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Jon McLoone

Is There Any Point to the 12 Times Table?

June 26, 2013 — Jon McLoone, International Business & Strategic Development

My government (I’m in the UK) recently said that children here should learn up to their 12 times table by the age of 9. Now, I always believed that the reason why I learned my 12 times table was because of the money system that the UK used to have—12 pennies in a shilling. Since that madness ended with decimalization the year after I was born, by the late 1970s when I had to learn my 12 times table, it already seemed to be an anachronistic waste of time.

12 times table

To find it being given new emphasis nearly 40 years later struck me as so odd that I thought I should investigate it a little more mathematically. Here is what I concluded.

Let’s start with a basic question: exactly why do we use times tables at all? (This is the kind of question my work on computerbasedmath.org has me asking a lot!)

I am going to claim that there are three basic reasons:

1) To directly know the answer to common multiplication questions.
2) To enable multiplication algorithms.
3) To enable approximate multiplication.

Let’s look at those in turn.

1) This reason is important. There are lots of small multiplication problems in day-to-day life, and there is no doubt that knowing the answer to these is useful. But knowing ANY answer to ANY question is useful. What’s so special about multiplying 1 to 12? Why stop at the 12 times table—why not learn 13, 14, 15, 16, and 17 times tables? Why not learn your 39 times table? As the table number goes up, the amount to learn increases as a square of the number while the commonality of encountering a problem that uses that table goes down. “Knowing” the answer to all possible questions is a big task and not worth the effort. This, after all, is why math was invented, so that we don’t have to know the answers to all possible calculations, but instead have a way to work them out when needed. We must draw a line somewhere and then move to a more algorithmic approach. The question is where.

2) There are many fancy computation algorithms, but most of us learn “multiplying in columns,” which involves operating on one digit at a time while managing number place and carrying overflows on to the next column. I still use it sometimes myself. By definition it needs the 0–9 times tables (and implicitly understanding the 10 times table), since it only takes one digit at a time, but any single digit could come up. Knowledge of 11 and 12 times tables is completely irrelevant. If this was the only consideration, we have a clear argument for where to draw our line—at the 10 times table. You can’t manage on less, and more is of no use.

3) But there is another useful algorithm, which is approximating numbers to a few significant digits. This might make a case for drawing the line higher.

Take as an example 7,203 x 6,892. If I want to know that exactly, then I reach for Mathematica (or if I absolutely have to, I reach for pencil and paper to apply multiplication in columns). But often I just need a rough answer, so I mentally convert this to 7,000 x 7,000 = 7 x 7 x 1,000 x 1,000 = 49,000,000. More formally I am converting the numbers the nearest approximation of the form k x 10n where k ∈ {the set of numbers for which I know times tables}. Then I use the times tables on the remaining significant digits and implicitly use the 10 times table to get the magnitude right. In this case the real answer is:

7203×6892

49643076

Giving me an error of 1.2%—good enough for lots of applications. Now if I knew my 72 times table, I could have made this 7,200 x 6,900 = 49,680,000. Only a 0.07% error.

So, now our “where do I draw the line” question becomes “how much better is a typical approximate calculation if I know up to the 12 times table compared to only knowing my 10 times table?” Let’s investigate. First I need to automate the process of approximating using a given lead number.

approximate[number_, lead_] := lead*10^Round[Log10[number/lead]] approximate[12345, 9] 9000

And extend that to finding the best approximation, if we have a choice of lead numbers.

approximate[number_, choices_List] :=    approximate[number, choices] = Nearest[approximate[number, #] & /@ choices, number][[1]];

For example, if I know only up to my 4 times table, then the best approximation for 18,345 is 20,000.

approximate[18345, {1, 2, 3, 4}]

Now our approximate product is just the product of the best approximations of each number.

approximateProduct[a_, b_, choices_List] := approximate[a, choices]*approximate[b, choices]

And the relative error can be found from the difference compared to the accurate answer.

relativeError[a_, b_, choices_] := Abs[approximateProduct[a, b, choices]/(a b) - 1.]

For example, working out 549 x 999 when you only know up to your 10 times table will give you a little over an 8% error.

relativeError[545, 999, Range[10]]

Now, let’s take “typical calculation” to mean a calculation involving uniformly distributed numbers between 1 and 1 million and take the “typical” error to be the average of 100,000 such calculations.

meanErrorUniform[n_] :=   meanErrorUniform[n] =    Mean[Table[     relativeError[RandomInteger[{1, 10^6}], RandomInteger[{1, 10^6}], Range[1., n]], {100000}]]

The typical error if you know up to your 10 times table is 9.4%.

meanErrorUniform[10]

But if you know up to your 12 times table, it is only 8.2%.

meanErrorUniform[12]

Here is the error as a function of how many of your times tables you have learned.

ListLinePlot of the error as a function

Interestingly, most of the improvement happens by the time you know your 7 times table. The odd bump at 10 is because the ability to approximate relies implicitly on knowing your 10 times table already (to be able to handle the trailing zeros).

We can work out how much relative improvement there is in the typical error for each extra table learned.

ListLinePlot of relative improvement

So the relative benefit gradually drops, in a cyclic way.

But the improvement in error from 9% to 8% comes at a price. Knowing up to your 10 times table requires recollection of 100 facts (OK, 55, if you assume symmetry). But knowing up to your 12 times table is 144 facts. Improving the error from 9.3% of the result to 8.1% is a relative improvement of 12% in the size of the error. But to achieve that you need to memorize 40% more information. That seems like a losing proposition.

Let’s look at the relative improvement in outcome, per extra fact memorized.

ListLinePlot per extra fact memorized

The “return on effort” drops very rapidly toward the 10 times table and then barely improves. It seems like a fairly compelling case for stopping our rote learning at 10. Indeed, if times tables were only for estimating, we would get the best return per effort by just looking at the orders of magnitude and using only the 1 and 10 times tables!

Of course numbers are not uniformly distributed. If you are in egg production, 6s and 12s probably come up a lot, just as they will if you happen to be a dealer in pre-decimal British coins! Context issues like these are hard to quantify, but one issue that is general is Benford’s law, which occurs in many real-world datasets. It says that if you look at real-life datasets that cover several orders of magnitude (e.g. populations of communities, or people’s debts, or file sizes on your computer), then the numbers are more likely to start with a 1 than with a 2, and more likely to start with a 2 than a 3, and so on. I don’t know if anyone has studied the distribution of second digits, so I will assume that is uniform. So here is a function that generates “real-world” numbers.

Function that generates real-world numbers

We can now repeat our analysis on these more realistic numbers.

meanErrorBenford[n_] :=   meanErrorBenford[n] =    Mean[Table[     relativeError[randomBenfordNumber[6], randomBenfordNumber[6], Range[1., n]], {100000}]]

ListLinePlot meanErrorBenford

Using these less uniform numbers gives poorer performance (making you more likely to need accurate computation rather than approximation). Improvement can still be achieved by knowing more tables, and this could be taken as an argument for learning beyond 12, but not when you take into account the return per extra fact learned, which makes an even stronger argument for stopping at 10.

Argument for stopping at 10

If you really are intent on some extra rote learning, there are better ways to spend your effort than learning 11 and 12 times tables. Learning all permutations of 1 to 10 together with 15 and 25 gives a better average result than 1 to 12 (since they more evenly approximate the numbers with a lead digit of 1 or 2).

Mean[Table[   relativeError[randomBenfordNumber[6], randomBenfordNumber[6], Range[1., 12]], {100000}]]

Mean[Table[   relativeError[randomBenfordNumber[6], randomBenfordNumber[6], Join[Range[1., 12], {15, 25}]], {100000}]]

Or, as Chris Carlson suggested to me, learn the near reciprocals of 100 (2 x 50 = 100, 3 x 33 = 99, 4 x 25 = 100, 5 x 20 = 100, 6 x 17 = 102, etc.), as they come up a lot. I would expect that learning squares and powers of 2 is also probably more useful than 11 and 12 times tables.

With no prospect of the pre-decimal money system returning, I can only conclude that the logic behind this new priority is simply, “If learning tables up to 10 is good, then learning them up to 12 is better.” And when you want to raise standards in math, then who could argue with that? Unless you actually apply some math to the question!

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Posted in: Education, Mathematics
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72 Comments


Phil Earnhardt

This is a wonderful little article, but I question its conclusion. Jon claims that the resources in our brain goes up as the square of the size of the multiplication table we memorize, but presents no evidence that that is the case. When I expanded my memorization of the multiplication tables from 12 to 16, I noticed that my fundamental understanding of multiplication shifted. I simultaneously had greater ease in the application of multiplication algorithms and in my approximation skills.

Functions in our brain are holistic. I suspect our brains are doing something far more interesting than looking up results in a table. Or our brain may do a table lookup for small values and shift to some other algorithm for larger ones. I would love to see some studies to see how students’ mathematical skills were impacted by learning multiplication tables to 15, 20, or even 30.

Posted by Phil Earnhardt    June 26, 2013 at 3:42 pm
    Jon McLoone

    There have been quite a few comments in Reddit with a similar theme that there are many fewer facts to learn for a table because people use patterns — 11s are easy for small numbers, 12s can be derived from the 3, 4, or 6 tables. But these, I believe are not “learning times tables” which is about fast factual recall, but are examples of some of the other “multiplication algorithms” that I refered in passing as Reason 2. There is lots of value in these. Working out that 12×12 can be arrived at using distributiveness 12×10+12×2 or factorisation 3x3x4x4 seems more useful than “knowing” that it is 144.

    Posted by Jon McLoone    June 27, 2013 at 9:55 am
    Jon McLoone

    Oops. Good catch. It doesn’t actually change the conclusion, but using 1-10 plus 15 and 25 actually gives a mean Benson error of 8.5%. I will get the article fixed.

    Posted by Jon McLoone    June 27, 2013 at 3:34 am
Moschops

Twelve is the first positive integer with more than two factors; 2, 3 and 4. It also has six divisors; 1, 2, 3, 4, 6, 12. This makes it a very useful number for many applications, as can be seen for example in the choice of 360 degrees in a circle, where knowledge of the 12 times table suddenly becomes very useful indeed. 360 is highly composite, and given that it’s an easy multiple of 12, knowing the 12 times table suddenly makes working with degrees a lot easier.

Posted by Moschops    June 26, 2013 at 4:09 pm
    Moschops

    No, wait! I meant to edit that! The bit at the start about factors isn’t meant to be there!

    Posted by Moschops    June 26, 2013 at 4:11 pm
Dan

One other reason you have not touched on here really is that learning times tables up to 12 can lead to a greater appreciation of the patterns involved. This certainly happens with tables from 2 to 9 but there are more patterns in 11 and 12…

Posted by Dan    June 26, 2013 at 4:51 pm
Soto

Interesting analysis.

This is complete speculation: Perhaps our society originally started teaching the 12 times table because of our use of the imperial system of measures. When you are working with 12 inches in a foot conversions are easy if you know the 12 times table.

I have never seen the time table displayed the way you have done above. My father made me memorize the times table, without ever writing anything down. On the drive to school each day (I went to Catholic school, so no bus) I was required to first recite the table, from 1×1 to 12×12 with everything in between. Then he would rapid-fire quiz me. I hated it at the time, but I now find it very useful (I am a professional scientist).

The combination of the times table with order of magnitude estimation, as you mentioned above, is definitely critical. After getting the table memorized, I was then taught that type of estimation. My teachers hated it. Instead of adding things in the standard ‘stacked’ manner, I would try to convert each number into a power of 10. From there all of the arithmetic is easy. As long as you keep track of the roundings, you can convert back to an accurate answer at the end.

Unfortunately, the school system never taught me to play with numbers, as if they were Lincoln Logs that I could manipulate in a myriad of ways. Instead, I was fortunate to have a father who did, since he was fascinated with numbers. I hope that this times table requirement in the UK is not an end, but rather a means to teaching children to play with numbers and gain an appreciation for mathematics.

Posted by Soto    June 26, 2013 at 5:08 pm
Bèr

I have no hard numbers toe back it up, but I would assume 12 turns up really often, because of its use in time and dates.

Posted by Bèr    June 27, 2013 at 12:46 am
guest

In the Netherlands we learn the 10 times table, I assume for the reasons mentioned. Interesting that this is different in the UK, I never would have thought of that. How about other countries?

Posted by guest    June 27, 2013 at 1:06 am
Robert

I don’t know about the UK, but America has 12 inches to a foot. so with my 12 times table I can easily know how many inches tall I am. of course most people are between 5 ft and 6 ft tall, so maybe Americans only have to remember 2 or 3 numbers from the 12 table.

or maybe it is because of clocks? how many hours do you have to study if your final is 4 days away? did you say 84? well you’re wrong. you know you are gonna procrastinate anyway and cram everything into the last hour.

Posted by Robert    June 27, 2013 at 1:15 am
twelve

The number 12 does appear quite often even after decimalisation.

12 hours on a clock face

12 months in the year

5 x 12 seconds in a minute, 5 x 12 minutes in an hour

30 x 12 degrees in a circle

12 inches in a foot. still used in the U.S. and occasionally in the U.K.

12, 24, 36, 48, 60, …., 120, .., 144 and the rest calculate if need be

Posted by twelve    June 27, 2013 at 2:10 am
    Damion

    Indeed, I was confused by the mention of old money when it’s obvious the old imperial measurement system is the cause. It’s still being staunchly stuck to by so many stalwarts, that I think this is basically the answer to the entire article. Despite how obvious and mathematically sensible stopping at 10 seems to be, knowing by heart the times-table to 12 makes using the stupid measurement system that just won’t die, continue.

    Posted by Damion    June 28, 2013 at 10:24 am
      Hugh Simpson

      I think the Gregorian calendar is quite clever…

      Posted by Hugh Simpson    July 21, 2013 at 7:54 am
Pseudonym

The analysis assumes that all parts of the multiplication table take the same amount of work to learn, which they clearly do not.

The 11 times table may only buy you a 1.8% improvement in relative error, but it’s basically free.

So this raises an interesting question. Assuming that you only ever care about approximate answers, are you better off learning your 9 times table or your 12 times table? (We will assume, of course, that you don’t “know” 9×12 either way.)

I will assume that 1,10 and 11 times tables are free, and that learning a times b is free if you already know b times a.

So let’s suppose that you have learned tables 1 through 11. There is a 12% relative error on multiplications, at a cost of learning 36 numbers.

Now let’s suppose you have access to tables 1 through 12 except 9. You still only have to learn 36 numbers, but the relative error on multiplications is 11%.

So if relative error is your measure, you are better off learning your 12 times table than your 9 times table. Of course, you don’t have access to all exact multiplications any more, assuming that you’re using the normal long multiplication algorithm.

Which, of course, you won’t. Subtraction is essentially no harder than addition, so you could alter the multiplication algorithm to “calculate” a multiplication by 9 as a multiplication by 10 minus a multiplication by 1. Since both multiplications are free, you’re probably ahead.

But if you’re going down that route, then all of the times tables between 6 and 9 are redundant in the same sense, since you could work in a balanced positional notation system with the digits -4 to 5. Given that the 1 times table is “free”, and if you know a times b then you also know -b times a and so on, you only need to learn 10 numbers in total. Plus you get the advantage that there’s less carrying when adding and subtracting.

Posted by Pseudonym    June 27, 2013 at 2:53 am
Jordan

I was always under the impression that the main reason to learn 12 times tables was because of:

- Degrees in a circle.
- Hours in a day.

Both of which is it hugely useful to be able to quickly use multiples of 12 for.

Posted by Jordan    June 27, 2013 at 3:02 am
Ash

One argument for the 12 times table might be that there are 12 hours in the day and 360 degrees in a circle, so 12 d

Posted by Ash    June 27, 2013 at 3:51 am
Ash

24 hours in a day, 360 degrees in a circle – so 12-based computation does pop up a bit

Posted by Ash    June 27, 2013 at 4:23 am
Hobbsy

learning my 12 times table was useful as a child for working out how many packets of Panini football stickers I could buy with my pocket money :) (’86/’87 season I think?)

Posted by Hobbsy    June 27, 2013 at 4:55 am
Christopher Bohn

Perhaps this is US-specific, but I still find the “twelves” useful when dealing with inches and with dozens.

Posted by Christopher Bohn    June 27, 2013 at 6:58 am
    Gabe Eckers

    I believe this is an excellent point. Although, I could see how someone could argue that it is simply part of reason 1.

    Posted by Gabe Eckers    June 27, 2013 at 8:39 am
      Jon McLoone

      Perhaps the US is rather big to include in “contect issues” but most of the rest of the world now uses metric decimal units. Oddly, in the UK we still mostly use feet and inches for measuring human heights, but any engineering length would be in meters,.

      Posted by Jon McLoone    June 27, 2013 at 9:26 am
        Don from B.C.

        Here in Canada, feet/inches are still used in construction and in medicine (human weight and height are often in ft/in and lbs). Multiplying by 12 is really a must, even though we use metric for pretty much everything else. Additionally, 16 can be helpful in construction as joists and studs are spaced in 16″ intervals.

        Posted by Don from B.C.    June 27, 2013 at 3:01 pm
Sunil

James Bond, who was 6 foot 4, contemplated the height chart which was labelled only in inches.

The army cook scratched his head. One hundred hungry soldiers needed their eggs and bacon. He had six boxes, a dozen eggs per box in store. Would be that be enough?

etc

Posted by Sunil    June 27, 2013 at 7:50 am
    Jon McLoone

    Given that you will probably break an egg or two, this seems a prime candidate for approximate calculation – “slightly more than 6 x (10 eggs) ~ 60 => hungry soldiers.

    Posted by Jon McLoone    June 27, 2013 at 10:56 am
Mike Tonge

You forget that a lot of engineers in the UK end up having to work in imperial standards at some point. Multiples and divisions of twelve are also useful for calculating times in your head.

Posted by Mike Tonge    June 27, 2013 at 7:58 am
Lennon

Interesting analysis, but I’m not sure I agree. It seems to assume that each times table is as difficult to learn as any other one, when several are much easier (5s, 10s, etc.). I would certainly argue that list includes 11s. Since they’re trivial to learn, I think they still come out as a net benefit. And maybe it’s a US thing, as Christopher points out, but enough things here come in dozens to make 12s worthwhile in my opinion. I certainly use it a good amount.

Posted by Lennon    June 27, 2013 at 8:37 am
One Thing

You’re forgetting time. 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day. All divisible by 12.

Posted by One Thing    June 27, 2013 at 9:05 am
    Jon McLoone

    See my comment to Phil Earnhardt, further up the page.

    Posted by Jon McLoone    June 27, 2013 at 9:56 am
Stewart C. Russell

Up until the 1950s, many Scottish schools taught up to 14×14, since stone (= 14 lb) and hundredweight (= 8 stone, or 112 lb) were in common use.

Posted by Stewart C. Russell    June 27, 2013 at 10:21 am
Jon McLoone

So there a lot of comments about 360 degrees, 60 minutes and 24 hours and how these are all divisible by 12. All of which is true. But it does not follow that the calculations are typically in multiples of 12. Comman angle multiplication calculations are multiples of 5,10,20, 30, 45 or 90. Common time calculations are multiply by 15, 30 or 60. If you are going the opposite direction, factors of 60 and 24, you can immediately pull 6 from knowing your 6 times table.

Posted by Jon McLoone    June 27, 2013 at 10:53 am
    Jordan

    While those are perfectly acceptable *if you have not memorized 12 times tables*, because I have I frequently use 12s instead.

    When I am given the number of hours until an even, for instance, (which happens more often than you might think), I look at the closest multiple of 12 to determine the approximate number of half days. Such as 46 hours is about 2 days (4 half-days).

    It’s also extremely useful for anything that deals with consumer credit, both loans and revolving credit, as terms are always in months and years occur in multiples of 12 months. You could use 6 month half-year intervals (and indeed, sometimes I do), but to pretend that simply because there *are* other, smaller divisors for multiples of 12 that knowing 12s isn’t useful enough to learn them is kind of silly, in my opinion.

    It is not difficult to learn, and it provides me significant practical benefit in day-to-day life. If I had not been taught 12s, I would have taught them to myself or memorized them through simple repetition of occurrence in life.

    Posted by Jordan    June 27, 2013 at 1:38 pm
    Steve

    In what field did you encounter these 20 degree angles? As a mathematician I’ve never had any interest in an angle that wasn’t a multiple of 15 degrees – or pi/12!

    Posted by Steve    June 28, 2013 at 2:24 pm
      Hugh

      As a mathematician, I find that the particular compound that forms 12 often makes it a useful low value example of ‘an interesting case’ in several fields. Can’t think of a^2*b that’s smaller :)

      Posted by Hugh    July 21, 2013 at 8:00 am
DexX

I would argue, though, that the 11 times tables don’t need to be memorised at all.

This is a case of remembering a rule instead of a sequence: multiplay by ten, then add the original number again. While 11 x 3,427,821 might make this hard, integers 1 to 12 are simple, especially 1 to 10.

Remembering then that x10 and x11 are simple rules rather than whole tables that need to be memorised, and of course that the x1 table needs no memorisation at all, means that times tables from 1 to 12 only require the memorisation of nine actual tables: 2 to 9 and 12.

On top of this, other patterns emerge which allow the memorisation of one table to support the memorisation of another, meaning that this synergy results in less brainpower being required. Good knowledge of x2, for example, means it’s easy to reverse engineer numbers into their halves. This in turn means x5 can be reduced to a rule as well. Similarly, x9 is x3 twice, and x12 is x3 then x4.

While I am in awe of the calculations above, I don’t think a strict mathematical approach is necessarily optimal in this case. There is just too much going on for simple formulae to capture.

Posted by DexX    June 27, 2013 at 11:05 am
    Jon McLoone

    There is greater value in creating patterns and understanding to enable answers than rote learning of those answers. By “times tables” I am really talking about the rote learning. There is value in it, but, as I have argued above, it runs out quite soon. But I certainly don’t want to take away from the exploration of how you can “work out” all kinds of answers including those that appear in the 12 times table.

    Posted by Jon McLoone    June 27, 2013 at 3:00 pm
hmjenkin

How does this change with different numbering systems (binary, base-10, base-12, hexadecimal, etc.)? Is there a numbering system that optimizes approximations with fewer lead numbers?

Posted by hmjenkin    June 27, 2013 at 12:41 pm
    Phil Hatchard

    Because 12 has many more factors than 10, more numbers “fit into” a base-12 numbering system than base-10. This means that recurring “decimals” would appear less frequently when converting from fractions, and more times tables would follow an obvious pattern in the last digits, as the 2 and 5 times tables do in decimal numbers.

    Using A and B as the extra 2 digits:

    2x table:
    2, 4, 6, 8, A, 10
    12, 14, 16, 18, 1A, 20

    3x table:
    3, 6, 9, 10
    13, 16, 19, 20
    23, 26, 29, 30

    4x table:
    4, 8, 10
    14, 18, 20
    24, 28, 30
    34, 38, 40

    6x table:
    6, 10
    16, 20
    26, 30
    36, 40
    46, 50
    56, 60

    Posted by Phil Hatchard    July 8, 2013 at 2:46 pm
      Phil Hatchard

      Although I’m not sure whether that would make a significant difference (if any at all) to the ease of making approximations.

      Posted by Phil Hatchard    July 8, 2013 at 2:48 pm
Dave Engh

Interesting. I started my schooling in Canada and always wondered why we *had to* go to 12 in the times table in primary school. Never realized it was (might have been?) a holdover from the Empire.

Posted by Dave Engh    June 27, 2013 at 1:06 pm
Cale Gibbard

Here is a plot of the number of divisors of n, for n from 1 to 1000.

http://i.imgur.com/FWPfDzz.png

The points marked in red are those which have more divisors than any previous number (and at least three), namely,

4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840

In here, we see the number of eggs in a carton, the number of hours in a day, the number of minutes in an hour (or seconds in a minute), and the number of degrees in a circle. :)

Every number after 12 here is (somewhat unsurprisingly) a multiple of 12, because the smallest number with k divisors will tend to have lots of small prime factors, and just a few larger ones — and all that’s required to be divisible by 12 is two 2′s, and a 3.

While I would argue that students should not be wasting time drilling multiplication tables at all (and should instead be given opportunities to need multiplication in order to induce that learning), it is nice to have a bit of facility with these highly divisible numbers.

Posted by Cale Gibbard    June 27, 2013 at 1:13 pm
andrew

i don’t have a problem with mental arithmetic myself, even with slightly more digits than these, but i find many of the comments interesting. do engineers and scientists have to do a lot of mental arithmetic in their daily work as they prototype or slog through equations? it seems a lot of commenters are pointing to their knowledge of the 12 times tables as the reason they are good as a scientist/engineer or that it’d be useful on a project.

“You forget that a lot of engineers in the UK end up having to work in imperial standards at some point. Multiples and divisions of twelve are also useful for calculating times in your head.”

“[I] now find it very useful (I am a professional scientist).”

*really*?

i guess someone else might say that “it’s for the people that will end up as a cashier, teller, stock clerk etc.” but in my opinion, most average people have trouble with anything but trivial multiplications. and if they need to actually get accurate results, in addition to the obvious calculator/scanner that they have for their job (which company allows their employees to derive answers off-head anyway?), they can use the algorithms we all learn and, therefore, it’s still not practical to teach tables higher than 11 (even though such knowledge is not bad).

remember, the issue is not whether it’s good or bad, useful or not. it’s whether we’re getting a bigger bang for the buck by forcing a generation of pupils to struggle through yet another useless section of the curriculum without adequately explaining why it’s practical (“you’ll be able to tell your height in inches if you’re ever in the u.s. and you’re walking out of a 7-11″).

Posted by andrew    June 27, 2013 at 1:26 pm
    Steve

    Don’t they learn this as well! I certainly did

    Posted by Steve    June 28, 2013 at 2:13 pm
Jon McLoone

There are a lot of comments about the relationship with sexagesimal units (base 60 from Babylonian origins) – 60 seconds in a minute, 60 minutes per hour, 360 degrees in a circle. But those are used more to do with their divisors than multiplications. Of course they are closely related, but not the typical number facts that you need multiplication for. How often is a time multiplication question start “a task takes 12 minutes…” or end “how long does it take to do that 12 times?”. No more than in any other unit system. The unit system will make questions more likely to involve multiplying by 60 or 24. And even the divisors that we need are typically, 1/4 or 1/2 or 1/8. As a result I know that a half hour is 30 minutes, 1/4 circle is 90 degrees, 1/8 is 45 degrees etc. 12 Doesn’t typically come up, even though it is a divisor of 60, and 360. There is more of a case for knowing the 24 and 60 times tables to support these units than 12.

Posted by Jon McLoone    June 27, 2013 at 3:32 pm
    Cale Gibbard

    To be honest, I think the tabular layout isn’t so much important as having a familiarity with numbers that have many small prime factors. A lot of numbers with 2, 3, and 5 as factors come up naturally in geometric settings. I could forgive someone for not memorising the fact that 7 * 8 = 56, if they knew by heart that 12*5 = 20*3 (this is involved in the fact that the dodecahedron and icosahedron are dual polyhedra).

    Posted by Cale Gibbard    June 27, 2013 at 7:58 pm
Ted

Well, there are 12 inches in a foot. So for US carpenters one could argue that it’s good to know that six feet is 72 inches, 8 feet is 96 inches … and as a carpenter in the U.S. you will get to remember those things eventually. Four feet are 48 inches, we use that one a lot. Carpenters also get good at 8ths and 16ths.

Twelve is a handy number. Eleven is a pain in the neck. Eleven, well, there aren’t eleven of anything in anything. I just don’t care about eleven. You can have it.

Posted by Ted    June 27, 2013 at 6:16 pm
jonathan

Instead of restricting your choiceLists to ranges from 1 to N, it could be interesting to see what happens if you let them range over ([1..10] ∪ {N}), i.e. if you’re going add just one more number to your 10s table, which will help you the most. My bet is on ’15′, especially if you use Benford.

Posted by jonathan    June 28, 2013 at 1:39 am
Alessandro Lensi

They should teach how to count by hand up to 2^10-1 using base2 :)

Posted by Alessandro Lensi    June 28, 2013 at 3:38 am
Max Allstadt

Why do I need to know the 12 times table cold?

Because I am a carpenter.

Next question?

Posted by Max Allstadt    June 28, 2013 at 4:01 am
Jonny

It’s to prepare them for dealing with the packaging industry.

Most of the required youth education in Britain today is designed to bring you to a level where, upon reaching required learning age (16), you know just enough to work in a shop… the most likely work destination of someone leaving school at 16 if they aren’t heading to college.

Following the decline in industry, the most likely career path is into retail. Working in a shop, you will soon realise that for some reason, most produce comes packed in boxes of 6, 12, or 24. knowing your 12 times table makes it easier to do stocktakes and quick auditing for shop staff busy running a shop. That’s why it’s still taught in schools in Britain – the expectation of a future career when you leave school. You get taught proper maths if you stay on.

Posted by Jonny    June 28, 2013 at 10:50 am
Steve

Have you considered that it’s not necessarily knowing the tables themselves that’s valuable, but the brain functions involved? The ability to memorise and recall information is valuable in itself, as is pattern recognition and the ability to reconstruct information by reference to the pattern – the shortcuts we learn to use when “memorising” our times tables.

If I asked you to work out, say, 29 x 15 you wouldn’t look it up in your memory bank and you wouldn’t perform multiplication by columns. You’d use the tricks of mental arithmetic that you probably first encountered when trying to recall the twelve times table.

Posted by Steve    June 28, 2013 at 2:08 pm
    Jon McLoone

    One problem is that when the goal is to “learn times tables” the teaching becomes disconnected from such side benefits. My daughter was given a poem to learn which starts “Cycling Suzie goes to Devon, one times seven equals seven” and continues similarly. Any cognitive relationship between the numbers or other tables or near computations is pushed aside for the most efficient way to coach the class on memorizing the facts. As I said, I do thinks some direct recollection of tables is just plain useful, but I would be all for getting the kids to learn them in meaningful context. But that still leaves the central question of my piece – where to draw the line?

    Posted by Jon McLoone    July 1, 2013 at 12:30 pm
Dragonfly

As a kid, math was always hard, and I *hated* multiplication. Later in life, though, when I worked in machine shops, arithmetic became easy, and then I began to see the patterns that showed up in the numbers I worked with — decimal equivalents to fractions being one example. I think if I’d been taught these “magic” things, and how you can get them to do neat things for you, math as a child would have been more interesting, and I might have had an easier time of it.

Posted by Dragonfly    June 29, 2013 at 1:46 am
Miles

Is not the inclusion of the tables up to x12 on the UK maths curriculum just the first move towards a reintroduction of pounds, shillings and pence as currency, together with going back to feet and inches and other Imperial measures?

Posted by Miles    June 29, 2013 at 8:49 am
John A

I’m afraid the basic reason for bringing back learning up to 12 is much worse than the one you suggest. The people concerned would never have dreamed that it was possible to take an active decision about when to stop; the logic is simply “that’s what we did a long time ago, and things were better then, so that’s what we’ll do again”. I am referring, of course, to the decision makers, not your respondents!

Posted by John A    July 1, 2013 at 6:26 am
Thomas

Very interesting post. Just to point out, though, that dozens still come in very handy in the wine trade.

But more handy is indeed the ability to split the multipliers down. A glass and a half for 264 people of mc

Posted by Thomas    July 1, 2013 at 1:29 pm
Andy

I seem to remember reading a history of numbers which related that the first use of base 12 was woth the summerians who tied knots in units of 60. Presumably this was related to the grain trade about 5 kyears ago It would be nice to know how the hours in a day and degrees in a circle originated thoug

Posted by Andy    July 1, 2013 at 5:04 pm
    Geoff

    If a grown man holds his arm outstretched and makes a fist …. It takes 12 fists to go horizon to horizon at the right latitude of course. Also, with arm outstretched, one pointer finger equals approx 1 degree. The fist is good approx of daytime hours at 44 th parallel at equinox) . You can try just sitting in your chair, it takes six fists to get to 90 degrees (noon). I was told this is a mariner trick, goes way back, and may be the origin of the 360 circle….and so twelves have been “handy” for a while.

    Posted by Geoff    July 2, 2013 at 10:21 pm
silverfin

Interesting discussion – always v happy to see people arguing about maths education.

However, there seems to be an assumption above that there is one theoretical ‘best’ quantity/selection of multiplication facts to be memorised by children (if it could only be agreed upon, or somehow ‘proved’). But individuals’ brains vary tremendously, and very much so regarding the time/effort required to (a) memorise strings of text and numbers so they can be recalled accurately at will and/or (b) become efficient at using number relationships for Derived Fact Strategies. So different arithmetical strategies, with different ratios of recall and calculation, will be more or less efficient for different students.

People above have already pointed out that some sets of multiplication facts are almost universally quicker to commit to memory than others, that some (e.g. 1, 10, 11) have a pattern so obvious it’s not necessary to memorise the facts individually at all, and that students generally do not learn the sets neatly in ascending order. For both those people who find memorisation of text/number strings relatively quick and easy and those who are quite the opposite, I suggest it makes sense to start with the easy ones, then add others in order of usefulness, up to a point where the input time/effort seems to outweigh the gains *for that individual*. Might be 2, 5 and 10 for one kid, everything up to 25^2 for another. Whereas everyone should be given the opportunity to become familiar with number patterns, and should certainly learn about the concepts involved in multiplication (e.g. commutativity, associativity).

Incidentally, a lot of supposed testing of number fact memorisation does nothing of the sort. Depending on the test subjects and the time allowed per question, fast DFS is often externally indistinguishable from direct recall, and people’s self-report of how they obtain answers is notoriously unreliable.

Posted by silverfin    July 6, 2013 at 12:31 pm
Iiss Marte Rthanu

As a Nigerian I find that I use 12 almost on a daily basis.

Posted by Iiss Marte Rthanu    July 7, 2013 at 2:20 am
Lucian

Multiples of 10 and 11 are self-evident, needing no actual memorization or calculation (you basically either double the digit, or add a zero to it). And multiples of 12 appear many times in real life (eitehr as hours of N days, or as months of N years) so that we become accustomed to them before having to learn them anyway…

Posted by Lucian    July 31, 2013 at 9:21 pm
Airplain

I’m an educator (US) who has maintained for nearly three decades that the teaching of times tables past the 10s is complete waste of time. The arguments advanced in this column for the usefulness of learning 12s could as well be advanced for learning one’s 24s table or even more so, ones 60s table.

But the fact is, we started the 12s tables in a time when imperial units seemed to lend some (limited, IMO) utility to the practice. But I need my students to be able to perform their times tables so that they can handle the algorithms of multiplication and fractions. I will even go so far as to say that I believe teaching to the 12s has a retarding effect on many students’ understanding of higher multiplication.

Posted by Airplain    August 7, 2013 at 3:17 pm
Pedro

I found your post really interesting. I’m a student from Spain and I came across the 12-times table while reading an article from The Guardian, and it caught my attention and immediately googled it (that’s why I ended up in your blog). In Spain we only learn up to the 10-times table, which I think is enough and sufficient for other operations (you can work out the solution to 12×4 the same way you multiply 568×45). And by the way, we are required to learn them by heart by year 2! I hope my point of view helps.

Posted by Pedro    December 9, 2013 at 11:55 am
Ajinkya

In India, we only learn up to 10 times tables, which I thought is good enough and sufficient. Don’t knew even that 12 times tables are existed.

Posted by Ajinkya    January 6, 2014 at 5:05 am
Tamara

I asked someone in my district about this. I question was not answered (forgotten or ignored?). As a teacher, at least in my school, there does not seem to be a lot of room to “play” with numbers. I would think that kids would have a lot less problems in math if they were given apply time to do this in schools. Instead however there is an emphasis on getting the “right” answer, and the product is what is judged rather that the process of thinking. I think thinking about numbers and how they work is so much more interesting and valuable, but that could be a function of my adult brain at work. If only we had the time have kids develop an understanding about the basic number facts and how they work.

Posted by Tamara    April 16, 2014 at 9:12 am


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