Perhaps because they're taught factorization rather than Euclid's algorithm, gradeschoolers are now being forced to say gcf ("greatest common factor") instead of gcd. Or maybe just to make it sound easier. So I asked a kid: 12 contains how many factors of 3? And how many factors of 2? And how many factors of 1? "Tell your teacher 1 can't be a factor of anything. 2 and 3 have no greatest common factor." "I'll get in trouble." A lot of stuff we're taught is oversimplified. We have to unlearn it and relearn it later. But this is ridiculous. Is there some way we funsters can gang up against this idiocy? --rwg They're also taught that improper fractions are not in lowest terms.
In the UK, `factor' is synonymous with `divisor' and the term `prime factor' is used to refer to the things that appear in the unique factorisation into irreducibles. Also, I haven't seen `gcf' anywhere, but children learn `hcf' (`highest common factor') -- which is even worse since it's defined in terms of the total order on Z, rather than merely the multiplicative structure. And nowhere (before undergraduate level) is unique factorisation ever justified. Sincerely, Adam P. Goucher
Sent: Monday, May 25, 2015 at 1:12 PM From: "Bill Gosper" <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] gcF??
Perhaps because they're taught factorization rather than Euclid's algorithm, gradeschoolers are now being forced to say gcf ("greatest common factor") instead of gcd. Or maybe just to make it sound easier. So I asked a kid: 12 contains how many factors of 3? And how many factors of 2? And how many factors of 1? "Tell your teacher 1 can't be a factor of anything. 2 and 3 have no greatest common factor." "I'll get in trouble."
A lot of stuff we're taught is oversimplified. We have to unlearn it and relearn it later. But this is ridiculous. Is there some way we funsters can gang up against this idiocy? --rwg They're also taught that improper fractions are not in lowest terms. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Leaving aside specific forms mis-teaching, many of which are avoidable, there's the general issue of how much the process of un-learning may be inherent to the nature of math, the mind, and learning. Could one create a 100% honest pre-college curriculum that wouldn't drive 99% of the students crazy with all its caveats and footnotes and the frustrating asides that end with "... but you won't get to learn about that till you're older"? Better, I think, for curriculum designers to frankly acknowledge that oversimplification and unlearning are part of the way human beings operate in every field of endeavor. Think of Tiger Woods, spending many months unlearning his 1990s-style swing so he could painstakingly develop a new, even better swing. Could he have developed that championship swing in his teen years, by thinking things through more clearly? Probably not; it took a Tiger Woods 1.0 to design a re-training regimen that would create a Tiger Woods 2.0. Similarly, when a young mathematician learns the axiomatic approach to the real number system (possibly via several steps, starting from the natural numbers), it's a process of re-construction. A student comes to this process already "knowing" a lot about the real numbers (otherwise the whole enterprise would be unintelligible and unmotivated), but she has to put aside her prior knowledge and intuitions in evaluating a proof for correctness. Unlearning facts and definitions is easy compared to unlearning the whole uncritical approach that dominates pre-college mathematics. On the other hand, my wife says it drove her nuts that her pre-college math teachers "kept changing the rules"; she would probably have appreciated something more like 100% honesty. And I think we can all agree that some of the lies math teachers tell are actually more cognitively indigestible than the truth! Also: math teacher preparation programs should be designed so that teachers who oversimplify are *aware* that they're oversimplifying. Jim Propp On Monday, May 25, 2015, Bill Gosper <billgosper@gmail.com> wrote:
A lot of stuff we're taught is oversimplified. We have to unlearn it and relearn it later. But this is ridiculous. Is there some way we funsters can gang up against this idiocy? --rwg
JPropp>And I think we can all agree that some of the lies math teachers tell are actually more cognitively indigestible than the truth! Also: math teacher preparation programs should be designed so that teachers who oversimplify are *aware* that they're oversimplifying. <Jim Propp Or worse, lying, apparently because "factor" sounds easier than "divisor". "How could these widely used textbooks be wrong? Who am I to contradict them?" So I'm asking you authority figures to help me (perhaps privately) petition this teacher that s|he is perpetrating a bad usage that will need to be unlearned, and should switch to "divisor" with the overwhelming support of teachers and practitioners of more advanced mathematics. --rwg And that reduced improper fractions can be in "lowest terms". I'll bet that many teachers imagine that further reduction of the fraction is possible after converting a reduced improper fraction to a mixed number! Ah, only now do I realize their insistence on mixed numbers leaves them with a smaller fraction to reduce. But if they'd just reciprocate that fraction, and iterate, they'd have continued fractions and Euclid's algorithm! I just stopped hating mixed numbers. On Mon, May 25, 2015 at 5:12 AM, Bill Gosper <billgosper@gmail.com> wrote:
Perhaps because they're taught factorization rather than Euclid's algorithm, gradeschoolers are now being forced to say gcf ("greatest common factor") instead of gcd. Or maybe just to make it sound easier. So I asked a kid: 12 contains how many factors of 3? And how many factors of 2? And how many factors of 1? "Tell your teacher 1 can't be a factor of anything. 2 and 3 have no greatest common factor." "I'll get in trouble."
A lot of stuff we're taught is oversimplified. We have to unlearn it and relearn it later. But this is ridiculous. Is there some way we funsters can gang up against this idiocy? --rwg They're also taught that improper fractions are not in lowest terms.
Which reminds me that I much enjoyed RWG's "rectangles as fractions" page, before a browser crash lost me the link ... does anybody still have it handy? WFL On 5/25/15, Bill Gosper <billgosper@gmail.com> wrote:
JPropp>And I think we can all agree that some of the lies math teachers tell are actually more cognitively indigestible than the truth!
Also: math teacher preparation programs should be designed so that teachers who oversimplify are *aware* that they're oversimplifying. <Jim Propp
Or worse, lying, apparently because "factor" sounds easier than "divisor". "How could these widely used textbooks be wrong? Who am I to contradict them?"
So I'm asking you authority figures to help me (perhaps privately) petition this teacher that s|he is perpetrating a bad usage that will need to be unlearned, and should switch to "divisor" with the overwhelming support of teachers and practitioners of more advanced mathematics. --rwg And that reduced improper fractions can be in "lowest terms". I'll bet that many teachers imagine that further reduction of the fraction is possible after converting a reduced improper fraction to a mixed number! Ah, only now do I realize their insistence on mixed numbers leaves them with a smaller fraction to reduce. But if they'd just reciprocate that fraction, and iterate, they'd have continued fractions and Euclid's algorithm! I just stopped hating mixed numbers.
On Mon, May 25, 2015 at 5:12 AM, Bill Gosper <billgosper@gmail.com> wrote:
Perhaps because they're taught factorization rather than Euclid's algorithm, gradeschoolers are now being forced to say gcf ("greatest common factor") instead of gcd. Or maybe just to make it sound easier. So I asked a kid: 12 contains how many factors of 3? And how many factors of 2? And how many factors of 1? "Tell your teacher 1 can't be a factor of anything. 2 and 3 have no greatest common factor." "I'll get in trouble."
A lot of stuff we're taught is oversimplified. We have to unlearn it and relearn it later. But this is ridiculous. Is there some way we funsters can gang up against this idiocy? --rwg They're also taught that improper fractions are not in lowest terms.
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On 2015-05-25 11:39, Fred Lunnon wrote:
Which reminds me that I much enjoyed RWG's "rectangles as fractions" page, before a browser crash lost me the link ... does anybody still have it handy? WFL
http://www.tweedledum.com/rwg/rectarith12.pdf --rwg
On 5/25/15, Bill Gosper <billgosper@gmail.com> wrote:
JPropp>And I think we can all agree that some of the lies math teachers tell are actually more cognitively indigestible than the truth!
Also: math teacher preparation programs should be designed so that teachers who oversimplify are *aware* that they're oversimplifying. <Jim Propp
Or worse, lying, apparently because "factor" sounds easier than "divisor". "How could these widely used textbooks be wrong? Who am I to contradict them?"
So I'm asking you authority figures to help me (perhaps privately) petition this teacher that s|he is perpetrating a bad usage that will need to be unlearned, and should switch to "divisor" with the overwhelming support of teachers and practitioners of more advanced mathematics. --rwg And that reduced improper fractions can be in "lowest terms". I'll bet that many teachers imagine that further reduction of the fraction is possible after converting a reduced improper fraction to a mixed number! Ah, only now do I realize their insistence on mixed numbers leaves them with a smaller fraction to reduce. But if they'd just reciprocate that fraction, and iterate, they'd have continued fractions and Euclid's algorithm! I just stopped hating mixed numbers.
On Mon, May 25, 2015 at 5:12 AM, Bill Gosper <billgosper@gmail.com> wrote:
Perhaps because they're taught factorization rather than Euclid's algorithm, gradeschoolers are now being forced to say gcf ("greatest common factor") instead of gcd. Or maybe just to make it sound easier. So I asked a kid: 12 contains how many factors of 3? And how many factors of 2? And how many factors of 1? "Tell your teacher 1 can't be a factor of anything. 2 and 3 have no greatest common factor." "I'll get in trouble."
A lot of stuff we're taught is oversimplified. We have to unlearn it and relearn it later. But this is ridiculous. Is there some way we funsters can gang up against this idiocy? --rwg They're also taught that improper fractions are not in lowest terms.
participants (5)
-
Adam P. Goucher -
Bill Gosper -
Fred Lunnon -
James Propp -
rwg