A Re-Introduction to Mathematical Thinking

Intro

“Math is hard,” Barbie once famously said. She was right, but math isn’t uniquely difficult. Playing the violin, hitting a baseball, or learning a second language—all are hard. What sets math apart, however, is the perception. When we struggle with the violin, we understand that it’s part of the learning process. But when we struggle with math, many of us retreat, thinking we’re not “math people.” In reality, there’s no such thing. People who excel at mathematics aren’t magically gifted—they’ve simply persisted through the struggle. Like any skill, progress in math comes from embracing confusion and frustration, knowing that this struggle is the path to growth.

- Paraphrased from the foreword of Living Proof

Too often, math courses—even at top institutions—don’t create the conditions for this kind of growth. The focus on grades, performance, and memorization can suffocate curiosity and discourage deeper understanding. Many courses are designed around fixed mindsets: you're tested on how much you can recall under pressure, not how much you’ve truly learned. This environment, where success is measured by a score rather than by progress, often leaves students frustrated and disconnected from the joy of learning. Even at prestigious schools like MIT, Stanford, or Carnegie Mellon, math is often reduced to a race for grades, with little room to experiment, fail, and grow.

That being said, Carnegie Mellon does one thing very well, and it’s something I’ve carried into this course: they focus on teaching fundamentals. The best parts of my education there came from classes that didn’t just teach formulas or techniques but explored the very nature of problem-solving. Instead of memorizing processes, we were encouraged to understand the intuition behind them—how these ideas were created and why they work. This deeper, conceptual understanding is what truly sticks, and it's this approach that forms the foundation of this course.

This course is designed from the ground up with a growth mindset at its core. Every element, from the structure of the lectures to the final exam, is carefully crafted to encourage exploration, collaboration, and, most importantly, meaningful learning. We focus not on getting the “right answer” as quickly as possible, but on developing the intuition behind mathematical concepts, so you can apply them creatively in new contexts.

Here’s how I do it:

Lectures: Rather than passive listening, our lectures are interactive. Each session blends instruction with group problem-solving, so you’re immediately applying what you’ve learned. Instead of simply learning how to solve a particular type of problem, you’ll explore why these problems are structured the way they are, building an intuitive understanding that goes beyond rote memorization. This echoes the best of Carnegie Mellon’s approach—learning to think, not just learning to solve. Each class is a mix of 15-minute lectures, 5 minutes of reflective note-taking, and 20 minutes of group problem-solving with help from me and the TAs. This structure, repeated 2-3 times in each session, allows you to immediately apply what you’ve learned and reinforce the material.

Growth-Oriented Assignments: In traditional settings, assignments often measure how well you can apply formulas, not how deeply you understand them. Here, assignments are designed to stretch your thinking and reward progress over perfection. Your growth in problem-solving, creativity, and conceptual understanding is what matters, and mistakes are embraced as essential stepping stones. Collaboration with peers is encouraged, not only to solve problems but to share approaches and thought processes that deepen learning.

The Exam: The final exam isn’t a high-pressure test of recall but an opportunity for self-assessment. Instead of racing against the clock to check boxes, you’ll reflect on how much you’ve grown and how your problem-solving approach has evolved. The focus is on progress: if you began the course unable to tackle certain problems and now find yourself approaching them with confidence and new insights, you’ve succeeded. This exam mirrors real-life problem-solving, where growth matters more than immediate results.

Research-Backed Learning Techniques: This course draws on proven learning strategies, like spaced repetition and active recall, to enhance retention and understanding. For example, instead of taking notes during lectures, you’ll reflect and write down your thoughts afterward—a technique shown to improve comprehension. Each topic is revisited at key intervals to ensure concepts become ingrained, not forgotten after a test.

Through this carefully designed approach, you’ll not only master key mathematical concepts, but you’ll also transform the way you think and approach problem-solving. Math will no longer feel like an obstacle; instead, it will become a tool for unlocking new perspectives and solving real-world challenges.

This course is a reflection of my own journey, where I realized that focusing on grades often stifled learning. By drawing on the best aspects of Carnegie Mellon’s fundamentals-focused approach and combining it with a growth mindset and cutting-edge teaching techniques, I’ve created a space where (I hope) you can embrace challenges, grow alongside others, and truly enjoy the process of learning.

Now is the perfect opportunity to take charge of your learning and growth. Enrolling in this course isn’t just about the material—it’s a personal commitment to embracing a growth mindset. There will always be reasons to delay or feel unsure, whether it’s time constraints or the fear of stepping out of your comfort zone. But growth happens when we choose to move forward despite those challenges. By joining today, you’re investing in a learning experience designed to help you overcome obstacles, unlock new ways of thinking, and make real progress—both in your problem-solving skills and your approach to life. Take that step for yourself, and start building the foundation for lasting growth.

Syllabus

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

Sep 22: What is a Proof?

Sep 29: Fun with Induction

Oct 06: Puzzles are just math

Oct 13: Probably probability

Oct 20: Famous fun problems that don't make sense

Oct 27: Final Exam (semi-collaborative)

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

About The Instructor

See Swarnim Kalbande

Logistics

6 classes from September 22nd - October 27th, 2024

Time: Sundays 10am-12pm EDT

Location: Online, but students in NYC may gather in Pier 57 Classrooms

Tuition: $200 (with need-based adjustment upon contact)

30 in-person students in NYC, shooting for a total of 50 including online students

Evaluation

Your grade will be a self-assessment, with 70% based on your effort on homework and 30% on how you feel you've grown throughout the course, particularly in relation to the final exam. The final exam will be graded on your ideas and approaches, not just on getting the right answers.

Final Exam - Problem-Solving Assessment:

• 30 mins: Work individually to brainstorm ideas and solutions for the problems.

• 60 mins: Collaborate in groups of 3, combining ideas to develop full solutions (without referencing the test).

• 30 mins: Individually write up the solutions to all problems, recalling insights from the group session.

After the exam, we'll discuss the solutions in follow-up office hours, where you'll have the chance to reflect on your growth. The last class will feature challenging math problems tackled in these three phases, ensuring you get the chance to collaborate, think creatively, and demonstrate how far you’ve come. Grades will be released shortly after the final exam.