Math and Physics Learning Strategies

“Learning is deeper and more durable when it’s effortful.”¹ Students know they have to put effort into math and physics, but how can your efforts be most effective? This tip sheet highlights three misconceptions reported by students when they are learning math and physics outside of class. Each misconception is paired with active learning or test-taking strategies to help you address them. Intentionally practice these strategies before test day so that you are comfortable applying them on test day. Combine these strategies with visiting office hours and working in study groups. High-performing students couple strong self-learning habits with collaboration and using their resources.

Misconception #1: “I Must Review My Notes Before Attempting Problems”

Many students claim that it’s futile to attempt practice problems before reviewing their notes. Even after attending lectures, you may assume that you lack the background knowledge to try practice problems. However, empirical research indicates that trying to solve a problem before seeing the solution results in better learning, even when mistakes are made in the process.¹ This is because retrieval practice–actively recalling concepts from memory–bolsters long-term retention and impedes forgetting by strengthening neural pathways in our brains.

Using retrieval practice to attempt practice problems is a more potent learning strategy than rereading lecture notes beforehand. Try self-quizzing shortly after lectures; you may be surprised by how much you can remember. Plus, actively re-constructing your own understanding of the lectures based on the pieces that stuck out to you is a more effective way of internalizing the lectures than passively re-reading them the way they were presented by someone else.

Suggestions:

• Give yourself a short amount of time (about 10-15 minutes) to struggle through a practice problem before referencing any resources.
• During this time, parse the problem for meaning, and use retrieval practice to connect it to something you vaguely remember from lecture. Write down what ideas and problem-solving strategies you think apply to the problem, even if you’re not sure, and try to connect as many dots as you can. Then, reference your resources to fill in the gaps.
• Take note of the parts of the problem you were able to intuit and the parts that you missed. Why did you miss them, and how will you recall the correct problem-solving strategies the next time you encounter a similar problem?

Misconception # 2: “I Must Do All Assigned Problems Before the Test”

Rather than solving all the assigned problems, selectively choose problems to solve. The notion of quality over quantity is especially true in math and physics classes, where professors typically assign a ton of practice problems. Each day, you have limited time and finite mental bandwidth, so you may not be able to solve every assigned problem without cutting into time reserved for other classes or sacrificing your work-life balance and burning out. What’s more, doing every single problem could trick you into believing that you know more than you actually know (a well-documented effect called the “Fluency Illusion”)², particularly if you’re solving the problems to check them off rather than asking metacognitive questions like “Why did the professor assign this problem?” or “Is the value of this problem worth the time and effort I think I would need to solve it?”

Suggestions:

• Before diving into practice problems, find the learning goals for each unit or exam and classify each assigned problem according to those goals. Learning goals can often be found on Canvas—in your course syllabi, lecture slides, or exam-review sheets provided by your TAs and professors.
• Then, organize the practice problems into groups. For example, maybe your calculus problems 2, 4, and 12-18 cover derivative rules, while problems 3, 6, 9, and 19-25 cover various applications of differentiation, like optimization and related rates.
• Once you’ve grouped the problems, commit to solving one from each group (under time pressure) to ensure exposure to all the testable topics and to gain practice with a variety of problem-solving strategies. You’ll learn, more quickly, which topics require further review, and now you’ll have a “problem bank” to draw from. Grouping problems will also help you identify patterns amongst problems in the same group, or across problems in different groups, reducing the total number of problems you may have to solve and saving you time. Study smarter, not harder!

Misconception #3: “I Can Solve All the Assigned Problems, So I’m Ready for My Test”

Being able to solve all (or even many) of the assigned problems is a great start—especially if you’re practicing the principle of “quality over quantity” explained above. But your math and physics tests in college will ask you to apply what you learned from the assigned problems to similar-type problems. The exam problems will involve the same fundamental concepts but will be different enough to feel brand new and unfamiliar. So, to score well on tests, you must be careful not to conflate memorizing or being able to reproduce the solution to a problem, with understanding. This is an example of the aforementioned “Fluency Illusion,” a common learning hazard that can be avoided by using certain strategies.

Suggestions: “Problem Manipulation”

• First, use an assigned problem to find a closely related problem (in a textbook or other resource from class) that treats a slightly different scenario or combines multiple concepts. Then attempt that problem and write down what is unique about the new problem. If the new problem is more challenging, ask yourself “why?” and list those reasons. Rinse and repeat with new problems if necessary.
• Next, compare problems on practice exams or review sheets—i.e., “exam-like” problems—to assigned problems from homework, lecture, or recitation. Re-solve some of these problems as a warm-up, then immediately attempt a few exam-like problems. Then ask yourself leading questions like, “Do the exam-like problems expect me to incorporate knowledge that I freely could look up when doing the homework problems? Do I need to know how to solve part (a) to do part (b)? What do I think my professor wants me to learn from the assigned problems in order to solve the exam-like problems, and what do I imagine my professor wants to me learn from the exam-like problems to get ready for test day?” Once you have an idea of how to solve the exam-like problems, ask yourself, “Could I reproduce the same steps under time pressure, or would I need to find a more efficient route?” Considering these types of questions will help you understand how professors are creating test questions from assigned questions.
  • You don’t have to try these first two steps on your own! Consider collaborating with peer(s) and pooling your collective knowledge. Maybe you are comfortable with problems on topics “A and B” while your peers understand topics “C and D.” Try working out solutions to various problems together and explaining the steps to each other. Afterwards, make sure that you can independently solve all the problems you discussed.
• Finally, try to create your own variation of at least one of the assigned problems. There are many ways to make variations of problems without creating entirely new problems. Here are a few strategies to consider, paired with examples (see image) to help you apply them:

• Change the notation.
  For example, if you’re used to seeing mathematical functions and their derivatives expressed as f(x) , d/dx [f(x)] (this is also written as f’(x), but the “prime” notation hides what variable you are differentiating with respect to, so the “d/dx” notation for derivatives is more clear), and d^2/dx^2 [f(x)] [this is also written as f’’(x)] in most of your calculus problems, try intentionally giving the functions and their derivatives different letters like g(t) , d/dt [g(t)] [this is also written as g’(t)], and d^2/dt^2 [g(t)] [or g’’(t)]. Now, the same mathematical function is called “g”, not “f”, and the independent variable is “t”, not “x”. But it’s still the same problem in disguise. Math and physics students often report that they could have solved a problem if it was written in notation that was more familiar to them.

• Example A: Understand the “base case.”
  Suppose you solved, in class, the standard physics problem of finding the minimum force F = 49.05 Newtons required to push a block (say, a mini-fridge) of mass m = 10 kilograms from rest up a ramp inclined at an angle theta = 30 Degrees above the ground. In this case, the surface between the fridge and the ramp is assumed to be perfectly smooth, so there is no friction present, and you are just pushing the fridge against one force: gravity.

• Example B: Change the scenario.
  Suppose that the ramp is inclined at a slightly steeper angle, say, theta = 45 Degrees. Since you’re pushing the fridge up a steeper slope, would you expect the force to be larger or smaller? Make a prediction, then re-calculate the value of the push force “F” using the new angle to test it. You’re thinking like a scientist!

• Example C: Solve the same problem without numbers, just symbols.
  Call the mass of the fridge “m” and the angle of the ramp above the ground “theta”. You can use the same algebraic steps as before to derive a mathematical expression F = mg Sin(theta) for the force written in terms of the symbols {m, theta, and g}, where g = 9.81 meters per second squared, is the acceleration of the fridge due to gravity.The expression F = mg Sin(theta) may look intimidating and cryptic because it’s written in terms of variables. So, how on Earth can you check that it’s correct when you don’t have numbers to work with?! Well, the expression must give the same force(s) you got before with m = 10 kilograms and theta = 30 Degrees or theta = 45 Degrees, right? Try it! This is the power of working with symbols – it teaches you that seemingly different problems are special cases of the same problem.

• Example D: Combine multiple concepts.
  Sticking with the fridge-on-a-ramp example, what else besides gravity could change how difficult it is to push the fridge up the ramp? Well, what if the surface between the fridge and the ramp wasn’t assumed to be perfectly smooth anymore? (Maybe your archnemesis put sandpaper on the ramp while you were sleeping.) Now, when you try to move the fridge, there is static friction between the ramp and the bottom surface of the fridge. Thus, you’re now pushing against two forces–gravity and friction. Go back to your steps in A–C; what parts of these steps must you change to account for the force of friction? Professors love to make test questions that add one or two more pieces to a question seen in lecture, recitation, or homework.

You’ve got this!

It’s okay if you have trouble with “Problem Manipulation” at first–the act of using your creativity and asking yourself leading questions will deepen your learning! Please use the examples above as guides.

Remember that you don’t have to reinvent the wheel! Notice how the fridge-on-ramp example took a standard physics problem and added variations in methodical steps. This is essentially what professors do when they write exams; they analyze the problems they’ve already assigned and then create variations of those problems that apply the same ideas and problem-solving strategies to new situations that mix multiple concepts. So, if you can learn how to do “problem manipulation,” you’ll teach yourself to think like a professor and your exams will feel more accessible. When you’re comfortable with “problem manipulation,” you can even try creating your own practice tests.

To learn how to apply these strategies in your math and physics classes, visit learningcenter.unc.edu to book an Academic Coaching appointment with one of our Academic Coaches, or a Peer Tutoring Appointment with a Peer Tutor. Strategy is the bridge between knowledge and performance.

Additional Resources

Works consulted

1. Brown, P., & Roediger III, H., & McDaniel, M. (2014). Make It Stick: The Science of Successful Learning. Belknap Press.

2. Carey, B. (2014). How We Learn: The Surprising Truth About When, Where, and Why It Happens. Random House.

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