with the blog post is the reference to ‘the most damaged’ learners. Maybe it’s my canadian ears, but the term seems harmfully degrading. Perhaps using the word ‘challenged’ would have been a better choice.

    • Michael pye says:

      Here are some more options.

      Students with extra needs.
      Students with educational needs.
      Students with processing difficulties.
      Alternatively students who find maths challenging.
      Low achieving students

      Issue is even these terms have negative conatations.

      David I do agree with Paul that most damaged students is less ideal then ideal.

      Still can’t think of a perfect replacement though.
      I try to avoided any terms if I can which is actually easier if your entire cohort is in that category.

      • David Didau says:

        I deliberated about the term for a while. I chose “damaged” because none of your or Paul’s substitutes convey quite how abnormal a child’s brain would have to be for it to be impossible for them to remember things. As I said, the ability to store items in long-term memory is in no way correlated with intelligence or ability. You would need to have experienced some form of severe brain damage for this to be impossible.

        • Michael Pye says:

          Understood, none of the substitutes convey the same meaning, just be careful of the context in which you use it. The old word retarded just meant slow (it is still occasionally used in physics).
          This is a completely accurate and objective description of a child with learning difficulties. Obviously no one would use it anymore as it is now emotionally charged.

          Going to read the Daily Mail now and purge myself. It will be nice to have a clear and perfect understanding of the world for an hour.

  11. “storing information in long-term memory – or learning” – this statement might be wrong. It assumes one type of memory – a memory of categories that can be tuned. I’m not sure this is the only type of memory “on board”.

  12. As I think, remembering facts greatly helps in learning. As papers for machine learning indicate, a learning process require a lot of data. A living organism sometimes cannot obtain such data in real time because a life is a very dynamic process sometimes. What helps to fix this issue – a memory, and a brain circuit that directs a “data” from memory back to the input of learning circuit. We name it “recall memories”. I think having this feature in a brain of homo sapience made him a great advantage over other species.

  13. […] to cope with applying and manipulating abstract knowledge. Thankfully – as I explained here – although we may be out of practice, remembering stuff is not only intellectually […]

Constructive feedback is always appreciated

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