On Sat, Oct 27, 2012 at 12:50 PM, Robert Hansen wrote: > > On Oct 26, 2012, at 11:11 PM, Paul Tanner wrote: >> >> Who says other than you? Name names. >> > > Just about everyone. You said yourself that you were ostracized because of > this idea. >
When I was talking about experiencing hostility to the idea of making things easier for students, I was not talking about elementary school kids, but about middle school and high school kids. In the thread
such that we simply perform "m over the bottom times the top" n times to quickly and easily directly obtain the numerator of the single fraction to be obtained from the adding-subtracting of any number of fractions.
I think that it is insane to be against people ever knowing these formulas, since knowledge of them could prove so useful in both arithmetic and algebra, since the patterns of are what is so important (see again the title of Keith Devlin's book published in a Scientific American series of science books, "Mathematics: The Science of Patterns").
But with respect to elementary school kids being exposed as soon as possible to using letters of the alphabet to represent numbers, it seems that "everyone" goes in the opposite direction: You seem to be against what is required of 4th graders in FL with respect to this use of letters of the alphabet.
> >> >> You think that they are able to grasp the idea of subtraction as the > inverse of addition if and only if it is not written using the letters >> of the alphabet? >> > > They don't understand what "inverse" means. >
Why not? I say this because some other countries treat their elementary school kids as able to handle these "big words" and "big concepts" of algebra (like "inverse" and the properties or laws like "commutative" and "distributive") earlier than you seem to think American kids are capable.
The idea of inverse operations is how Korean elementary school kids are taught subtraction and division - they actually introduce subtraction *as soon as* they introduce addition by teaching subtraction to be the operation that is the inverse of addition, and they actually introduce division *as soon as* they introduce multiplication by teaching division to be the operation that is the inverse of multiplication.
And we these "big words" and "big concepts" in Liping Ma's book being used by Chinese elementary school teachers when explaining things to their kids - they start this even in the 1st grade.
> > > Can't at least many of them understand that 10 - 3 = 7 is true when > and only when and 10 = 7 + 3 is true? > > > They don't understand what "true" means. They might agree with you, but they > don't understand it. >
Wow. They don't know what "true" and "false" means?
Hmmm. So you claim that elementary school kids are not capable of being taught that it's bad to lie? How old do Americans have to be before they are capable of being taught this?
But, if you insist, use the clause "10 - 3 = 7 when and only when and 10 = 7 + 3".
And to anticipate, again:
East Asian elementary kids are taught what you seem to think young people at this age incapable of being taught:
In 4th grade Chinese kids are even taught that dividing by a number is equivalent to multiplying by the reciprocal of that number, and they are taught this *as soon as* they are introduced to fractions in the 4th grade.
> >> >> Can't at least many of them understand that we can replace these > numbers in such examples with letters of the alphabet a,b,c to express > the idea being true for all numbers, that a - b = c is true when and >> only when a = c + b is true? >> > > They certainly don't understand what "true for all numbers" means.
Why do you think that they not capable of understanding that formulas always will give the right output, no matter what numbers they plug in? See the above again about East Asian kids of elementary school age being treated by their teachers as more capable then you seem to think young people of this age capable.
>> And if so, can't they all see the visual attribute of the written > expression of the definition of subtraction as the inverse of > addition, that a number or variable looks like it moves back and forth >> the equality symbol and the sign of the number changes as it does? >> > > No, they cannot understand this.
I said "see" and here did not mean "understand" - it's something that anyone can see. When comparing "a - b = c" and "a = c + b" by going back and forth between the two equalities, anyone can see that it looks like "b" is sliding back and forth over the equality symbol while the sign in front of it changes. Knowing that is looks like this is quite useful later beginning even in even pre-algebra when beginning to solve for variables.