(1) Euclid.s algorithm for finding the greatest common divisor (GCD) of
two integers is based on the idea that if you subtract one from the
other, you don.t change the GCD. So 1989 – 867 = 1122, so GCD(1989,867)
= GCD(1122,867). Repeat as needed until you get it down to GCD(51,51) =
51. Thus to reduce the fraction 1989/867, divide both by 51, getting
39/17.

(2) Modular arithmetic is based on subtraction. Why is it that a number
is divisible by 3 if and only if the sum of its digits is divisible by
3? Modular arithmetic answers that right away.

Don.t mistake the _algorithm_ for subtraction (involving borrowing,
etc.), which is a _process_, for subtraction itself, which is not a
process.

Elementary school children can understand what prime numbers are. How
many prime numbers exist? Do they go on forever, or not? The
patternlessness of primes seems to put the answer out of reach of
mortals. But Euclid succeeded in proving that there are infinitely many.
One seeking beauty in arithmetic shouldn.t miss that one.

Don.t seek to be successful in every unit. Don.t teach .units.. Don.t
require those not interested in math to work on it. At most, tease them.
I suppose standardized tests imposed by Caesar might get in the way of
that, but that.s the ideal.

(What I remember of my first- and second-grade math instruction is that
when I mentioned negative numbers, my second-grade teacher said .THERE.S
NO SUCH THING AS NEGATIVE NUMBERS IN THE SECOND GRADE!!!.)

(1) Euclid’s algorithm for finding the greatest common divisor (GCD) of two integers is based on the idea that if you subtract one from the other, you don’t change the GCD. So 1989 – 867 = 1122, so GCD(1989,867) = GCD(1122,867). Repeat as needed until you get it down to GCD(51,51) = 51. Thus to reduce the fraction 1989/867, divide both by 51, getting 39/17.

(2) Modular arithmetic is based on subtraction. Why is it that a number is divisible by 3 if and only if the sum of its digits is divisible by 3? Modular arithmetic answers that right away.

Don’t mistake the _algorithm_ for subtraction (involving borrowing, etc.), which is a _process_, for subtraction itself, which is not a process.

Elementary school children can understand what prime numbers are. How many prime numbers exist? Do they go on forever, or not? The patternlessness of primes seems to put the answer out of reach of mortals. But Euclid succeeded in proving that there are infinitely many. One seeking beauty in arithmetic shouldn’t miss that one.

Don’t seek to be successful in every unit. Don’t teach “units”. Don’t require those not interested in math to work on it. At most, tease them. I suppose standardized tests imposed by Caesar might get in the way of that, but that’s the ideal.

(What I remember of my first- and second-grade math instruction is that when I mentioned negative numbers, my second-grade teacher said “THERE’S NO SUCH THING AS NEGATIVE NUMBERS IN THE SECOND GRADE!!!”)

You have containers and each container can only hold 10 items. Say you have 17 items, so you need to fill one container (you’ll have to make it a rule to full one first, so they don’t divide them 8 and 9, as opposed to 10 and 7,) and you want to take away 9 objects.

Take away everything you can from the smaller one, since our goal is to use as few containers as possible. But… you can only take 7, so you need to dip into the full one and pull out two more to give you 9–leaving 8. There you have borrowing, and it’s exactly what you’re doing with the numbers. Alternatively, to make it even more like the typically taught process, literally combine the two groups, since they know that 9 is bigger than 7. Be creative.

This is a very simple example, but hopefully it may show you that, even with basic subtraction, there’s more to it than “cross out the tens place, reduce by one, stick a one in front of the ones place.” (I’m using two digit number for simplicity.)

- Somebody who loves the process (usually I only work with students algebra and higher as a tutor.)

Teach both. Don’t neglect one for the other.

I would say yes. If you give context to the math or include it in some kind of game, crosswords for example:

Is it possible I enjoyed first and second grade math?

http://neoparaiso.com/imprimir/crucigramas-de-tablas.html

The Kamii books are available through Prospector at the public library.

In the last year I have just started to understand some of the formulae that seemed totally illogical to me almost a half century ago. In fact, the area of a triangle just became comprehensible to me a few minutes ago. You can do it.

Let me describe a typical math problem coming out of PCMI (some teachers and school districts have started adopting curriculum models that incorporate math problems like this): Suppose you have to assemble a train of length n, and you can combine train carts from length 1 to n to make up the required length. How many ways can you assemble this train?

A simple problem like this can easily take a whole week to teach. “Isn’t this very inefficient?” Well, not if you consider all the math and learning skills that goes into solving this problem: tactile learning, enumeration, tree diagram, number sequence, algebraic proof… The best part is, there are multiple levels and points of entry, so students with different ability levels can all gain accomplishment by different means.

I only wish every teacher could go to PCMI at least once… and it’s much more relaxing than TFA Institute!