What to do when you get stuck

academics High School mathematics

Students in classes such as linear algebra and calculus often ask what to do when they get stuck.

Asking yourself the following questions will unstick you surprisingly often:  

1. Am I stressed?

If so, try to relax. Math cannot hurt you, and staying  calm will help you think clearly.   

2. Do I know what the question means?

Look at each mathematical word in the problem and try to remember its exact definition. If you don’t remember the definition of a word, underline it, then look up its meaning in your notes or textbook. Once you know the meaning of each word, make sure that you also understand what the entire question is asking. 

3. Have I seen a problem that looks like this before?

In classes like precalculus, calculus, and linear algebra, problems on homework and exams almost always look like examples that were covered in the textbook or lecture. Look through your brain, notes, and textbook for such examples, and use them as a guide. 

If you’ve been through those three questions and are still stuck, then you’re in a good spot: this is an opportunity to expand and apply your knowledge in a new setting.

This is more common in higher-level math courses, where exercises may not closely resemble things you’ve seen already. In this situation, possible courses of action include:  

1. Work out a small example.

This is especially useful for true/false questions, or questions where you are asked to prove something. You should start with the smallest, simplest example you can think of, then increase complexity if needed.  Working out an example carefully ensures that you understand all the objects in the problem, and often reveals essential features that you might not notice when thinking purely abstractly. 

2. Rephrase the question

In the best case, you’ll be able to rephrase the question to look like something you’ve seen before. Failing this, try changing perspective (e.g. if you have a geometry question, try thinking of it algebraically or vice versa) or even just writing everything in the problem out as explicitly as you can. 

3. Formulate and solve a simpler version of the problem

By imposing extra assumptions, you may be able to simplify the problem enough to make it solvable. Understanding whether and why the methods you develop for the simplified version fail can be helpful for gaining insight into the original problem.  

4. Break the problem into pieces

Can you do the problem case-by-case? Does something in the problem factor? Is there a series of smaller problems that you can solve that together will solve the main problem? 

5. Talk to someone

“Someone” might be a classmate, a TA, a tutor, a professor, a user of sites such as Math Stack Exchange, a random friend, or even a rubber duck. The real value here is often in the process of explaining the difficulties that you’re encountering, which can help you to understand and overcome them. 

6. Think hard, then walk away and come back later

Counterintuitively, not working on a problem is sometimes what unlocks it. For this method to work, you must spend some time thinking intensely about the problem: rephrase the problem. If you reach a point where you are truly making no progress, stop and do something else. When you return later, you may have fresh ideas  

Difficult problems will require many cycles of problem-solving effort.

If you are working on an exercise in a book or course, then a reasonable solution does exist and you can keep plugging away. In the real world, all bets are off, and you should frequently reevaluate whether you’re attacking the right problem in the first place. Best of luck!

Connor studied Mathematics at Cornell, and is now completing a PhD in Math at the University of Wisconsin Madison, focusing on combinatorics and algebraic geometry.

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