👨‍🏫 Instructor: Swarnim Kalbande

👨‍🏫 TAs: Kruti Sutaria, Michael Hollander

🗓 6 classes from June 1st - July 6th, 2024

🕰️ Saturdays 2pm-4pm

🗺 Pier 57 Classrooms

💰 Tuition: $100-$300 sliding scale (with need-based adjustment)

📋 Apply here for the in-person classes: https://forms.gle/XRkBFKPKLFMKebQ78 - there may be a waitlist and you will be notified via email if you make it into the course

👥 30 students

About

“Math is hard,” Barbie famously declared. Well, Barbie was right, but math is not uniquely hard. Playing the violin is hard, hitting a baseball is hard, and learning a second language is hard. What seems to make mathematics different from playing the violin or learning Chinese is that the struggle to play violin doesn’t make people feel defeated and dumb. Somehow, when we encounter difficulties in mathematics, our natural tendency is to retreat, to think it’s too hard, we’re not smart enough, or we’re not “math people.” We allow ourselves to be defeated by the difficulty. We understand that learning to play the violin requires making many, many hours of horrible screeching sounds, that learning to speak Chinese means making error after error and not being understood. But, somehow, when it comes to mathematics, we fear making mistakes. We imagine that there are “math people” to whom it is all transparent and, if it doesn’t come to us immediately, we must not be one of them. There are no such people. People who succeed in mathematics, like people who learn a musical instrument or a new language, spend a lot of time not understanding and feeling frustration. The path to understanding in mathematics necessarily involves, in the words of Steve Klee (4), being “willing to struggle.” It is strange that people do not understand this about mathematics when it is commonplace in essentially every other field of human endeavor. The people whose stories are in this book clearly understand this fact. Some of them, for example Lola Thompson (7) and Laura Taalman (8), avidly embrace the struggle, they seek out the experience of frustration and confusion because they have realized that persistence in the face of difficulty leads to the rewards of learning and growth.

- From the foreword of Living Proof

This is a course that focuses first on collaboration and friendship amongst peers and then second on having fun. Math is just an excuse to do that and also learn something cool together along the way.

I loved math (and hated it sometimes too) because it provides valuable tools to our mind and helps make thinking itself easier and more powerful. There's the obvious ways in learning the basics of proof writing which helps us learn how to concisely and strongly present our argument for any true fact by building upon other true facts and axioms, which is analogous to how we logically deduce, argue and discuss anything in life. Similar to any other skills, practicing this skill will help train your brain with this infrastructure until it becomes second nature and your thinking and talking becomes clearer, more concise and cohesive.

The less obvious ways are by learning problems in real life whose solutions can be figured out with math more often than you'd think, and building an intuition that helps these small calculations become second nature so they can augment your thought process and take fewer cores of your brain while doing so to free up your thinking. Similar to how when you first learn chess you're constantly thinking about how each piece moves but once that becomes second nature your brain frees up to actualy think of the strategy and tactics while you play.

This course takes my favorite fun and practical topics from various courses I took at Carnegie Mellon, math concepts that more directly help with everyday life and thinking rather than learning integration and differentiation and memorizing formulae.

By the end up of this course you will have developed a better intuition for the unintuive and see a path to solving challenging mathematical problems - all the while having fun with others doing the same, supporting each other in this struggle.

Each session of this class is designed with collaboration and rententive learning, with the lecture being intermingled with solving problems in groups with concepts you just learned, taking notes after from memory and not during the lecture, grabbing snacks or coffee with your classmates and even the final exam in format you've probably never seen (which the final exam only exists to show you what you've learned and you grade your own efforts and improvement rather than what fraction of the test you were able to do - if before you couldn't even solve one of these problems and now had the right ideas for multiple of them even if they didn't end with a solution that's an A).

Syllabus

Each class will be sets of 15min lecture then 5 mins of note taking and 20 mins of group problem solving with mine and TA help, with 2-3 such sets in each 2hr class. I will change the groups each set so you get to know every single person in the class well by the end of it.

See Appendix for example problems from each module to get a glimpse of what you will be learning.

Reference Materials used for this class, all publicly available at below links:

• https://www.math.cmu.edu/~jmackey/151_128/bws_book.pdf
• An Introduction to Probability Theory and it’s Applications by William Feller
• Carnegie Mellon’s 15-151 and 15-251 Courses
• Textbook for 15-151: An Infinite Descent into Pure Mathematics by Clive Newstead

Inspired by professors John Mackey of 15-151 and Anil Ada of 15-251

Evaluation

Final exam - problem solving assessment:

• 30 mins - Individual ideas on solutions for problems
• 60 mins - Combine ideas in groups of 3 to start forming full solutions (but with test put aside)
• 30 mins - Individually write up the solutions to all the problems on the test, recalling from group session
• Followup office hours: Class discusses solutions to all the problems

The last class will consist of a final exam with really hard math problems that’s done in 3 phases - first you spend 30 mins trying the problems yourself and writing down your ideas on solving them. Then you will be assigned groups of 3 and combine your ideas to start forming solutions to some of the problems for 60 mins. Finally, in the third phase everyone goes back to finishing their test individually using the ideas from the group discussion. Grades will be released shortly after the final exam.

About me

Swarnim Kalbande

skalband@alumni.cmu.edu

Email me with any questions, issues or if you just wanna grab a coffee!

zirus23 on discord

I'm Swarnim, a 24-year-old Software Engineer with a background in CS from Carnegie Mellon. Beyond my professional life, I'm passionate about a range of hobbies from writing fantasy novels and running D&D games, to cooking butter chicken and hosting parties. What I value most are the connections I make and cherish spending time with friends and new acquaintances alike. I first encountered Fractal in an AI research course I took here and was captivated by its community of peers teaching each other. I've always had an intuitive grasp of math, which helped me coast through college without needing to delve deeply into studying. However, this approach made me miss out on the joy of learning. I want to revisit those missed opportunities by teaching a course on subjects I wish to revisit myself. Outside of work, I’m deeply involved in worldbuilding for my novels and D&D campaigns, aiming to create a comprehensive fantasy universe. This pursuit not only fuels my creativity but also gives my other hobbies like sketching/drawing some ground roots. I'm always looking for people to collaborate on creative pursuits with (including designing this course!), if you're interested I'm always available to chat over a coffee.

Appendix - Example Problems

Q. We have two identical glasses. Glass 1 contains x ounces of wine, glass 2 contains x ounces of water (x > 1). We remove one ounce of wine from glass 1 and add it to glass 2. The wine and water in glass 2 mix uniformly. We now remove 1 ounce of liquid from glass 2 and add it to glass 1. Prove that the amount of water in glass 1 is now the same as the amount of wine in glass 2.

Q. Prove that n^3 − n is divisible by 3 for all all natural numbers n (hint: use induction)

Q. Mr. Jones has two children; at least one is a boy. What is the probability he has two boys? hint: it’s not 1/2 or 1/4 or 3/4

Q. There are 251 Bad Guys and 251 Good Guys. The 251 Bad Guys line themselves up in a room, each carrying a white hat and a black hat. One by one, Good Guys come into the room. When Good Guy i comes in (1 ≤ i ≤ 250), one of the hatless Bad Guys waves his hand. (The Bad Guys get to choose who waves.) Good Guy i then gets to tell the waving Bad Guy to put on either his black hat or his white hat. Following this, the Good Guy leaves, never to be seen again (i.e., he cannot communicate with the future incoming Good Guys). After the first 250 Good Guys have entered the room the game changes. At this point, only one Bad Guy is hatless—let’s call him the Bad Captain. The Bad Captain must put on a hat now, but he gets to decide the color. Finally, the last Good Guy enters the room—let’s call him the Good Captain. In some sense, his task is to guess—based on the hat colors he sees—who the Bad Captain is. More precisely, the Good Captain must announce some subset J of the Bad Guys, and J must be guaranteed to contain the Bad Captain. The Good Captain then pays a price of |J| dollars. For example, the Good Captain is allowed to pick J to be all 251 Bad Guys. However, this costs a lot: $251. It’s possible for the Good Guys to do better. The Good Guys are trying to use a strategy that guarantees they’ll never pay more than some C dollars. (They can decide on a strategy together before the game starts.) The Bad Guys are trying to use a strategy that forces the Good Guys to pay as much as possible. Find and prove a strategy for the Good Guys that keeps their cost C as small as you can. You certainly do not have to prove your C is as small as possible (in fact, we personally don’t know how to do this). You should just strive to get C small. Getting it in the low triple-digits is a reasonable start. Getting it in the low double-digits would be a lot better.

Q. You have a jar filled with marbles. Some are orange and some are blue. You also have a supply of marbles of each color outside the jar. We carry out the following process. We reach in and pick two marbles out of the jar. - If they are both orange we remove them and add a blue marble from our external supply of marbles. - If they are both blue we put one back and remove the other. - If one is blue and one is orange we put the orange one back and remove the blue one. We continue until there is only one marble left. What is the color of this last marble? hint: The answer depends on the initial situation. What we want is a precise description of the circumstances under which the last marble is guaranteed to be blue and a description of the circumstances under which the last marble is guaranteed to be orange.