I would like to make a complaint really. School math(s) can be the most boring way to learn: sitting down and rote learning binomial expansion or the volume of a cylinder is just not interesting. It seems that schools don't teach the interesting way, with plenty of variety and proof. I have some very basic facts that students are never taught explicitly, perhaps because of the rigidity of education. Instead, teachers seem to count on a mistake being made. My specific example is the fact that
ab is not equal to
a*b, but (a*b).
We are all taught of the order of operations, however the rule stating that
ab can be expressed as (or is shorthand for)
a*b, is wrong. I was pulled up when I tried to solve a linear-style expression like
20/2a=4 like a smart-alec by first assuming that this is equal to
20/2*a=4. This is wrong, right? Apparently, terms are always in their own little group, and this effects the order of operations.
20/2a=4 is the same as
10/(2*a)=4. The mistake would not have been made by using TeX style, maths notation.
My questions are: Am I correct in saying
ab = (a*b)? Why are these things generally overlooked? Are there any other typical errors involving the order of operations, especially when using linear notation? Are these things simplified by teachers as to avoid bombarding the students with 'special cases'?