When should a kid have memorized the multiplication table?

Question

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?

Answer 1

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, in the 1970's. In the U.S., this would be around age 8 or 9 years old.

Now, as someone who teaches community college remedial math, I'll note that 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 likely aren't involved in teaching algebra or higher-level subjects (including work with fractions, factoring, etc.). Students who haven't memorized the times tables are close to helpless when they get to one of our algebra classes.

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/

Answer 2

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.

Answer 3

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.

Answer 4

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