On rote memorization and antiquated skills

When I write that we should focus on teaching useful skills, and avoid deliberate rote memorization… I get far more opposition than I expect. To my surprise, there is an abundant supply of teachers and parents openly supporting rote memorization and antiquated skills.

  • When I object to having kids being drilled in doing complicated but useless operations like the long division… I am told that “you need to do it the long way in your early education so you understand what’s going on under the hood.”

    Take a pen and some paper and compute 123.422 divided by 12.4. This is a real problem that grade 6 or grade 7 kids are being asked to solve. If you are anything like me, you will probably pause and realize that you have forgotten how to do it. Of course, it will come back… but my point is that it is a dead skill that only a tiny minority of adults use. And most of these adults are teachers.

    If we are trying to teach what is under the hood… what is the hood?

    How processors do divisions is not quite like pen-and-paper long division. In fact, most mathematicians have no idea how a processor does a division. Though it is an interesting process that might be worth teaching to some high school kids… it never gets taught…

    Let me repeat that: these educators telling you that kids need to know “what is under the hood” most probably could not design a circuit to compute a division without access to the Internet.

    “Knowing what is under the hood” is overrated. Honestly, I barely know what is under the hood of my car. I do not need to. And even though I have been programming for over 30 years, and I am a published computer scientist… there is much that I do not know or understand about modern microprocessor technology. At a lower level, though I know a bit about circuits, but I do not really understand how a transistor works. It is not that I could not understand… it just never proved useful. I could look it up on Wikipedia, I guess, but I am not even motivated to do that.

    How come millions of people use computers, but they do not know how transistors works? Well, because knowing how transistors work is not useful to them. Useful facts are relative: to a few people in the world, knowing about the inner workings of transistors is really important, but for most of us, it is irrelevant knowledge. Irrelevant knowledge is not automatically useful by some magical process.

    So what makes us so certain that all kids worldwide, millions of them, need to know how to divide 12.45 by 45.1 by hand using pen and paper…?

    For a long time, in Europe and North America, privileged kids had to learn latin. The intuition was that since European languages were derived from Latin, learning Latin made you better at these languages. Though this is true, the effect is small. You are much better off learning useful skills (like learning Spanish or German or French) than learning Latin. And that is why few kids learn Latin today.

    Just like we stopped teaching Latin, we should stop teaching long divisions and related antiquities.

    As far as I can tell, kids are probably better off playing Minecraft.

  • When I object to rote memorization for its own sake, people tell me that what is being memorized is useful. This, somehow, justify forcing kids into rote memorization routines.

    Let me restate my objection: rote memorization on its own is theory divorced from practice. If you work a lot with numbers, you will learn to multiply and divide them reasonably quickly.

    Deliberate rote memorization is an attempt to take a shortcut in the learning process… Instead of having people learn important facts by themselves through practice, we decide once and for all what the important facts are, we delay practice, and start with the memorization of the “important facts”.

    And, of all the important facts we could teach kids, we decide that 6 times 9 is one of them. In this world of ubiquitous computers, we have somehow concluded that doing arithmetic super quickly in one’s head is critical.

    Let me repeat that I have never memorized what 6 times 9 is… instead, I do 6 times 10 is 60, minus 6 is 54. It takes me a few extra milliseconds I suppose… but my approach scales… to do 15 times 9… I do 10 times 9 is 90, plus half that is 135…

    In a country like Canada where we are supposed to leave 15% tips, I am often amazed at how few people can compute the tip quickly. I suppose that is because the multiplication tables only go up to 12… but why 12 and not 15 or 25 or 50? Of course, multiplying by 15 is trivial once you stop trying to memorize and just apply simple algorithms. Maybe more people could do it if we stressed computational thinking instead of rote memorization.

    I pity the kids who were forced by their parents into memorizing these tables… I will not discourage my kids from using algorithms to solve problems.

    Is it any wonder that kids worldwide conclude that mathematics is boring?

    And what kind of model of learning does this rely upon? That the brain is some sort of crude data bank to be filled?

    Our brain memorizes many important facts because they are useful to us. I know the names of my kids. I can also recall when they were born. I know two natural languages: French and English. I know about the fundamental theorem of algebra. I know about Galois fields. I know how to make excellent bread. I know the exact recipe to make good yogourt. My two sons appear to know a lot about Minecraft. But none of this memorization is deliberate rote memorization. Rather, it is natural or organic memorization… by repeated practice, you memorize important facts. Babies do not learn languages by deliberate rote memorization.

    Yes, learning mathematics is hard (for everyone!), but it simply requires practice. Nobody ever became a great mathematician, or even a good one, by relying on rote memorization.

    I have learned many computer languages through my life… The latest one is Go. I think I must have learned the basics of Go in a week or so. How did I do it? I started programming. I absolutely did not start by rote memorizing the grammar of the language. I just started programming and when I got stuck, I looked up the answer. And some facts just stuck with me over time because I kept looking them up. For example, I have memorized that fmt.Println(“…”) will print a string on screen. It is just something that anyone programming in Go will get to know. The objective is never to memorize fmt.Println per se. That string is not important… we just have to memorize it, as a side-effect of our practice.

    Now, of course, if you enjoy rote memorization, then there is nothing wrong with it. There are entire competitions dedicated to rote memorization. It is a fine sport, but a lousy general-purpose learning strategy… especially when you apply it to kids.

16 thoughts on “On rote memorization and antiquated skills”

  1. Thanks for this post! Even in college learning, there rote memorization and fact retrieval all over the place… in part because of the ease with which it can be tested by machines as opposed to real assessment of real learning, open-ended and messy.

    Of course, usefulness is in the eye of the beholder. For me, learning Latin was useful; I use it all the time for my own learning: http://bestlatin.blogspot.com/

    Yet like ANY new set of language skills, Latin cannot be learned by rote memorization (although, sadly, that is often how it is taught).

    So too with writing skills, which is what I teach. For example, memorizing spelling lists will not necessarily lead to good spelling. That does not mean that good spelling is not important. It just means we need to think about how to learn that skill in the absence of it being naturally produced as a skill through oral communication. And that means teaching students explicitly about how to use a spellchecker and also spellchecker limitations (so they need to know how spellchecker actually work), teaching them how to use dictionaries efficiently, and, most importantly, giving good timely feedback about their writing in a HOLISTIC way, where the real focus is on their success as a writer. Spelling is part of that, but mere spelling cannot and should not be separated from its purpose: helping writers to write effectively, just one of many skills they need. Along with lots… and lots… of practice (just as you say), accompanied by accurate, honest, meaningful, useful feedback.

    You might enjoy this chart which I find very helpful and inspiring; it is about teaching writers versus teaching writing, but it can be applied to many other fields as well! πŸ™‚


  2. Funny you mention your own multiplication algorithms, because you just made me realize I use the same πŸ™‚ I started programming around age 9 and I remember giving algorithmic answers to math test questions I could not do the “correct” way in 7th grade. Man, that did not fly, even though the answer was correct. I think Hamming’s wise words resonate even more nowadays: “The purpose of computation is insight, not numbers.”

  3. Teaching Latin was outright harmful for English, b/c people believed that some grammar rules valid for Latin (e.g., not ending sentences with prepositions) should have also be valid for English. So, one must be super-careful with respect to avoiding antiquated skills.

  4. Subtract $20.25 – $10.23. This is a real example that (judging from personal experience) actual college students working as cashiers are unable to solve without pulling out their calculators. If they have so little sense of numbers, how do you expect them to learn any higher mathematics? But if you don’t drill them on arithmetic, this is the result. Which is not to say that I think they should only do rote learning, or that doing multi-digit long division on paper is a useful activity for them in real life, but I wouldn’t want to see basic arithmetic (e.g. memorization of multiplication up to 12×12) go away.

  5. Yes, Leonid, and even worse: still to this day, Latin teachers will defend the teaching of Latin as something that is good for the students’ use of English. Totally crazy-making. If we want to help students with their English, well, let’s teach them some English! Not Latin. πŸ™‚

  6. @Eppstein

    First, I agree that lots of college students are weaker in mathematics that they should be.

    But how sure are you that having been drilled in tables make it easier for you to compute $20.25 – $10.23 ?

    I have a 9-year-old boy. I just asked him and he answered right away.

    (I did for real in case you are wondering.)

    So if college students cannot do it… there is a deeper problem.

    A better challenge would be to ask students to compute 15% of $10,23.

    My youngest might not be able to solve this one because he may not not about percents yet, I did not check, but my 11-year-old can do it easily… I do not even need to check… and I can tell you that he was not drilled into learning his tables…

  7. I agree with you. I drilled and chanted multiplication tables as a schoolboy in Scotland, but you are quite right that anything beyond 12x was terra incogita, and that an algorithmic approach would have been superior there.

    And to in support of your assertions that the most useful memorisation comes from repeated use in practical situations, let me offer this anecdote:

    When I was a student I worked in a small corner shop in the north of England whose stock was entirely arrayed around two walls of the small square customer area. I stood behind a counter facing these walls. Customers would enter, select things from the shelves, place them in a basket, and then present the basket on the counter for the items to be checked out and paid for. I generally used a conventional electronic till to perform this duty.

    But one way that I found to amuse myself in what was really quite a boring job was to observe the customer as they placed items in the basket, and mentally calculate their total cost before they approached the counter. I would then glance at the basket, and casually state the exact total.

    I’d then use the till to confirm the result, to the customer’s amazement. This was quite fun, and a few months of it had the side effect that I’m now much better at mental arithmetic than my school years alone would have given me any right to deserve.

    There was no call for multiplication and division in the corner shop trick, but the “oral memory” nature of the practice helped me to create a mental space that holds numbers, and enabled these to be developed later. I still check stuff on paper the way they taught me at primary school though, and I don’t regret learning that solid method before branching out into party tricks.

    And yeh; sod it, calculator, which is glumly getting the bus, when you could have had a nice walk and a laugh.

  8. > learning Latin made you better at these languages. Though this is true, the effect is small. You are much better off learning useful skills (like learning Spanish or German or French)

    This is what Steven Pinker pointed out in his “How to Get Inside a Student’s Head”:


    > Even if learning music were shown to enhance math skills, that doesn’t mean it is as effective as the same number of hours spent learning math.

    But I’m not sure if it’s a good idea to put learning German/French/Spanish in the “useful skills” category.

    Here’s linguist Geoffrey Miller (3-part series):

    More on How to Argue for Foreign Language Instruction (2/3)

    And here’s Freakonomics:

    Is Learning a Foreign Language Really Worth It?

    (P.S. I’m currently learning Spanish.)

  9. @Mark

    One can reasonably ask whether learning French, Spanish or German is worth much to an English-speaking individual. I am not advocating compulsory foreign languages, except for non-English speakers.

  10. Learning a foreign language is most valuable when you have the time to develop real fluency. Unfortunately, the goal in teaching Latin is often not fluency at all, but instead a continuous emphasis on “translate into English,” which results in little/no actual Latin fluency, an elaborate secret decoding game. I learned Latin because it was very useful to me (I was studying Renaissance Polish literature, and I was interested in bilingual Latin-Polish authors), and I needed real fluency because I was reading Latin texts which were available in Latin only, not in any translation. Usefulness is very much in the eye of the beholder; it’s not an objective thing that is inherent in a topic. It’s about how the learner applies it.

    College language requirements that demand students do two or three semesters of foreign language are usually a waste of time because no real fluency is gained. Studying linguistics itself is often a much better choice, although by requiring a foreign language, we are denying students the option to make that choice for themselves unfortunately.

  11. @Daniel, even though British, Australian, and US folks don’t really have to learn languages (you can’t remember them anyways if you don’t use them in your daily life), most people on Earth have to bilingual, trilingual, etc. For them, knowing another language is an important survival skill.

    So, for the majority of population on Earth, studying a second/third language is super important. However, can you avoid rote memorization if you started learning the language say in high school?

    I suspect that the answer is not. Ideally, you learn the language via immersion. However, this is not practically feasible in many cases. So, at least some memorization, is inevitable.

  12. @Daniel

    What (if anything) do you think should be compulsory?

    I can’t think of anything other than basic literacy (including computer literacy), numeracy, perhaps some basic Economics.

    If kids have very long mandatory school years, schools/teachers can afford to spend a large fraction of them making kids do lots of meaningless things, perhaps including rote memorization of multiplication tables.

    Signaling in K-12

  13. Disclaimer: I hold the opinion that comment streams are not a good medium for making progress in a conversation. The fact that this comment is both very long and still incomplete shows that this particular conversation has surpassed the capacity of online comments.

    Long division is not about memorization – it’s a technique. (let’s remove judgment words like “rote” and “antiquated”) The two deserve different treatments.

    Memorization is just another way of saying “knowing”. There are things you must know, and things you can derive. The more that is on your “know” side, the more things you will be able to derive. Can it be taken too far? Yes. Do elementary math classes take it too far? Yes. But it’s not worthless.

    If yu haf tu luk up tha speling ov evry wurd yu tayp, you’ll either end up writing things that slow down the reader, or you’ll greatly slow down yourself as the writer.

    Long division is a technique that’s quite departed from any understanding, and I can agree that it’s not needed. However, a bare-human should be able to divide two numbers. Any reasonable technique is acceptable, and they don’t have to be a pro at it. But removing the ability to do basic arithemetic with paper and pencil would be like removing the requirement to ever write a sorting algorithm for computer scientists.

    Having both instant access to a lot of data in your brain and an understanding of how new data is generated are both vital to making intuitive as well as deep progress.

    As someone who has published novel research, I know that for you to have published your works, it required being so familiar with the previous works that you were able to devote enough brain power to make a leap and discover the next step. You could say that people who only need to use what is already known can use it by looking it up. But I would then disagree that there are two classes of people – those who use and those who understand. This is a philosophical distinction. Everyone should understand what they are doing and how their work is used — but to different degrees. Thanks to TurboTax, I don’t have to understand anything about how taxes are computed. But knowing some of the details helps me know which way to answer ambiguous questions in the program.

  14. @Aner

    However, a bare-human should be able to divide two numbers.

    There are many things that we could argue a bare human being should be able to do… and few of these are forced on all children.

    Build a bow, hunt, cook bread, fire a rifle, light a fire without a match…

    All of these are more likely to be useful than knowing how to divide 124.4 by 11.2 using pen and paper. Yet none are required from our children.

    Thanks to TurboTax, I don’t have to understand anything about how taxes are computed. But knowing some of the details helps me know which way to answer ambiguous questions in the program.

    On that basis, we could argue that accounting is a fundamental skill that should be taught to all kids.

    The problem is that it is a never ending list of things that might be beneficial, potentially.

    Some people think that everybody should know how to play an instrument and read music. I do not. It is not that I disagree that there are benefits to learning an instrument… but there are so many things that I “should” learn while my life is finite.

    Yes, I have published research… but even in my field of interest, I know very little. There is simply too much.

    If you are going to force everyone to learn something… then you should have the burden of the proof that it is indeed highly beneficial.

    And that is the problem: all we have are anecdotal remarks… there is no solid evidence that drilling the students so that they can divide 13.4 by 1.4 is generally helpful.

  15. I’ve come to believe that the purpose of an education is to acquire skills and that the actual knowledge is secondary.

    Rote memorization, especially in elementary and high school serves to teach kids how to have the ability to consume a lot of information at once and regurgitate a fair amount of it. Maybe I don’t need to rattle off the state capitals very frequently, but I do need to read thousands of lines of code with accurate recall. I attribute years of doing similar mental exercises in my youth to this ability–my brain is shaped to do this, and I’ve trained myself to do it efficiently. We also need methods of verifying that people are smart enough to use their brains and time efficiently, otherwise we wouldn’t know who to hire (especially when hiring unproven blank-slates, like kids right out of college). Maybe we don’t need someone who memorized all the Latin conjugations and declensions, but we do need someone who *can* use their brain in this way. This skill is secondary to the ability to sit down, concentrate, work through failure, and be naturally intelligent.

    Similarly, we need people who can do mathematical processes–people who can think algorithmically. Teaching our kids long division is similar to teaching quicksort–a good software developer doesn’t *need* to have the algorithm memorized, but a great engineer probably does–or at least has a pretty good intuitive understanding (because she did have it memorized at some point). Maybe we should be teaching kids sorting algorithms instead of long division (though doing so may that would put the numerical analysis folks in bad mood).

    Teaching how to assimilate information, learn processes, when combined with critical thinking skills and the ability to create an argument produce smart, creative people who can tackle difficult problems.

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