Learn the formula for studying math and physics

What research papers and U of T professors say on how to best study quantitative subjects

Is there such a thing as having a ‘math brain’?

It may seem like most people in your classes were born with a profound knowledge of vector space axioms and the ability to do impeccable epsilon-delta proofs in their sleep. This might be true to some extent — research from John Hopkins University suggests that there are in fact people who are able to grasp mathematical concepts quicker, something the researchers attribute to an innate “number sense.” But not having this ability doesn’t mean you can’t become proficient at math.

Dr. Alfonso Gracia-Saz, an associate professor at U of T’s Department of Mathematics, compared developing your mathematical abilities to training for a marathon. Just like “almost all healthy adults [would] be able to run a marathon with training and discipline,” he wrote to The Varsity, “almost all healthy adults are able to excel at and enjoy math.”

I can personally testify to this. I used to be a quintessential social sciences girl who dreamed of attending the Munk School of Global Affairs and Public Policy, and eventually work on Parliament Hill; I was convinced that math and physics were way beyond my reach.

The environment I was in wasn’t helping either — my grade 10 science teacher preferred talking about his homegrown tomatoes and missionary trip to East Africa over actually teaching.

Fast forward three years, and I’m a math major.

In her book Mindset: The New Psychology of Success, Dr. Carol Dweck, a professor at Stanford University’s Department of Psychology, argues that our beliefs about whether our abilities are completely innate or able to be improved can play a pivotal role in our success.

The “growth mindset is based on the belief that your basic qualities are things you can cultivate through your efforts,” she wrote. Realizing that your math and physics skills can be boosted with effort and practice may help you change your relationship to encountering challenges and setbacks while learning.

Rethink your learning

There is no doubt that grasping unfamiliar concepts or solving difficult math problems can be incredibly challenging and frustrating, but researchers argue that struggling and making mistakes is a crucial part of the learning process. In other words, it may actually be beneficial for your brain and help you become a better learner.

Cristina Peter, a learning strategist at the Koffler Student Services Centre, spoke to The Varsity about difficulties with learning material. She elaborated that “most challenging concepts are built on basic fundamentals,” and encouraged students “to unravel the basic concepts that the challenging concept is built upon.”

Gracia-Saz agrees with the importance of embracing the struggle, adding that being patient with your own learning is also crucial to getting to the bottom of a new concept. Some approaches to solving unfamiliar problems may not work at first, but eventually you will find the one that works, and this process of trying and failing “will help you develop good intuition and solidify your understanding.”

A few days after my first midterm of PHY151 — Foundations of Physics I, an introductory course to calculus-based physics, my professor Dr. Stephen Julian told us an anecdote from his undergraduate career, which he kindly recounted in an email to The Varsity.

“I was pretty ignorant when I arrived at university. Probably many of my classmates had properly learnt their high-school math and physics, but for me it was all new. When the first set of midterm exams came up… the subject was special relativity, and I thought that the way to study was to read a book (it wasn’t even our textbook) about special relativity the night before the exam.”

“Of course, when I got into the exam I couldn’t actually do anything and I failed miserably,” he continued. “It was only in my third year that I really came to understand that in order to do well you need to learn the material as you go along (again, after every lecture you need to review and think about the material), and you need to practice solving problems.”

“Learning the concepts without the math is not going to work,” he added.

Why physics is so hard to learn, and how to handle it

Unfortunately, the prevailing view of physics is that it’s a notoriously hard subject which only a lucky few can master. Even physics students who love the subject often struggle to get to the core of new material.

Dr. Edward Redish, a professor at the University of Maryland’s Department of Physics explained in a research paper on the cognitive science of teaching physics that one of the characteristics of studying cognitive science is that we tend to organize our observations and experiences of a particular subject into mental models or patterns, which we have to build upon to maximize our learning. This usually means that passively reading and re-reading the textbook, and going to lectures may not be enough to build a solid understanding of the subject.

Peter suggests taking a moment after class to reflect on what you’ve just learned and identify gaps in your understanding.

Other researchers also suggest testing yourself repeatedly on each topic and making a set of criteria for applying new concepts to ensure that you know when and how to use your knowledge. It’s also beneficial to verbalize your thoughts to check if you truly understand the material — you might find it helpful to pretend you’re talking to a younger sibling — and to try to relate it to your prior knowledge or real-life examples.

Julian highlighted the importance of giving yourself time to get your head around a new concept — he recommended “thinking about the concept as soon as it is introduced in lectures (or indeed the day before if you are assigned pre-class readings).”

He also emphasized that “you need to keep going back to it over the following days and weeks. With time, and patience, the concept will take root and grow in your mind.”

Solve problems like a pro

The expert-backed approaches to solving math and physics problems in this non-exhaustive list may help you power through your next problem set with a lot more ease.

If you find yourself stuck on a question, Julian recommends thinking “of a simpler version of the same question, and keep simplifying until you find something that you can solve, [then trying] to work your way back to the original question.”

According to Gracia-Saz, looking at many solved examples is a “trap” that won’t replace actually going through the problem-solving process yourself, which is a valuable learning tool. He also stressed the importance of perseverance in learning math — “nothing is as satisfying as solving a good math problem after hours of struggling, except perhaps chocolate.”

Peter recommends rationalizing every step of solving a problem, such as creating a chart with your steps on one side and the explanation of your actions on the other.

George Pólya’s famous problem solving method is also worth looking into.

Getting help

Your professors and teaching assistants will most likely be happy to help you with the course material. When taking advantage of this, Peter recommends planning your questions in advance to maximize your learning.

If you are interested in learning more about time management, note-taking, and problem-solving strategies, you can book an appointment with a learning strategist at your faculty or college, or visit them during their drop-in hours at the Sidney Smith Commons. You can also check out on-campus math and physics resources, such as the Math Learning Centre, Vic Peer Tutors, the Statistics Aid Centre, and college-specific math help centres.

If you’re looking for a community to learn with, you should also consider joining or starting a study group, where you can make new friends and review concepts.

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