Which if I was ever told it in school I’d forgotten! That’s damn clever- I was going to try every prime up to the square root of 323 one at a time.
Euclid’s algorithm computes the gcd of two numbers (without needing to factor them!) and is simple to perform. If you want to factor integers into their prime factors, that is a lot more work.
Eta so, never factor two integers to compute their greatest common factor, and never ever teach children to do it! Do teach them to factor small integers in other contexts.
Yeah, but you are only teaching students to take tests at that point, not to actually understand the math and concepts that they are using.
Like I said, it’s an NP problem, which means that there is no “efficient” way to factor the numbers.
You think it is efficient to use the “having already memorized it” method, and it may be faster, but it will only work if it is one that you have memorized and not forgotten.
I go 391-323=68
68 is divisible by 4, and gives me 17.
There could have been more steps in there, if I wanted to break it down to the teaching level, and sure, knowing that 68 was divisible by 4 was useful, but it did not come from times tables, I could have used addition instead to build that.
And to be clear one should be taught to read off the greatest common divisor and least common multiple from the prime factorization, but not to start factoring 6-digit numbers for that express purpose.
What does “NP problem” mean?
You could have used addition, but it would have been too slow, IMHO.
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Maybe math instruction locally is “doing it wrong” and they should be teaching other things altogether and not “teaching kids to take tests”, but that doesn’t help students currently in the local school system. Hypotheticals don’t matter on the ground level of all this.
Oversimplification: An NP problem is one where it’s not necessarily easy to find a solution, but it is easy to check a solution once it’s found. For example, “Factor 468287” is a reasonably hard problem; but if I give you the answer (569*823) it’s relatively easy to check that it’s correct.
The Simple English Wikipedia article might be a good place to start for a somewhat more detailed explanation.
Sorry, it means non-polynomial.
It is the difficulty in solving problems. Multiplying is polynomial. If you increase the size of the numbers, the difficulty in solving it does not go up by very much.
Factoring is (probably) non-polynomial, as the numbers in the problem get bigger, the difficulty in solving the problem grows faster than linearly.
This is the basis for encryption. It is actually pretty easy to multiply 2 huge numbers, even 100s of digits, but factoring those back out, that’s a bit harder.
True, I was thinking that something didn’t look right there, but was too lazy to google and brush up.
People seem to be oddly misinterpreting what I’m saying, or the context towards which I’m saying it, or something.
In expressing that I think schools should have differently focused math curricula, I am of course ALSO expressing that I think they should grade students on the differently focused tasks of the differently focused math curricula, not continue grading students on the arbitrary manual arithmetic drills of the current curricula. And in expressing that I think things should be done differently, I am of course not denying that things currently are not done differently (tautologically)… I am expressing a wish for a counterfactual present or altered future world.
If your students have to make good grades in math right now by displaying high performance at otherwise pointless tasks, and are not currently taught in ways which ensure them good grades by ensuring their high performance at these otherwise pointless tasks, that’s unfortunate, but also, all the more reason society ought consider doing things differently in the future. Some aspects of current life suck for arbitrary capricious reasons, and sometimes I note that and note how I wish things were, and could be, better.
The way we were taught showed us how to solve specific problems in a linear fashion. Many of us became overwhelmed when we were later presented with abstract concepts, and this was simply accepted as some kids "not being mathematically minded.
Elementary school kids today are being prepared for higher maths at the same time they are learning addition and subtraction. They are taught to estimate, and to solve by units (ones, tens, hundreds . . .) in order to build the concepts and mental pathways that lead to an understanding of the relationships of numbers.
They are being taught math in a way that builds a level of comprehension which only geniuses used to develop, because they had to get there on their own.
Don’t be afraid to let your kids surpass your understanding. But if it really bothers you, you can work the mathematics courses at khanacademy.com for free. I recommend starting at kindergarten and working your way up, even though the early ones will be insanely easy for you. It’s a different way of thinking about numbers, and the easing into it helps.
I did it in order to be able to help Celtling with her homework, and suddenly the problems I had with binary equations started to come clear.
Every time someone tells a story like this, it only illustrates that people can approach arithmetic mindlessly. How is memorizing a multiplication table not equally mindless? A person could just as well have taken 30 seconds to do that multiplication out with pen and paper following some grade school algorithm. Developing a sense of what is and isn’t conceptually obvious (or obviously incorrect, as in the other genre of anecdote like this) is just as separately necessary with manual arithmetic algorithm running as with having the calculator do it for you; developing a sense of what is and isn’t conceptually obvious (or obviously incorrect) is just as possible with the calculator as with manual arithmetic.
What’s more, this 30 seconds lost no one anything. The story is told as though this slight delay was of great importance, but it wasn’t. (They got the right answer, and you’re still criticizing them?)
If this same story came up with the person using a calculator to take a square root, would it have proven children need to be drilled in manual square root digit extraction, and that something has gone terribly wrong in a society which does not do so?
Or what if it had been 14,517 pounds over weight at $0.13 a pound and someone wished an exact answer? The numbers here are suspiciously round. The rough “number sense” of rounded numbers could be developed either way, and for the exact answer… a calculator would be convenient.
I’m not against “number sense”; I’m saying calculators can be used to develop number sense equally (indeed, ultimately, more efficiently) as well as all these historical pen-and-paper drills. Children absolutely should be taught what multiplication is, and shown how they COULD do it by hand. Then, instead of worrying about drilling them into highly efficient fast error-less calculating machines, running mindlessly through steps of some algorithm (over whose details different camps bicker and bicker), we could just…let them use the pre-existing highly efficient fast error-less calculating machines to run mindlessly through steps of some algorithm.
Within 10%? If you know your arithmetic perfectly, shouldn’t you get it perfectly? Ah, but you’re not worried about precise values, and precise additions. You’re just using rough number sense. Which is a separate matter from being drilled on carrying out additions by manually running through an addition algorithm. This kind of number sense can be developed in various ways even using a calculator (or shall we get rid of the register’s calculator too?).
This kind of number sense is different from “Here are some mechanical instructions you can run through mindlessly, executing a computer program by hand. Now go do 50 of these, quickly, without error” drilling, which makes children rightly hate “math” as a tedious chore. That’s what my point is, or is in the vicinity of.
Actually by 30 seconds later, the conversation had moved on, and his interruption was a bit of a non-sequitur. And, it’s not a matter of mindlessly remembering the times table. I have in fact criticized the memorizing of the times table. It’s a matter of not recognizing that this was an easy problem that just needed a decimal place shift. This is because he had no “number sense”, and that was because he relies on a calculator to do easy stuff for him.
Square roots are not used in life as often as adding is.
If you need to pull out a calculator every time you need a square root, you are going to pulling out a calculator may be a few times a year.
If you are going to pull out a calculator every time you need to add or subtract two numbers, then you are going to spend quite a bit of time with a calculator in your hand.
The numbers were round because those were the numbers that he gave us. The 0.10 per pound is definitely correct, but I think the overage was like 10,something, but was related in the story as “about 10,000 pounds over”. As far as your separate example there, well yeah, that’s a bit harder, and takes me a few seconds to come up with a reasonable estimate, and slightly longer to do in my head than to bring up a cal app if I need it precise.
I really don’t think so. If you type two numbers into a calculator, and get a third, what have you learned? You have not learned anything about “number sense” or anything like that. You may at some time start remebering some of your more common calculations, but doing it by hand in a number of different methods does give you a better idea of what exactly it is you are doing.
No, I’m not using rough number sense, I’m using rough addition. Something is 2.83, so I call it 3, something else is 5.31, so I call it 5, a third thing is 8.45, so I call it 9. I keep track of the significant digit, and loose track of the cents, rounding up when I’ve accumulated a few of them.
The register’s calculator needs to be precise, not an estimate, otherwise they are going to be overcharging customers (not cool) or running at a loss.
One of the main reason for changing the way that math is taught is that it was rote and mindless memorization and repetition. The entire point of these “new” methods is to teach a better understanding of math, how the numbers work, and avoid the rote and repetition that you correctly criticize.
As to the overall argument you are making here, you could make the exact same argument to explain why children shouldn’t need to learn to read, and that they can listen to books on tape instead. And even more so in the future, when Siri will be able to take dictation and read anything they need to have read to them.
Horrible misuse of terminology.
A problem is in NP if an answer to the problem can be checked in “reasonable” time. “Reasonable” here means the run time is polynomial in the length* of the input. Another way of thinking about it is if you double the size of the problem, the running time goes up a most a constant factor. E.g., doubles, quadruples, etc.
Note that factoring is trivially in NP. It is not “maybe” an problem in NP.
Given the alleged factors of a number you just multiply them to see if you get the original number.
There is a special subset of problems in NP called NP-complete which are all “equivalently hard” in a formal sense. An example of this is trying to pack stuff of different weights into a given number of trucks with certain max. weight capacities. Surprisingly hard. But, again, trivially in NP because if someone shows you which package goes in what truck you can easily check that they all fit.
No reasonable person working in the field believes factoring is NP-Complete. There’s a lot of known facts that strongly imply it isn’t. E.g., prime testing is now known to be polynomial.
Now, as to GCD (not GCF), there’s Euclid’s algorithm. Which as you can tell from the name is a few thousand years old. That’s what I was taught, what should be taught, and what should be used for mundane purposes.
- For numbers this means the number of bits, not the value of the number itself.
In theory, they are. (Also, this isn’t new, they tried to prepare us for higher math when I was in elementary school in the 80s when they tried to teach set theory to elementary school students; there were lots of worksheets with dots.) In practice, not always so much.
My experience is with tutoring kids who end up not just not building the conceptual framework, they haven’t learned the linear methods either. They definitely do not see the connections between the skills, methods, and concepts. (If I had a dollar for every time I heard “____ was last year/month/week. I don’t remember how to do it. Besides, we’re learning this now,” I could have a nice dinner out. With drinks. And tip.)
The problem & method in the OP are fine; I just see kids who don’t quite know how to do that and don’t quite know how to use the borrowing method, and couldn’t explain either one (nor any other method for that matter)
GCD and GCF both seem fine to me. Greatest Common Divisor or Greatest Common Factor; well, divisor and factor mean the same thing in this context, so, whatever.
What does irk me slightly is LCD. What’s the D here? I’d want to say LCM, Least Common Multiple. But I suppose people are thinking of “Least Common Denominator” or some such thing, imputing some fractional context (though of course one might have reasons for thinking about LCMs that have nothing to do with fractions).
My no-cite opinion, for the nickel it’s worth, is that such comprehension is not truly teachable. To me, trying to teach that is like trying to teach all people to run world-class 100-meter sprint times.
I can’t imagine ever being convinced that the rote memorization of small-number addition/subtraction facts and multiplication tables is flatly unnecessary for effective elementary-level mathematical instruction.
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amarinth – well said.
Why not make kids rotely memorize their exponentiation tables, too? If it’s so bad to think of 7 * 3 as 7 + 7 + 7, why is it ok to think of 7^3 as 7 * 7 * 7?
The whole point of NP problems is that the computation is polynomial, but it could be that only verifying the solution can be done deterministically in polynomial time and finding it is hard. This is the most important unsolved problem in computer science.
Factorizing integers is not presently known to be P. It’s obviously NP.