[Winstonm]

gwnn, on 2011-February-10, 06:10, said:

I meant to say that it's way too advanced for winstonm

No, I knew the order of numbers was defined.

[helene_t]

Let's try to prove that 2+2 = 4

2+2 = 2+(1+1) because 2 is defined as 1+1
= (2+1)+1 we have to assume by axiom that we are allowed to move brackets in sums. Makes sense, if I have two apples and get one more and later get one more again, it doesn't matter if the two latter apples arrive as one batch or not
= 3+1 because 3 is defined as 2+1
= 4 because 4 is defined as 3+1

[Winstonm]

You may find it hard to believe, but nowhere along my educational route did anyone ever simply stop and say, these are axioms. An axiom is.... We use axioms because...This is how it works...

Instead, math was always presented as "do it this way because the me and the book say to do it this way".

For me, at least, having an understanding of the why seems almost an intergral part of my ability to learn.

[wyman]

Winstonm, on 2011-February-10, 09:32, said:

You may find it hard to believe, but nowhere along my educational route did anyone ever simply stop and say, these are axioms. An axiom is.... We use axioms because...This is how it works...

Instead, math was always presented as "do it this way because the me and the book say to do it this way".

For me, at least, having an understanding of the why seems almost an intergral part of my ability to learn.

This is a weird pedagogical thing. One would like to give children the building blocks so that they have full understanding; however, two problems:
1) sometimes it's easier (and often more effective -- people in general do poorly with abstraction) to give them something half-built, not talk about the internals, and allow them to continue building, swap out parts, mimic what others have built, ask questions about why -- with these pieces -- we can't build X, Y, or Z; and
2) sometimes teachers don't understand the building blocks well enough to teach them. Certainly I don't trust the vast majority of elementary educators to do any justice to mathematics.

It is striking to me that when I ask what you learned in math between, say, 3rd grade (when people usually get into times-tables and basic multiplication) and 8th grade (possibly 7th), where you start (pre-)algebra, most people can only come up with "long division" or "we multiplied 3 digit numbers by 2 digit numbers. Then we multiplied 3 digit numbers by 3 digit numbers." This has to be a function of how kids are learning at those ages; retention just seems to be really bad, so we have to pound it in there by rote. Part of me says "there has to be a better way," but part of me says "they're kids, and I think in elementary school, it's more important to teach them to be good people and let them develop socially than it is to teach them long division."

Squeezes seem so easy in comparison.

[helene_t]

Winstonm, on 2011-February-10, 09:32, said:

You may find it hard to believe, but nowhere along my educational route did anyone ever simply stop and say, these are axioms. An axiom is.... We use axioms because...This is how it works...

Instead, math was always presented as "do it this way because the me and the book say to do it this way".

For me, at least, having an understanding of the why seems almost an integral part of my ability to learn.

I didn't come across axioms in junior hi either. There were a couple of proofs or pseudo-proofs but whether the fundaments on which they were based were axioms, lemmas or just common sense wasn't clear. It doesn't really matter, either. In my "proof" that 2+2=4, you may wonder whether the "axiom" I invoke, namely that I am allowed to move brackets, is really part of the axiomatic basis for number theory as it is taught in universities, or if it was something I made up for the purpose of this "proof". But it doesn't matter, since I am not going to provide a coherent account of number theory here anyway. Not did your teachers at high school.

If you take maths to a serious level (I mean, taking the graduate classes made for students who specialize in maths for its own sake, rather than as a tool to use in subjects), at some point you will run into assertions that are not so obviously true. And then it may be of interest to know (and maybe even to be able to prove!) that the axiom of choice, for example, is something that can't be proven so although it is intuitively obvious (to most of us), it is something we just have to assume by axiom, and we could also assume its negation, and it may be of interest to explore how maths would look like if based of the negation of the axiom of choice.

It relies a lot of intuition, and this intuition is used in two ways: to decide which axioms we chose (we chose the axiom of choice rather than its negation because it is more intuitive), and to make short-cuts in the proofs (we don't bother to prove everything, most of the time we accept something as proven as soon as it is intuitively obvious that it can be proven).

But for everyday purposes, maths is about what just has to be true by sheer logical necessity.

[Winstonm]

helene_t, on 2011-February-10, 10:23, said:

I didn't come across axioms in junior hi either. There were a couple of proofs or pseudo-proofs but whether the fundaments on which they were based were axioms, lemmas or just common sense wasn't clear. It doesn't really matter, either. In my "proof" that 2+2=4, you may wonder whether the "axiom" I invoke, namely that I am allowed to move brackets, is really part of the axiomatic basis for number theory as it is taught in universities, or if it was something I made up for the purpose of this "proof". But it doesn't matter, since I am not going to provide a coherent account of number theory here anyway. Not did your teachers at high school.

If you take maths to a serious level (I mean, taking the graduate classes made for students who specialize in maths for its own sake, rather than as a tool to use in subjects), at some point you will run into assertions that are not so obviously true. And then it may be of interest to know (and maybe even to be able to prove!) that the axiom of choice, for example, is something that can't be proven so although it is intuitively obvious (to most of us), it is something we just have to assume by axiom, and we could also assume its negation, and it may be of interest to explore how maths would look like if based of the negation of the axiom of choice.

It relies a lot of intuition, and this intuition is used in two ways: to decide which axioms we chose (we chose the axiom of choice rather than its negation because it is more intuitive), and to make short-cuts in the proofs (we don't bother to prove everything, most of the time we accept something as proven as soon as it is intuitively obvious that it can be proven).

But for everyday purposes, maths is about what just has to be true by sheer logical necessity.

It appears to me that one of the basic difficulties in teaching mathematics is that we are dealing with abstractions rather than real objects, and as children we are used to objects. That to me explains why it is easier to get kids to understand that 2 apples + 2 apples = 4 apples. It is not so easy to grasp the concept of -2 apples + -2 apples = -4 apples when there are no minus appole objects to point to.

The former we can illustrate - the latter we can only conceptualize with a substitution for actual minus objects, the number line.

I could be wrong about all this, though.

[wyman]

Winstonm, on 2011-February-10, 11:38, said:

It appears to me that one of the basic difficulties in teaching mathematics is that we are dealing with abstractions rather than real objects, and as children we are used to objects. That to me explains why it is easier to get kids to understand that 2 apples + 2 apples = 4 apples. It is not so easy to grasp the concept of -2 apples + -2 apples = -4 apples when there are no minus appole objects to point to.

The former we can illustrate - the latter we can only conceptualize with a substitution for actual minus objects, the number line.

I could be wrong about all this, though.

Well, if we had a scale that measured in apples (of course, we could use 1 lb weights), +apples would go on one side and -apples would go on the other. The sum is always the net reading on the scale. This is the number line with something concrete illustrating it. This probably doesn't help with -A x -B = A x B, but it should help with adding negatives to positives and negatives to negatives. This should show that taking one away from one side is the same as adding one to the other; armed with this knowledge we can also talk about what 3 - (-4) is. We start with 3 on the + side of the scale, and then we take away -4; that is, we take away 4 from the negative side. Taking from one is the same as adding to the other, so we lump 4 more on the + side for a net of +7.

Idk, stream of consciousness

[BunnyGo]

Winstonm, on 2011-February-10, 11:38, said:

It appears to me that one of the basic difficulties in teaching mathematics is that we are dealing with abstractions rather than real objects, and as children we are used to objects. That to me explains why it is easier to get kids to understand that 2 apples + 2 apples = 4 apples. It is not so easy to grasp the concept of -2 apples + -2 apples = -4 apples when there are no minus appole objects to point to.

The former we can illustrate - the latter we can only conceptualize with a substitution for actual minus objects, the number line.

I could be wrong about all this, though.

I agree, it's a very tricky problem. Part if it is that the teachers don't understand the abstraction usually.

That said, shapes are pretty concrete and contain a lot of the abstractness (this also requires a teacher who really "gets it" so maybe it wouldn't be that applicable). But you could give a child a grid and 9 checkers and ask them to make a square. Count the sides of the square, etc. Then with 16 checkers. What about 17? Huh...can't do it? How about 12? Still no? Can you make a rectangle with 12 pieces? How long are it's edges.

In the end, you can really develop numbers from geometry...but it takes a bit of work. Would be interesting to try though.

[hrothgar]

BunnyGo, on 2011-February-10, 13:15, said:

That said, shapes are pretty concrete and contain a lot of the abstractness (this also requires a teacher who really "gets it" so maybe it wouldn't be that applicable). But you could give a child a grid and 9 checkers and ask them to make a square. Count the sides of the square, etc. Then with 16 checkers. What about 17? Huh...can't do it? How about 12? Still no?

I can make a square with 12 checkers...

[Winstonm]

Quote

I can make a square with 12 checkers...

Sounds like the geometry version of "Name That Tune".

[BunnyGo]

hrothgar, on 2011-February-10, 13:26, said:

I can make a square with 12 checkers...

On a grid with no leftovers?

[hrothgar]

BunnyGo, on 2011-February-10, 13:58, said:

On a grid with no leftovers?

Yes, the square wouldn't be filled in, but it would still be a square, and even have an equal number of checkers on each side

[PassedOut]

I can make one with 17 checkers, and I even have a checker left for the center.

[gwnn]

I think he means a 5x5 square which has 12 white and 13 black squares?

[PassedOut]

gwnn, on 2011-February-10, 14:44, said:

I think he means a 5x5 square which has 12 white and 13 black squares?

Yes, otherwise the center checker would not be on the grid.

[kenberg]

This will only appeal to mathematicians: The University gives a mathematics competition for high school students. The first round is a flock of mostly but not entirely straightforward problems. We take the best fifty or so and invite them to take the second round where the problems are tougher. Back when I was involved with this we gave a problem involving a checkerboard. One kid's solution began "Consider a checkerboard as a free abelian group on 64 generators..." His solution was correct.

[Winstonm]

kenberg, on 2011-February-10, 15:30, said:

This will only appeal to mathematicians: The University gives a mathematics competition for high school students. The first round is a flock of mostly but not entirely straightforward problems. We take the best fifty or so and invite them to take the second round where the problems are tougher. Back when I was involved with this we gave a problem involving a checkerboard. One kid's solution began "Consider a checkerboard as a free abelian group on 64 generators..." His solution was correct.

That must have really pissed off the first 63 generations of Free Abelians.

[mycroft]

I'm sorry, but if anybody can build a square out of any number of checkers, they're not using the same sets I grew up with. I find mine always tend to have those wavy edges that my drafting teacher tended to mark down as "don't freehand, use a ruler."

I'm sorry, couldn't resist.

[PassedOut]

mycroft, on 2011-February-11, 14:18, said:

I'm sorry, but if anybody can build a square out of any number of checkers, they're not using the same sets I grew up with. I find mine always tend to have those wavy edges that my drafting teacher tended to mark down as "don't freehand, use a ruler."

I'm sorry, couldn't resist.

You rich kids! When I was a kid we had to make our own checkers, and it was easiest to make them square.

[Winstonm]

PassedOut, on 2011-February-11, 14:58, said:

You rich kids! When I was a kid we had to make our own checkers, and it was easiest to make them square.

Luxury! We had to get up three hours before we went to bed, hue checkers from the bones of our own legs, then walk barefooot miles and miles in 12 foot drifts of snow on top of furnace grates until the soles of our feet were crisscrossed enough from the char to be used as a checkboard.