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To contextualize: I know someone who is ten years old, and needed to use repeated addition to compute $4 \times 8$, i.e., needed to calculate it as $4 \times 8 = 8+8+8+8$.

Question: By what age should a kid have memorized the multiplication table?

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    $\begingroup$ This is impossible to answer in my opinion. Do you mean when does someone understand that or is able to recall the fact? I know students that understand but struggle to recall. $\endgroup$ – Karl Sep 26 '15 at 10:54
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    $\begingroup$ Can the 10 year old quickly add the 4 eights, or does s/he struggle with adding the 4 eights? I think that makes a difference in whether you have a cause for concern. $\endgroup$ – Amy B Sep 27 '15 at 0:00
  • $\begingroup$ Preferably never... your calculator (today phone) does arithmetic faster and more accurately. $\endgroup$ – vonbrand Sep 27 '15 at 12:45
  • $\begingroup$ @AmyB It takes her a time to adding eights. $\endgroup$ – Pichi Wuana Sep 28 '15 at 10:01
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    $\begingroup$ I'm 49 years old, have a PhD in physics, and teach math and physics for a living. When I need to know 4x9, I do it as 18+18. I don't have it memorized that it's 36. I've done it this way since I was a kid, and I do it very quickly. Why is this a problem? $\endgroup$ – Ben Crowell Sep 30 '15 at 0:53
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As an example of one curriculum, the Common Core standards say that students, "By the end of Grade 3, know from memory all products of two one-digit numbers." (http://www.corestandards.org/Math/Content/3/OA/C/7/) This matches my personal experience, too, for what it's worth (back in the 1970's).

Now, as someone who teaches community college remedial math, in practice there are large cohorts of any age who have not actually memorized the times tables (age 18, 20, 30, etc.). This skill is taken as the fundamental prerequisite for our lowest-level arithmetic courses. As an informal observation, people who think that times table memorization is optional (a) don't truly understand why the base-10 place value writing system was architected the way it is, and (b) aren't involved in teaching algebra or higher-level subjects (including work with fractions, factoring, etc.). Students who haven't memorized times tables are close to helpless when they get to an algebra class.

To support my students in college remediation I established a website with timed quizzes, having times tables as the first and most fundamental skill. For some students it's the first occasion that they've been made aware that automaticity in "knowing times tables" is not the same as "being able to repeatedly add". See: http://www.automatic-algebra.org/

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    $\begingroup$ For common core, curious why you didn't just link to the source article [Operations & Algebraic Thinking] (corestandards.org/Math/Content/3/OA/C/7) $\endgroup$ – JTP - Apologise to Monica Sep 27 '15 at 14:34
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    $\begingroup$ I have to say my experience is different. I teach many a student who is not "close to helpless" in algebra even though they don't know all their times tables. The student who knows many of them and can get to the others is usually very successful. Also, your order of ops activity in the automatic algebra quiz sometimes marks wrong an order that will actually still produce the correct result. (Though in general I like the tool you've made.) $\endgroup$ – DavidButlerUofA Sep 27 '15 at 14:35
  • $\begingroup$ @DavidButlerUofA The order of operations has has been pretty thoroughly tested for over a year now, so I'd be pretty skeptical of that. Do you have an example that you think is incorrect? If you want to take it offline (and I'd be interested) email me dcollins at superdan dot net. Thanks for checking it! $\endgroup$ – Daniel R. Collins Sep 27 '15 at 17:10
  • $\begingroup$ Email sent, @DanielR.Collins $\endgroup$ – DavidButlerUofA Sep 27 '15 at 20:58
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    $\begingroup$ I was about to mark my calendar to acknowledge this in the summer of 2020. But reconsidered . $\endgroup$ – JTP - Apologise to Monica Apr 11 '18 at 14:08
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I hope I can contribute an answer with a story. I failed one grading period in 3rd grade math because I stubbornly refused to memorize the multiplication tables. At that age I understood that multiplication was explained as repetitive addition. My reasoning was "why memorize that which I can derive?"

I failed because quizzes were timed, and I couldn't complete the repetitive addition fast enough. My teacher could see what I was doing, since the I did the addition in the margins. It still amazes me that she never pulled me aside and explained WHY memorization was important.

Why is memorization important? I can still only think of two related reasons: (1) It's needed for the fluency required for higher math; and (2) division (and other kinds of factoring) requires the immediate recognition afforded by memorization.

So when should students memorize multiplication tables? 1. After they have mastered addition. 2. Before they are introduced to division.

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    $\begingroup$ This was added as an anonymous suggested edit which got rejeced: " So, to answer the question: What age? Whatever ages are in 3rd grade. I think it would be good to find out why this student is using repetitive addition. If it's because s/he didn't do homework and is falling back to addition as plan B, that's one thing. But if it's because the student wants to understand concepts rather than mindlessly applying rote memorization, they should get an atta-boy. . . But then be redirected to memorize anyway. " $\endgroup$ – Joonas Ilmavirta Sep 30 '15 at 6:43
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    $\begingroup$ I see this sometimes (marginal sequential additions) among my remedial college students. But let me one-up that: at least once I've seen someone who had to make individual tally marks in order to find 8×9. $\endgroup$ – Daniel R. Collins Sep 30 '15 at 8:20
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Age is not a good marker of mathematical ability as abilities can differ so much. In addition I'd add that having a good memory obviously an advantage in the mathematical development of a child but it is not a prerequisite nor a guarantee of success.

On to 'knowing' your times tables. It is tempting to say that someone that recalls $8\times 4 = 32$ knows that times table but the reality is more complex. A child that knows their times tables should be able to answer questions like, " Which times table is $32$ , $56$ and $96$ in? " Could there be more than one? Why is $8\times 4=2\times 16$ ? There are other examples but the point is that recalling the facts isn't enough to 'know' them.

To be clear recalling multiplication facts is an integral part of future development but caution is required or frustration can set in.

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  • $\begingroup$ I have a masters in Mathematics and I can't answer that first question. Or rather not without factorization or thinking for quite a while. As to the the second question I'm not even sure what the correct answer would be. Pointing out both are the same power of 2? Computing their "value"? $\endgroup$ – DRF Aug 4 '17 at 10:22
  • $\begingroup$ @DRF: Re the second question: You have failed to see it because it is too obvious. Some students don't realize that you can "move" factors between the multiplicand and the multiplier without changing the value of the product, because they conceptualize multiplication as a black box that just takes two numbers and spits out a different number according to a memorized "times table." They don't think of it as a semantically meaningful operation over the ring of integers. $\endgroup$ – Kevin Apr 29 '18 at 1:47
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The answer of this question might be found in the question: "When are multiplications used in primary education?", and for answering that question, let's have a look at following example: the multiplication of numbers containing multiple digits, like in this example: 789*567.

We are performing this multiplication as follows:

9*7=63
80*7=560
700*7=4900
=> result 5523
9*6(0) = 54(0)
80*6(0) = 480(0)
700*6(0) = 4200(0)
=> result 47340
9*5(00) = 45(00)
80*5(00) = 400(00)
700*5(00) = 3500(00)
=> result 394500
Add everything : 5523 + 47340 + 394500
3 => result 3
80+40 = 120
600+300+500 = 1400
5000+7000+4000 = 16000
40000+90000 = 130000
300000 = 300000
=> result =  447363

Total amount of calculations (how many times did we calculate (add, multiply, ...) something?):

9 multiplications
6 additions
=> 15 calculations

Now do the same but replacing a*b by a+a+...+a (b times)

9*7 = 9+9+9+9+9+9+9 = 63
80*7=80+80+80+80+80+80+80 = 560
700*7=700+700+700+700+700+700+700 = 4900
=> result 5523
9*6(0) <- 9+9+9+9+9+9  = 54, add a (0) = 540
80*6(0) <- 80+80+80+80+80+80 = 480, add a (0) = 4800
700*6(0) <- 700+700+700+700+700+700 = 4200, add a (0) = 42000
=> result 47340
9*5(00) <- 9+9+9+9+9 = 45, add two (0) = 4500
80*5(00) <- 80+80+80+80+80 = 400, add two (0) = 40000
700*5(00) <- 700+700+700+700+700 = 3500, add two (0) = 350000
=> result 394500
Add everything : 5523 + 47340 + 394500
3 => result 3
80+40 = 120
600+300+500 = 1400
5000+7000+4000 = 16000
40000+90000 = 130000
300000 = 300000
=> result =  447363

Total amount of calculations (how many times did we calculate (add, multiply, ...) something?):

0 multiplications
53 additions
=> 53 calculations

In other words, in case the kik doesn't know the multiplications by heart, (s)he will need to do 3 times the amount of calculations, just for multiplying two numbers of three digits. This, in its term, will ensure that (s)he will need 3 times more time for finishing his/her homework, tests and exams, leading to disappointing school results.

I guess this simple example clearly shows that, once the kid will need to multiply numbers of at least three digits, (s)he needs to know the multiplication tables by heart.

Good luck

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    $\begingroup$ This does not answer the question. The question is not why one should memorization multiplication tables, but when. $\endgroup$ – Daniel R. Collins Aug 1 '17 at 21:13
  • $\begingroup$ The when explains the why: at the moment that the kid will learn to do multiplications of numbers, containing different digits, he'll need to be able to know the multiplication tables by heart. The exact age might differ from country to country, so this exact number is impossible to state. $\endgroup$ – Dominique Aug 2 '17 at 6:50
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    $\begingroup$ Maybe you could edit your question to include the content of you comment; possibly somewhat towards the start to avoid the misunderstanding. (Possibly the when and why are interchanged.) $\endgroup$ – quid Aug 2 '17 at 10:20
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    $\begingroup$ Thanks, @quid, I just followed your advise. $\endgroup$ – Dominique Aug 2 '17 at 10:32

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