Establishing a rigourous and effective mathematical curriculum is difficult, to say the least. As stated in the last section, I think students are rushed through calculus. I do see the reason for this: Calculus is the basis of most modern science and engineering techniques; providing calculus at an early age should give them an edge. The problem is that it doesn't.
It is unanimously agreed that Calculus 2 is the hardest calculus course. Why? It isn't because integrating and differentiating trig functions is difficult; it isn't because integration by parts is difficult. So what is it? Trig identities, infinite series, partial fractions. These all have something in common: they are algebra, not calculus. Perhaps if students were given a thorough understanding of algebra before taking calculus, calculus professors could spend their time explaining calculus and proving the theorems, instead of teaching algebra. This way, calculus classes would prove to be the joke they are; hell, you could even get rid of calc 3: it's just calc 1 and 2 with extra variables. If you understand algebra, this is mind-numbingly tedious.
Imagine if history teachers talked about WWII in high school, but didn't mention the Treaty of Versailles, the Weimar republic, or the rise of Totalitarian States until college! It's absurd, isn't it? Well, that's what math instructors do.
I think my tangent is over, let me outline my curriculum. It contains a wide range of useful (to engineering) mathematical topics, as well as some absolutely necessary mathematical knowledge.
Kindergarten/1st grade: Introduction to numbers. Adding and subtracting. Basic shapes. Interesting properties.
2nd Grade: Adding/subtracting continued. Basic multiplication. introduction to simple division. Intro to decimal system. Intro to fractions.
3rd Grade: Multiplication and division. Remainders, decimal expansions. Basic fraction arithmetic.
4th Grade: More on arithmetic. More on shapes. Fractions.
5th Grade: Intro to generalizations and abstraction. Exponents. Order of operations. Intro to shapes. Intro to counting.
6th Grade: Cartesian Plane. More on abstraction. pre-algebra.
7th Grade: Cartesian Plane, graphing. Pre-algebra, algebra.
8th Grade: Mathematical logic. Sets. Basic Probability. Basic proofs. Algebra.
9th Grade: Geometry. Intro to trigonometry. Proofs.
10th Grade: Functions, algebra, variables. Sums. Sets. Probability.
11th Grade: Trigonometry. Stats. Intro to graph theory. Intro to game theory. Modelling.
12th Grade: Infinite series and sequences. Limits. Fundamentals of number theory and combinatorics.
Note that 10th, 11th, and 12th grade represent a foundation for all higher level math.
Next: Brief "lectures"
Arbtirary thoughts on nearly everything from a modernist poet, structural mathematician and functional programmer.
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