Traditionally, math begins with addition, subtraction, and the multiplication tables in the 3rd grade or so. Then, there is an introduction to geometry, algebra, and calculus in high school. Finally, there is usually a course in differential equations in college, and maybe some complex analysis if you're in a STEM field, and somewhere along the way there is also probability and statistics.
Scruffy says, this is not enough. To be competent in the modern world, a college graduate should understand differential geometry and tensor calculus (same thing, kind of), topology, and abstract algebra including but not limited to the theory of groups, rings, modules, and categories. Waiting for graduate school to understand these things is a guarantee of difficulty. These topics should be well studied prerequisites for any kind of graduate work.
Scruffy proposes to condense the course work at the high school level, as the beginning of a new mathematics program. BEFORE we start learning about triangles, we need an exposure to surfaces. Otherwise, there is no motivation for triangles, trigonometry, or conic sections. 7th grade is the perfect time for a "survey of mathematics" course. Why are we learning abstract algebra? What good is it? Why is calculus important? What is topology, and what is differential geometry?
In 6th grade, is when we want to cover basic algebra, combinatorics, and probability.
The 8th grade algebra course should include systems of linear equations, matrix math, and change of coordinates in addition to polynomials.
The 9th grade geometry course should include manifolds and an introduction to dot product, cross product, and tensor product.
The 10th grade calculus course should include multi variable calculus and de-emphasize the memorization of integrals and derivatives. It should specifically include an introduction to differential equations. Students should understand Stokes' and Green's theorems by the end of the course.
All this will prepare the student for the typical 11th grade introduction to physics, which should include Hamiltonian and LaGrangian mechanics (introductory dynamics, the LaPlacian, what is a wave equation, and so on). Math in 11th grade should be fun, not rigorous. It should include topology, abstract algebra, and probability as an introduction to quantum science. It should also specifically include an introduction to non-Euclidean geometries. It should provide motivation for the continuing study of math in college.
This is an example of a math curriculum that provides motivation for continuing study:
en.wikipedia.org
Then, by the time we get to college, we ALREADY know what an orbit is, before we start taking first year physics and chemistry. We know what polar coordinates are BEFORE we have to solve problems involving angular momentum, and thus we can easily and intuitively understand concepts like quantum spin, and we can already start to manipulate them using spinors and quantum computing gates.
First year math in college should be hard core differential equations. (Since we already learned linear algebra and multi variable calculus in high school). It should also include an introduction to stochastic equations (like Brownian motion, which fits right in with first year chemistry and physics).
Second year math should be differential geometry and topology, with a focus on tensor calculus. By the end of the second year in college, students should be able to seamlessly transition between coordinates systems, both geometrically and algebraically. The course should include an introduction to measure theory and invariances, so we understand measure preserving and conformal transformations, and it should also include the mechanics of distances and angles on curved surfaces.
After all this, third year college math should focus on abstract algebra and algebraic topology, starting with immersions and including fiber bundles, sheaves, foliations, simplexes with graph theory, and so on.
Finally, senior level.math in college should once again be fun and motivational. Dimensionality is a good topic, fractional dimensions, dusts, space filling curves, fractal geometries, and so on. Electives might include a course in quantum field theory, for instance the creation and annihilation operators tie right in with dimensionality and also pertain to neural networks and machine learning. Advanced probability theory could be another elective - the Ito, Langevin, and Malliavin calculuses, how to solve non-equilibrium equations in open systems, information geometry, stochastic dynamics and stability, and entropy.
Anyone who graduates from college should be conversational with mathematics. It doesn't mean you have to pull equations out of your butt, it just means you have to be familiar with the principles. You don't have to know what a Lorentz boost is unless you're a physicist, but you should know enough about coordinates systems so someone could explain it to you in 5 minutes.
This business of graduating from college without knowing how to balance a checkbook has to stop. Scruffy says: if you want to graduate, you WILL learn math. If you can't do it, go to trade school instead. (Where you'll learn it anyway lol).
Scruffy says, this is not enough. To be competent in the modern world, a college graduate should understand differential geometry and tensor calculus (same thing, kind of), topology, and abstract algebra including but not limited to the theory of groups, rings, modules, and categories. Waiting for graduate school to understand these things is a guarantee of difficulty. These topics should be well studied prerequisites for any kind of graduate work.
Scruffy proposes to condense the course work at the high school level, as the beginning of a new mathematics program. BEFORE we start learning about triangles, we need an exposure to surfaces. Otherwise, there is no motivation for triangles, trigonometry, or conic sections. 7th grade is the perfect time for a "survey of mathematics" course. Why are we learning abstract algebra? What good is it? Why is calculus important? What is topology, and what is differential geometry?
In 6th grade, is when we want to cover basic algebra, combinatorics, and probability.
The 8th grade algebra course should include systems of linear equations, matrix math, and change of coordinates in addition to polynomials.
The 9th grade geometry course should include manifolds and an introduction to dot product, cross product, and tensor product.
The 10th grade calculus course should include multi variable calculus and de-emphasize the memorization of integrals and derivatives. It should specifically include an introduction to differential equations. Students should understand Stokes' and Green's theorems by the end of the course.
All this will prepare the student for the typical 11th grade introduction to physics, which should include Hamiltonian and LaGrangian mechanics (introductory dynamics, the LaPlacian, what is a wave equation, and so on). Math in 11th grade should be fun, not rigorous. It should include topology, abstract algebra, and probability as an introduction to quantum science. It should also specifically include an introduction to non-Euclidean geometries. It should provide motivation for the continuing study of math in college.
This is an example of a math curriculum that provides motivation for continuing study:
Erlangen program - Wikipedia
Then, by the time we get to college, we ALREADY know what an orbit is, before we start taking first year physics and chemistry. We know what polar coordinates are BEFORE we have to solve problems involving angular momentum, and thus we can easily and intuitively understand concepts like quantum spin, and we can already start to manipulate them using spinors and quantum computing gates.
First year math in college should be hard core differential equations. (Since we already learned linear algebra and multi variable calculus in high school). It should also include an introduction to stochastic equations (like Brownian motion, which fits right in with first year chemistry and physics).
Second year math should be differential geometry and topology, with a focus on tensor calculus. By the end of the second year in college, students should be able to seamlessly transition between coordinates systems, both geometrically and algebraically. The course should include an introduction to measure theory and invariances, so we understand measure preserving and conformal transformations, and it should also include the mechanics of distances and angles on curved surfaces.
After all this, third year college math should focus on abstract algebra and algebraic topology, starting with immersions and including fiber bundles, sheaves, foliations, simplexes with graph theory, and so on.
Finally, senior level.math in college should once again be fun and motivational. Dimensionality is a good topic, fractional dimensions, dusts, space filling curves, fractal geometries, and so on. Electives might include a course in quantum field theory, for instance the creation and annihilation operators tie right in with dimensionality and also pertain to neural networks and machine learning. Advanced probability theory could be another elective - the Ito, Langevin, and Malliavin calculuses, how to solve non-equilibrium equations in open systems, information geometry, stochastic dynamics and stability, and entropy.
Anyone who graduates from college should be conversational with mathematics. It doesn't mean you have to pull equations out of your butt, it just means you have to be familiar with the principles. You don't have to know what a Lorentz boost is unless you're a physicist, but you should know enough about coordinates systems so someone could explain it to you in 5 minutes.
This business of graduating from college without knowing how to balance a checkbook has to stop. Scruffy says: if you want to graduate, you WILL learn math. If you can't do it, go to trade school instead. (Where you'll learn it anyway lol).
