Here are a number of ideas
(1) We have allowed the logical roots of mathematics to intrude upon the teaching of basic mathematics. These roots answer questions that beginning students never have. It is true that set theory, analysis, abstract algebra are necessary for a professional mathematician to understand. But the percentage of students who will be professional mathematicians is well under 1%. The tools needed by a professional mathematician are not necessary to learn how to apply arithmetic or algebra or even calculus.
We ask high school students to memorize the definitions and names of the associative and commutative laws of addition and multiplication and then never do anything with them. Every student simply assumes on the basis of experience that they are true about numbers and never even dreams about non-abelian groups. Experience with numbers soon convinces children that
7 * 2 * 3 = 14 * 3 = 42 = 7 * 6 and
3 + 4 = 7 = 4 + 3.
Giving names to such facts takes up room in text books and gives teachers something “new” and “deep” to talk about, but it teaches a first year algebra student nothing that the student does not already know. Then we teach PEMDAS and avoid talking about non-associative and non-commutative operations that students are familiar with such as subtraction and division. So while we could have an interesting but unnecessary conversation about different varieties of mathematical operation, we don’t. The whole topic is just drudgery and make-work for students.
(2) We still spend too much time teaching mechanics and not enough time teaching how to apply mathematics to problems not posed in mathematical terms (“word problems”). I seldom if ever calculate 397 times 821 by hand. I know how to do it because I was educated before hand calculators, but I suspect I would not be hindered if I had to rely solely on a calculator. But not knowing when to multiply and what to multiply would have stopped my business career before it began.
(3) We used to teach proofs, but then gave it up because proofs that meet the demands of modern rigor are far too subtle for kids to cope with. We can‘t expect them to grasp Hilbert‘s geometry so we don’t make them grapple with Euclid‘s cruder version. Proof is an exercise in extended logical thinking, and we don’t bother to spend much time on it.
(4) The examples we use to make math relevant are too frequently absurd. Students doubt rightly that if they know they have a total of $17.49, they will have any interest in the number of quarters, dimes, nickel, and pennies that make up that sum.
(5) We ignore very important things. For example, the lurking Platonism of most mathematicians means that most people have no clue that the estimated difference between two roughly equivalent estimates has a very wide range of error.
1000 - 990= 10.
But suppose the 1000 and 900 are estimates accurate to within plus or minus 1%. Those are pretty good estimates, but the true difference might be anywhere between 1010 - 80.1 = 29.9 or 990 - 999.9 = - 9.9 a range of error of plus or minus 299%, and we cannot even sure of the sign of the difference. For many purposes, that makes the estimated difference perfectly useless.
