Here's my question: Does anybody know what is developmentally appropriate at each age? If math is developmental, is there a chart to describe the development somewhere? Is it really true that most adults are not developmentally ready for calculus?
I have some further questions for the science/engineering experts here.
What is the appropriate level to introduce calculators for use in classroom math and should their use be allowed without restriction from that point forward?
Its a lot faster to do computations in your head for many numbers rather than use a calculator. Its also a lot easier to check work if you know numbers well. There are the basic number things like multiplication tables and division - memorization of up to 20 x 20 or 99 x 99 is not a bad goal. Beyond that, knowing the log tables for base 10 and e will allow you to do division and multiplication for much larger number sets. Knowing sine ( and by implication - cosine ) by tenths of a radian is a good target as well. Knowing the numerical methods to derive other numbers is a good target, too. Knowing primes between 1 and 1000. I could go on. You just never know where you will find a number that you know..
When to use calculators? I say from the very beginning. But with the caveat that you should be able to walk away from the calculator and still be able to do the numbers to a good approximation quickly as a rough estimate at least. Calculators will have to be used if there are a lot of numbers to process - ie stats or a very long problem.
Can most adults do calculus? The answer is no. By calculus I mean a word problem involving rates that must be solved using integration. One needs to know Geometry, Analytical Geometry, and Algebra fairly well to even set up and solve most problems. And one must be able to hold all this information in one's mind and decide from the list of stuff what to use and how to use it.
Calculus is just one small branch of Mathematics, too. Most mathematicians consider Calculus to be trivial.
What is developmentally appropriate? If someone is good at the basic prerequisites for that subject, then they are ready for the subject. Some kids are ready for Algebra when they are nine and others when they are 14 - and others when they are 18. Some - never. Once one has Algebra down and Geometry down, then Analytical Geometry comes next, then Calculus. Mathematically mature students could jump Analytical Geometry and do Logic then go into Real Analysis then back track to Calculus.
