Teaching Teenagers Graduate-Level Math — Part 2/7
How to guide high schoolers through advanced mathematical concepts
In the previous part, I described my experience with teaching students at a summer academy. I have described what methods I used to keep students engaged, what fun exercises and debate topics we did, and how I made them really understand difficult concepts. Now I am getting to the content of the individual lessons.
Setting
I taught six lessons, 90 minutes each. I gave each lesson a special topic and each lesson corresponded to 1–3 mathematical disciplines. I combined debates, exercises, theory, and more exercises/examples.
Lesson 1
- Name: Math is not just numbers. From sets to structures
- Disciplines covered: Set Theory, Algebra, Mathematics overall
How the lesson went:
Exercise
Our first “exercise” was an ice-breaking activity at the beginning (which I forgot) and introducing each other.
Exercise
Next, each student gets a few minutes to write the best possible definition of Math on paper. Then we talked about them and debated. I did a follow-up and introduced Mathematics as a discipline.
Theory
We learned the concepts of set, and “extra features” on sets.
The extra features are basically what can be done with elements of a given set. Examples:
- You can do the operations +, -, *, : with the set of natural numbers
- You can compare elements of natural numbers
- You can determine the distance between elements of natural numbers. We can tell the number 3 is more distant from 1000 than from 10
Exercise
Students got examples of well-known sets, e.g. Natural numbers, Real numbers, odd numbers, even numbers, a circle (as a set of elements with equal distance from a given point), or a family tree (as a set of family members with relations to each other).
They were to think about what “features” (e.g. operations, relations, or distances) these sets had.
Theory
I introduced the term “Mathematical Structure”, which is a set equipped with some extra “features”. (Btw “Mathematical Structures” is the exact name of my Master’s degree study field.)
I told students that different mathematical structures are studied in different mathematical disciplines (corresponding to courses at university).
I started with one of such disciplines — Abstract Algebra which studies algebraic structures. That is a lot of known and super-important “things”, like groups, rings, fields, modules, vector spaces, lattices, and algebras over a field.
I explained groups first as it is one of my favorite mathematical structures.
Exercise
I have shown students a few simple examples of groups. We went through the group axioms and checked that they apply.
- Group of integers (Z) where the elements are integer numbers, and the group operation is the addition
- Associativity: e.g., (2 + 10) + 3 = 2 + (10 + 3) etc. for any three elements of Z
- Identity Element: The identity element is 0 because a + 0 = a for any integer a.
- Inverse Element: The inverse of an integer a is -a, as a + (-a) = 0.
2. Symmetric Groups where the elements are permutations of the group, and the group operation is the composition of permutations.
What the hell is “permuation”?
Imagine a rubic cube. Each move (“permutation”) you can do with the group. (Check Rubik’s cube group).
- Associativity: Permutation composition is associative because it doesn't matter if you first do moves “A” and “B” together and then move “C”, or if you make a move “A”, and then moves “B”, and “C”.
- Identity Element: The identity element is the permutation that leaves every element unchanged, e.g. you just don’t do anything to the Rubik’s cube.
- Inverse Element: The inverse for each move is the opposite move that returns the cube to the same state.
Exercise
I then showed the students other structures (e.g. Natural numbers) with a given operation and asked them why they are NOT groups.
Theory
Finally, I talked about the origins of Group theory and why was it invented.
Mathematicians tried to solve whether polynomials had equations, when, and how to tell that they do. (It is a longer story why that is important).
Imagine you have a simple quadratic equation like x^2 − 5x + 6 = 2. The roots of this equation are the values of x that make the equation true. In this case, the roots are x=2 and x=3.
Galois observed that the solvability of a polynomial equation is somehow related to whether the set of its solutions behaves like a group. (That was before the term “group” existed).
This topic is out of the scope of my lesson, but it’s good to know that Group theory is important for other parts of Math.
Teaching Teenagers Graduate-Level Math Series
- Part 1: Intro, methodology, context
- Part 2 (this one): Sets, Structures, Algebra, Groups
- Part 3: Geometry
- Part 4: Mathematical Analysis
- Part 5: Topology, Graph theory
- Part 6: Set Theory, Logic
- Part 7: Applied Math
