This a short but versatile guide to solving mathematically oriented problems. I have used this framework on math tests, math competitions, and throughout my undergraduate education in engineering. Although it is conceptually simple, expressing intuitive ideas adds valuable clarity. Throughout the guide, I will use “equations” as a catchall for concepts, ideas, and anything else you learn.s
Step 1: What does the question tell me?
The first step is to simply read the question and note down all the information that you know. Sometimes it is useful to write the information as the variables that you anticipate you will end up using.
Step 2: What else do I know?
The second step is to brainstorm what else you know about the situation. Initially, you should assume that any real-world knowledge or intuitions aren’t just fair game – these deductions are encouraged! Often, an additional intuitive assumption makes the difference between the student who is limited to straightforward applications of a concept and a learner who aces the most difficult questions and learns to love mathematics as they do so. The best thing that could happen is that your intuition is wrong. Why this is will be addressed in “Step 5: What did I learn?”.
Step 3: What do I want?
Now look at what the question is asking you and work backwards from the methods you have and think about what you can do. Are there equations you typically apply to solve similar questions? Note them down. If there is no obvious way to the solution that looks like it will work, it is worth the time to slow down and brainstorm multiple different routes to the solution.
Step 4: Which equations/techniques apply?
By now, you should have some points of departure and some lines of reasoning that you know will lead to the solution. Now, the question becomes about finding some link, usually through applying the course concepts, that takes you from the initial information to the final solution. You can think of steps 1+2 as building the start of the bridge, step 3 as building the end, and step 4 as applying course concepts to build the middle of the bridge. While doing this, remember that every equation is premised on a set of assumptions – make sure to verify that those assumptions apply before relying on the equation.
Step 5: What did I learn?
Now compare your solution against the solution manual. If you got the question wrong, note where your reasoning was wrong and the reasons behind those mistakes. If you got the question right, review the model answer closely for additional assumptions or ideas that you inadvertently skipped over. If you failed to reach a solution, learn the type of reasoning the solution used and add that to your repertoire.
Crucially, this is where you either validate or refute your intuitive assumptions from Step 2. If your intuition is incorrect, make sure you understand precisely why it is wrong. If your intuition is right, make sure you know precisely how that intuition is built into the equations and concepts you have learned. Do not be afraid to take time to go on tangents and explore concepts. This is the entire point of doing practice questions in the first place.
Do not skip out on this intuition step. It is where the learning happens – every other step, including doing the actual question, was merely preparing your mind for this reflective exercise. Choosing not to reflect on the question is like showing up to swimming class but refusing to get in the water.