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I think, a lot of students are bothered by learning multiplication tables by heart, in particular when it comes to numbers greater than 10.

Why should one learn (or not learn) these things by heart?

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    $\begingroup$ There seems to be a big conflict there (at least in the US), compare en.wikipedia.org/wiki/… $\endgroup$ – Markus Klein Mar 17 '14 at 7:48
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    $\begingroup$ In my opinion, it is unnecessary. I never did it myself and never felt the use of it. $\endgroup$ – user774025 Mar 17 '14 at 14:01
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    $\begingroup$ Is this question similar to asking whether the alphabet should be memorized? $\endgroup$ – Gerald Edgar Mar 17 '14 at 15:19
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    $\begingroup$ @GeraldEdgar I don't think so. I agree with mostly all people that multiplication tables for numbers $\leq 10$ are very much used, but I would say most people don't know more than this and I personally don't see the point in knowing exactly a multiplication table for some weird big number. I think, it's even more important how to approximate the size of some multiplication (here you need tables for numbers $\leq 10$), but I don't see an application so important that you really need the actual result and not some approximation (in such a case, you will then maybe use a calculator?) $\endgroup$ – Markus Klein Mar 17 '14 at 15:51
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    $\begingroup$ At most one needs to memorize up to $8\times8$. The $10\times n$ entries are trivial, and the $9\times n $ entries can be done as $10n-n$. Personally I don't have $8\times 7$ or $8\times 6$ memorized; I calculate these as $2\times(4\times 7)$, etc. The real issue here is that many kids simply hate learning their times tables, and it sours them on math forever. Maybe we should have kids memorize a $5\times5$ table and break down problems like $26\times 14=(25+1)\times(10+4)$. The algorithm would be more complicated, but they could learn the $5\times 5$ table in an afternoon. $\endgroup$ – Ben Crowell May 20 '14 at 15:05

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There is some relevant research (and bibliography) in this paper:

"Improving basic multiplication fact recall for primary school students" (Wong and Evans, 2007) http://link.springer.com/article/10.1007/BF03217451

For me the most striking thing is how low the scores reported in their Figure 2 are (36.89/60 at best), even after interventions that they consider to be valuable. I had no idea that teaching of multiplication tables was so unsuccessful. The study was done with 10 year old pupils in inner city schools in Sydney; I do not know how far it would generalise.

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    $\begingroup$ The link is to an article costing $40. Is there a summary or free access somewhere? $\endgroup$ – JoeTaxpayer Mar 17 '14 at 11:57
  • $\begingroup$ The article does not have a short summary that could usefully be cut and pasted. Home pages for the authors are goo.gl/YZjc5d and goo.gl/fP4H2Z, but they do not seem to have posted copies there. I will post something on meta about this general issue. $\endgroup$ – Neil Strickland Mar 17 '14 at 12:06
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    $\begingroup$ Here is the article, courtesy of myself and my university. $\endgroup$ – Jānis Lazovskis Mar 17 '14 at 12:31
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    $\begingroup$ The paper is really background for your question rather than a direct answer. It shows that the pupils in the study do not know their tables very well, and it measures the (positive, but limited) success of various methods to improve the situation. It refers to a large number of other papers, some of which probably address your question more directly. It was the most relevant paper that I could find after searching Google Scholar for 15 minutes; a more exhaustive search would probably find other things. $\endgroup$ – Neil Strickland Mar 17 '14 at 14:59
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    $\begingroup$ Late comment (thanks to jlv above for the article). Note that the metric is a timed 1-minute attempt at 60 questions. The pretest for the paper-based group started at an average of 22.63 correct per minute, so the post-test 36.89 represents a 63% improvement (p. 98). Granted that the target for mastery is 40 per minute (p. 92), the intervention went from about half that to nearly mastery success, which seems pretty good for less than 3 hours of intervention time (11 sessions × 15 minutes each, p. 95). $\endgroup$ – Daniel R. Collins Sep 27 '15 at 23:27
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I'm teaching a course of math for future primary school teachers and discussed with them a little bit this point.

In my opinion teaching multiplication table is necessary up to 10 and completely useless afterwards.

The point is that the usual multiplication algorithm is devised exactly with this purpose in mind: you cam multiply every pair of numbers as long as you know how to multiply two numbers which are both less than 10. So if you know the multiplication table up to 10 by heart you know everything you need to multiply (and in fact also to divide) "everything". I think this is an useful conceptual point. I do not mean you necessarily say something of this kind to first grade pupils but that you can, in such way, make them understand alows the power of certain algorithms.

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    $\begingroup$ Great answer. The one nuance I would add is that students had better be able to multiply any arbitrary number by 10. There are a surprising number of community college students in remediation who don't know about that. $\endgroup$ – Daniel R. Collins Sep 27 '15 at 2:52
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Reasons for having automaticity with single-digit times tables (from the perspective of a community college lecturer with many remedial courses):

  • Long multiplication algorithm
  • Long division algorithm, and thus:
  • Convert fractions to decimals
  • Understand why rational numbers have repeating decimal expansions
  • Understand the proof why $\sqrt{2}$ is irrational
  • Factor integers, and thus:
  • Understand the Fundamental Theorem of Arithmetic
  • Reduce fractions by finding the LCD, and thus:
  • Add, subtract, multiply, divide, and compare fractions
  • Factor polynomials, and thus:
  • Reduce rational expressions
  • Solve higher-degree equations
  • Understand the Fundamental Theorem of Algebra
  • Estimations to double-check technology output

Perhaps on a deeper level I'd say that the base-10 place value writing system was architected specifically to easily support these operations, assuming that single-digit elementary operations were memorized (like phonemes) -- so if someone hasn't done that, they're really not using the language correctly.

Students should certainly know how to multiply any arbitrary number by 10. For products of two numbers above 10, I would agree that it's not really critical -- although the 11-table is trivial and knowing the 12-table may be handy when dealing with clocks, inches, and units in dozens (and $12^2$, one gross, does pop up a lot).

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Someone already mentioned this but I don't think people understand the importance: The purpose to know things off by heart in Maths is to recognize the inverse applications.

I explain to my students that recognizing 121 as the square of 11 is like spotting a friend in a crowd. If you did not know his face he will just be another number.

This is true for multiplication tables (even up to 12 in my opinion) and the powers $n^m \text{ for } \left \{ n,m\in Z| n^m<1000\right \}$ and especially for formulas and theorems. I especially see it with factorization in Algebra, all Euclidean Geometry and the compound angle formulas in Trigonometry. So if the person asking this is a primary school teacher, please understand that by teaching them to know these things off by heart is introducing them to the friends that will help them with Maths later on in life.

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As I tutor students and watch those with proficiency in up to 10x10, there's a clear difference in their ability to get through the problems in a timely manner. It's a distraction to use a calculator for the simple math (sometimes) required in algebra problems, say, 6 X 7.

I'm not suggesting a cause/correlation of intelligence. Teachers around me are split on the calculator issue, those who are in favor clearly don't share my concerns.

My claim is this: Take two sets of students, those who mastered the tables and those that didn't. But screen those that didn't to produce those who are otherwise proficient, fitting the teacher's claims that the calculator is a tool and it's the solving process that counts. As these students move forward, the calculator is a crutch and time waster. This results in the calculator student getting slowed enough to had a disadvantage over the non-user.

Disclaimer - I am strictly speaking of calculator for the pieces of longer problems where I would naturally just do the math in my head. Single digit math, and simple math on larger numbers. When I see a student keying in "7+5" on a calculator, yet producing a B or A grade exam, I ask myself how that student will fare on the SATs (The college entrance exam in the US, where Math was, in my day, scored on a scale to 800 being high score). Or even on exams in college, where some amount of the manipulation is simple as part of the larger problem.

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  • $\begingroup$ The trouble is that nice, round, single-digit numbers appear only in the problems teachers need solutions for. The explanation of that mistery surely merits another question. $\endgroup$ – vonbrand Mar 17 '14 at 16:52
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    $\begingroup$ Let's assume for this discussion that's true. And my answer is not about real life, but about test taking. If tests were untimed and SATs didn't matter, the answer would be less relevant. I am at odds with those who think learning the method but using calculators for what I believe trivial isn't a concern. My experience tells me that if for no other reason, knowing the 10x10 table gives the student an advantage. $\endgroup$ – JoeTaxpayer Mar 17 '14 at 21:13
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The point of learning multiplication tables is not to get the product of two numbers. It is to be able to quickly recognize that a larger number can be factored into smaller parts that are easier to handle.

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Let me add my 2 cents in. As someone who grew up in Latin America (where the tables were drilled into us, or else) and who now tutors mathematics, more than once I've had the experience of students were AMAZED that I could recall simple multiplication faster than they could type it in their calculator.

I personally like to think we memorize the tables for the same reason we learn to drive, or for the same reason Feynman reportedly advised students to be so familiar with derivatives they should be able to do them in their sleep. If we were always hesitant about driving, if we were always thinking about the individual tasks driving requires, we would never enjoy the actual purpose of driving (getting to nice places). Similarly, if we don't memorize the times tables we will be too lost in the trees to see the actual mathematical forest.

It is my honest opinion that math reforms swing the pendulum too far away from drill. Drill and memorization have their place in mathematics. It is only after we have become so proficient at a task that it becomes automatic that we can begin to fully appreciate it and tinker with it in creative ways.

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I always thought that multiplication tables above 10 were used in English-speaking countries because of non-decimal units: in Italy nobody bothered with them.

This said, I think tables up to 10×10 are useful in real life, even if they are no more used for long multiplication; so they should be learned by heart.

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    $\begingroup$ As a student in the early '70s learning multiication, the table was 12x12. But I recalled seeing one go to 25x25 and thinking that was coming in a few more grades. This was before calculators were invented, and was just for reference. When my daughter learned the table, it was already 10x10 only. $\endgroup$ – JoeTaxpayer Mar 17 '14 at 11:54
  • $\begingroup$ Multiplication by 11 is easy; multiplication of small numbers by 12 is often useful for things packaged as dozens, I'm not sure how often 7x12=84 or 9x12=108 would get used (as a programmer, 8x12=96 comes up often). I don't think it's vital to have all the products memorized, though, if one has enough memorized that one could, if one had to compute seven or nine dozen, quickly say "six dozen is 72, so adding another dozen would be 72+12=84" or "ten dozen is 120, so subtracting one would yield 120-12=108". If someone encounters "seven dozen" often enough that it would be useful... $\endgroup$ – supercat Jun 20 '14 at 15:10
  • $\begingroup$ ...to have it memorized, the person will pretty quickly start associating "seven dozen" with "eighty-four". Having the 10x10 table memorized will make it easier to multiply numbers by hand--a skill which isn't needed terribly often, but is sometimes necessary. $\endgroup$ – supercat Jun 20 '14 at 15:24
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From A Brief History of American K-12 Mathematics Education in the 20th Century by David Klein:

Some proponents of the Activity Movement [of the early twentieth century] did not even acknowledge that reading and learning the multiplication tables were legitimate activities. As in the 1990s, there was public resistance to the [traditional] education doctrines of this era. Among the critics were Walter Lippman, one of the nation's most widely respected commentators on public affairs, and literary critic, Howard Mumford Jones.

In the 1940s it became something of a public scandal that army recruits knew so little math that the army itself had to provide training in the arithmetic needed for basic bookkeeping and gunnery.

Klein Cites Judging Standards for K-12 Mathematics, In: What's at Stake in the K-12 Standards Wars: A Primer for Educational Policy Makers in the last sentence. It appears that the progressive ideals emerging from the late twentieth century that include teaching only "useful" or "necessary" mathematics and omitting the rote memorization of multiplication tables, like those similar initiatives that came nearly a century earlier, will certainly result in a public that lack basic skills necessary for homeland defense.

If the army found that these recruits were unable to carry out basic bookkeeping and gunnery, their classmates with the same deficiencies certainly had difficulties with the following basic but important household tasks:

  • keeping and updating a home inventory
  • calculating the cost of a grocery bill with many items
  • checking the accuracy of a bill or financial statement
  • calculating long term costs of services billed at regular intervals
  • planning to provide food or beverage items for an event with many attendees
  • preparing short- or long-term emergency provisions for a household
  • estimating the rate of speed one must travel to arrive at a destination in a given time
  • estimating the time required to travel between two distant locations
  • estimating the long term consequences of making changes to a budget
  • modifying a recipe to accommodate more or less people
  • estimating the cost of home improvements
  • estimating the cost of a series of medical appointments
  • planning or estimating the cost of travel for a household

While it may not be absolutely necessary for all students to memorize multiplication tables because of technology or reference tables, or because the internet culture allows households to "outsource" these tasks, the prudent choice is to teach multiplication tables and require students to memorize them, if only because we've already seen that omission of such basic skills results in less educated and less able graduates. Teachers and administrators should evaluate their goals for education: if the goal is independence, individuation, and empowerment of the student, and freedom from necessary assistance for life tasks, then multiplication tables should be taught and memorized, or students at least should be comfortable quickly calculating the products of single digit numbers mentally.

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As someone who moved from computer science and programming into teaching, I always think of times tables as lookup tables. There's nothing there that you couldn't work out using repeated addition, but it's faster if you can simply recall that "6 times 7 is 42".

Considering how fundamental single-digit multiplication is to other algorithms (multi-digit multiplication, factorisation, etc.), the speed gains add up.

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    $\begingroup$ And speed is probably important in the context of working memory when solving problems. $\endgroup$ – user1527 Dec 14 '15 at 16:25

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