Is there any point to the 12 times table?
blog.wolfram.com
blog.wolfram.com
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/res5... [2] http://www.computerconservationsociety.org/resurrection/imag...
Thanks for additional info on the machinery involved.
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.
And if there's some domain-specific application, like in carpentry, then let those people use their own units.
* Scientists and engineers culturally select tasteful units of their choice to represent time, like 1300 milliseconds or 134 hours... and everything else as well: $1.34 billion or $1.34 dollars, not $1 billion and 340 million or $1 dollar and 34 cents.
* When the number representation get too big, people just do X * 10^n where n is a tasteful choice.
* In business, days and hours are distinct concepts and should not be mixed together or confused. A person who worked for 5 days did not work 120 hours.
* Mathematicians could care less.
* Many culturally common units of time are bigger than 12 * 12.
* Calculations that are big are working-memory bound. This is the biggest point. Working memory is probably the strongest factor to large and fast calculation of any kind.
* The bigger the calculation, the more people tend to 10^n multiplicands with the distributive property, rather than 12^n, since most people work in base 10. Some engineers use 2^n or 3^n (I offer that 3^n is efficient without further explanation).
I propose the hypothesis that children who memorized up to 10 * 10 are no more likely to join or do well in the STEM fields than those who memorized up to 12 * 12.
I argue that calculations are working-memory bound, and that common uses of time go beyond 12 x 12.
For time, we often go by units of 5, 10, 15, 30, 45, 60, of these only 60 can find 12 as a divisor. You don't get to claim usefulness for all time. For the rest of these units, it's working memory doing arbitrary integer computation for us.
If we're talking about months in a year -- what's the use case for fast mental calculation? Is this the little sphere where we say 12 * n is useful?
We're talking about a policy for education for all children here. Your use cases sound really limited, if it at all impacts later cognitive performance outcomes.
It's working memory that's going to be the differentiating line between "fast mental math" and "busting out calculator" or "punching in numbers into SAS or SPSS". It's difficult to believe that Taiwanese children are going to have trouble versus American children in time based operations, if that at all continues to be a practice under the common core.
I can't imagine asking some Taiwanese business man, "Quick, what's 6 * 12 in a business context", and he goes, "Let me quickly grab a calculator."
Can you do 15 * 14? I think you can. Why can we do it? Is it because we learned 12 * 12? No. It's working-memory bound. We're doing capricious calculations with distributive property via 10^n multiplicands.
How about 24 * 17? I think you can do that too. Why can we do it? Working memory. Not because we learned 12 * 12 when we were little.
In business? In the STEM fields?
how many times do you calculate "X per year" in your head? i do it multiple times per day, and not just at work.
Seldom enough that I 'calculate it in my head' (as your comment suggests you do too) rather than memorise the 12x table.
But then again I grew up in metric country that only bothered with drilling kids up to the 10x table at school. Never had to worry about feet to inches conversions very often either.
Meanwhile, reviewing my post it reminded me of my US History teacher in middle school telling me that the Conquistadors in the Americas were after the 5 Gs: Gold, Gold, Gold, God, and Glory. I listed five things: Time, Time, Time, Degrees and Factors. I'd be thrilled if someone could say the same thing but with all of the words starting with the same letter.
Thanks to the seconds and minutes counting in 60s and hours and months (somewhat) counting in 12s knowing base 2, 3, 4, 6 and 12 is exactly what someone needs to work up and down.
Many baking items are also sold in units of 12 (E.G. eggs, 144 being that packet size that businesses buy in).
Finally there's the learning aspect. Learning the 12s is a lot like extending the factor tables beneath it in a way that makes sense to analog brains. It's actually not /that/ much more costly. 13 however is a whole new prime, which isn't that useful in daily life; all of the larger numbers are big enough to be handled as estimates or 'long math' precision numbers.
PS: If I could magically force all humans to learn, know, and use a new base (and that's the only option I had) I'd probably pick Octal. Hex (for the above reasons) is useful, and great for computer addressing, but less great for actual human-analog computation.
We’ve solved that in Europe, too – they’re sold in packages of 10, and businesses buy in packages of 100.
There are 12+ loaves in a baker's dozen (the exception that proves the rule)
The point on #5 is critical. Before we had a solid currency system or notion of debt, we commonly had to divide things up from a group. 12 gives a lot of flexibility in that regard. In fact, selling 12 units of something became so common that certain industries added an extra unit just to be sure that the number 12 was given (baker's dozen).
Regardless, your observations seem to come down to "12 is highly composite and we use highly composite numbers for time"—valid, but interesting?
Meanwhile, your response got me to look up whether there's a metric equivalent to degrees or radians. Apparently that exists and it's a measure called Gradians. There are 400 Gradians in a circle, and 400 is not divisible by 12. I guess that idea went the way of the French Republican Calendar.
And apparently, in what I have to assume is a related use:
> In surveying, the gradian is the default unit of angles in many parts of the world.
( https://en.wikipedia.org/wiki/Gradian )
If you ever fooled around with your calculator in school, you probably noticed that it provided angle measurement conversion between degrees, radians, and gradians; the idea has certainly not gone the way of the French Republican Calendar.
Integer math.
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.
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.
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.
3 and 9 are also fast though you get either a remainder or a repeating number. So, really the advantage is 6 vs 5.
say in hex you have a number 0x56aF900000, you know that your divisors can have no more than 5 non-zero (hex) digits. This is not so for decimal numbers; 32 * 125 = 4000
With a base 12 version of everything you would have 1/12 of a day being 2 hours, 1/144 of a day is 10 minutes, and 1/1728 of a day is 50 seconds. These units would give us both easy divisibility and are close enough to existing time units to make sense.
This change would mean we have to memorize a 12 times table rather than a 10 times table. But in base 10 a 10 times table has easy patterns for 1, 2, 5, 9 and 10. A 12 times table has repeating patterns for 1, 2, 3, 4, 6, 8, 9, 11 and 12. The result is that a 12 times table in base 12 is actually less work to memorize than a 10 times table in base 10.
In the long run this transition would be a clear win. But it isn't enough of one compared to the transition to ever make sense to initiate.
It is easier for the computer to adapt to us, rather than vice versa. And over time things get easier and easier for the computer, while staying the same for us. Which is why over time we wind up interacting with computers in ways that are more and more adapted to us rather than them.
We already have acceptably good algorithms to convert to/from different representations, and roundoff errors are generally acceptable. (If they weren't, we wouldn't default to using floating point for most things.) Where precision matters, we have reasonably efficient implementations of exact integer arithmetic and exact fraction arithmetic. The performance hit from using these is less than the hit from using a high level language like Python versus a low-level language like C.
The main challenge is getting programmers to use the right representation for the purpose that they are using it for. Which is mostly challenging exactly because floating point works so well in practice that it is easy not to learn what its limitations are.
In physics it is uncommon to have exact measurements for much of anything except integers. In physics, measurement error is usually a bigger issue than roundoff errors from floating point, so the non-existence of an exact floating point representation of the stated measurement is a non-issue.
In the real world, most cases that I'm aware of where we need exact base 10 calculations have to do with dealing with money in a single currency. You would be amazed how often square roots don't come up in this context!
Fractions show up as soon as you start doing currency conversions. (In fact the Euclidean algorithm for finding the GCD is believed to have been first developed by merchants for exactly this problem.) They are easy to add to a programming language, but you'd only use them in contexts (like finance) where square roots make no sense.
If you want to go beyond fractions to add square roots and so on, you can do algebraic numbers. This is a fairly specialized mathematical need, but Mathematica has done it perfectly well for decades.
For arbitrary real numbers, we're sort of stuck. We can come up with representations that can cover any real number that we can talk about. But we can't take 2 arbitrary representations of a real number and in finite time decide whether or not they are actually the same number. That said, Mathematica again solves this problem "well enough" for most practical purposes. See http://mathworld.wolfram.com/Ferguson-ForcadeAlgorithm.html for more.
This is a complex topic, with ongoing research across multiple fields of study.
That said, floating point is a "good enough" default for a surprisingly wide variety of situations.
In the long run, trinary gives you the best radix-economy, being closest to e: https://en.wikipedia.org/wiki/Radix_economy
In base-N, you can accurately write any fraction of (1/y) provided that y can be expressed using the same prime-factors found in N.
For example, 10 has the prime factors of 2 and 5, leading to the requirement that y=(2^a × 5^b) .
This means base-10 can accurately represent (1/80), because 80 = (2^4 × 5^1). In base-2, you're limited to stuff in the form (2^a).
Finally, the, uh, infinity-of-integers that matches (2^a) is always going to be "smaller" than the infinity which can be matched by (2^a × 5^b), since the latter "contains" the former when b=0.
In this fashion, primes p seem "better" as they increase, because for a given size limits on k and n, you get more 'numbers' per unit space on the real line. Composite numbers are 'even better', but there are a lot of pains in the ass with composite numbers, stemming from Z/nZ not being a field if n not prime.
Ultimately, though, your internal representation is binary, using anything not two produces other inefficiency. That's why IEEE 754 BCD is not base-10 but base-1000 (since 2^10 = 1024 which is the closest a power of two gets to a power of 10, for bit-wise representation efficiency). We don't have bi-quinary computers.
In fact, I think it exemplifies the upper bound for the number of digits you'd need for "related fractions". For example, we know "one eighth" is expressible in decimal as 0.125. Even if you pick a really big N value, N * (1/8) should never need more than three digits to the right of the decimal point.
But there are two approaches that we can use.
The first is to point out that in real life it is more common to encounter small numbers than big ones. So if you choose a probability distribution for what denominators you expect in practice, and sum up the values of all of the exactly represented numbers, you can get to a fixed probability of even division. And it won't be zero.
Unfortunately your choice of the relative probability of having a 2 in the denominator versus a 3 is extremely arbitrary. So while the approach makes sense, the numbers you get really will be made up. (Though no matter how you make them up, your general statement is guaranteed correct.)
The second approach is to just look at how the count of exactly representable numbers below N scales with both N and your set of primes. This has been studied. The exact counts are a mess, but if you have a set S of prime divisors of size k, then the count is (1+o(1)) * 1/k! * [product over p in S of (log(N)/log(p)].
The upshot is that for large N, more prime factors always wins. And for specific prime factors, you just look at the ratio of logs.
For instance with this approach we can say that log(3)/log(5) = 1.46497352071793... times as many denominators are exactly representable in base 12 as in base 10. (The log(2) factors cancel out.) Therefore base 12 is better.
Of course the second approach says that base 6 is as good as base 12, and it is smaller, so why not use it? Well, the reason why we keep on winding up with base 12 in practical situations is that divisibility by 4 comes up a lot. It isn't just exactly representable that matters, efficiency matters.
It's just that top to bottom very few people are willing to trade the loss of speed for the increase in accuracy (not that decimal is great either, you still run into plenty of repeating fractions)
1 2 3 4 5 6 7 8 9 a b
so 10 would be 12
I did not otherwise specify, so I represented my numbers in base 10 everywhere even though I was talking about base 12.
(With TeX formatting implied)
In fact we mostly indicate the base when there is a good possibility that it isn't 10, and any stray numbers fail to have a base represented, they should be assumed to be base 10.
(Mathematical notation mostly uses sensible defaults except for differential geometry.)
The only country to actually use the calendar was, of course, France for 16(!?) years until Napoleon repealed it during his imperial reign.
https://en.wikipedia.org/wiki/Metric_time
Anyone, feel free to correct me. I, literally, just read about this (fascinating) subject a couple nights ago. I haven't had time to independently follow up on the topic, but I plan to do so soon.
Amendum: A better link, https://en.wikipedia.org/wiki/Decimal_time#France
If anything, the way the French "pushed their influence" was by conquest: The first countries that adopted metric were forced to when they were conquered by Napoleon. Many of them reverted after Napoleon was beaten, but then gradually they started converting to it again,
Since you're mentioning Napoleon I assume you just got your periods mixed up. There were indeed many crazy suggestions for calendars after the French revolution...
Most famous of all is the Eighteenth Brumaire, when Napoleon overthrew revolutionary institution and became the de facto dictator: https://en.wikipedia.org/wiki/Coup_of_18_Brumaire
The date was made even more famous by one Karl Marx, who used it in the evocative title of his essay https://en.wikipedia.org/wiki/The_Eighteenth_Brumaire_of_Lou... (or maybe the essay got famous because the title was evocative, don't know).
Also quite famous was the https://en.wikipedia.org/wiki/Thermidorian_Reaction which saw Robespierre and the Jacobins overthrown.
For other slightly less known events, see https://en.wikipedia.org/wiki/Glossary_of_the_French_Revolut...
(Also, there's the old rule-of-thumb from Grace Murray Hopper that "pi seconds equals a nanocentury".)
https://en.wikipedia.org/wiki/Roman_calendar#Nundinal_cycle
"Because the term of office of elected Roman magistrates was defined in terms of a Roman calendar year, a Pontifex Maximus would have reason to lengthen a year in which he or his allies were in power, or shorten a year in which his political opponents held office. For example, Julius Caesar made the year of his third consulship in 46 BC 445 days long."
Of course that would mean I would have to memorize the entire multiplication table again: because 55 in duodec would be 65 in dec. And 5x18 would be 84.
Multiply by 36 (12*3) then divide by 10?
But it isn't the "move the decimal point" of other metric unit conversions.
I think so, by multiplying by 12 and then 3.
But I only have 10 fingers!
On a practical note, wouldn't that mean inventing 2 new symbols to represent 10 and 11 as single digits, otherwise I could see that getting very confusing.
You could't really use A and B, as then some people would be unable to find the correct seat on an airplane.
That's why I mentioned counting on the finger-joints :-)
> On a practical note, wouldn't that mean inventing 2 new symbols to represent 10 and 11
Already been done, over a hundred years ago, and they're even in Unicode already[1]!
10 is a rotated 2 (ᘔ)
11 is a rotated 3 (Ɛ)
"Finger-counting systems in use in many regions of Asia allow the counting to 12 by using a single hand."[0]
[0] https://en.wikipedia.org/wiki/Finger-counting
(Edit: silly me and trying to use Markdown)
I agree about the symbols, though.
You can count to 10 with 5 fingers with not much practice (look up Chisanbop); I think most kids could pick up counting to 12 on their finger joints pretty quickly.
I'd have to run the numbers. It would be interesting to consider different subsets. I also think adding in 15 would probably worth the extra number for real-world datasets.
Isn't that backwards? 7 is the hardest, so you spend more time on it, not less.
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.
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!
I suspect getting the 12x table added to the NZ maths curriculum was probably the main deliverable of the 2001 Knowledge Wave Conference.
Personally I strongly believe that this is so important that it should be enforced by rote memorisation very early on (as in my case).
For some reason far too many people see no problem with insisting children learn their alphabet to ensure literacy but suddenly lose interest when it is times tables in order to ensure numeracy.
Fortunately we have just had a (wrongly much derided) Education Secretary in Government called Michael Gove who has done an excellent job at demanding that these basics are covered and as a result is now widely (and wrongly) hated.
He has also insisted that school IT courses should involve teaching programming. Up to now, they have just been about to create a Microsoft Powerpoint presentation, or how to use Microsoft Word. Yes, seriously. That bad. They squandered the early lead the UK had with IT by dumbing down everything to such an extent that no bright kid would want anything to do with computers.
Now 11, that's a total mystery. Other than it being between 10 and 12, I see no reason to memorize 11s.
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.
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
1571427This 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.
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.
Though either way, you are quite right that the US has little influence on the UK in this respect.
There are perhaps about three good reasons to push metric units, and the vast majority of us don't find them compelling enough to care. It's truly a non-issue.
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.
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-2400x... [2] http://www.wickes.co.uk/Wickes-General-Purpose-OSB3-Board-18... [3] http://www.wickes.co.uk/Wickes-Studwork-%28CLS%29-38x144x240... [4] http://www.wickes.co.uk/Wickes-9006-Wallpaper-Embossed-White... [5] http://www.wickes.co.uk/Wickes-Skipton-Internal-Softwood-Doo... [6] http://www.wickes.co.uk/Blue-Circle-Extra-Rapid-Cement-25kg/...
https://en.wikipedia.org/wiki/Preferred_number#Buildings
Which is about a foot!
And the other 94% of the world population…?
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-ti...
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.
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.
It's easier to speak in terms of days and halves of days. Often this is accurate enough.
"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...
Gove is no longer the Education Secretary...he is now the Justice Secretary. God help us.
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.
I am not and never have been a Tory, but he is the one Tory minister I can respect. (The rest are dreadful).
They are always trying so hard to sell Mathematica in these blog posts, but this is the lowest I've seen.
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.
It sounds weird that someone opens an IDE and waits ten seconds looking at a splash screen to do a simple multiplication.
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
If you memorized all times tables up to 99, you could so long multiplication in base 100:
1234
4567
----
1234
4567
----
22
78
1234
4567
----
82678
1234
4567
----
82678
15
30
1234
4567
----
82678
55530
-------
5635678
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".
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.
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.
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
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.
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
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.
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.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
We worked behind the counter in a builder's merchant. She is no longer working for us.
I hope that person you worked with can at least work a calculator.
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.
Two more from the garden center:
I was looking for a pump for an ornamental pond, and the pumps were advertised by the amount of gallons of water they displaced per hour. I didn't know the conversion between ft3 and gallons and I didn't have my phone so I couldn't look it up. I told one of the attendants my pond was about 60 cube feet, and asked if she could help me look up how much gallons that was. She said; "I don't do numbers like that. The only numbers I know how to count is money wink". Weird. I tried to explain; all I need is the conversion factor. To no avail. I just ended up buying the biggest one. My neighbors complimented me on how soothing that burbling water is when they go to bed. Passive aggressive jerks :)
I wanted some dirt delivered for a raised flower bed. I needed about 30 cube feet. The voice on the phone told me they only deliver by the scoop (from that big shovel on their bulldozer). Fair enough, but I asked if he had more or less an idea how large such a scoop was, in some unit I could understand, since I didn't want to end up with too much dirt. He was pretty curt in his response; "We're not going to start do that, all those units and measurements, we only do scoops". Wait what, like, how do landscape architects do this? Are scoops a thing?
I asked people if it was unreasonable to expect more in each situation. Most of them were along the lines; "yes, you're being obnoxious, just get the damn scoops".
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.
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.
> 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.
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.
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.
Lots of commodities are sold in dozen multiples.
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
Really, if I could have come out of school with the 20x20 table memorized, I wouldn't mind now at all.
I think that's been dropped from most US elementary school curricula now, bit of a shame.
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!
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.
Plus inches in feet.
On a more serious note, the world does not end with 10.
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
[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.