Factoring polynomials is also an excellent example of the sort of thing students might as well have a computer do for them; conceptually, there is nothing to it (find the polynomials which, multiplied together, yield this), but as a computational task, well, it’s pointlessly tricky and heuristic for humans, and also basically never comes up outside of math class in almost anyone’s life. It’s a contrived skill to worry about. Those for whom it does come up… often use computers to do factoring (of integers, even!).
Can you expand upon this? I’m interested in what was done and how ASA was used, so I can explain it to my son when he asks “What use is this?”
I don’t know why and I don’t care. My kids have to make good grades in math right now – historical inertia or not. These issues are staring my family in the face right now … this isn’t hypothetical for us.
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An elementary school math task kids can’t do without rote learning of multiplication tables: Greatest Common Factor problems. Another: Least Common Denominator problems. Yet another: multiplying fractions. All of that comes in well before formal algebra classes.
My children started getting problems like this in fourth grade, and needed to be able to solve them quickly without a calculator:
687 x 43 =
1,293 x 87 =
5,216 x 315 =
Absolutely no way to solve those quickly without knowing the multiplication tables by heart.
Calculators are great, and using them will accelerate any math based tasks you may have.
But, you do not always have a calculator in your hand. You don’t always have it as convenient to put down what you are doing to pull out a calculator to do simple arithmetic.
I will use as an example, a story from a couple months ago. A friend of mine in landscaping had cut down an enormous tree, and wanted to take it over to the sawmill. On he way over, he got waved into a weighing station, and they found that he was 10,000 lbs. over weight, then pointed out that the fine was $0.10 a pound. Most of us grimaced, ouch, right?
About 30 seconds later, one person piped up, looking up from the calculator display on his phone, and said, “Wow, that’s like a thousand dollars”.
We use basic arithmetic every day, and shouldn’t need to get out a calculator for it every time we need to manipulate numbers, especially if precises values are not important. When I go grocery shopping, I total up my cart as I am putting stuff in, and my mental total is well within 10% of the register’s total, usually closer. If I had to use a calculator, it may be more accurate, but it would be far more work, so, I probably just wouldn’t do it at all.
Sometimes I go on long car trips. I see that the exit that I want is exit 97, and I am currently at mile marker 143. Getting out a calculator to figure out how many miles it to my destination exit is inconvenient while I am driving, so being able to do it in my head is easier. (Note: My method for that particular subtraction problem is to count 97 up to 100 [3], add that to 143 [146], remove the 1 in the hundreds place((subtract 100)), and get 46.)
So yeah, making sure that people can do simple math in their head is actually teaching a very important life skill, one that if is lacking will cause frustration and inefficiency at best, and can cause real financial problems for those who don’t grasp how numbers move, and especially how they add up.
Those other functions, the roots and sines and logs and such, those are not used on nearly as everyday basis. If you are in a situation where you actually need to know the log of x, for instance, you are most likely not in a situation where a calculator is inconvenient to use.
Actually, one of my pet peeves about STEM education is that mirrors are actually intuitive, until society beats that correct understanding out of kids. Your typical five-year-old has a better understanding of mirrors than your typical college freshman.
Both averaging and GCF (and by extension, LCD) are done mostly using subtraction, not multiplication or division (at least, if you do them sensibly). If you think that you know of a method for finding GCF that involves knowing your times table, I challenge you to find the GCF of 104813 and 185741 using your method.
If you knew your times tables up to 7 digits, that would be easy. 
Which is why knowing your times tables up to 144 makes knowing the gcf of 48 and 54 trivial, but doesn’t do you any good for anything above what you have memorized and can repeat by rote.
GCF and LCD with small numbers was done using multiplication-table knowledge when I was a kid, and still seems to be done the same way with my kids work. Not concerned with finding factors for six-digit numbers … that’s not the challenge in front of my kids.
Finding the GCF of, say, 12 and 54: how is that done readily without knowing that both numbers appear on the 6 times table? The kid is expected to do the following, and quickly:
- Write out [1, 2, 3, 4, 6, 12] for 12 and [1, 2, 3, 6, 9, 18, 27, 54] for 54.
- Pick out the largest factor the two have in common – in this case, 6.
54 = 124 + 6
12 = 62
so the gcd(54,12) = gcd(12,6) = 6.
Hard to think of a much speedier algorithm than this classical one. You could start from 54 and repeatedly subtract 12, and so on, but that would involve more separate “steps”.
You would definitely not write down all the factors of the two numbers for this problem; what for? That is an important number skill, just not for this problem.
Right, times tables make doing it the hard way slightly less hard. The easy way is still easier:
Add up 12s until you get close to 54. 12+12+12+12 = 48, that’s pretty close. So now I need to find the GCF of 48 and 54. So I subtract them: 54-48 = 6. So now I need to find the GCF of 48 and 6. Add up 6s until you get close to 48, and whoa, I can get to exactly 48 that way! So 6 is the GCF.
I can speed that up slightly by noticing right off the bat that 48 is a multiple of 6, but I don’t even need to know what 48 ÷ 6 is. And even without that speed-up, it’s still a lot quicker and more reliable than listing out all of the factors of both (and are you sure you didn’t miss one?).
EDIT: And I see that I actually made it even more complicated than I needed to, because I forgot that we started with 12, not 48. So there isn’t even any step where the times table helps at all.
As for factoring numbers, that reminds me of the puzzle someone posted here: find all solutions of sqrt(a) + sqrt(a+b) = sqrt(a+21), where a and b are positive integers. Not appropriate for really young kids since you need to know some basic algebra, but after some manipulation it all comes down to listing all the factors of (IIRC) 441.
Recall of the parts in blue still essentially requires rote learning of tables.
The only method of finding GCFs I ever learned requires it. And whenever the teacher requires that all factors be written out … that too.
EDIT: the trouble with using repeated addition or subtraction is that it’s too slow.
Same here.
I’m sorry, I’m not getting the multiple reference to mirrors.
Good to know I’m not the only one,
Factoring is an NP problem (probably), so yeah, there is no “fast” way to consistently do it.
The shortcuts you get from “knowing” the answer is great, for when the answer is one that you know.
But if that’s the only way that you factor, and you don’t know how to do it by addition and subtraction methods, then you don’t know how to determine the factors of any number you have not memorized the factors for.
Try a simpler one, just 3 digits. What’s the GCF of 323 and 391?
Your memorized multiplication tables will do you no good here, and if you do not know the proper method, then your only method is going to be guessing and checking. (Which to be fair, would not be all that difficult on this problem.)
No worries – the fourth- and fifth-grade math tests never give numbers even as high as your examples. The worst you might see is a layup (for adults) like “Find the GCF for 10 and 120” or something along those lines.
Much more typical is having to find the GCF of two two-digit numbers – commonly, one of the two will be less than 20.
Now that I’m thinking about it … these types of GCF questions might be contrived to specifically test for knowledge of multiplication tables. That wouldn’t be so bad if the groundwork were laid out in earlier grades and memorization of multiplication tables were part of the curriculum. But it’s not 
Euclid’s algorithm, which is a repeated subtraction method for finding the GCD is one of the oldest algorithms in use and is efficient.
It was new math in 300 BC
That’s great. My cohort was never taught that nor tested on it. Neither have my own children.
“Repeated subtraction” and “efficient” don’t seem to go together, though … unless the student has a lot of small-number addition/subtraction facts memorized. Working on fingers or drawing/crossing out dots on paper won’t be quick enough. There’s no escaping some amount of rote learning, it seems to me.