Tips for Best Usage

Learning is not a spectator sport; the more you engage with the material in front of you, the better!

For lesson-oriented worksheets and videos, I strongly recommend you notes as you work, write down questions in the margins that you identify while going through the material, and most importantly, try out examples yourself before you read or watch the solutions. After you have finished, take a couple minutes to write down what you consider the most important takeaways from the lesson! Use these, along with any lingering questions as your starting point the next time you sit down to study.

When trying out practice problems, don’t give up the minute you get stuck. I do not give busy work; you’re likely encountering something novel that takes time to grapple with. Looking at solutions can often spoil the thought process; identifying the first step is usually the most important part of a problem!

What to do When You’re Stuck

When you’re stuck on a problem, here’s some questions to ask yourself:

1. Have I seen a problem like this before? What was the approach for that problem?

2. What exactly is this problem asking me to do? What information is given directly in the problem?

3. What are the key definitions, formulas, and concepts that seem to be at play in this problem? (If you are feeling unsure of these, look back in your notes or review the readings.) Can I use any of these to identify more information I could proceed from?

4. Are there ways I can simplify this problem? This could be breaking it down into component parts, plugging in specific numbers (for a variable or constant) to see what happens, or backtracking from the desired answer to see what tools you need to get to that point (if it’s not a one or two step solution).

If, after asking yourself these questions and reviewing the material, you are still stumped, make a note and move to a different problem. I will typically give myself 10-20 minutes to sit with a problem I’m feeling completely stuck on before moving on. You may need to return to the problem with a fresh set of eyes, or with newfound knowledge from solving other problems!

After You’ve Solved a Problem

Finding the solution to a problem doesn’t mean you’re done working with it! Here are some questions to reflect on once you’ve figured out a solution to a problem, that help tremendously to move towards mastery:

1. What was the general process to solve this problem? Could I confidently talk a classmate through my approach and why it works?

2. How can I check that my answer is correct? Does it seem like a reasonable answer based on the information in the problem? (E.g., if you find that a random event happens 200% of the time, or that you’re getting 0 for an answer when it should be a positive number, something probably went wrong in your solution.)

3. Where did I get stuck on this problem? What piece of information, computational tool, etc did I need to recognize to get unstuck? How did I figure out what additional information I needed? (Was it by rereading the problem, going back to previous examples, etc?)

4. What other approaches could I have tried for this problem? Can I use these to get to the same answer? If not, where does the alternate approach fail, or why do these answers disagree?

5. What mistakes am I (or other students) likely to make on another problem of this type? What can I do to remind myself of them next time?

6. Challenge: How can I modify this problem to make it require a different approach? (E.g., would changing the power from a square to a cube make a big difference, or changing sampling with replacement to sampling without replacement?) This is often how professors will write exam questions; being able to do so yourself, and figure out the solution to a novel variation, is a sign of deep mastery of the material! (Keep in mind doing so may create problems that are unexpectedly far too difficult, so don’t worry if you get stumped here.)