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A Proof

How to Understand Mathematical Proofs (Step-by-Step Guide for Students)

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If you’ve ever looked at a mathematical proof and thought, “I have no idea what’s going on,” you’re not alone.

For many students, how to understand proofs is where math starts to feel completely different. It’s no longer about solving equations or getting a final answer; it’s about understanding why something is true, and that shift can feel confusing, frustrating, and sometimes even discouraging.

This is something I’ve thought about a lot, especially after presenting my research at TCURC, where I focused on explaining mathematical ideas to a broader audience. One of the biggest challenges isn’t the math itself : it’s how we approach understanding it. 

In this guide, I’ll show you how to understand mathematical proofs step by step, using a practical approach you can apply to any proof-based course.

Quick Summary:

• Proofs show why something is true, not just that it is

• Focus on understanding each step, not memorizing

• Ask “why?” at every line

• Rephrase arguments in your own words

Why Do Proofs Feel So Hard?

The first thing to understand is that proofs aren’t supposed to feel easy at first.
Here’s why they often feel difficult:

• They’re not calculation-based
• Steps are often skipped or compressed
• They require understanding, not memorization

Unlike typical math problems, you can’t just follow a formula. Instead, you need to understand the reasoning behind each step. 

And then there are different kinds of proofs, which can read more about here.

A key idea:

Proofs don’t just show that something is true, they show why it must be true.

What a Proof Is Actually Doing

One way I like to explain how to understand proofs, especially to students, is by comparing them to essays.

Think of a proof like this:

• Thesis: What you’re trying to prove

• Body: The logical steps that support your claim

• Conclusion: The final statement that confirms the result

Just like in writing, each step should follow logically from the previous one. If a step doesn’t make sense, the whole argument starts to break down.

This perspective makes proofs feel less like random symbols and more like structured reasoning. You can find a nice flow-chart I made for my TCURC poster involving proofs and human thinking.

Step-by-Step: How to Read a Proof

Instead of trying to understand everything at once, break proofs into smaller, manageable steps.

1. Identify the Goal

Start by asking:

What is this proof trying to show?

Don’t move forward until you’re clear on the end goal.

2. Understand the Assumptions

What are you given?

This might include:

• definitions
• known theorems
• initial conditions

Knowing what you’re allowed to use is crucial.

3. Break It Into Chunks

Don’t read a proof from start to finish in one go. Instead:

• read a few lines
• pause
• make sure you understand that section

Think of it like reading paragraphs, not a wall of text.

4. Ask “Why?” at Every Step

This is the most important habit. For each line, ask:

Why is this true?

If you can’t answer that, that’s where your understanding needs work.

5. Rephrase in Your Own Words

After reading a section, try to explain it simply.

If you can’t explain it, you don’t fully understand it yet, and that’s okay. It just tells you where to focus.

A Simple Example

Let’s look at a basic idea:

Claim: The sum of two even numbers is even.
Step 1: Start with definitions

An even number can be written as:

• 2a and 2b, where a and b are integers.

Step 2: Add them

2a + 2b = 2(a + b)

Step 3: Interpret the result

Since a + b still an integer, the result is of the form 2 times an integer, which means it’s even.

What matters here isn’t the result; it’s the reasoning.

At each step, you should be asking:

• Why can we write numbers this way?
• Why does factoring work?
• Why does this form imply the result?

That’s how you actually understand the proof.

Common Mistakes Students Make

If you’re struggling with proofs, chances are you’re doing at least one of these:

• Reading passively instead of engaging with the logic
• Skipping steps that feel “obvious”
• Memorizing proofs instead of understanding them
• Not questioning each step

These habits make proofs feel harder than they actually are.

How to Get Better at Proofs

The good news is that proof skills are learnable.

Here’s what actually helps:

• Practice writing your own steps, even if they’re messy
• Compare your reasoning to solutions
• Focus on understanding, not speed
• Talk through proofs with others

From my experience as a TA, one of the biggest differences between students who improve and those who struggle is this:

The best students actively engage with the logic instead of just reading it.

Why This Actually Matters

It’s easy to think proofs are just an academic exercise, but they build something much more important:
structured thinking.

The same skills used in proofs: clarity, logical flow, and justification, apply far beyond math:

• problem-solving
• decision-making
• communicating complex ideas

There are even tools like LEAN that formalize proofs to verify correctness, showing how this type of reasoning extends into computing and real-world systems.

Final Thoughts

Understanding proofs isn’t about being naturally “good at math.”

It’s about learning how to:

• break things down
• question each step
• think logically and clearly

It takes time, and it can feel uncomfortable at first, but that’s part of the process.

The more you practice, the more proofs start to feel less like a mystery and more like a structured way of thinking.

If you’re currently taking a proof-based course like MAT157 or similar, or just starting to encounter proofs, don’t worry if it feels difficult at first. I myself did not understand proofs fully until my second year. 

You can read here about my experience in a proofs-based course here.

But that’s completely normal, and it’s something you can get better at. 

Most people struggle with proofs not because they aren’t capable, but because they were never taught how to think through them.

 

Once you start treating proofs as a process instead of a mystery, everything changes.

TCURC 2026

Presenting My Research at TCURC: Explaining Mathematical Proofs to a Broader Audience

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Presenting My Research at a Conference

I recently had the opportunity to present my mathematics research at an undergraduate research conference, and it turned out to be one of the most rewarding academic experiences I’ve had so far. It was a great opportunity not only to share my work, but also to hear about projects from students across many different disciplines.

Undergraduate research conferences are interesting because they bring together people from fields that normally don’t interact much. You might walk past posters on biology, psychology, computer science, economics, or philosophy, all in the same room. That also means you must think carefully about how to explain your work to people outside your field.

For mathematics, that can be especially challenging.

Research Summary

  • Conference: Trinity College Undergraduate Research Conference (TCURC)

  • Field: Mathematics

  • Topic: VERIFYING MATHEMATICS BY MACHINE – TYPE THEORY AND COMPUTER CHECKED PROOFS

  • Focus: Computer-checked proofs enhance mathematical rigour by eliminating hidden assumptions and enforcing complete logical transparency.

Presenting my mathematics research poster at the Toronto Conference for Undergraduate Research (TCURC)
 
Explaining Mathematical Proofs to a General Audience

One of the questions I was asked during my poster session was how my research connects to fields beyond mathematics. A comparison I used was between mathematical proofs and essays.

In a typical essay:
• The thesis is the claim you want to support
• The body paragraphs provide evidence and reasoning
• The conclusion summarizes what you have demonstrated

A mathematical proof follows a very similar structure:
• The theorem is the statement we want to prove
• The logical steps support the argument
• The final statement confirms that the theorem has been proven

The key difference is that mathematics requires every step to follow strict logical rules. In that sense, proofs are essentially structured arguments built entirely on logic.

Because of this, the logical thinking developed in mathematics connects to many other areas, from computer science to the social sciences.

Explaining complex ideas clearly is an important skill in mathematics. I’ve also seen how valuable this is while teaching as a mathematics teaching assistant, where communicating logical reasoning to students is essential.

When Math Appears in Everyday Life

Another example I discussed during my presentation involved sphere packing, a classic mathematical problem known as the Kepler conjecture.

The question is surprisingly simple:

What is the most efficient way to pack spheres together in three-dimensional space?

The answer turns out to be the familiar pyramid-style arrangement you might see when oranges are stacked in a grocery store. This structure packs spheres together as densely as possible.

What makes the story fascinating is that although the conjecture was proposed in 1611, mathematicians spent centuries trying to prove it rigorously. Eventually, computer-assisted proofs and formal verification tools helped confirm the result. This means that something as ordinary as the way fruit is stacked in a store reflects deep mathematical reasoning.

As I joked during my presentation:

“So the next time you see a box of oranges stacked like that, math is the reason why.”

Conversations That Made the Experience Memorable

One of the most rewarding parts of presenting was the conversations with people who stopped by the poster.
Many asked thoughtful questions, including where research like this might go in the future. That question is actually harder to answer than it sounds, because mathematics often opens many different paths at once.
Research related to logic and proofs can lead to areas like:

• automated theorem proving
• formal verification of mathematics
• computer-assisted proofs
• connections with computer science and artificial intelligence

Thinking about these possibilities made me realize how broad the landscape of mathematical research really is.

Why Undergraduate Research Matters

Events like TCURC highlight the value of undergraduate research. They allow students to:

• share their work with a broader audience
• practice explaining complex ideas clearly
• receive feedback and new perspectives
• see how research connects across disciplines

For me, the experience reinforced something important: mathematics is not just about solving problems, but about communicating ideas and exploring possibilities. And sometimes, those ideas show up in places you might never expect, even in a box of oranges at the grocery store.

What I Learned from Presenting at TCURC

Presenting my research at TCURC reinforced how important communication is in mathematics. Explaining proofs to people outside the field required finding intuitive comparisons and everyday examples. Conversations with attendees also highlighted how mathematical logic connects to many different disciplines, from computer science to the social sciences.

Final thoughts

Presenting at TCURC was both exciting and rewarding. It was great to discuss ideas with people from different backgrounds and to see the wide range of research being done by undergraduate students.
Experiences like this remind me that research is not just about the final result, it’s about the conversations, questions, and curiosity that move ideas forward.

Grading_Math

How Math Assignments Are Really Graded (From a TA’s Perspective)

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University math grading can feel mysterious.

Many students assume grading is rigid, purely answer-focused, or even arbitrary. After grading hundreds of assignments across multiple first-year calculus sections, I can confidently say that this assumption is wrong.

Grading in mathematics is not just about correctness. Many students don’t fully understand how math assignments are graded at the university level. It’s about clarity, reasoning, structure, and interpretation.

As a Teaching Assistant at the University of Toronto, I’ve graded hundreds of assignments per term, primarily in first year calculus. This is a perfect opportunity to see high school thinking vs college thinking. Patterns emerge quickly. Certain mistakes appear repeatedly. Certain strengths stand out immediately. And the difference between an average solution and a great solution is often much smaller, and much simpler, than students think.

How Math Assignments Are Graded in University

Why Clear Structure Matters in Math Assignments

One of the biggest misconceptions about university math grading is that complexity earns marks.

It doesn’t. Clarity does.

When grading math assignments, I’m asking:

  • Can I follow the student’s reasoning?
  • Does each step logically follow from the previous one?
  • Is the solution organized in a coherent way?

A well-structured solution doesn’t jump from step 1 to step 7 without explanation. Even in computational courses, graders need to see how you move from setup to execution.

You don’t need more steps.

You need clearer ones.

Why Correct Setup Often Matters More Than the Final Answer

In computational and interpretation-based courses, the setup is often where most of the marks live.

If a student:

  • Identifies the correct formula
  • Sets up the mathematical expression properly
  • Applies the appropriate method

they usually receive substantial credit, even if a small arithmetic mistake happens later.

On the other hand, if the setup is incorrect but the final answer happens to be right, that’s usually coincidence. And coincidence does not earn marks.

The grading process rewards mathematical reasoning, not lucky outcomes.

Why Showing Your Work Matters in University Math

Another common misunderstanding about grading in mathematics is that writing less makes your solution look more “confident.”

In reality, graders need to see your reasoning.

If too many steps are skipped, we’re left wondering:

  • Did you understand the rule you used?
  • Was this a correct application?
  • Is the conclusion justified?

Showing your work doesn’t mean writing paragraphs. It means making your logic visible.

Math is not just about being correct, it’s about being convincing.

Interpretation Questions Are Not Just Computation

In applied math courses, interpretation questions are often where students lose marks unnecessarily.

Students compute the number… and stop.

But interpretation questions are asking something different:

  • What does this result mean?
  • What does it tell us about the situation?
  • What conclusion can we reasonably draw?

For example, if you compute a rate of change, you should explain what that rate represents in context.

Strong math assignments connect calculation back to meaning.

That connection is often what separates average work from excellent work.

Common Mistakes I See When Grading Math Assignments

After grading hundreds of papers, a few recurring patterns stand out:

  • Writing only the final answer with no work shown
  • Plugging numbers into formulas without explaining why that formula applies
  • Ignoring what the question is actually asking
  • Failing to interpret results in context
  • Losing clarity through messy notation

None of these issues require more intelligence to fix.

They require more attention to communication.

Because at its core, a math assignment is structured communication of reasoning.

What Separates a great Math solution

The strongest assignments are rarely the longest.

They are:

  • Cleanly structured
  • Logically ordered
  • Clear in notation
  • Explicit in reasoning
  • Concise but complete

The best students don’t necessarily write more, they communicate better.

There’s a quiet confidence in well-organized mathematics. Every step has purpose. Nothing is decorative.

Clarity reflects understanding.

How Grading Changed the Way I Approach Mathematics

Grading math assignments has changed how I write mathematics myself.

It has made me more attentive to:

  • Structure
  • Logical flow
  • Clear setup
  • Explicit interpretation
  • Writing for an audience

Clear thinking produces clear writing. And clear writing often signals clear thinking.

Mathematics is not just about arriving at the right answer ; it’s about demonstrating how and why that answer makes sense.

That’s what graders are really looking for.

Frequently Asked Questions About Math Grading

Do math graders only care about the final answer?

No. Most grading schemes allocate significant marks to correct setup and reasoning. The final answer is important, but it is rarely the only thing that matters.

Do small arithmetic mistakes ruin your grade?

Not usually. If your setup and method are correct, you typically receive substantial partial credit.

How can I improve my math grades quickly?

Focus on structure, clear reasoning, and interpretation. Make your thinking visible. Don’t treat math like a guessing game; treat it like an argument.

Final Thoughts

If you’re trying to improve your math grades, shift your mindset.

Instead of asking:

“How do I get the right answer?”

Ask:

“Can someone else clearly follow my reasoning?”

That small shift often makes a measurable difference.

If you found this helpful, consider sharing it with someone taking a math course this term.

And if you’re interested in how structured thinking applies beyond the classroom, you might also enjoy my project analyzing Spotify data using regression on this site.

302 pic

Dancing to the Data: Exploring Spotify Danceability with Linear Regression 

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A STA302 Project at the University of Toronto 

Dancing to the Data: Exploring Spotify Danceability with Linear Regression 

Danceability is one of the most fascinating—and subjective—qualities of music. Why do some songs make people want to move instantly, while others don’t, even when they share similar tempos or genres? With the growth of music streaming platforms like Spotify, we now have access to detailed audio features that allow us to explore this question using data. 

This blog post summarizes a final project completed for STA302 (Methods of Data Analysis II) at the University of Toronto, where my friends and I used multiple linear regression to investigate which musical and popularity features are most strongly associated with a song’s danceability.  

Project Motivation 

Spotify defines danceability as a score based on musical elements such as tempo, rhythm stability, beat strength, and overall regularity. While this metric is widely used in music recommendation systems, it raises an interesting analytical question: 

Is danceability driven more by musical structure or by popularity-related factors? 

Using statistical modelling, this project aimed to determine which features truly matter when predicting how danceable a song is. 

 

Data Collection 

The analysis used the Top Spotify Songs 2023 dataset, sourced from Spotify’s public Web API and published on Kaggle. The dataset contains 952 unique songs, each with: 

  • Machine-generated audio features (e.g., danceability, valence, energy, BPM, acousticness, speechiness) 
  • Popularity metrics (streams, playlist counts) 
  • Consistent, complete measurements suitable for regression analysis 

Because the data are publicly available and contain no personal information, the project posed minimal ethical risk. 

 

Exploratory Data Analysis (EDA) 

Initial exploratory data analysis focused on: 

  • Assessing variable distributions and skewness 
  • Examining correlations between predictors 
  • Identifying influential observations and potential data quality issues 

This step informed later modelling decisions, including transformations and predictor selection. 

 

Methodology 

To model danceability, we fit a multiple linear regression (MLR) model following a structured, statistically rigorous workflow taught in STA302 at the University of Toronto: 

Predictor Selection 

  • Removed non-significant predictors using t-tests on regression coefficients 
  • Applied subset selection, comparing: 
  • Adjusted R² 
  • AIC / AICc 
  • BIC 
  • Considered forward, backward, and stepwise selection methods 
  • Compared nested models using partial F-tests (ANOVA) 

Model Assumptions 

Key linear regression assumptions—normality, linearity, and homoscedasticity—were evaluated using residual diagnostics. When violations were detected, we applied Box–Cox and log transformations, resulting in well-behaved residuals. 

 Here’s the Rmd file with our code.

Results 

The final model demonstrated strong overall significance: 

  • F(8, 943) = 46.58, p < 2.2 × 10⁻¹⁶ 
  • Adjusted R² = 0.277 
  • All retained predictors were significant at p < 0.01 
 

Key Findings 

  • Valence and speechiness were strong positive predictors of danceability 
  • Acousticness, liveness, energy, BPM, mode, and log(playlists) showed significant negative associations 
  • Popularity metrics contributed relatively little explanatory power compared to musical and emotional features 

These results suggest that emotional tone and rhythmic qualities play a central role in how danceable a song feels. 

 Here’s our poster summarizing our findings:

 

Screenshot 2026-01-04 at 19.46.16

Conclusions and Limitations 

This STA302 project demonstrates how linear regression provides an interpretable framework for understanding complex, subjective outcomes like musical danceability. The findings align with prior research, reinforcing the importance of valence and rhythm-related features. 

However, the model has limitations: 

  • Potential nonlinear relationships not fully captured by linear regression 
  • Skewed popularity metrics 
  • A dataset limited to already-popular 2023 tracks 

Despite these constraints, the analysis shows that musical structure and emotional content matter more than raw popularity when predicting danceability. 

Final Thoughts 

This project highlights the power of classical statistical methods in modern data science contexts. By combining Spotify’s rich audio features with rigorous modelling techniques learned in STA302 at the University of Toronto, it’s possible to gain meaningful insights into how we experience music—and why some songs make us want to dance. 

One of my many notes

MAT157 at the University of Toronto: An Honest Review from a UofT Mathematics and Statistics Student

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If you’re considering taking MAT157: Analysis I at the University of Toronto, you’re probably already aware it has a bit of a reputation. Whether you’re a math enthusiast, a theoretical dreamer, or just someone looking to challenge yourself, MAT157 is not your average first-year calculus course, and I’m here to break down exactly what to expect.

As a UofT student who took the course in 2023-2024, here’s my honest review of MAT157, what it covers, how it’s structured, and whether it’s worth the hype (or the headache).

Quick Summary:

• Difficulty: Very high  

• Workload: Heavy (proof-based weekly assignments and exams)  

• Recommended for: Strong interest in pure math  

• Not ideal if: You prefer computation over theory

What is MAT157?

MAT157 is the highest first-year calculus course at UofT. It’s officially titled Analysis I, but many students know it as “the proof-heavy calculus class.”

Course focus: you can find the official description here, but roughly covers real analysis fundamentals and proof-based approaches to calculus concepts.

If you’re coming from a high school calculus background and expecting more of the same, prepare for a big shift. I’m telling you, if you’re not prepared, this course will hit you like a sack of bricks.

Who Should Take MAT157?

I’ll be blunt: this course is not for everyone, and that’s okay. MAT157 is aimed at students who:

  • Are considering a math specialist
  • Love abstract thinking and proofs
  • Are willing to spend serious time learning how to write and understand rigorous mathematics

If you’re someone who enjoys solving puzzles, finding elegant solutions, and thinking deeply about why things work, this course might be fun. If not, you’re going to struggle A LOT.

Course Content & Structure

MAT157 covers a lot, and it does so in a very formal way. Topics include:

  • Logic and proofs
  • Sequences and limits
  • Continuity
  • Differentiation and integration (with rigorous definitions)
  • Theoretical underpinnings of calculus, including constructing real numbers (Yes, we had to PROVE that real numbers are indeed real).

Instead of applying formulas, you’re proving they work.

Lectures:
These were usually theoretical and fast-paced. Lecture attendance (and attention) is key here, since even missing one can set you back a lot. However, the professor was incredible; I truly admire how she was able to do such elegant proofs with novel ideas that I wouldn’t even begin to consider.

Problem Sets:
Expect weekly problem sets that require you to write full, formal proofs. They’re time-consuming but incredibly rewarding once you get the hang of it. The difficulty ranged from not too bad to absolutely mind boggling (Question 6, if you know, you know).

Exams:
Often proof-based, with a few computations later on. They test your understanding, not just your memorization. Don’t expect standard plug-and-chug questions; be prepared to write a full rigorous proof.

Pros of Taking MAT157

Deep understanding of math
Great foundation for upper-year pure math courses
Incredible professors (may be subject to change) 
Looks impressive for grad school/math-focused paths

Cons & Challenges

Workload is heavy – Problem sets can take hours, or even days
Not computational – If you want practical calculus, this isn’t it
Steep learning curve – Especially if you’ve never done proofs before
Can hurt your GPA if you’re not fully committed

My Personal Take

I won’t sugarcoat it: MAT157 was the hardest course I took in my entire academic career (so far). But it was also the most intellectually rewarding. It completely changed how I think about math, and honestly, about problem-solving in general.

For those who plan to take it or are taking it now, don’t feel discouraged if things don’t start well. I actually failed my first midterm and barely passed the second one. I even almost failed the final, but I still ended with an above average grade.

I have zero regrets about choosing this because it gave me a sense of mathematical maturity and a lot more respect for mathematics in general. The entire teaching team was also incredible and really supportive every step of the way, so you’ll be in good hands.

Final Verdict: Should You Take MAT157?

Yes, if:

  • You love proofs and abstract reasoning
  • You’re thinking about a math specialist or grad school in math
  • You want a serious challenge and are ready to put in the work

No, if:

  • You prefer computational or applied math
  • You’re not comfortable with a heavy workload in first year
  • You’re just trying to get a math requirement out of the way

Want more honest reviews like this? Let me know which UofT courses you’re curious about next, or if you’ve taken MAT157, drop your experience in the comments.

An excerpt from a book on statistics

Why I Chose Math and Statistics at UofT

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For me, choosing to study mathematics and statistics wasn’t so clear; for 18 years, I thought about being a doctor. But what led me to this path, and why did I feel drawn closer to these fields? In this blog post, I’ll share my personal journey and the reasons why math and statistics became my academic passion.

The Fascination with Problem-Solving

From a young age, I was always intrigued by the process of solving problems. Whether it was puzzles, games, or real-life challenges, I found great satisfaction in breaking down complex issues into smaller, more manageable parts. This analytical mindset naturally led me towards studying mathematics and statistics, as both disciplines are deeply rooted in problem-solving.

Math is all about patterns, logic, and structure, while statistics provides the tools to make sense of data, predict outcomes, and analyze real-world phenomena. Together, these subjects offer an ideal foundation for anyone interested in understanding the world through numbers.

The Growing Importance of Data in Today’s World

Another key factor in my decision was the increasing role of data in nearly every industry. In today’s digital age, data is everywhere — from healthcare and finance to technology and social media. Understanding how to collect, analyze, and interpret data is a skill that has never been more valuable.

By studying math and statistics, I felt I could position myself at the forefront of this data-driven revolution. I realized that a solid grasp of statistical techniques and mathematical theories would open doors to a wide variety of careers in fields like data science, machine learning, and research.

The Versatility of a UofT Math and Stats Degree

One of the most attractive aspects of studying math and statistics is the versatility it offers. A degree in these subjects doesn’t just prepare you for one specific career path; it equips you with critical thinking and problem-solving skills that are highly transferable to many professions. UofT offers a specialist combining these two fields, and I immediately knew this was what I wanted to do.

Whether you want to work in finance, healthcare, technology, or government, math and statistics are foundational to understanding complex systems. The ability to work with large datasets, analyze trends, and make data-driven decisions is a sought-after skill in virtually every industry.

The Thrill of Discovering New Insights

What really excites me about math and statistics is the constant opportunity to uncover new insights. Math is a vast and ever-evolving field, with new theories and techniques emerging all the time. Statistics, too, is constantly advancing as new data collection methods and computational tools become available. I am lucky enough to have had professors working on incredible research, and hearing them speak so passionately about them inspires me.

Through my studies, I’ve learned how mathematical models can help explain everything from precious metals to predicting credit card usage. The thrill of applying mathematical and statistical theories to real-world situations is truly rewarding, and I look forward to the many challenges and discoveries ahead in my academic journey. 

A Passion for Logical Thinking and Precision

Mathematics has always felt like a natural language for me — one that is clear, logical, and precise. Unlike some other disciplines where answers can be subjective, math and statistics provide a concrete framework for finding solutions. This sense of clarity and certainty is something I value deeply.

Statistics allows for precision in decision-making. Whether it’s determining the effectiveness of a new drug or predicting future trends in the economy, the ability to make informed decisions based on data is incredibly empowering. It’s a field that blends creativity with rigorous analysis, and that’s what makes it so compelling.

The Future of Math and Stats: Endless Possibilities

As technology continues to evolve, the demand for professionals with expertise in math and statistics is only set to increase. The rise of artificial intelligence, big data analytics, and predictive modelling means that the skills I’m developing will be even more relevant in the future.

Whether I end up working in academia, business, or tech, the opportunities available to someone with a background in math and statistics are vast. The ability to understand and analyze data is becoming essential in every aspect of modern life, and I’m excited to be part of this evolving field.

Conclusion: The Perfect Fit for My Interests and Career Goals

Choosing math and statistics at university was a decision that felt right for me. It was driven by my passion for problem-solving, my interest in data, and my desire to be part of a field that has the power to shape the future. As I continue my studies, I’m excited to explore the many ways in which math and statistics can be applied to real-world problems and contribute to society.

If you’re considering a similar path, I encourage you to explore the possibilities that math and statistics offer. Whether you’re drawn to the analytical side of things or the thrill of discovering new insights, there’s a world of opportunities waiting for you in these fields.

Bell

UofT Course Review-STA257

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Taken: Fall 2024

Prerequisites/Co-requisites: First year calculus (137/157), linear algebra and multivariable calculus (237/257)

Course Average: B-

How I Did: Above Average

Workload: Average

Content Covered

A mathematically rigorous introduction to probability, with applications chosen to introduce concepts of statistical inference. Probability and expectation, discrete and continuous random variables and vectors, distribution and density functions, the law of large numbers. The binomial, geometric, Poisson, and normal distributions. The Central Limit Theorem.

Lectures

Lectures were twice a week and moderately paced. The professor was incredible; he had clear organized notes as he went through each topic and explained in a very concise way. There were also a few online polls during each lecture, but more on that later. His office hours were also fantastic; he was a great guy to talk to not just about the course but beyond, he’s extremely passionate about what he does.

Tutorials

Each week, we had a tutorial right after the lecture where we went through a few questions related to the week’s topic. To be honest these were a bit hit or miss; some were really useful while others were a bit boring and redundant. But the TAs were good at explaining and had lots of office hours closer to exam times.

Evaluation Methods

Poll Participation, Midterm Exams, Final Exam.

Poll Participation

Every lecture, we had a few polls in the form of multiple-choice questions about questions based on certain topics. We had a few minutes to discuss the question with our neighbours before inputting our answer. He would then go through the question and give the answer. We had a little over 100 questions total by the end of the semester. 

Midterm Exams

We had 2 midterms throughout the semester. Each were 100 minutes long and tested material up to that point. They were a mix of computational and proof based, but a little skewed towards computations. The first one was not too bad, all the questions were not bad at all. The second midterm though, well, let’s just say that it wasn’t the best (integrating that absolute value pdf will forever haunt me), but I ended doing better than what I expected.

Final Exam

The final was 3 hours long and was a dead average between the first and second midterms in terms of difficulty. All the questions were very fair, and I am forever grateful that the one topic I did not like had a pretty simple question on it which I was able to do.

My thoughts on the course and how to do well

This is my favourite course taken to date; I love the entire theory of probability and studying the odds of things happening, and this course definitely gave me a strong foundation in that. The one topic I mentioned that I didn’t like was the idea of convergence, more specifically ‘Convergence in Probability,’ which has made a few appearances later in my statistical journey, so I had to get used to it, but now it’s not that bad. Don’t be scared by the description of ‘rigourous’; it does have some faster paced sections, but overall, it’s very manageable if you do all the practice problems and go to all the lectures. An absolutely goated course that I would recommend to anyone who needs stats for their degree.

5 Things I Wish I Knew Before Starting University

5 Things I Wish I Knew Before Starting University

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I remember 20-year-old me showing up on the first day of classes, eager to get the university experience he dreamed of. Now, almost halfway through my undergrad, I can confidently say that things did not go according to plan, but that’s OK.

I learned not only a lot of math and stats in 2 years, but a lot about myself and life in general. Whether you’re a prospective student or just about to graduate, you should read on and see if you relate.

1. You don’t have to have it all figured out.

There’s so much pressure to figure out your major, your residence, making friends having a set plan for when you graduate. But I can confidently say that my plans went straight out the window after the first 2 weeks; I had to modify my program because I realized I really hated overly theoretical material (ironic considering I chose this path) , and my social life was drier than my sense of humour. I was really thrown at first, but I soon learned that sometimes the best plan is having no plan.

I’m not endorsing throwing caution to the wind on every occasion, but be open to other avenues, whether it be academic, social or otherwise. I’m almost finished with my second year, and I can definitely say I’m very happy with how things turned out.

2. You will have bad days.

There will be some days where you’ll be off your game. You may not be able to focus on everything in lecture, you may not be able to participate in tutorials like you usually would (because WE all participate happily and voluntarily) and no matter how much you try to study, nothing seems to stick.

Let me tell you, all of this is normal. Whenever I have a bad day, I firstly always acknowledge that I’m having a bad day and try to just accomplish the necessary things. After that, I do absolutely no work for the rest of the day (I know if you’re a UofT student, this could mean a death sentence depending on your program). But I’m telling you, taking that rest is definitely important for moving forward, which leads to my next point.

3. What works for one person may not work for you.

There is no ‘one size fits all’ approach to anything in life, and anyone who says otherwise is lying. We are all unique human beings at the end of the day, and, while there may be some overlap, we all cope and relax in our own way.

I always hear stories from my friends who pulled all-nighters at Robarts drinking lethal amounts of Red Bull, yet I can’t study past 10:00 without spacing out. While some people take a walk or do sports to relax, I like to write blog posts while listening to random Eurovision songs.

My point is, do what makes you happy and you’ll be refreshed to go again.

4. You’ll meet a lot of people, but you’ll also feel lonely sometimes.

University is a huge place. I can confidently say I have met hundreds, maybe thousands of people over the past 2 years here, but I still feel lonely sometimes. Honestly, I don’t have the stereotypical close-knit friend group you see in college movies, the ones you see having a picnic on a sunny day on the campus grounds. I’m not a social butterfly who clicks with everyone right away, and no matter how hard I try, most don’t seem to work out or last.

Unfortunately, some people can be cliquey and downright awful, but the right people, who appreciate you for you will find you eventually. While this may sound cheesy, it was definitely true in my case; I’ve found a few friends that I am grateful for everyday and who knows who I may meet later down the line.

5. Put yourself out there but don’t lose yourself in the process.

In university, no one tells you what to do. It is up to you and you alone to decide who you want to be later down the line. You may even find yourself changing in some ways; the way you speak, the way you act, who you hang out with and more. This is completely fine and is a fundamental part of growing up, but don’t lose yourself in the process. Don’t lose your quirks, your soul, your identity just because a few other people may not understand it. Realize that you can never please everyone.

This was probably the harshest lesson I learned, because I was so desperate to fit in like everyone else. But I soon realized that I didn’t feel like myself and didn’t like the person I was becoming. So, I took a step back and re-evaluated myself and moved forward. I mentioned I’m not a social butterfly, but I try to make small talk where possible to try to get to know people. I’m still a work in progress, but better than what I was before.

So, there you have it. I’m still learning so much every day, and I hope these lessons would help you out.

ACT

UofT Course Review-ACT230

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I’ll be reviewing one of my favourite electives, a finance course for non-actuaries. It gives a general introduction to financial concepts which I think everyone should be at least familiar with since financial literacy is one of the most important life skills. Once again, this is my personal experience and is affected by the exact instructor teaching and specific assessment methods.

Taken: Fall 2024

Prerequisite: Any first-year calculus course

Course Average: C +

How I Did: Above Average

Workload: Average

Content Covered

Introduction to financial mathematics, interest measurement, present value calculation, annuity valuation, loan amortization, consumer financing arrangements, bond valuation.

Lectures

Lectures were once a week where we learned a new concept roughly each week and did a lot of practice questions related to the topic. The instructor was really good; very straightforward, clarified questions with ease and went through all the different tips and tricks with certain topics. The best part was we were told what kind of questions would appear on the exams, which I’ll get to in a bit.

Tutorials

Each week, we had an online tutorial where we went through a few questions related to the week’s topic. The TA’s were awesome at explaining and really broke down questions step by step to get to the answer. I liked that these were online so they could be done from anywhere without the hassle of rushing to class.

Evaluation Methods

Quizzes, Midterm Exams, Final Exam.

Quizzes

Every 2-3 weeks, we had an online quiz with 3 questions related to the previous weeks’ material. These quizzes were single attempt with a 30-minute time limit, so speed and accuracy were important. We had 5 in total, with the best 3 being counted towards our final grade. The questions were on the easier side, being straightforward with the computations and all relevant information given. They were also in the multiple-choice format, so you could plug in and verify the answer.

Midterm Exams

We had 2 midterms throughout the semester. Each were 50 minutes long and tested material up to that point. The first one was 6 long answer questions, which was quite difficult to finish in that time frame. The questions themselves weren’t too bad, but the time was too short in my opinion given we had to show all working. Luckily, I was able to finish in time, but with only 3 minutes to spare at the end to check over. The second one was better, with 3 multiple choice (still had to show all working) and 2 longer questions which were a lot more manageable in 50 minutes. It also helped that we were allowed a formula sheet, which was half of 1 side of an A4 page.

Final Exam

The final was 2 hours long and had 10 questions with some parts. The questions were for the most part straightforward but a few challenging and time-consuming ones. This time, we were allowed a full side of an A4 page for formulae, which was quite helpful (all formulae could fit on 1 side). I finished with quite a bit of time to spare and was able to redo all of the computations, so no complaints.

My thoughts on the course

This was one of the best courses I have taken. I gained a better understanding of basic financial principles such as interest rates, loans and bonds which are very applicable to real life. The formula sheet was a lifesaver since memorizing all the formulas would have been pretty difficult. My favourite topic was loan amortization, since I got to see applications to credit cards (my latest project, you should read it!) and mortgages, giving me a better insight into how monthly payments are calculated (useful in the future if you take out a mortgage/loan to make sure they’re not scamming you). The instructor and TAs were kind and very helpful with a lot of support though office hours and patience answering questions. I highly recommend this course to anyone (it is computation heavy, so be prepared!).

CC

Love At First Swipe: A brief look into credit cards

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Today’s project will be about one of my favourite things: credit cards. I think of them as a debit card with an extra step: instead of the money coming from your checking account, it comes from the bank, and you just pay it off using your checking account. Of course, this assumes account holders have enough self-control as to not spiral out of control and accrue massive debts by overspending. But, if you know how to use them responsibly, they can be one of the best things ever. From perks, rewards to securing bigger lines of credit in the future, I think they’re worthwhile if one can be responsible.

Methods

I created a decision tree model which predicts a person’ s credit utilization ratio is high or low based on different factors such as their credit limit and revolving balance of their account. Most credit bureaus state that a person should not use more than 30% of their available credit, so that’s what I used to create the High-Low split of the model. I then analyzed this model by creating a confusion matrix of predications and assessing the model’s accuracy as well as any limitations. I also did a financial evaluation using credit card interest rates and an amortization schedule for paying off credit card debt to see just how long it takes.

Analysis

The graph shows that there is an overall negative relationship between credit limits and average utilization ratios. This is unsurprising a fixed amount spent with a lower limit will have a higher ratio compared to a higher limit.

The decision tree created had an accuracy of 96.8% when classifying new data, which is quite high. The corresponding confusion matrix yielded a false positive rate of 3.58% and a false negative rate of 2.45%, both very low when compared to the sensitivity and specificity, both above 95%. This implies that the model is a very good predictor of whether someone has a high or low credit usage based on their credit limit and total revolving balance at any given point in time. However, it is important to note that this model would need further testing on other datasets to confirm its applicability as it may conform to this data, but not to others.

Based on the amortization, we see that it takes 35 months, or approximately 3 years, to pay off a $5000 debt assuming consistent monthly payments of $200 are made at the end of each month. This is a quite some time which assumes that a person can put aside that money for this specific purpose. We also see that as more of the principal is repaid, the lower the interest accrued becomes, suggesting a negative association.

If the monthly payments were higher or payments were made on a different time basis, such as fortnightly, this would take less time as less interest is accrued, so the balance decreases to zero much faster. But if the monthly payments are less of if you miss a month or two, this will set you back even further as more interest accrues.

 

Conclusion

Credit cards can be very useful if used responsibly. If not, they can accrue massive debts that will take many months, even years, to pay off.

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