/sci/ - Science & Math

I don't know about the rest of the curriculum, but when it comes to calc 1/2 and parts of analysis I would use spivaks calculus

>>7113825
Isn't spivak's calculus harder for people who aren't really math oriented?

>>7113829
that's true, but I like it because its great by moving students away from computation to more proof based things. I believe that thinking that way is important in both STEM and non-STEM fields

>>7113833
Sweet, I'll definitely check out the book then. I've always been somewhat bad at math due to a poor highschool education. I have to take math classes now for my CS degree, but my professor is terrible and barley shows up to class.

I'm trying to get better at math so that I can hopefully double major or at least minor in it.

>>7113835
You'll definitely struggle with it if it's your first foray into proof based math, but if you want to gain a better understanding of math rather than just recite formulas its a good book. Ebook versions of the book and solutions manual are available online.

>>7113812

>1st grade: counting numbers; negative integers; addition and subtraction and multiplication, simple word problems
>2nd grade: fractions and decimals; division; factors and prime numbers
>3rd grade: basic algebra and starting word problems (word problems will continue throughout school); probabilities; basic geometry
>4th grade: more complicated operations such as squares and exponents; square and cube roots; quadratic equations
>5th grade: introduction to Cartesian coordinates and graphing; introduction to functions and graphing functions; learning about integers, rational and irrational numbers, real numbers etc.

>6th grade: learning about more complicated functions, and finding roots, asymptotes, singularities, etc; domain and range of functions; radians, trigonometry and graphing trig functions; simple conic sections, higher-level geometry
>7th grade: more advanced algebra, learning to factor polynomials, polynomial division, rational roots theorem; introduction to complex numbers and the complex plane, properties, simple examples of functions extended over the complex numbers
>8th grade: simple logic and proofs; introduction to calculus, through defining limits and using these to define differentiation and integration; applications of these; introduction to series, infinite series, taylor expansions, convergence of series, etc. (Calculus-based physics must be taken in this year.)

>9th grade: slightly more formal logic, basics of set theory and notation; real analysis and maybe basic introduction to topology
>10th grade: introduction to abstract algebra, applications both in pure math and in the real world, including rigorous definitions of vector spaces
>11th grade: multivariable calculus, applications; introduction to complex functions
>12th grade: First semester is introduction to differential equations, but with assumed knowledge of abstract algebra; Second semester is linear algebra, with assumed knowledge of formal definitions of vector spaces

>>7113838
I don't mind struggling as long as I gain a better understanding. A lot of my high school teachers were all about learning formulas and making us do problems over and over again rather than telling us math worked.

CS gave me a peek of what math is capable of and now I really just want more. I also just want to punch my past self in the face for thinking that math was useless.

>>7113841

>>7113841 (continued)

Possible math electives in high school:
>introduction to formal logic
>formal introduction to set theory
>introduction to partial differential equations (prereq: multivariable calc, 11th grade)
>introduction to number theory (prereq: abstract algebra, 10th grade)
>introduction to topology (prereq: abstract algebra, 10th grade)
>introduction to Lie groups (prereq: differential equations, 12 grade; it would be for advanced students who took 12th-grade math early)

Did I forget anything? What do you think?

>>7113854
I think this is sweet, I'm gonna attempt to teach myself things from this list that I don't know yet

>>7113841
I think you are overestimating what an average 1st, 2nd, and 3rd grader can comprehend.

>>7113877
Oh come on, I was able to learn those things, and I'm of average intelligence.

By the way, I forgot to mention: kids can drop out of the math curriculum in 8th grade.

>>7113862
Good idea! Some of the topics may be difficult if it's been a long time since you've done math, though.

>>7113903
The best part of these threads is when people have ridiculously high expectations for young children.

>>7113911
Once again, I learned this stuff, and I'm just of average intelligence.
Any average kid can learn it, and if they can't they'll just be behind one year.

Or if you want, you can call the one-year-behind track "normal" and the regular track "advanced".

My Math Curriculum. It's rigorous, but I honestly think it's realistic for non-genius kids. stretch everything out for 1 year longer in elementary school and then it's definitely realistic for any good student.

Grades 1-5: Everything a Kid should know to be able to take today's SATs. All phrased and taught in a way that lends itself towards later abstraction.

Grade 6: Set theory, logic, how to write a proof. Study more types of functions than you need to know for SATs (SATs don't even have trig). Can Manipulate and solve algebraic problems, and can prove really basic facts (sqrt(2) is irrational, infinitely many primes, etc.).

Grade 7: Proofs in Plane Geometry, Trigonometry, etc. All that BS high school level algebraic and geometric stuff you learn.

Grade 8-9: Precalculus, with Proofs. Two years because this will probably be the hardest part. More set theory, proving things about real numbers, and doing proving basic facts about pre calc (all the stuff before differentiation). This takes 2 years because not only are we proving things, but we are still training them to be good at computation. Still lots of annoying computing because it is kind of useful to be good at it.

Grade 10: Calculus, Spivak. Homeworks will not include too many difficult proofs and there ill still be a half emphasis on being able to compute. Basic Calc I + Calc II college course.

Grade 11: Multivariable Calculus, Linear Algebra each for half a year. Again, halfway between rigorous and computational. Think decent college level course. All with proofs.

Grade 12: Half a year of topology, ending that half a year with reviewing Spivak material with more rigor (baby rudin level). Half a year of abstract algebra.

They can learn number theory and combinatorial arguments in math team. average person doesn't need to know them. Math olympiads love elementary number theory and combinatorics.

Based on what I would have liked to have known before undergrad.

ITT: autism

>>7113928
That's the only post in the thread that is grounded in reality. 10/10

>>7113924
>set theory and logic in 6th grade
Way too difficult.

>two years of precalc in grades 8-9
Way too simple.

>>7113924
>Based on what I would have liked to have known before undergrad.

My didn't teach you calculus unless if you were in honors courses. I really wish I had some experience of calculus in high school.

>>7113924
>abstract algebra before topology
Don't students need to know abstract algebra already in order to understand, for example, homotopy groups?

>>7113924
>>7113854
>>7113841
>topology

I feel like people don't know what they are talking about when they use that word.

Honestly you guys have to be joking me if this is how you imagine a better math curriculum to be.

I would personally concentrate on math exploration and logic/set theory for the first 8 years. Then once they hit highschool, don't force them to take math. Those that do will get filtered easily into pure or applied depending on their tastes, and then you can start on a more rigid curriculum, guiding students through foundations formally first. After that you have the applied and pure mathematics groups split and they should take courses that interest them, rather than being forced to take a ladder.

Ladders suck.

>>7113812

>>7113924

I was this guy, but fuck
I like this better:

>>7113841

Then, for standard undergrad curriculum (I don't believe in going full sophistication in undergrad. I think undergrad is the best time to explore. Think books like Stein & Shakarhi's series, or Artin, instead of Folland and Lang).:

1st Semester: Really rigorous real, complex analysis ala baby rudin and conway or stein. Rigorous but nothing crazy. This gets people whose high schools weren't as good up to speed. Think of it as Calc I/Calc II in College now. The good kids can Skip it.

Abstract Algebra: More advanced than what's done in HS. More fast paced. Groups, Rings, Modules, Algebras, advanced linear and multilinear algebra.

Topology: Finish Munkres, use Hatcher for AT section.

2nd Semester: Real Analysis finish up advanced calc type stuff, some intro to rigorous study of ODEs and PDEs, End with an introduction to measure, integration. Get up to measure theoretic differentiation theorems.

Algebra: brief Introduction to homological algebra, representation theory, lie groups ,and galois theory.

Topology: Standard Smooth Manifolds course.

Taken Any time in 2nd-4th years (but still required)
Riemannian Geometry
Algebraic Topology
Commutative Algebra
Lie Groups
Differential Topology (more advanced than smooth manifold course. Think characteristic classes by Milnor).
Introduction to Algebraic Geometry
Functional Analysis
Fourier Analysis
Homological Algebra
Riemann Surfaces
Representation Theory
Also, some required physics courses and mathematical physics courses (think Takhtajan's Quantum Mechanics for mathematicians).

This is 12 required courses over 6 semesters--not too crazy.

Then, some electives. This is people's opportunity to take classes on foundations subjects, number theory, dynamical systems, PDEs, or whatever they're interested in.

This all aims to give students a good idea of what they would want to do if they wanna apply to grad school

>>7113934
It's because they have never taken topology and don't know what it is: that's why they put it there.

>>7113948
>Then once they hit highschool, don't force them to take math.
Yeah, that's what I said here: >>7113903
And as you said, we could create different tracks; a physics/math track, a pure math track, an applied track, etc.

>I would personally concentrate on math exploration and logic/set theory for the first 8 years.
Kids can't and won't focus on foundations for 8 years. Teach them how to do and use math first.

>>7113948
When I say Topology, I mean cover a lot of Munrkes. And teaching foundations formally for that long is the dumbest thing I've ever heard. No one will learn anything that way because you don't have the ability to think that abstractly at that point in your life. Why the hell would you think rigorous formalism is the best way to teach mathematics. Logicians gtfo and let the real mathematicians talk.

>>7113961
>exploration AND logic/set theory.
Of course it will not take 8 years to teach them predicate logic and naive set theory. But you'd introduce it little by little. The majority of the 8 years is math exploration.

>>7113958
But I'm the guy you responded to, and I'm the first one who mentioned topology.

>>7113965
see >>7113966

>>7113934
>>7113958
Actually, my research area is algebraic topology. And by topology I obviously mean point set topology, which requires no abstract algebra. Show me one class called 'topology" that's not a point set topology class. I know a point set class usually gets to algebraic topology by the end, but this is a bit slower. Think first part of Munrkes.

>>7113932
I mean naive set theory: thinking about sets, subsets, power sets. Thinking about mathematical objects abstractly as sets. I did this in 7th grade.

And yeah, I may have made too much for precalc. I got scared because that's where I planned on really jumping up the formalism. The grades 8-9 are when kids are becoming more mature and probably have better ability to think abstractly.

>>7113972
very few individuals believe that you are an algebraic topologist

I'm surprised that no one has complained about my putting logic, sets, and abstract algebra before differential equations and linear algebra here: >>7113841

The reason I did that was that I think it would be much easier to understand the latter subjects after knowledge of the former.

>>7113983
Ask me anything I couldn't look up in a book/online and I'll answer you. Nothing personal because active researchers in algebraic topology today isn't so numerous. It'd be incredibly easy to doxx me if I even said anything I've worked on.

Some kind of basic logic class should be a possible course for high school students. Doesn't have to go very deep. Might cover things like:
* propositional logic
* validity
* truth tables
* what a well-formed argument is

Then you might cover some more specific things, such as:
* epistemology, philosophy of science & maths
* a history of logic, some relevant philosophy from people like Aristotle, Russell
* maybe first-order logic and set theory, some basic number theory (i'm talking stuff like proving even * even = even, irrational * rational = irrational).
* logic gates, binary, some other practical applications of logic to engineering
* dissection of arguments in politics and philosophy (perhaps focusing on things like law and ethics)

It doesn't have to include all of the above. Such a class would be enormously important and give students a crucial starting point for multiple disciplines, such as: philosophy, maths, engineering, law, polsci, literature, science.

>>7113999
That's not even a math class though. What subject would that fit under?

>>7114003
autism 101

>>7114003
It's not specifically Maths but I think its content is realistic enough for an average high schooler to learn something over the course of a year, and would be useful to Maths students, among other things.

Apologies if it's not feasible for the American schooling system; I am not American.

>>7113999
I think this course should be split up into separate courses for different subjects.

>>7113995
Not the original guy but I would be interested in hearing about some stuff of Seifert Pairing.

I remember being in 1st grade, and asking my teacher to tell us what's above a million. She didn't want to go over billions, or trillions, or anything like that.
I think about how much the US education system held me back and I wonder if I would have been useful if I was educated at the pace I wanted to go.

>>7113841
9th grade+ is actually pretty close to what I did in highschool. It was fan-fucking-tastic. Before abstract algebra I want super in to math. By the time I got to linear algebra I wanted to mathterbate all day

What do we even teach in Math in the US before 7th grade or so? I don't really remember, but I apparently learned how to do basic arithmetic at some point. I think we started doing pre-algebra in 7th grade, or something?

>>7114355
Oh man, math before 7th grade was complete crap.

Literally every year was 95% stuff we learned the previous year, 5% new and very easy stuff.

I only survived the system because I had parents willing to help me learn math outside of school. It's not hard to understand why kids in US can't do math for shit. If you aren't ahead in math by the time you get to 7th grade, you're kind of fucked.

>>7114127
lel, rekt
they took me all the way up to quadrillion in first grade
now i enjoy 300k+ starting with double phds in the quasi-topology of singularities by the Ricci flow of Barnett integrable function spaces

>>7114127
>hey teacher what's above a million
>a billion
>and after that
>a trillion
>and after that
>a quadrillion
>and after that
>a quintillion
>....
What's wrong with learning multiplication

>>7114514
The more numbers you know helps you in math.