We Don’t Need to Memorize the Multiplication Table

Octal Multiplication TableI’ve had something on my mind for a long time that greatly bothers me: the multiplication table. This is because I think that education should be fun — full stop. Memorizing the multiplication table is not fun. But more than that, I believe they are the source of most people’s hatred of math. At the same time, I’ve long thought that one should probably know them. And that created a conflict in me because I thought that it was really only possible to know them by memorizing them. But the other day it occurred to be what hogwash that is.

To begin with, I’m really good at math. Yet I’m pretty flaky on the multiplication table. I remember in the sixth grade, I had this teacher who seemed more like a drill sergeant. And he used to do this thing where he would go desk by desk asking each student one question, “Seven times eight?” He went sequentially through the table and the desks, so I always calculated which question I would get so I could figure it out and say it the moment he asked me. In that case, there was clearly some mathematics education going on for me, but it had nothing to do with the multiplication table.

I have never memorized what 6×7 is. Instead, I know that 6×6 is 36, and so I know 6×7 must be 6 more than that, which is 36+6, which is 42.

The rationale for knowing your multiplication table is that if you know them, then you can do things like multiply and divide arbitrarily large numbers. But if that’s true, why were we taught the table from 1 through 12? It should have been 1 through 9, right? You can calculate 11×12 — you don’t need to memorize that it is 132. And I think the reason for it is that it was a way to fill the curriculum. It didn’t take long enough to force the students to learn up to 9×9, so they went up to 12.

And as you get into higher math, you don’t even much deal with numbers. And for people like me who are naturals at math, it never is about numbers but rather elaborate games you make up. You think about things like the way the nines go down by one as you go up the table, and elevens go in the opposite direction. If you are lucky, you get introduced to octal and start seeing numbers in a whole new way. But you are unlikely to ever play with numbers if they are forced on you in the same way that the ordering of the alphabet is. (I am still quite fuzzy about which letters go before others without the help of the song.)

Multiplication is just a matter of counting. I have never memorized what 6×7 is. Instead, I know that 6×6 is 36, and so I know 6×7 must be 6 more than that, which is 36+6, which is 42. I don’t find this fun, but I did think it was a nifty trick when I was a kid — one I was never taught. And I fear that many if not most children memorize their multiplication table without having a profound sense that all they are doing is counting. That’s sad.

As everyone should know, math is extremely abstract. That’s why I say it is the closest thing to theology, and certainly it is the closest I ever get to God. But the mindless memorization of the multiplication table is not just boring. For many students, it turns math into alchemy. Sure, it’s abstract. But much more, it is mysterious in the same way God is: unknowable. And why even try?

There is no need for children to learn the multiplication table. I have a simple proof:

  1. What is 9×9?
  2. 9×9 = 9+9+9+9+9+9+9+9+9.
  3. QED.

20 thoughts on “We Don’t Need to Memorize the Multiplication Table

  1. Math is not my best subject. I had trouble learning factoring in high school. And you can’t do algebra without that core mechanic. So I had to drop and repeat algebra as a sophomore. I liked geometry. I should say, it came easily for me. Things usually do, and I get frustrated and quit when they don’t. Geometry was similar to the symbolic logic class I took in college. By the time I got around to addressing the required math for my degree program I had was 26, so I had to take a remedial algebra class that didn’t count as a math credit. And the teacher used a system for factoring I hadn’t been taught before. And it was so easy. I took the course for the math 1 credit the next semester, which was summer semester. It started at 7:40 am, and I lived on the other end of the city from ASU. So that didn’t help. I think I got a B- in that. I do remember before an exam the instructor asked if there was anything anyone wanted to review first. I asked him to go over changing log base. He said that wasn’t on the test. And it was. To be fair, it might be the case that you could solve that problem without it and I just didn’t know how. My other math credit was a statistics class. I actually use that in my work sometimes. Apart from that I don’t use or remember any algebra I hadn’t learned by middle school. All this is what prevented me from pursuing a career in engineering.

    • That’s something that makes me mad. For hundreds of years, people have been coming up with great ways to teach math and science. Yet mostly, we get the same approaches used by everyone. When I started teaching physics I discovered “the book.” It was an exhaustive text on all the ways people had come up with to teach basically every difficult subject in basic physics. It really helps to have 5 different ways to explain things. Too often, and instructor will just repeat the same thing.

      I also think that for math we should only have a mastery system. Math builds. If you are a little unclear on the last thing you learned, you won’t be able to get full mastery on what comes next. But we don’t do that because it makes educating students harder because they all have to be treated as individuals. And we want education to work like a factory.

      But I think by far the biggest thing is that math teachers have to love the subject. Sadly, a lot of math teachers not only don’t really like it, they are afraid of it. Math is awesome. And teachers should think that — especially the ones teaching very young students.

  2. I’m guessing the tables go up to 12 because once upon a time people worked with pounds, shillings and pence, and the only way to do the accounts quickly was with memorised times tables. But here in Australia we recently marked the 50th anniversary of our switch to decimal currency, which you can make of what you will. But I can say they do come in useful for me, because most of the time if you present me a basic multiplication question the answer comes to me instantly, just by rote. Worth it? I have no idea.

    • Ah, that makes sense!

      I’m not against memorizing the multiplication tables. I’m against forcing students to do it. I think that if you need it, you will naturally memorize it. I was always amazed that my second wife couldn’t do the most basic math. But if it had to do with money, she was great. “What’s 20×1.1?” She wouldn’t have a clue. But she could tell you what 20 time $1.10 was in her head instantly. A lot of it is just about being engaged.

  3. I do the “I remember this multiplication is X and if I add this one additional number then that is what it is.”

    I am awful at math and I hated it pretty much since I can’t remember the rules.

  4. @ Frank & Lawrence — I had this horrible calc teacher I still have nightmares about. She insisted on teaching her own special method for solving particular equations. You couldn’t use the textbook method on tests for these equations; you had use use her method or you’d get no credit for your answer.

    Well me and a friend developed our own method, a variation/simplification of hers, and presented it to her outside of class. Surely she could appreciate the creativity, since she was so proud of her method. Oh, hells, noes! She warned us never to use our method on a test. My friend was so appalled he dropped out of school and got a GED (he’s now got a math PhD.) I simply stopped going to class. I was a senior, I’d already been accepted to college, calc wasn’t a graduation requirement. Gimme an “F,” I don’t care.

    So she went to the dean and made sure I wouldn’t get my HS diploma unless I passed her class. I had to go over to her friggin’ house to do all my makeup exams! What a loon.

    • That’s a great story. I would question you, but I’ve run into that kind of teacher. Some of them are amazingly dogmatic. And I’ve noticed a lot of math teachers try to systematize things like solving equations when it really doesn’t work that way. It’s kind of like music: there are rules but you can only take it so far. That’s one reason I’ve always loved differential equations. There is little to it that you can systematize. You just need to play around with it long enough that you start to get it. Of course, the vast majority of differential equations have no solution, so there’s that too.

    • I can’t believe that in the less than one hour I had it down I got two different people complaining! It’s kind of impressive actually. It’s nice to have commenters.

  5. Sure, you can write what 9×9 is, but how would you write 5×3? Some math teachers insist it’s 3+3+3+3+3, and not 5+5+5. See http://www.patheos.com/blogs/friendlyatheist/2015/10/21/why-would-a-math-teacher-punish-a-child-for-saying-5-x-3-15/ . The comments section went on forever. The odd thing was that Hemant Mehta, a math teacher, was saying that the modern way of teaching was to only allow the one way of expanding it, although most of the time he says modern teaching means describing many ways of solving problems.

    I agree multiplication tables don’t need to go beyond 9, but I think they’re pretty useful in everyday life (“4 potatoes each for 7 people? That’ll be 28”). I had to learn far more things than multiplication tables by heart – Latin principle parts, for instance (“dico, dicere, dixi, dictum; ferro, ferre, tuli. latum …”). Times tables have been far more useful.

    • I agree that it’s useful. It’s also useful to be able to do reasonably large calculations in your head — which I can do (although not as well as I once could). But my point is that this isn’t critical to understanding math. And given that I think math is the most basic kind of thinking, I think that understanding math in a creative sense is critical to education. I think I said in another comment that I’m not against the multiplication table. I’m just against making kids memorize it. If you really need it, you will memorize it naturally.

      • Then teachers need to present projects and situations where students find need to ‘memorize’ the tables.

        They should ‘memorize’ them by repeatedly doing repeated addition. Over and over. Students who don’t learn the times tables usually don’t become good at any quantitative subject. Instead, they rely on calculators and have no insight into numbers. You get students turning on the calculator to divide 0.5 by 0.25.

        It’s scary though; I was a full-time private tutor for years. Elementary education was the only major I ever experienced in which people would say they’re doing it because they ‘can’t do math’.

        • But dividing .5 by .25 really has nothing to do with numbers. It’s proportions. You could teach it to someone who didn’t know multiplication or even addition. Take half an apple, cut it in half. How many pieces do you have?

          I posit the correlation you present is causal in the opposite direction. I am good at math because I have always been able to play with math. I see no reason why all children can’t do the same thing. As I went through school, I was in all the advanced math classes and all the remedial English classes. English was never taught to me in a way that made sense, just as math is not taught to most students in a way that they can understand, appreciate, and love.

          But what you mention at the end there is exactly my experience. Most of my grammar school teachers were terrified of math. Is it any wonder that they don’t create students who love math?

          • I really think you overestimate the degree to which most people love anything with mental effort.

            Anyway, people in non-English Europe learn their times tables just fine, and not just people who take quantitative majors in university. What they do differently in elementary education, I don’t know. I do know that the Germans and French don’t force people to go to university as the only realistic route to a job that pays more than minimum wage.

            The fact is that people who don’t learn times tables are unable to see anything quantitative, or logical, or proportional, as anything but formal mumbo-jumbo that has nothing to do with them, 99% of the time. Almost always, people who don’t know times tables can’t see the 0.5 as an apple, 0.25 as half. Not because there is any necessary connection between times tables and halving something, but simply because they are unpracticed.

            I’m all for mathematical play, but people who don’t know times tables can’t do it. I don’t really care what the causal direction is.

            But I only learned them up to 9.

            • I don’t think I overestimate the degree to which most people love this kind of stuff. Maybe I do, of course. But I used to be part of the GLOBE program and it seemed to me that every 4th grader was fascinated with everything. Things changed for the worse as you got to higher grades and high school students should just be put on an island somewhere.

              But my complaint about the multiplication tables is that it is route memorization. I’m terrible at that. Just the same, I don’t think that’s how I learned them. I learned them as infinite series. But I don’t recall them being taught that way. And this may be why I still am sketchy on the 7s. I still feel there is something fundamentally wrong with the number 7.

              I don’t think knowing my multiplication tables helps me in life — ever. Knowing math helps me all the time. I can recreate the tables. I don’t generally remember 6*7 and 8*7, but I do know 7*7. And it is never the case that I don’t have the time to add or subtract 7 to/from 49.

  6. My daughter had a hard time learning the multiplication table.
    So I wrote an android application to help her :)

    Each exercise you solve helps you to feed the cute panda.
    You can get daily reminders to feed the panda (solve exercises) and you get rewarded with trophies upon goals completion.

    This application helped her to learn the multiplication table without noticing :-)

    * Currently the application is only for Android phones *
    https://goo.gl/9QutZx

  7. Allow me to say to everyone that, if you google “Handy Multiplication Table” you will find a book with powerful techniques that enable kids to learn all the multiplication table quickly and without memorization. I hope this piece of information can help some.

    • I’ve tried to find it, but all I find is Mr. President’s Handy Multiplication Table. It sounds awesome, but for all I know, it is an attack on President Obama. I’m going to have to buy it now and review it. There is no reason for memorization. If your math depends upon memorization, you are doing it wrong. Other than 0-1-2-3-4-5-6-7-8-9, there should be no memorization. Thank you for introducing me to this!

      Update: I just bought the Kindle version. This will be fun!

  8. Hi Frank! You are welcome! It is my pleasure to share this information. Yes, “Mr. President’s Handy Multiplication Table” is the title. I think you will have fun with those handy techniques. From Amazon, here is its description:

    “Mr. President’s Handy Multiplication Table is a powerful and fun tool that will enable kids and adults to learn the whole multiplication table quickly and without memorization. Readers will reap the many benefits from facing their fears and mastering the multiplication table.”

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