29 ms·

Is there any point to the 12 times table?

185 points by kesselvon 11y ago

sandworm101 11y ago
Forget the stats and calculus. Think carpentry, measuring and cutting wood products. If you are building things in the US/UK/Canada then you are using feet and inches. 12 inches to a foot. It's a tiny thing to learn and will serve kids well in any number of professions. Now 11, that's a total mystery. Other than it being between 10 and 12, I see no reason to memorize 11s.
- nsxwolf 11y ago
  I remember that memorizing the 11 column was trivial and fun, up to times 10 anyway.
- x1798DE 11y ago
  11s are practically free, though, in base 10. The algorithm "repeat the non 11 number twice" works up till 10 x 11, where the "add a 0" algorithm for 10 kicks in. So you're just really memorizing 11 x 11 = 121 and 11 x 12 = 132.
  - BillTheCat 11y ago
    A simple trick: For 11 x a 2-digit number you can simply take the first digit of the 2-digit number then the 2 digits of the number added together then the last digit of the 2-digit number. Ex: 11 * 12 = 132 or 1, 1+2, 2. 11 * 45 = 495 or 4, 4+5, 5. For numbers which sum to more than 10 add the carry to the first number ex: 11 * 59 = 649 or 5, 5+9 = 14 so add 1 to the initial 5 and keep the 4, 9.
    - pc86 11y ago
      I'm 29 and never heard this before. Neat!
      - deleted 11y ago
        [deleted]
      - the-dude 11y ago
        This is golden: I am just over 40 and I now know that I knew nothing at 29!
    - gshubert17 11y ago
      This generalizes to multiplying any number times 11, https://en.wikipedia.org/wiki/Trachtenberg_system#Multiplying_by_11 https://en.wikipedia.org/wiki/Trachtenberg_system#Multiplyin... Take a number, say 142857. Prepend a 0 on the left. Underneath each digit write the sum of the digit above and the one to the right. If you work from the right, it's easier to keep track of the carries. 0142857 1571427
  - derefr 11y ago
    But it's also "practically free" to just mentally do "10x + x", which I can actually add faster than my brain will return an answer from the post-10 part of the times table. (Not that I didn't memorize the 11s and 12s all the same at the time, but they got a lot less use and so a lot less reinforcement.)
- deleted 11y ago
  [deleted]
- zanny 11y ago
  That is only yet another reason in literally infinite ones why the US needs to actually push metric units. We bleed our stupidity into the UK and Canada while the rest of the world makes sense.
  - btilly 11y ago
    Makes sense..until you talk about time. Time makes no sense.
    - sandworm101 11y ago
      If you want to terrify an american, ask them about switching to metric time. https://en.wikipedia.org/wiki/Metric_time https://en.wikipedia.org/wiki/Metric_time This comes up all the time for me as americans use a different date format. I've had contract negotiations grind to a halt because someone was doing math based on 06/05/15 rather than 05/06/15.
      - web007 11y ago
        Use something like 12/24/48 (US) or 48/12/24 (UK) if you want to "force" compliance, then you'll at least have an easy sanity check for each field. The other benefit is you can check Y2038 compliance at the same time, but you should at least specify 4-digit years for whatever you're spec'ing, so you don't create the next Y2K.
      - kstrauser 11y ago
        I write everything either ISO or spelled out. Your dates would be 2015-06-05 or "June 6, 2015". It's an easy quick that everyone seems to be comfortable with.
      - mark-r 11y ago
        Switch to ISO format. There's less confusion in 2015-05-06. Although the one time I tried to push for it, at a company with plants in both the US and Ireland, I couldn't quite get agreement and we settled on 2015-MAY-06.
  - 11y ago
- jjp 11y ago
  > If you are building things in the US/UK/Canada then you are using feet and inches. In the UK you are more likely to be using millimetres if working from any kind of design. Working in inches and feet would depend on your age and perhaps whether you are working on an older property that was designed in inches.
  - michaelt 11y ago
    In the UK your designs and plans will be in mm, but your materials will vary between metric-rounding-of-imperial, (2400x1200mm) [1] imperial-but-labeled-in-metric (1220x2440mm) [2] and imperial-in-one-dimension-metric-in-the-other (38x144x2400mm) [3]. Wallpaper? 10m x 520mm for sure [4] but doors? 1981x762mm [5]. Cement? Back to metric, with 10kg and 25kg bags [6]. You'll also probably measure long distances in miles but short distances in meters; and weigh your ingredients in grams but your body in stone. I guess what I'm saying is if you want a rationally designed measurement system, don't copy us Brits and especially don't copy our builders. [1] http://www.wickes.co.uk/Knauf-Plasterboard-Square-Edge-2400x1200x12-5mm/p/104123 http://www.wickes.co.uk/Knauf-Plasterboard-Square-Edge-2400x... [2] http://www.wickes.co.uk/Wickes-General-Purpose-OSB3-Board-18x1220x2440mm/p/110517 http://www.wickes.co.uk/Wickes-General-Purpose-OSB3-Board-18... [3] http://www.wickes.co.uk/Wickes-Studwork-%28CLS%29-38x144x2400mm-Single/p/101857 http://www.wickes.co.uk/Wickes-Studwork-%28CLS%29-38x144x240... [4] http://www.wickes.co.uk/Wickes-9006-Wallpaper-Embossed-White-10m/p/610194 http://www.wickes.co.uk/Wickes-9006-Wallpaper-Embossed-White... [5] http://www.wickes.co.uk/Wickes-Skipton-Internal-Softwood-Door-Clear-Pine-4-Panel-1981x762mm/p/206690 http://www.wickes.co.uk/Wickes-Skipton-Internal-Softwood-Doo... [6] http://www.wickes.co.uk/Blue-Circle-Extra-Rapid-Cement-25kg/p/224662 http://www.wickes.co.uk/Blue-Circle-Extra-Rapid-Cement-25kg/...
  - twic 11y ago
    You will almost certainly be working in millimetres, but because of ISO 2848, you will very likely be working in multiples of 300 millimetres: https://en.wikipedia.org/wiki/Preferred_number#Buildings https://en.wikipedia.org/wiki/Preferred_number#Buildings Which is about a foot!
- creshal 11y ago
  > in the US/UK/Canada And the other 94% of the world population…?
  - wtetzner 11y ago
    So they don't have to learn it. Not sure what the problem is.
- arethuza 11y ago
  "If you are building things in the US/UK/Canada then you are using feet and inches." I don't think that's been the case for a pretty long time in the UK - certainly DIY supplies are metric e.g. here is the Homebase site: http://www.homebase.co.uk/en/homebaseuk/diy/timber/planed-timber http://www.homebase.co.uk/en/homebaseuk/diy/timber/planed-ti...
SixSigma 11y ago
I recently worked with a person who had to do her three times table using her hands, starting from 3. She would still struggle at about 6 or 7 times. We worked behind the counter in a builder's merchant. She is no longer working for us.
- theworstshill 11y ago
  Why not?
  - pc86 11y ago
    Presumably the inability to do basic math quickly.
  - Symbiote 11y ago
    It's exactly the kind of job where someone will phone and ask "do you have 30 widgets I could pick up in an hour?". They come in packs of 4 widgets, and there are only 7 packs in stock.
  - SixSigma 11y ago
    Approx 300mm to the foot. All products in mm, half the customers say "6ft widget". Getting the calculator out to do 6 x 3 is unprofessional.
- xlm1717 11y ago
  You know, whether or not children should learn up to 12 by the age of nine is debatable, but by the time you're an adult, assuming no learning disabilities you should be able to do mental math proficiently enough that you don't need to memorize any tables, you can just do it. I hope that person you worked with can at least work a calculator.
- lordnacho 11y ago
  I had a job once where we hired a derivatives trader who couldn't multiply 8 and 4. In the job interview. At my next job we hired another trader who came up to me and said "what does the little 2 in a superscript mean?". As in what does 3^2 mean? Both very friendly people.
Amorymeltzer 11y ago
>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. This is absolutely worthwhile, easy approximation of 1 or 100 divided by 6 or 8 in particular occurs regularly, and not enough people know it offhand. Hugely helpful. After 1-10, your energy is probably better spent on cool patterns and "tricks" within those sets (final digits of 9, division by 7, etc.); much greater return.
- thaumasiotes 11y ago
  Ok, but what's with the weird phrasing? This is the same set of facts as (1/2 = 0.50, 1/3 = 0.33, 1/4 = 0.25, 1/5 = 0.20, 1/6 = 0.17, 1/7 = 0.14 ...), but phrased to sound like something you'd have to put effort into.
ck2 11y ago
They should probably learn HEX at age 9
- walshemj 11y ago
  Back in the 70's (UK) I remember doing non base 10 maths at about that age plus basic set theory much more useful than rote leaning your times table.
  - PeCaN 11y ago
    Did that in the US in the mid 2000s, though spent hardly a week on either IIRC. Something like base 20 Mayans, base 60 Sumerians, and base 2 computers. Do math in each and convert them to/from base 10. I think that's been dropped from most US elementary school curricula now, bit of a shame.
    - Zikes 11y ago
      Was base 60 Sumerians accompanied by cuneiform, by chance? I got to learn some of that in a World History course in high school, which was pretty fun.
    - schoen 11y ago
      I believe it was introduced as part of some New Math curricula. https://en.wikipedia.org/wiki/New_Math https://en.wikipedia.org/wiki/New_Math Richard Feynman was pretty harshly critical of this in his stories of working as a textbook reviewer. I really appreciated that my elementary school did it, but I had a sense that most students didn't quite understand the significance (or remember it afterward if they weren't into computer programming). I think it has a lot of potential value in terms of understanding what place value is and where a place value system comes from. When I've tried to teach programming, I've found that most people who aren't already programming enthusiasts don't remember the binary system or say they never learned it at all. Edit: the Wikipedia article I linked to also has several examples of people making fun of non-base-10 arithmetic and kids being taught it in the New Math, including a song by Tom Lehrer!
- 11y ago
kitwalker12 11y ago
he got off lucky. back in India, we had to cram tables upto 20.
zeveb 11y ago
Twelve has more divisors than ten (1, 2, 3, 4, 6 & 12 vs 1, 2, 5 & 10). If we were really smart, we'd switch from base 10 to base 12: many more 'decimals' (really duodecimals) would be non-repeating. One can very quickly count by twelves on the joints of one's fingers, using the thumb as an index (0-143 is a much larger range than 0-10, and it's easier to hold one's hands in the shape necessary). If we were really smart, we'd switch from base 10 to base 12: many more 'decimals' (really duodecimals) would be non-repeating. And then there are measures like a gross (144) and a great gross (1,728). Part of the reason for these traditional measures is that they are more flexible than base-10 measures: an eighth-gross or a third-gross are both integer quantities, unlike an eighth-hundred or third-hundred.
- panglott 11y ago
  Base-60 math is even superior.
  - noobermin 11y ago
    The Sumerians had it right?
    - deleted 11y ago
      [deleted]
- Retric 11y ago
  The first 3 prime numbers are 2, 3, 5. Base 10 and base 12 both use two of them. (12/4=3 yay, but 10/4 = 2.5 so meh) The same is true of base 6 or base 15. Every other prime number repeats so switching is pointless, unless you divide by 3 vastly more often than by 5. PS: Several ancient systems use base 60 which uses 2, 3, and 5 and may have been created by joining a base 6 system with a base 10 system. However, base 30 also works well for this and is thus a clear step up from base 10 or 12 while being simpler than base 60.
  - zeveb 11y ago
    > Every other prime number repeats so switching is pointless, unless you divide by 3 vastly more often than by 5. It's highly probable that in duodecimal one would divide by 2, 3, 4, 6, 8 & 9 as often as one divides by 5 in decimal, and avoid 5 as much as one avoids 3 & 7 in decimal.
panglott 11y ago
Most of the multiples of eleven are really, really easy, and probably just has simple pedagogical value. Multiples of 12 come up a lot because dozens are used a lot in real-world counting of length and time. Perhaps less in a metric country, but hours and days are still divisible by 12.
- slavik81 11y ago
  Eggs, donuts and beers also still come in multiples of 6 or 12.
twothamendment 11y ago
Is there any point? Depends on where you live. It doesn't hurt to know that 8' is 96 inches, 4' is 48, etc. Now if I were in a country on metric system, I don't know if I could come up with a reason or time that I needed to know them.
Animats 11y ago
Only in non-metric countries. Back in the British era of pounds, shillings, and pence, hardware had to be built to do arithmetic in that system. This resulted in one of the strangest, and most complex, purely mechanical computing devices ever built - the McClure Multiplying Punch [1], from Powers-Samas. This device came out in 1938. The comparable IBM machine was the IBM 602 Multiplying Punch, but IBM only did decimal multiplies. The McClure machine had a mechanical multiplier for pounds, shillings, and pence. It contained a physical multiplication table, made out of brass plates, and the machinery to use them for multiplication by table lookup. There's a picture of the "Pence x 7" plate.[2] That's one row of a multiplication table, and it's 12 wide, from 0 to 11. Sixpence x 7 = 3 shillings 6 pence. The brass column heights reflect that. [1] http://www.computerconservationsociety.org/resurrection/res59.htm http://www.computerconservationsociety.org/resurrection/res5... [2] http://www.computerconservationsociety.org/resurrection/images/images59/res59k.jpg http://www.computerconservationsociety.org/resurrection/imag...
- myth_buster 11y ago
  The author is arguing in the context of UK post decimalization of units. Thanks for additional info on the machinery involved.
- deadowl 11y ago
  I disagree. 1. We have time. Days are 24 hours, a multiple of 12. 2. We still have time. Hours are 60 minutes, a multiple of 12. 3. Yet more time. Minutes are 60 seconds, a multiple of 12. 4. We have circles. There are 360 degrees in a circle, another multiple of 12. 5. The numbers 1, 2, 3, 4, 6 and 12 itself divide into 12 evenly. The next smallest number that has more factors than 12 is 24, which manages to be a multiple of 12.
  - threatofrain 11y ago
    Those seem like minor use cases for disagreement. I'm not sure there should be an education policy or tradition just because our current unit of time is divisible by 12, or just because circles can be represented with the arc degree, especially when a lot of students go on to use radians, even in non-metric countries. And if there's some domain-specific application, like in carpentry, then let those people use their own units.
geofft 11y ago
Even in metric countries, there are still 24 hours in a day, so being able to have a sense of what 36 hours is or what 96 hours is is useful.
- waxjar 11y ago
  I don't think this comes up that often. It's easier to speak in terms of days and halves of days. Often this is accurate enough.
KhalilK 11y ago
Obligatory Numberphile video: https://www.youtube.com/watch?v=U6xJfP7-HCc https://www.youtube.com/watch?v=U6xJfP7-HCc
noobermin 11y ago
Am I going to be the only one to comment that the article actually was pretty cool? He put up some really nice plots in a masterful way, which is better than most of the hand-waving I see here. The best attempt he put forward to deal with the distribution of numbers one would see in daily life (a generalization of the "non-metric" argument) is using Benford's Law[0], which is as good a blog post that doesn't turn to hardcore statistics (of all numbers used, if any such thing actually exists) can do. [0] https://en.wikipedia.org/wiki/Benford's_law https://en.wikipedia.org/wiki/Benford's_law
squozzer 11y ago
Yes we would not have this cute little ditty. http://youtu.be/_uJsoZheTR4 http://youtu.be/_uJsoZheTR4
VeilEm 11y ago
It's really not a big deal for a child to memorize this table. I also think memorizing the squares up to 25 is worth it.
- TheCowboy 11y ago
  If we ignore that it takes "144 facts" compared to "100 facts" for a 10 times table, and take it as a single item of learning, it may not be a big deal. The problem is that it's not the only piece of distracting trivia that is heaped on students as something they should know that likely holds little consequence to achieving big picture learning goals like critical thinking and better math skills. A more American example would be memorizing the capital cities of every US state, along with other state trivia. Memorizing things like the state capitals just gives the illusion of education, but is actually a waste of educational resources and learning time. Time, along with the attention span of a child, is a scarce resource. It's good to trim the curriculum when we identify information that has weak relative value, and it's too easy to find something else to replace it.
myth_buster 11y ago
@Dang, article from 2013.
kazinator 11y ago
There is some use in knowing subsets of the time tables beyond 9: specifically M x N combinations <= 100. For instance, if you know that 8x12 = 96, you know that 1/12 is approximately 0.08. And since 0.96/12 is exactly 0.08, you know that the remainder is 0.04/12 which gives you the .003333... in 0.0833333... Essentially, it supports numeric intuition. These higher times tables can be useful in long division. In long division you have to form hypotheses about how many times the divisor goes into the partial dividend, to extract the next digit of the partial quotient. So for instance, something like this comes up: ________ 12 | 980 Now 12 doesn't go into 9, so we try 98. How many times does 12 go into 98? If you know your 12x12 times table by rote, you might realize in a flash that 12x8 is 96, so put down an 8. If you memorized all times tables up to 99, you could so long multiplication in base 100: 1234 4567 ---- You would instantly know that 67 x 34 is 2278, so ... put down the 78 and carry the 22: 1234 4567 ---- 22 78 and then 67 x 12 is 804. Bring down the 22 into it and we get 826: 1234 4567 ---- 82678 Damn, that was fast! :) Then we keep going with the 45 similarly: and we can move by two places to the left. 45 x 34 = 1530; put down 30; carry 15: 1234 4567 ---- 82678 15 30 and then 45 x 12 = 540, plus carry is 555: 1234 4567 ---- 82678 55530 ------- 5635678 Totally awesome, and we only had to memorize 4950 product combinations from 1x1 to 99x99. Contrast that with plodding through it one digit at a time, with four partial product rows to then add together.
- thaumasiotes 11y ago
  > For instance, if you know that 8x12 = 96, you know that 1/12 is approximately 0.08. You get this same intuition in a much better way by knowing that "since 1/8 is approximately 0.12, 1/12 is approximately 0.08".
lordnacho 11y ago
There's lots of questions thrown up by this. If people need to know times tables, do they also need to know powers of 2? They come up quite often if you deal with computers. What about squares? Anyone doing polynomials later on will be using them constantly. My sense is you will end up memorising the things that come up anyway, so why be so strict? You'll end up teaching kids that math is a memory game. So for fun, what do I remember, off the top? 210 = 1024. Anything higher I keep doubling. 132 = 169, biggest non derived square in my head. 1 more than 7 * 24 (hours in a week). 86400 = secs in a day? Large prime number... 10243 (Useful for demonstrating Diffie Hellman on paper) Ramanujan's Cab... argh... is it 1729? 3.14159 (Guess I'm not winning any contests) 2.71 (See above) 6E23 Add your own random numbers that are in your head.
- schoen 11y ago
  You're right about 86400 and 1729. I know up to 2¹⁷=131072 and also the special cases 2²⁰=1048576 and 2²⁴=16777216 (the 7s in the middle make it easier to memorize, and it seems to come up a lot). I would really like to remember 2³²=4294967296 (I always just remember "4.2 billion") and should probably get around to that. I learned a lot of pi in middle school but even then I knew that it wouldn't be useful for anything, and it hasn't, other than knowing a lot of pi. The start of e goes 2.718281828, so it's not a whole lot of effort to go a little beyond your 2.71 if you wanted to. Here are some Unicode superscripts ¹²³⁴⁵⁶⁷⁸⁹⁰ in case anyone wants to use them in this thread.
  - lordnacho 11y ago
    4.2 Billion is useful, it's the addressable memory in 32 bits.
    - schoen 11y ago
      Yes, exactly (and similarly the number of IPv4 addresses) -- I'd like to memorize the precise value sometime.
- weinzierl 11y ago
  9,81 m/(s*s) 3E8 m/s 6.022E23 1.602E-19 As 9.109E-31 kg 1.381E-23 J/K 1/137 (Rarely needed it, it’s just so easy to remember) 8h*19d*12m is ca. 1800 h/y 65 is ASCII A I know the numbers in my sleep but always have to think twice about the units. Somehow I never can remember the gravitational constant and the Planck constant.
sageikosa 11y ago
I suggest learning them up to 16, or at least learn the 16th column-row (some of the others not so much).
- thomasahle 11y ago
  Yup, that would be great to know, when doing hex. Really, if I could have come out of school with the 20x20 table memorized, I wouldn't mind now at all.
Spooky23 11y ago
To calculate dozens Lots of commodities are sold in dozen multiples.
tempestn 11y ago
I would argue that the 11s don't really count, since knowing them is trivial. The only one that really requires memory is 11x11, and it's probably not strictly worth knowing, but then kids get the accomplishment of knowing a whole extra row essentially for free. I agree that the 12s aren't worth memorizing though. They're pretty easy to mentally calculate, being a combination of 1 and 2, and after doing that for a few years, the memorization will often happen naturally. No real need to force it.
mserdarsanli 11y ago
> Take as an example 7,203 x 6,892. If I want to know that exactly, then I reach for Mathematica They are always trying so hard to sell Mathematica in these blog posts, but this is the lowest I've seen.
- gknoy 11y ago
  I don't know that it's meant to shill the product -- rather, that's the tool that he has handy. I work all the time with Python, and frequently open an IPython shell to do some quick math ("what's the average of 4d6 pick 3?" "what is X% of my paycheck?" "how much do I need to contribute to max out my $FOO?"), simply because it's easier than a calculator. I mean, sure, it's advertising Mathmatica, but it doesn't seem to be unecessarily doing so. It's more that it's ${programing-language-of-choice} for the author.
  - mserdarsanli 11y ago
    I don't work with Python, but when I want to multiply some numbers, I type `python` into terminal and use it as a calculator, as it allows me to do such operations quickly. It sounds weird that someone opens an IDE and waits ten seconds looking at a splash screen to do a simple multiplication.
    - aaron695 11y ago
      Dogfooding
spyckie2 11y ago
1-10 you memorize. 13-100 you calculate it out. 100+ you use a calculator. 11-12 is half memorized/half calculated. I think of them as introducing you to how to calculate with larger numbers you haven't memorized. Some of them you know - 12 * 5? 60. But what about 12 * 7? 10 * 7 + 14. or 12 * 5 + 12 * 2. Dealing with 11 and 12 in the times tables gives you good practice for those calculation tricks that you use for numbers greater than 12. It's not worth it to memorize an additional 44 rules but it is worth it to know how to do math.
clarkmoody 11y ago
This type of post is perfect Hacker News material! 1) Challenge an existing assumption about something we all do 2) Hand-wave some first-order guesses as to why we do it 3) Get nerdy with code and graphs and come to your conclusion, along with helpful suggestions for bettering the reader 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!
- Scarblac 11y ago
  But do we all learn tables up to 12? In the Netherlands it's only up to 10 for as far back as I know.
  - antod 11y ago
    Same in New Zealand. The only times I ever saw anything with 12x tables it had escaped from the US education market and had exercises involving nickels, dimes and quarters etc.
    - TimJRobinson 11y ago
      Interestingly in Australia we learnt all the way to 12.
    - Widdershin 11y ago
      My experience in New Zealand differs, we learned the 12x tables at primary school and there was nothing vaguely imperial about it. This was around about a decade ago.
      - antod 11y ago
        Interesting - my experience was in the early 80s. I suspect getting the 12x table added to the NZ maths curriculum was probably the main deliverable of the 2001 Knowledge Wave Conference.
mdip 11y ago
Since I grew up in the United States, learning 12 in our multiplication tables (which was also required) had a lot of value. Being able to multiply by 12 makes handling inches/feet conversions a mental operation and that's something an average US citizen encounters in normal life regularly enough. I suspect in the UK this is partially due to legacy and "everyday life" usage of imperial units[1], combined with the idea that "it's really just one more number"[2]. "Officially" they are metric, but its usage varies (except temperature, which nobody can mentally calculate when I tell them it's "90 degrees out" in Fahrenheit). In the broader sense, I'd prefer that a lot more time be spent on number theory than on arithmetic. I am rarely without a smartphone to the point where its in my hands for even basic math that I could do in my head (but which, due to infrequent use, I would spend a few seconds questioning whether or not I really did get the right answer). The schools start with algebra and other concepts much earlier, now (at least my district does), but I started teaching my children formulas and basic algebra from the point they started learning to add and subtract (kindergarten in my district) along with binary and hexadecimal positional numerical systems. I remember struggling with that in my teens when I started learning software development my entire exposure to numbers was in decimal (10 is the number after 9, always!). In the first grade, my kids didn't have that mental block and I found they grasped the idea far more quickly than I expected. As they've learned multiplication and division, I've shown them the same in binary and hexadecimal and illustrated how they relate to one another. Everyone struggled being introduced algebra so late when I was growing up. And it progressed so quickly, going from Algebra (2 yrs), Geometry (1 yr), Trigonometry (1 yr) and pre-calculus/calculus that students fell mostly into either "memorize the formulas/algorithms and regurgitate the test answers or struggle like crazy and never get to Trig/Calculus until college. I left with an understanding of math but that was mostly due to my taking up programming at that time and having to learn what it all really meant because I wanted to do things in code that required me to understand it. [1] The British Metric Society "The Mess We're In" http://www.metric.org.uk/the-mess-we-are-in http://www.metric.org.uk/the-mess-we-are-in [2] The reality is that you're only asking students to memorize one more number since 10 is "add a zero and multiply by one" and 11 is an obvious pattern with single digits.
switch007 11y ago
It's moved from "should" to must. "Every child in England will be expected to know their times tables before leaving primary school from next year. Pupils will be tested against the clock on their tables up to 12x12 in new computer-based exams that the Department of Education (DfE) said were part of the government’s “war on innumeracy and illiteracy”." http://www.theguardian.com/education/2016/jan/03/pupils-face-new-tests-to-ensure-they-know-times-tables-by-age-11 http://www.theguardian.com/education/2016/jan/03/pupils-face... Gove is no longer the Education Secretary...he is now the Justice Secretary. God help us.
- TheOtherHobbes 11y ago
  I like the idea of declaring war on innumeracy. That totally works as a metaphor for improved teaching. I'm not sure people need to know the 12 x 12 table, so much as knowing how percentages and compound interest work. How long will it take someone to pay off a credit card debt if the APR is 39.9% and they can't afford to clear the balance every month? If they don't understand why the answer is "Probably forever unless they get lucky with an inheritance" they shouldn't be using a credit card.
- Wildgoose 11y ago
  Gove did a fantastic job as Education Secretary and that is why he is now hated by a teaching profession being forced to do their job. I am not and never have been a Tory, but he is the one Tory minister I can respect. (The rest are dreadful).
legulere 11y ago
In Germany you usually learn up to 10×10 in school and there are usually 10 eggs in a box (4, 6 and 25 are also common sizes though and there's also a big set that's 20×20 which you don't usually learn though) > And when you want to raise standards in math, then who could argue with that? The problem is that you can only spend so and so much time with children in class. Learning 12×12 is 80% more combinations to learn compared to 9×9. That's time you could teach them something else that might be more useful.
- kijin 11y ago
  Actually, it's only ~32% more combinations to learn, because nobody really needs to memorize the 10's and 11's (except 11 x 11). You memorize all the combinations up to 9 x 9, compute the 10's and 11's on the fly, and memorize the 12's. Edit: Corrected the number. This is how you get 32%: The 9 x 9 multiplication table contains 8 x 8 = 64 combinations to memorize. The 1's don't count. The 12 x 12 multiplication table contains an additional 20 combinations to memorize, namely: - 10 combinations from 12 x 2 to 12 x 12, not counting 12 x 10 - 9 combinations from 2 x 12 to 11 x 12, not counting 10 x 12 - 11 x 11 = 121 So the total is 84 combinations, which is ~32% more than 64.
jheriko 11y ago
politics is a component too... i wouldn't go looking for purely intelligent decision making from a government selected by popularity contest. the perception that school is getting easier, a criticism of the government that is politically undesirable, can be addressed by a 'return to traditional methods' in superficial ways... even if there is not really a concrete problem to address. politics isn't about doing or achieving anything in terms of education or quality of life for others - its about making other people happy.
justinhj 11y ago
Whenever my son has a new math teacher, I taught him to ask if he can change the base when reciting the times tables. For example the 12 times table in base 12 is 10, 20, 30, 40 and so on. It usually gets a laugh or a tut before he has to do it the right way.
cjensen 11y ago
I grew up in the Bay Area in California. We only memorized up to 9x9. Though 12 is used often in measurements, I can't say as that I've missed having rote knowledge of the multiples of 12.
- Namrog84 11y ago
  Would you say then that you don't have any x10 memorized and just consider it trivially easy to compute any number?
Bulkington 11y ago
We're overthinking. Think, instead, as a school marm with real chalk and a switch in hand, or even when kids used slate tablets: 10x is easy. 11x is easy up to 10x. 12x requires a little work and pulls all the multiples together, but 13x just screws with everything we've learned so far. So, stop at 12. Or is a calculator in hand the default these days? As one who was proud to finally get to 12 x 12 = 144, I say make 'em suffer.
potatote 11y ago
Not sure if someone already has pointed this out, but one small caveat I'd add is that learning the table of 11 is not that difficult. I think the extra effort required to learn 11 and 12 tables are, therefore, a bit overestimated.
amelius 11y ago
Let's just switch to binary, and let them learn the 1-times table.
contravariant 11y ago
As someone who never learned the 12 times table, and only uses metric, I'm not sure if it would have been better if I had learned the 12 times table, but for some reason I have actually memorized most multiples of 12 (without any conscious effort). They do seem to turn up a lot for various reasons.
aidenn0 11y ago
I always assumed it was because we'd forget the largest couple of rows we memorized, so 20 years later we could still do up to 10
yason 11y ago
Memorisation is caching. If you need 11-times or 12-times tables, you will calculate them enough many times that your cache will become "hot" and you remember them out of the box. If you don't, then you don't have enough use for those multipliers.
dahart 11y ago
The reasons given for learning multiplication tables all focus on multiplication, but one of the best reasons is for division. Doing division on paper or in your head is made so much easier if you have your single digit multiplication table memorized. But why 12 instead of 10? I think: - There are loads of common everyday uses that overflow 10 by just a little. - Clocks - Eggs - 11 is almost as trivial as 10, it would be a shame to stop at 10 when 11 is free. - 12 is really super easy, and combined with the above reasons has a good case. - 13s are significantly harder to remember and work with than 11 or 12, nothing comes in 13s, its the perfect place to stop. :) (edit: course, as soon as I wrote that I remembered the baker's dozen, which was super common to hear when I was a kid, not so much anymore for me. But I think the point still stands, because with a baker's dozen you only pay for 12, you never had to multiply by 13.) * edited for formatting
djrogers 11y ago
I can think of about a dozen reasons why this is important...
oska 11y ago
> Download this post as a Computable Document Format (CDF) file. Hmm? Ah. > CDF files can be read using a proprietary CDF Player with a restrictive license, which can be downloaded free of charge from Wolfram Research. [1] Fuhgeddaboudit. [1] https://en.wikipedia.org/wiki/Computable_Document_Format https://en.wikipedia.org/wiki/Computable_Document_Format
protomyth 11y ago
12 x 12 is 144 and if we're not worried that our children will not have the mathematical skills to count and divide a gross of bottle rockets, then this is truly not America. Plus inches in feet. On a more serious note, the world does not end with 10.
greggarious 11y ago
Can I please point out this is the point where many people get turned off from math, never to return? I HATED memorization. I was able to do the math in my head up through 10, but 11x11, 11x12 and the rest of the 12s eluded me, and I just really, really was not able to sit still and just do flashcards. I ended up only chancing back on math through programming and a course in logic in undergrad. Everything else was mostly self taught as needed.
tempestn 11y ago
Here's a good example I just came across of why it's useful to be able to do approximate multiplication in your head (perhaps even while intoxicated): http://news.nationalpost.com/full-comment/robyn-urback-sorry-about-your-1000-uber-bill-but-the-internet-isnt-idiot-proof http://news.nationalpost.com/full-comment/robyn-urback-sorry...
stewdio 11y ago
Jon McLoone really seems to be approaching this topic from a place of frustration—rather than from an overwhelming sense of beauty. I think both mathies and visual designers can agree that 12 is a very beautiful number. In fact the ONLY single thing that makes sense about the imperial system is the division of feet into 12 inches, something mirrored by picas and points in typography: https://en.wikipedia.org/wiki/Pica_(typography) https://en.wikipedia.org/wiki/Pica_(typography) This beautiful number is evenly divisible by 1, 2, 3, 4, 6, and itself—a very high percentage (50%!) of the positive integers leading up to and including itself. (To be sure 60 is also a beautiful number but its ratio of factors to non-factors is of course lesser.) I can see you gearing up to argue that 6 is more beautiful, and I get that. I really do. But the ability to evenly divide into quarters is just so ... pleasant :) Does the ability to divide evenly in several ways serve as a soapbox from which to argue that it is mathematically important to memorize the first 12 numbers of which 12 is a factor? I suppose not. But could we simply do 12 the honor of learning these two extra columns of the multiplication table because it is beautiful? (And really you’re just learning ONE extra column because 11 is such a simpleton. Silly 11.) Of course dividing things into 12 units can SOUND pleasant as well. Western music is founded on dividing an octave into 12 units—though the exact method of division can take many forms. Here’s just a sample: https://github.com/stewdio/beep.js/blob/master/beep/Beep.Note.js#L268 https://github.com/stewdio/beep.js/blob/master/beep/Beep.Not... So I love 12. Join me in loving 12. Together we can praise 12. Full disclosure, I have been known to previously adore the number 12. On December 12, 2012 I proposed this “Cleaner Grid of Time” beginning with the months in a year: http://stewd.io/b/2012/12/12/twelve http://stewd.io/b/2012/12/12/twelve

185 points by kesselvon 11y ago

Forget the stats and calculus. Think carpentry, measuring and cutting wood products. If you are building things in the US/UK/Canada then you are using feet and inches. 12 inches to a foot. It's a tiny thing to learn and will serve kids well in any number of professions. Now 11, that's a total mystery. Other than it being between 10 and 12, I see no reason to memorize 11s.

nsxwolf 11y ago
I remember that memorizing the 11 column was trivial and fun, up to times 10 anyway.
x1798DE 11y ago
11s are practically free, though, in base 10. The algorithm "repeat the non 11 number twice" works up till 10 x 11, where the "add a 0" algorithm for 10 kicks in. So you're just really memorizing 11 x 11 = 121 and 11 x 12 = 132.
- BillTheCat 11y ago
  A simple trick: For 11 x a 2-digit number you can simply take the first digit of the 2-digit number then the 2 digits of the number added together then the last digit of the 2-digit number. Ex: 11 * 12 = 132 or 1, 1+2, 2. 11 * 45 = 495 or 4, 4+5, 5. For numbers which sum to more than 10 add the carry to the first number ex: 11 * 59 = 649 or 5, 5+9 = 14 so add 1 to the initial 5 and keep the 4, 9.
  - pc86 11y ago
    I'm 29 and never heard this before. Neat!
    - deleted 11y ago
      [deleted]
    - the-dude 11y ago
      This is golden: I am just over 40 and I now know that I knew nothing at 29!
  - gshubert17 11y ago
    This generalizes to multiplying any number times 11, https://en.wikipedia.org/wiki/Trachtenberg_system#Multiplying_by_11 https://en.wikipedia.org/wiki/Trachtenberg_system#Multiplyin... Take a number, say 142857. Prepend a 0 on the left. Underneath each digit write the sum of the digit above and the one to the right. If you work from the right, it's easier to keep track of the carries. 0142857 1571427
- derefr 11y ago
  But it's also "practically free" to just mentally do "10x + x", which I can actually add faster than my brain will return an answer from the post-10 part of the times table. (Not that I didn't memorize the 11s and 12s all the same at the time, but they got a lot less use and so a lot less reinforcement.)
deleted 11y ago
[deleted]
zanny 11y ago
That is only yet another reason in literally infinite ones why the US needs to actually push metric units. We bleed our stupidity into the UK and Canada while the rest of the world makes sense.
- btilly 11y ago
  Makes sense..until you talk about time. Time makes no sense.
  - sandworm101 11y ago
    If you want to terrify an american, ask them about switching to metric time. https://en.wikipedia.org/wiki/Metric_time https://en.wikipedia.org/wiki/Metric_time This comes up all the time for me as americans use a different date format. I've had contract negotiations grind to a halt because someone was doing math based on 06/05/15 rather than 05/06/15.
    - web007 11y ago
      Use something like 12/24/48 (US) or 48/12/24 (UK) if you want to "force" compliance, then you'll at least have an easy sanity check for each field. The other benefit is you can check Y2038 compliance at the same time, but you should at least specify 4-digit years for whatever you're spec'ing, so you don't create the next Y2K.
    - kstrauser 11y ago
      I write everything either ISO or spelled out. Your dates would be 2015-06-05 or "June 6, 2015". It's an easy quick that everyone seems to be comfortable with.
    - mark-r 11y ago
      Switch to ISO format. There's less confusion in 2015-05-06. Although the one time I tried to push for it, at a company with plants in both the US and Ireland, I couldn't quite get agreement and we settled on 2015-MAY-06.
- 11y ago
jjp 11y ago
> If you are building things in the US/UK/Canada then you are using feet and inches. In the UK you are more likely to be using millimetres if working from any kind of design. Working in inches and feet would depend on your age and perhaps whether you are working on an older property that was designed in inches.
- michaelt 11y ago
  In the UK your designs and plans will be in mm, but your materials will vary between metric-rounding-of-imperial, (2400x1200mm) [1] imperial-but-labeled-in-metric (1220x2440mm) [2] and imperial-in-one-dimension-metric-in-the-other (38x144x2400mm) [3]. Wallpaper? 10m x 520mm for sure [4] but doors? 1981x762mm [5]. Cement? Back to metric, with 10kg and 25kg bags [6]. You'll also probably measure long distances in miles but short distances in meters; and weigh your ingredients in grams but your body in stone. I guess what I'm saying is if you want a rationally designed measurement system, don't copy us Brits and especially don't copy our builders. [1] http://www.wickes.co.uk/Knauf-Plasterboard-Square-Edge-2400x1200x12-5mm/p/104123 http://www.wickes.co.uk/Knauf-Plasterboard-Square-Edge-2400x... [2] http://www.wickes.co.uk/Wickes-General-Purpose-OSB3-Board-18x1220x2440mm/p/110517 http://www.wickes.co.uk/Wickes-General-Purpose-OSB3-Board-18... [3] http://www.wickes.co.uk/Wickes-Studwork-%28CLS%29-38x144x2400mm-Single/p/101857 http://www.wickes.co.uk/Wickes-Studwork-%28CLS%29-38x144x240... [4] http://www.wickes.co.uk/Wickes-9006-Wallpaper-Embossed-White-10m/p/610194 http://www.wickes.co.uk/Wickes-9006-Wallpaper-Embossed-White... [5] http://www.wickes.co.uk/Wickes-Skipton-Internal-Softwood-Door-Clear-Pine-4-Panel-1981x762mm/p/206690 http://www.wickes.co.uk/Wickes-Skipton-Internal-Softwood-Doo... [6] http://www.wickes.co.uk/Blue-Circle-Extra-Rapid-Cement-25kg/p/224662 http://www.wickes.co.uk/Blue-Circle-Extra-Rapid-Cement-25kg/...
- twic 11y ago
  You will almost certainly be working in millimetres, but because of ISO 2848, you will very likely be working in multiples of 300 millimetres: https://en.wikipedia.org/wiki/Preferred_number#Buildings https://en.wikipedia.org/wiki/Preferred_number#Buildings Which is about a foot!
creshal 11y ago
> in the US/UK/Canada And the other 94% of the world population…?
- wtetzner 11y ago
  So they don't have to learn it. Not sure what the problem is.
arethuza 11y ago
"If you are building things in the US/UK/Canada then you are using feet and inches." I don't think that's been the case for a pretty long time in the UK - certainly DIY supplies are metric e.g. here is the Homebase site: http://www.homebase.co.uk/en/homebaseuk/diy/timber/planed-timber http://www.homebase.co.uk/en/homebaseuk/diy/timber/planed-ti...

SixSigma 11y ago

I recently worked with a person who had to do her three times table using her hands, starting from 3. She would still struggle at about 6 or 7 times. We worked behind the counter in a builder's merchant. She is no longer working for us.

theworstshill 11y ago
Why not?
- pc86 11y ago
  Presumably the inability to do basic math quickly.
- Symbiote 11y ago
  It's exactly the kind of job where someone will phone and ask "do you have 30 widgets I could pick up in an hour?". They come in packs of 4 widgets, and there are only 7 packs in stock.
- SixSigma 11y ago
  Approx 300mm to the foot. All products in mm, half the customers say "6ft widget". Getting the calculator out to do 6 x 3 is unprofessional.
xlm1717 11y ago
You know, whether or not children should learn up to 12 by the age of nine is debatable, but by the time you're an adult, assuming no learning disabilities you should be able to do mental math proficiently enough that you don't need to memorize any tables, you can just do it. I hope that person you worked with can at least work a calculator.
lordnacho 11y ago
I had a job once where we hired a derivatives trader who couldn't multiply 8 and 4. In the job interview. At my next job we hired another trader who came up to me and said "what does the little 2 in a superscript mean?". As in what does 3^2 mean? Both very friendly people.

Amorymeltzer 11y ago

>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. This is absolutely worthwhile, easy approximation of 1 or 100 divided by 6 or 8 in particular occurs regularly, and not enough people know it offhand. Hugely helpful. After 1-10, your energy is probably better spent on cool patterns and "tricks" within those sets (final digits of 9, division by 7, etc.); much greater return.

thaumasiotes 11y ago
Ok, but what's with the weird phrasing? This is the same set of facts as (1/2 = 0.50, 1/3 = 0.33, 1/4 = 0.25, 1/5 = 0.20, 1/6 = 0.17, 1/7 = 0.14 ...), but phrased to sound like something you'd have to put effort into.

ck2 11y ago

They should probably learn HEX at age 9

walshemj 11y ago
Back in the 70's (UK) I remember doing non base 10 maths at about that age plus basic set theory much more useful than rote leaning your times table.
- PeCaN 11y ago
  Did that in the US in the mid 2000s, though spent hardly a week on either IIRC. Something like base 20 Mayans, base 60 Sumerians, and base 2 computers. Do math in each and convert them to/from base 10. I think that's been dropped from most US elementary school curricula now, bit of a shame.
  - Zikes 11y ago
    Was base 60 Sumerians accompanied by cuneiform, by chance? I got to learn some of that in a World History course in high school, which was pretty fun.
  - schoen 11y ago
    I believe it was introduced as part of some New Math curricula. https://en.wikipedia.org/wiki/New_Math https://en.wikipedia.org/wiki/New_Math Richard Feynman was pretty harshly critical of this in his stories of working as a textbook reviewer. I really appreciated that my elementary school did it, but I had a sense that most students didn't quite understand the significance (or remember it afterward if they weren't into computer programming). I think it has a lot of potential value in terms of understanding what place value is and where a place value system comes from. When I've tried to teach programming, I've found that most people who aren't already programming enthusiasts don't remember the binary system or say they never learned it at all. Edit: the Wikipedia article I linked to also has several examples of people making fun of non-base-10 arithmetic and kids being taught it in the New Math, including a song by Tom Lehrer!
11y ago

kitwalker12 11y ago

he got off lucky. back in India, we had to cram tables upto 20.

zeveb 11y ago

Twelve has more divisors than ten (1, 2, 3, 4, 6 & 12 vs 1, 2, 5 & 10). If we were really smart, we'd switch from base 10 to base 12: many more 'decimals' (really duodecimals) would be non-repeating. One can very quickly count by twelves on the joints of one's fingers, using the thumb as an index (0-143 is a much larger range than 0-10, and it's easier to hold one's hands in the shape necessary). If we were really smart, we'd switch from base 10 to base 12: many more 'decimals' (really duodecimals) would be non-repeating. And then there are measures like a gross (144) and a great gross (1,728). Part of the reason for these traditional measures is that they are more flexible than base-10 measures: an eighth-gross or a third-gross are both integer quantities, unlike an eighth-hundred or third-hundred.

panglott 11y ago
Base-60 math is even superior.
- noobermin 11y ago
  The Sumerians had it right?
  - deleted 11y ago
    [deleted]
Retric 11y ago
The first 3 prime numbers are 2, 3, 5. Base 10 and base 12 both use two of them. (12/4=3 yay, but 10/4 = 2.5 so meh) The same is true of base 6 or base 15. Every other prime number repeats so switching is pointless, unless you divide by 3 vastly more often than by 5. PS: Several ancient systems use base 60 which uses 2, 3, and 5 and may have been created by joining a base 6 system with a base 10 system. However, base 30 also works well for this and is thus a clear step up from base 10 or 12 while being simpler than base 60.
- zeveb 11y ago
  > Every other prime number repeats so switching is pointless, unless you divide by 3 vastly more often than by 5. It's highly probable that in duodecimal one would divide by 2, 3, 4, 6, 8 & 9 as often as one divides by 5 in decimal, and avoid 5 as much as one avoids 3 & 7 in decimal.

panglott 11y ago

Most of the multiples of eleven are really, really easy, and probably just has simple pedagogical value. Multiples of 12 come up a lot because dozens are used a lot in real-world counting of length and time. Perhaps less in a metric country, but hours and days are still divisible by 12.

slavik81 11y ago
Eggs, donuts and beers also still come in multiples of 6 or 12.

twothamendment 11y ago

Is there any point? Depends on where you live. It doesn't hurt to know that 8' is 96 inches, 4' is 48, etc. Now if I were in a country on metric system, I don't know if I could come up with a reason or time that I needed to know them.

Animats 11y ago

Only in non-metric countries. Back in the British era of pounds, shillings, and pence, hardware had to be built to do arithmetic in that system. This resulted in one of the strangest, and most complex, purely mechanical computing devices ever built - the McClure Multiplying Punch [1], from Powers-Samas. This device came out in 1938. The comparable IBM machine was the IBM 602 Multiplying Punch, but IBM only did decimal multiplies. The McClure machine had a mechanical multiplier for pounds, shillings, and pence. It contained a physical multiplication table, made out of brass plates, and the machinery to use them for multiplication by table lookup. There's a picture of the "Pence x 7" plate.[2] That's one row of a multiplication table, and it's 12 wide, from 0 to 11. Sixpence x 7 = 3 shillings 6 pence. The brass column heights reflect that. [1] http://www.computerconservationsociety.org/resurrection/res59.htm http://www.computerconservationsociety.org/resurrection/res5... [2] http://www.computerconservationsociety.org/resurrection/images/images59/res59k.jpg http://www.computerconservationsociety.org/resurrection/imag...

myth_buster 11y ago
The author is arguing in the context of UK post decimalization of units. Thanks for additional info on the machinery involved.
deadowl 11y ago
I disagree. 1. We have time. Days are 24 hours, a multiple of 12. 2. We still have time. Hours are 60 minutes, a multiple of 12. 3. Yet more time. Minutes are 60 seconds, a multiple of 12. 4. We have circles. There are 360 degrees in a circle, another multiple of 12. 5. The numbers 1, 2, 3, 4, 6 and 12 itself divide into 12 evenly. The next smallest number that has more factors than 12 is 24, which manages to be a multiple of 12.
- threatofrain 11y ago
  Those seem like minor use cases for disagreement. I'm not sure there should be an education policy or tradition just because our current unit of time is divisible by 12, or just because circles can be represented with the arc degree, especially when a lot of students go on to use radians, even in non-metric countries. And if there's some domain-specific application, like in carpentry, then let those people use their own units.

geofft 11y ago

Even in metric countries, there are still 24 hours in a day, so being able to have a sense of what 36 hours is or what 96 hours is is useful.

waxjar 11y ago
I don't think this comes up that often. It's easier to speak in terms of days and halves of days. Often this is accurate enough.

KhalilK 11y ago

Obligatory Numberphile video: https://www.youtube.com/watch?v=U6xJfP7-HCc https://www.youtube.com/watch?v=U6xJfP7-HCc

noobermin 11y ago

Am I going to be the only one to comment that the article actually was pretty cool? He put up some really nice plots in a masterful way, which is better than most of the hand-waving I see here. The best attempt he put forward to deal with the distribution of numbers one would see in daily life (a generalization of the "non-metric" argument) is using Benford's Law[0], which is as good a blog post that doesn't turn to hardcore statistics (of all numbers used, if any such thing actually exists) can do. [0] https://en.wikipedia.org/wiki/Benford's_law https://en.wikipedia.org/wiki/Benford's_law

squozzer 11y ago

Yes we would not have this cute little ditty. http://youtu.be/_uJsoZheTR4 http://youtu.be/_uJsoZheTR4

VeilEm 11y ago

It's really not a big deal for a child to memorize this table. I also think memorizing the squares up to 25 is worth it.

TheCowboy 11y ago
If we ignore that it takes "144 facts" compared to "100 facts" for a 10 times table, and take it as a single item of learning, it may not be a big deal. The problem is that it's not the only piece of distracting trivia that is heaped on students as something they should know that likely holds little consequence to achieving big picture learning goals like critical thinking and better math skills. A more American example would be memorizing the capital cities of every US state, along with other state trivia. Memorizing things like the state capitals just gives the illusion of education, but is actually a waste of educational resources and learning time. Time, along with the attention span of a child, is a scarce resource. It's good to trim the curriculum when we identify information that has weak relative value, and it's too easy to find something else to replace it.

myth_buster 11y ago

@Dang, article from 2013.

kazinator 11y ago

There is some use in knowing subsets of the time tables beyond 9: specifically M x N combinations <= 100. For instance, if you know that 8x12 = 96, you know that 1/12 is approximately 0.08. And since 0.96/12 is exactly 0.08, you know that the remainder is 0.04/12 which gives you the .003333... in 0.0833333... Essentially, it supports numeric intuition. These higher times tables can be useful in long division. In long division you have to form hypotheses about how many times the divisor goes into the partial dividend, to extract the next digit of the partial quotient. So for instance, something like this comes up: ________ 12 | 980 Now 12 doesn't go into 9, so we try 98. How many times does 12 go into 98? If you know your 12x12 times table by rote, you might realize in a flash that 12x8 is 96, so put down an 8. If you memorized all times tables up to 99, you could so long multiplication in base 100: 1234 4567 ---- You would instantly know that 67 x 34 is 2278, so ... put down the 78 and carry the 22: 1234 4567 ---- 22 78 and then 67 x 12 is 804. Bring down the 22 into it and we get 826: 1234 4567 ---- 82678 Damn, that was fast! :) Then we keep going with the 45 similarly: and we can move by two places to the left. 45 x 34 = 1530; put down 30; carry 15: 1234 4567 ---- 82678 15 30 and then 45 x 12 = 540, plus carry is 555: 1234 4567 ---- 82678 55530 ------- 5635678 Totally awesome, and we only had to memorize 4950 product combinations from 1x1 to 99x99. Contrast that with plodding through it one digit at a time, with four partial product rows to then add together.

thaumasiotes 11y ago
> For instance, if you know that 8x12 = 96, you know that 1/12 is approximately 0.08. You get this same intuition in a much better way by knowing that "since 1/8 is approximately 0.12, 1/12 is approximately 0.08".

lordnacho 11y ago

There's lots of questions thrown up by this. If people need to know times tables, do they also need to know powers of 2? They come up quite often if you deal with computers. What about squares? Anyone doing polynomials later on will be using them constantly. My sense is you will end up memorising the things that come up anyway, so why be so strict? You'll end up teaching kids that math is a memory game. So for fun, what do I remember, off the top? 210 = 1024. Anything higher I keep doubling. 132 = 169, biggest non derived square in my head. 1 more than 7 * 24 (hours in a week). 86400 = secs in a day? Large prime number... 10243 (Useful for demonstrating Diffie Hellman on paper) Ramanujan's Cab... argh... is it 1729? 3.14159 (Guess I'm not winning any contests) 2.71 (See above) 6E23 Add your own random numbers that are in your head.

schoen 11y ago
You're right about 86400 and 1729. I know up to 2¹⁷=131072 and also the special cases 2²⁰=1048576 and 2²⁴=16777216 (the 7s in the middle make it easier to memorize, and it seems to come up a lot). I would really like to remember 2³²=4294967296 (I always just remember "4.2 billion") and should probably get around to that. I learned a lot of pi in middle school but even then I knew that it wouldn't be useful for anything, and it hasn't, other than knowing a lot of pi. The start of e goes 2.718281828, so it's not a whole lot of effort to go a little beyond your 2.71 if you wanted to. Here are some Unicode superscripts ¹²³⁴⁵⁶⁷⁸⁹⁰ in case anyone wants to use them in this thread.
- lordnacho 11y ago
  4.2 Billion is useful, it's the addressable memory in 32 bits.
  - schoen 11y ago
    Yes, exactly (and similarly the number of IPv4 addresses) -- I'd like to memorize the precise value sometime.
weinzierl 11y ago
9,81 m/(s*s) 3E8 m/s 6.022E23 1.602E-19 As 9.109E-31 kg 1.381E-23 J/K 1/137 (Rarely needed it, it’s just so easy to remember) 8h*19d*12m is ca. 1800 h/y 65 is ASCII A I know the numbers in my sleep but always have to think twice about the units. Somehow I never can remember the gravitational constant and the Planck constant.

sageikosa 11y ago

I suggest learning them up to 16, or at least learn the 16th column-row (some of the others not so much).

thomasahle 11y ago
Yup, that would be great to know, when doing hex. Really, if I could have come out of school with the 20x20 table memorized, I wouldn't mind now at all.

Spooky23 11y ago

To calculate dozens Lots of commodities are sold in dozen multiples.

tempestn 11y ago

I would argue that the 11s don't really count, since knowing them is trivial. The only one that really requires memory is 11x11, and it's probably not strictly worth knowing, but then kids get the accomplishment of knowing a whole extra row essentially for free. I agree that the 12s aren't worth memorizing though. They're pretty easy to mentally calculate, being a combination of 1 and 2, and after doing that for a few years, the memorization will often happen naturally. No real need to force it.

mserdarsanli 11y ago

> Take as an example 7,203 x 6,892. If I want to know that exactly, then I reach for Mathematica They are always trying so hard to sell Mathematica in these blog posts, but this is the lowest I've seen.

gknoy 11y ago
I don't know that it's meant to shill the product -- rather, that's the tool that he has handy. I work all the time with Python, and frequently open an IPython shell to do some quick math ("what's the average of 4d6 pick 3?" "what is X% of my paycheck?" "how much do I need to contribute to max out my $FOO?"), simply because it's easier than a calculator. I mean, sure, it's advertising Mathmatica, but it doesn't seem to be unecessarily doing so. It's more that it's ${programing-language-of-choice} for the author.
- mserdarsanli 11y ago
  I don't work with Python, but when I want to multiply some numbers, I type `python` into terminal and use it as a calculator, as it allows me to do such operations quickly. It sounds weird that someone opens an IDE and waits ten seconds looking at a splash screen to do a simple multiplication.
  - aaron695 11y ago
    Dogfooding

spyckie2 11y ago

1-10 you memorize. 13-100 you calculate it out. 100+ you use a calculator. 11-12 is half memorized/half calculated. I think of them as introducing you to how to calculate with larger numbers you haven't memorized. Some of them you know - 12 * 5? 60. But what about 12 * 7? 10 * 7 + 14. or 12 * 5 + 12 * 2. Dealing with 11 and 12 in the times tables gives you good practice for those calculation tricks that you use for numbers greater than 12. It's not worth it to memorize an additional 44 rules but it is worth it to know how to do math.

clarkmoody 11y ago

This type of post is perfect Hacker News material! 1) Challenge an existing assumption about something we all do 2) Hand-wave some first-order guesses as to why we do it 3) Get nerdy with code and graphs and come to your conclusion, along with helpful suggestions for bettering the reader 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!

Scarblac 11y ago
But do we all learn tables up to 12? In the Netherlands it's only up to 10 for as far back as I know.
- antod 11y ago
  Same in New Zealand. The only times I ever saw anything with 12x tables it had escaped from the US education market and had exercises involving nickels, dimes and quarters etc.
  - TimJRobinson 11y ago
    Interestingly in Australia we learnt all the way to 12.
  - Widdershin 11y ago
    My experience in New Zealand differs, we learned the 12x tables at primary school and there was nothing vaguely imperial about it. This was around about a decade ago.
    - antod 11y ago
      Interesting - my experience was in the early 80s. I suspect getting the 12x table added to the NZ maths curriculum was probably the main deliverable of the 2001 Knowledge Wave Conference.

mdip 11y ago

Since I grew up in the United States, learning 12 in our multiplication tables (which was also required) had a lot of value. Being able to multiply by 12 makes handling inches/feet conversions a mental operation and that's something an average US citizen encounters in normal life regularly enough. I suspect in the UK this is partially due to legacy and "everyday life" usage of imperial units[1], combined with the idea that "it's really just one more number"[2]. "Officially" they are metric, but its usage varies (except temperature, which nobody can mentally calculate when I tell them it's "90 degrees out" in Fahrenheit). In the broader sense, I'd prefer that a lot more time be spent on number theory than on arithmetic. I am rarely without a smartphone to the point where its in my hands for even basic math that I could do in my head (but which, due to infrequent use, I would spend a few seconds questioning whether or not I really did get the right answer). The schools start with algebra and other concepts much earlier, now (at least my district does), but I started teaching my children formulas and basic algebra from the point they started learning to add and subtract (kindergarten in my district) along with binary and hexadecimal positional numerical systems. I remember struggling with that in my teens when I started learning software development my entire exposure to numbers was in decimal (10 is the number after 9, always!). In the first grade, my kids didn't have that mental block and I found they grasped the idea far more quickly than I expected. As they've learned multiplication and division, I've shown them the same in binary and hexadecimal and illustrated how they relate to one another. Everyone struggled being introduced algebra so late when I was growing up. And it progressed so quickly, going from Algebra (2 yrs), Geometry (1 yr), Trigonometry (1 yr) and pre-calculus/calculus that students fell mostly into either "memorize the formulas/algorithms and regurgitate the test answers or struggle like crazy and never get to Trig/Calculus until college. I left with an understanding of math but that was mostly due to my taking up programming at that time and having to learn what it all really meant because I wanted to do things in code that required me to understand it. [1] The British Metric Society "The Mess We're In" http://www.metric.org.uk/the-mess-we-are-in http://www.metric.org.uk/the-mess-we-are-in [2] The reality is that you're only asking students to memorize one more number since 10 is "add a zero and multiply by one" and 11 is an obvious pattern with single digits.

switch007 11y ago

It's moved from "should" to must. "Every child in England will be expected to know their times tables before leaving primary school from next year. Pupils will be tested against the clock on their tables up to 12x12 in new computer-based exams that the Department of Education (DfE) said were part of the government’s “war on innumeracy and illiteracy”." http://www.theguardian.com/education/2016/jan/03/pupils-face-new-tests-to-ensure-they-know-times-tables-by-age-11 http://www.theguardian.com/education/2016/jan/03/pupils-face... Gove is no longer the Education Secretary...he is now the Justice Secretary. God help us.

TheOtherHobbes 11y ago
I like the idea of declaring war on innumeracy. That totally works as a metaphor for improved teaching. I'm not sure people need to know the 12 x 12 table, so much as knowing how percentages and compound interest work. How long will it take someone to pay off a credit card debt if the APR is 39.9% and they can't afford to clear the balance every month? If they don't understand why the answer is "Probably forever unless they get lucky with an inheritance" they shouldn't be using a credit card.
Wildgoose 11y ago
Gove did a fantastic job as Education Secretary and that is why he is now hated by a teaching profession being forced to do their job. I am not and never have been a Tory, but he is the one Tory minister I can respect. (The rest are dreadful).

legulere 11y ago

In Germany you usually learn up to 10×10 in school and there are usually 10 eggs in a box (4, 6 and 25 are also common sizes though and there's also a big set that's 20×20 which you don't usually learn though) > And when you want to raise standards in math, then who could argue with that? The problem is that you can only spend so and so much time with children in class. Learning 12×12 is 80% more combinations to learn compared to 9×9. That's time you could teach them something else that might be more useful.

kijin 11y ago
Actually, it's only ~32% more combinations to learn, because nobody really needs to memorize the 10's and 11's (except 11 x 11). You memorize all the combinations up to 9 x 9, compute the 10's and 11's on the fly, and memorize the 12's. Edit: Corrected the number. This is how you get 32%: The 9 x 9 multiplication table contains 8 x 8 = 64 combinations to memorize. The 1's don't count. The 12 x 12 multiplication table contains an additional 20 combinations to memorize, namely: - 10 combinations from 12 x 2 to 12 x 12, not counting 12 x 10 - 9 combinations from 2 x 12 to 11 x 12, not counting 10 x 12 - 11 x 11 = 121 So the total is 84 combinations, which is ~32% more than 64.

jheriko 11y ago

politics is a component too... i wouldn't go looking for purely intelligent decision making from a government selected by popularity contest. the perception that school is getting easier, a criticism of the government that is politically undesirable, can be addressed by a 'return to traditional methods' in superficial ways... even if there is not really a concrete problem to address. politics isn't about doing or achieving anything in terms of education or quality of life for others - its about making other people happy.

justinhj 11y ago

Whenever my son has a new math teacher, I taught him to ask if he can change the base when reciting the times tables. For example the 12 times table in base 12 is 10, 20, 30, 40 and so on. It usually gets a laugh or a tut before he has to do it the right way.

cjensen 11y ago

I grew up in the Bay Area in California. We only memorized up to 9x9. Though 12 is used often in measurements, I can't say as that I've missed having rote knowledge of the multiples of 12.

Namrog84 11y ago
Would you say then that you don't have any x10 memorized and just consider it trivially easy to compute any number?

Bulkington 11y ago

We're overthinking. Think, instead, as a school marm with real chalk and a switch in hand, or even when kids used slate tablets: 10x is easy. 11x is easy up to 10x. 12x requires a little work and pulls all the multiples together, but 13x just screws with everything we've learned so far. So, stop at 12. Or is a calculator in hand the default these days? As one who was proud to finally get to 12 x 12 = 144, I say make 'em suffer.

potatote 11y ago

Not sure if someone already has pointed this out, but one small caveat I'd add is that learning the table of 11 is not that difficult. I think the extra effort required to learn 11 and 12 tables are, therefore, a bit overestimated.

amelius 11y ago

Let's just switch to binary, and let them learn the 1-times table.

contravariant 11y ago

As someone who never learned the 12 times table, and only uses metric, I'm not sure if it would have been better if I had learned the 12 times table, but for some reason I have actually memorized most multiples of 12 (without any conscious effort). They do seem to turn up a lot for various reasons.

aidenn0 11y ago

I always assumed it was because we'd forget the largest couple of rows we memorized, so 20 years later we could still do up to 10

yason 11y ago

Memorisation is caching. If you need 11-times or 12-times tables, you will calculate them enough many times that your cache will become "hot" and you remember them out of the box. If you don't, then you don't have enough use for those multipliers.

dahart 11y ago

The reasons given for learning multiplication tables all focus on multiplication, but one of the best reasons is for division. Doing division on paper or in your head is made so much easier if you have your single digit multiplication table memorized. But why 12 instead of 10? I think: - There are loads of common everyday uses that overflow 10 by just a little. - Clocks - Eggs - 11 is almost as trivial as 10, it would be a shame to stop at 10 when 11 is free. - 12 is really super easy, and combined with the above reasons has a good case. - 13s are significantly harder to remember and work with than 11 or 12, nothing comes in 13s, its the perfect place to stop. :) (edit: course, as soon as I wrote that I remembered the baker's dozen, which was super common to hear when I was a kid, not so much anymore for me. But I think the point still stands, because with a baker's dozen you only pay for 12, you never had to multiply by 13.) * edited for formatting

djrogers 11y ago

I can think of about a dozen reasons why this is important...

oska 11y ago

> Download this post as a Computable Document Format (CDF) file. Hmm? Ah. > CDF files can be read using a proprietary CDF Player with a restrictive license, which can be downloaded free of charge from Wolfram Research. [1] Fuhgeddaboudit. [1] https://en.wikipedia.org/wiki/Computable_Document_Format https://en.wikipedia.org/wiki/Computable_Document_Format

protomyth 11y ago

12 x 12 is 144 and if we're not worried that our children will not have the mathematical skills to count and divide a gross of bottle rockets, then this is truly not America. Plus inches in feet. On a more serious note, the world does not end with 10.

greggarious 11y ago

Can I please point out this is the point where many people get turned off from math, never to return? I HATED memorization. I was able to do the math in my head up through 10, but 11x11, 11x12 and the rest of the 12s eluded me, and I just really, really was not able to sit still and just do flashcards. I ended up only chancing back on math through programming and a course in logic in undergrad. Everything else was mostly self taught as needed.

tempestn 11y ago

Here's a good example I just came across of why it's useful to be able to do approximate multiplication in your head (perhaps even while intoxicated): http://news.nationalpost.com/full-comment/robyn-urback-sorry-about-your-1000-uber-bill-but-the-internet-isnt-idiot-proof http://news.nationalpost.com/full-comment/robyn-urback-sorry...

stewdio 11y ago

Jon McLoone really seems to be approaching this topic from a place of frustration—rather than from an overwhelming sense of beauty. I think both mathies and visual designers can agree that 12 is a very beautiful number. In fact the ONLY single thing that makes sense about the imperial system is the division of feet into 12 inches, something mirrored by picas and points in typography: https://en.wikipedia.org/wiki/Pica_(typography) https://en.wikipedia.org/wiki/Pica_(typography) This beautiful number is evenly divisible by 1, 2, 3, 4, 6, and itself—a very high percentage (50%!) of the positive integers leading up to and including itself. (To be sure 60 is also a beautiful number but its ratio of factors to non-factors is of course lesser.) I can see you gearing up to argue that 6 is more beautiful, and I get that. I really do. But the ability to evenly divide into quarters is just so ... pleasant :) Does the ability to divide evenly in several ways serve as a soapbox from which to argue that it is mathematically important to memorize the first 12 numbers of which 12 is a factor? I suppose not. But could we simply do 12 the honor of learning these two extra columns of the multiplication table because it is beautiful? (And really you’re just learning ONE extra column because 11 is such a simpleton. Silly 11.) Of course dividing things into 12 units can SOUND pleasant as well. Western music is founded on dividing an octave into 12 units—though the exact method of division can take many forms. Here’s just a sample: https://github.com/stewdio/beep.js/blob/master/beep/Beep.Note.js#L268 https://github.com/stewdio/beep.js/blob/master/beep/Beep.Not... So I love 12. Join me in loving 12. Together we can praise 12. Full disclosure, I have been known to previously adore the number 12. On December 12, 2012 I proposed this “Cleaner Grid of Time” beginning with the months in a year: http://stewd.io/b/2012/12/12/twelve http://stewd.io/b/2012/12/12/twelve