Let's throw the bathwater out with the baby

Revamp of Mathematics?


Should fractions be taught?

That’s jumping the gun. Should counting numbers be taught?

Your question is ill defined because what does “being taught” entail? But, yes, counting numbers should be introduced.
In the first place, like it or not, our society counts things. It is my understanding that Native Americans weren’t all that into counting numbers. There are expressions like “a lot”, “a few”, “as many as there are leaves on the trees”, and so on that give numerical information without getting as fussy as our society does.
I remember that as a TA I was involved in the mass teaching of Calculus I. The four (I think.) tests were given at night, the answers were marked on punch cards and machine graded. I always thought it was interesting that the course started off with analytic geometry based on two and three dimensional inner product spaces; it defined the angle, x, between two vectors A and B in terms of the inner product. And then we gave a multiple choice test.
The scores would eventually be posted and lines were drawn across the printout. Above the first line was an A, below it was a B. The line’s placement wasn’t so much a function of the score as the number of lines above it. One point would be the difference between an A and a B.
I could look at a worked out problem and decide if it was worth an A, B, C, D or F. And the student could too. I would say, “This is a B solution to the problem.”or “This is a C test paper.” They would look at it and know I was right. If they argued, they usually had a good case and I could see that they were right.
On the other hand when you give points you have to decide between 6 and 7 on a 10 point problem. I think this is foolish. When I started, and a long way into, teaching I used to give points until I realized that it was crazy. It is an example where counting numbers (God forbid fractional credit) are not useful.
Our particular culture seems fascinated with whole numbers. (I have discussed this in my Calculus Book that is available on this blog.

Why is it hard for kids to remember 9 x 7?

Because they can’t visualize 67 things. (Just kidding.)
I can think of three things and I have a gut feeling for what 3 means. 6,872,970 means about 7 million which means more than I can count. Each expression has its use.
I accept the fact that there are times when accuracy is needed and the people who need it will learn how to carry more significant figures. But the very fact that we round off says that we have to handle something less than exactness. Going from 6,872,970 to 6.8 x 10*6 is just a step on the road to “as many as there are blades of grass in the park”.

Aren’t whole numbers of any use to the non-mathematician?

Of course they are of some use. They have to know which concepts correspond to which symbols. (Five corresponds to the symbol 5 and both correspond to ***** stars) I would present the basic ideas of practical arithmetic and would limit the numbers to three digits. If the student understands and has some facility with three digits they can use more digits if the need arises.
I would also introduce exponents and logs. I’m not sure when but certainly after the first grade. (Most kids aren’t as precocious as I was.) I have heard that the brain doesn’t stop growing until a person is in their twenties. Apparently the front part is where reason and cognitive functions happen and that’s the last part to develop. I suppose all this is well known. Anyway, if that is true I don’t see any purpose in trying to introduce abstraction before the brain of a student is ready to process abstract ideas.
There was an old algebra book on my grandmother’s bookshelves and when I was in the 4th grade I tried reading it. The x’s made no sense to me at all and trying to make sense out of x + 2 = 4, x = 2 frustrated me. By the time I was in the 7th grade and tried again, it seemed obvious.
So, just for the sake of argument, I would start logs and exponential when the student was mature enough, about 12 years.
The logarithm is an example of a transform like the Fourier transform or LaPlace transform. The LaPlace transform changes a differential equation into an algebraic equation, in some way the transform changes the differential equation into an easier equation to solve. The logarithm changes a multiplication problem into a simpler addition problem.
So the logarithm is an example of a much more general concept, changing a given problem into a simpler problem.
There aren’t all that many basic principles and because they are basic they can be introduced fairly early. As one progresses, the student sees that new topics are really just variations on themes.
So what are counting numbers good for?

Well, they are an introduction to unending sequences. I remember my dad telling me about the whole numbers and how there was no largest when I was 6. Knowing nothing else about whole numbers, I could see that whole numbers were a heavy concept.

I don’t know for sure what mathematics should be taught but I do know that the mathematics curriculum should be given a serious reorganization. As far as I can see, K-12 is taught a whole lot different now than it was a hundred years ago. I guess they don’t teach Hooke’s method of extracting square roots any more.
I hate the hand held calculator but I have to realize that they exist and that they are a part of society. After teaching that A divided by B is the number of times I can subtract B from A plus a remainder, I would then give three digit practice and how to approximate higher digit divisions. Then let the long tedious divisions be done on a calculator.
The most important thing is to give the student a way of looking at mathematics that gives them a facility in learning mathematics.
Another idea that has occurred to me is to put all the money into more and smaller classes, say (one teacher /ten students) let the teachers teach how they want to. I have heard that an NC in the number on a small airplane means that if you go into a spin, you can take your hands and feet off the controls and the plane will right itself. Maybe if we took people who had some knowledge of a subject, told them to teach it and then let them teach, the system might right itself.