Advice on Proof-based Math Topics

[Diaz Lilahk]

Hello Everyone,

I have a handful of high school students that are all prospective math/physics majors and have pooled their resources to hire me to teach them a proof based math course because it has become apparent to them in my physics class that proof and derivations are important. Basically I meet with them 2 hours a week and run it like a socratic method or students have to prove the theorems with my limited guidance and so far I have covered the following topics:
1) GCD
2) Euclid's Lemma
3) Well Ordered Principle and Mathematical Induction
4) Fundamental Theorem of Arithmetic
5) Theorems of Elementary Arithmetic a*0=0, (-a)(-b)=ab etc.
6) Arithmetic mean - Geometric Mean Inequality
7) Pythagorean Theorem
8) Cauchy Schwartz Inequality
9) Irrationality of sqrt(p) where p is prime

I will have only a limited amount of time with these students and I need to decide on what topics would be most valuable for them to be exposed to, here is the list of topics, which do you think would be most valuable:

1. Conic Sections - going from the geometric definitions to the algebraic representations of conics
2. Proofs of Archimedes: Areas of Circle, Quadrature of the Parabola, On the Sphere and Cylinder
   
3. Exploring the completeness property of reals
4. Sets, Nested Intervals and the Uncountability of the Reals
   
5. Exploring sequences and series - in particular using telescoping series to derive Σi, Σi^2 , etc
6. Area under curves Riemann Sums
7. Counting and Binomial Theorem
8. Sets and the Axioms of Probability
9. Limits, Continuity (delta-epsilon proofs)
10. Differentiability
11. Properties of Exponential and Logarithmic Function
12. Vectors, Vector Spaces, Linear Operators

I am leaning towards not doing too many higher level topics because that is what a college class is for, and I don't want to give them a false impression that they really understand the material. Thank you, your help will be really appreciated.

Zaid Khalil

 

[Fredrik]

I would include 11 and 12, and probably 9 and 10. If there's time, I would include the basics of set theory (notation and simple proofs) and maybe some logic (notation and truth tables).

9. The high school course in Sweden doesn't define limits of functions or limits of sequences. I would like to start with limits of sequences, because they're easier, and discuss things like 0.999...=1. Then limits of functions.

11. Suppose that there's a way to define $a^x$ for all real numbers x. Prove $a^x\geq 0$, $a^x\neq 0$, $a>0$. (Conclude that there can't possibly exist an adequate definition when $a\leq 0$). Prove that if $a>0$, we have $a^0=1$, $a^{-x}=1/a^x$ and $a^{x-y}=a^x/a^y$. Tell them that the rule $(a^x)^y=a^{xy}$ holds as well, but can't be proved by these methods. (I think the way to prove it is to prove that it holds for all x,y in a dense subset of the real numbers, and then use the fact that there's a unique continuous extension to the set of real numbers; this is too difficult for high school students).

Cover the basics of logarithms. Explain that "the number of particles decreases exponentially" means that there exist real numbers $a$ and $N_0$ such that $0<a<1$, $N_0>0$ and $N(t)=N_0 a^t$ for all $t\geq 0$. Show how to rewrite the right-hand side as $N_0 2^{-t/T}$ and explain how to see that T is the half-life.

12. The high school course in Sweden includes the standard definitions of addition and scalar multiplication on $\mathbb R^2$, but doesn't prove, or even state, the properties of those operations (i.e. the vector space axioms). I feel that this is an excellent topic for students who are starting to learn proofs. You can prove a few of the axioms, and leave the rest as exercises. You don't have to mention the words "vector space axioms" if you don't want to. You can present them as theorems to be proved.

 

[Fredrik]

Another thing that can be fun is to use the real number axioms to prove elementary theorems like:

For all x,y,z, if x+z=y+z then x=y.
For all x, 0x=0.
For all x, -x=(-1)x.
For all y, if yx=x for all x then y=1.
For all y, if x+y=x for all x then y=-x.
For all x, -(-x)=x.
(-1)²=1
1>0

I would start with a brief discussion about "axioms". I would say that a list of statements defines a branch of mathematics: the branch in which those statements are true. I would emphasize that axioms aren't called that because they're "obvious" or anything like that. The statements that we call axioms are simply the ones that appear on the list of statements that defines the branch of mathematics that we have chosen to work with. This raises the question of how a statement can be considered true in one context and false in another. A simple example can answer that: $\forall x\exists y~ y^2=x$. This is true if our "for all" and "there exists" refer to positive real numbers, but false if they refer to real numbers (since -1 doesn't have a square root that's a real number).

I would also explain that the axioms can be motivated by examples from geometry. We're choosing this branch of mathematics, because of what we want to be able to apply it to. For example, commutativity of multiplication can be explained by saying that we want xy to be the area of a rectangle with bottom edge length x and side edge length y, and we want this rectangle to have the same area if it's rotated 90 degrees. The distributive law can be explained by cutting a rectangle in two pieces and saying that we want the area of the rectangle to be equal to the sum of the areas of the pieces.

 

[Diaz Lilahk]

Another thing that can be fun is to use the real number axioms to prove elementary theorems like:

For all x,y,z, if x+z=y+z then x=y.
For all x, 0x=0.
For all x, -x=(-1)x.
For all y, if yx=x for all x then y=1.
For all y, if x+y=x for all x then y=-x.
For all x, -(-x)=x.
(-1)²=1
1>0

I would start with a brief discussion about "axioms". I would say that a list of statements defines a branch of mathematics: the branch in which those statements are true. I would emphasize that axioms aren't called that because they're "obvious" or anything like that. The statements that we call axioms are simply the ones that appear on the list of statements that defines the branch of mathematics that we have chosen to work with. This raises the question of how a statement can be considered true in one context and false in another. A simple example can answer that: $\forall x\exists y~ y^2=x$. This is true if our "for all" and "there exists" refer to positive real numbers, but false if they refer to real numbers (since -1 doesn't have a square root that's a real number).

I would also explain that the axioms can be motivated by examples from geometry. We're choosing this branch of mathematics, because of what we want to be able to apply it to. For example, commutativity of multiplication can be explained by saying that we want xy to be the area of a rectangle with bottom edge length x and side edge length y, and we want this rectangle to have the same area if it's rotated 90 degrees. The distributive law can be explained by cutting a rectangle in two pieces and saying that we want the area of the rectangle to be equal to the sum of the areas of the pieces.

We have covered these. This is actually how I started the class and it was almost identical to the style that you mention. It's kind of funny because I have a student that took AP BC Calculus last year in a very memorization based course, and he was considering taking a similar Vector Calculus this spring so that he can have a head start on college and I have convinced him that it will only be more crap that he will memorize and then have to unlearn in order to learn it properly.