Teaching Teenagers Graduate-Level Math — Part 1/7
How to guide high schoolers through advanced mathematical concepts
Can anyone learn difficult mathematical concepts? Is high school math enough to grasp the concept of infinity, find homeomorphisms between spaces, or apply mathematical analysis to problems? Based on my experience last summer, I am saying YES.
I got the amazing opportunity to teach at Discover Academy — a Summer school where I led a group of selected high school students through six 90-minute lessons on advanced mathematics.
Discover is a program designed for talented Czech and Slovak students that brings in fascinating people (…and me) from top universities and with interesting life stories to lead courses on a variety of topics.
Teaching someone who just learned logarithms how to prove logical inconsistencies in Math or how to compare bigger and smaller infinities is a challenge for sure. For anyone else facing a similar one, I am sharing here how my course was structured, how I kept the high schoolers engaged, and what I learned along the way.
1. Overthink the structure.
My first take is to really think through how you organize the course. There are centuries of Math to cover, plus you have to understand the basic definition to get to proving really cool things.
My goal was to show high schoolers that blindly inserting numbers into formulas, a few equations, and two-dimensional coordinate charts doesn’t accurately represent what Math really is about.
Of course, it is impossible to divide Math into mutually exclusive collectively exhaustive fields.
For simplicity, I’ve narrowed down what we covered to “top picks” for Algebra, Geometry, Analysis, Topology, Set Theory, Logic, and Applied Math, excluding many other gems like Category Theory, Representation Theory, or Probability Theory.
I structured the course such that in each lesson, students got to learn about several mathematical disciplines.
Course outline
- LESSON 1: “From sets to structures” (covering Abstract Algebra, Group Theory, the terms “structure”, “set”, “operations”)
- LESSON 2: “When Euclidean geometry is not enough” (covering Geometry)
- LESSON 3: “Some infinities are bigger than others” (covering Mathematical Analysis, Basic proof techniques)
- LESSON 4: “Bridges, donut cups and more” (covering Topology and Graph Theory)
- LESSON 5: “Math is not perfect” (covering Set Theory, Logic, Axioms and limitations of Mathematics)
- LESSON 6: “Is math found, or discovered?” (covering Applications of Math, Theory of Relativity — the topic students wanted really hard, and debate on the philosophy of Math)
In the next article, I will get into more detail with content of each of the lessons, including the practical examples, exercises, and students reception.
2. Make it interactive.
During each lecture, I encouraged discussions and debates. I frequently asked their opinions on the topics and how the given topic fits into what they are learning in high school.
⬛️ EXAMPLE: In the final lesson, they were to divide into two groups and debate about whether mathematics was discovered in nature, or invented by humans. They used arguments and examples from the history of Math that they learned along the way in my course.
I provided as many practical examples and exercises as possible (take “practical” with a pinch of salt regarding math :)).
⬛️ EXAMPLES: When discussing non-Euclidean geometries, students searched for its presence in nature. In the Topology lesson, each student received a city map containing bridges over rivers and was tasked with solving the Königsberg Bridge Problem.
3. Ask for frequent feedback
After each lesson, I asked students to write me anonymous feedback on paper notes. I told them to include questions they were too afraid to ask or problems they wanted to solve in the next lesson.
In the next lesson, I tried to incorporate their feedback (which was both positive and negative).
If students asked questions during a lesson and I skipped some due to time limitations or had to think more about the answer (which happened more than I expected with high schoolers), I always opened the next lesson by getting back to the questions.
⬛️ EXAMPLE: I taught students Euclid’s Postulates (which are important in Plane Geometry), and they had trouble understanding one of the postulates, so for next lesson, I brought visual examples of that axioms.
4. Travel in time.
When going through education, I always try to look at the “bigger picture” and keep the perspective in mind.
To keep the broader context in mind, I occasionally added to the beginning of a lesson a quick exercise where I sorted students into groups and gave them papers with dates of what we learned in the previous lessons, for example: “Proof that the square root of two is not a rational number” or “Beginning of group theory”.
5. Be ready for adding and removing content
It is challenging to prepare content that covers 100% of the lesson. Not 86%, not 140%. You have to have in mind, which topics are the crucial ones, and which are the backup.
Sometimes students get caught up in interesting discussions about something they except the least.
6. Be inter-disciplinary
I believe it is most efficient to study different disciplines (or subjects at school) together in context, not as separate entities because you lose important interconnections that way.
When going through education, I always try to look at the “bigger picture” and keep the perspective in mind.
Hence, I occasionally added a quick exercise to the beginning of a lesson where I sorted students into groups and gave them small paper notes with dates of important events we learned in the previous lessons, for example: “When was it proven that the square root of two is not a rational number” or “Beginning of group theory.” Students attempted to sort these events chronologically. This way, we summarized the development of Math and connected it with other subjects and historical periods that had reasons behind certain discoveries.
I also connected Math with other disciplines.
⬛️ EXAMPLES: In the last lesson, I showed students the highly demanded use of Math in Special Relativity. I also talked about the first proofs done by computer software or applying Math in cryptography.
7. Appeal to different types of learners
You should use a variety of teaching styles to appeal to different types of learners. Remember that some students are more extroverted and like to ask questions and discuss, some may need more time (even time alone) to deeply understand things, some prefer visual examples.
⬛️ EXAMPLES
From what I observed among mathematicians, you can use a proxy and divide people doing Math into
“People with high computational power” and
“People attracted to abstract nonsense.”
The first type usually calculates very quickly but struggles when dealing with higher-dimensional spaces and concepts needing a great level of abstraction.
The second type makes 10 mistakes on 3 lines of calculations, but can easily visualize any bizarre mathematical space in their head.
I tried to include topics and exercises for both of these thinking styles. (Indeed this is just my own proxy and personal observation).
Here are illustrations of different exercises for people who prefer different styles of thinking in Math:
Conclusion
The students grasped the math content surprisingly well and posed complex questions.
Reflecting on my high school days, I half-expected eye rolls or laughter. Maybe it was the selection bias (how many students voluntarily choose to spend summer days taking courses), but they were really proactive, asked advanced questions, and overall were nice humans.
I definitely wanna do this again.
