Your ability to solve problems is infinite. This is not something you are born with. Instead, you can grow your problem-solving skills by working hard and working smart. In this post, I share questions virtually guaranteed to help make progress on any problem you face. If you leverage the strategy of asking yourself these questions and repeat these practices over long periods of time, you will transform yourself into an expert problem solver.
Below is a list of over 60 questions to get your problem-solving juices flowing. I organize these questions into four different phases of problem solving. To generate this list, I relied heavily on what I call the five-volume bible of problem solving:
- How to Solve It: A New Aspect of Mathematical Method
- Mathematics and Plausible Reasoning, Volume 1: Induction and Analogy in Mathematics
- Mathematics and Plausible Reasoning, Volume 2: Patterns of Plausible Inference
- Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving, Volume 1
- Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving, Volume 2
All five of these books were written by George Pulya. In this introductory post, we focus on strategies from the four-step process to solve problem outlined in How to Solve It.
I encourage you to use this list as a check list. As you embark on each problem-solving adventure, choose a question from the appropriate phase of problem solving. Then, write as much as you can in response to this item. After you’ve exhausted your mind on a specific item, move to some other question from that phase. Slowly and methodically work your way through these questions. Do your best to practice leveraging all the questions in all the stages. This doesn’t mean you have to practice all the questions on a single problem. But, as you continue to solve problems, make sure that you are leveraging all the questions on this list. There are great lessons and unique insights that come from meditating on each question. Together, these form the foundation of a problem-solving habit that can catapult you to being able to solve almost any problem you can imagine.
STEP 1: UNDERSTAND THE PROBLEM
- What is unknown?
- What are the data?
- What is the condition?
- Is it possible to satisfy the condition?
- Is the condition sufficient to determine the unknown?
- Separate the various parts of the condition.
- What types of example(s) can you generate to learn more about the problem?
- How can you visualize this problem?
- What pictures or diagrams can you draw to illuminate key features?
- Draw at least one diagram to describe the problem.
- Label the diagram(s) with key variables.
- What notation or symbol(s) makes sense to use for this problem?
- Why do you like that notation?
- Define each variable verbally.
- What questions come up for you as you meditate on this problem?
- Describe the problem in abuelita (simple, intuitive) language
- Describe the problem in nerdy (formal, technical) language
STEP 2: DEVISE A PLAN
- What problem(s) might be related to this one?
- Have you seen this problem before?
- Have you seen this problem in a slightly different form?
- What related problems have you solved before?
- How can you use your previous results, solutions, or methods on those related problems here?
- Look at the unknown(s). What problem(s) do you know that have the same/similar unknowns?
- What auxiliary elements can you introduce to allow you to use your work solving other problems on this problem?
- What theorem(s) could be useful?
- Write the theorem statement(s) out in full.
- Translate the notation of the problem into the notation of the theorem or vice versa.
- What definition(s) do you know that could be useful?
- Write the definition statement(s) out in full.
- If so, write the definition(s) out in full. Make sure to explicitly connect the notation of the problem to the notation in the definition.
If you can’t solve the proposed problem, create and solve any related problem(s):
- How can you simplify the problem?
- What is a more accessible, related problem?
- What is a more general problem?
- What is an analogous problem?
- Identify different parts of the problem?
- What can you do to solve the parts of the problem that you find?
- What if you keep only a part of the condition and drop the other part(s)?
- Identify subproblems within the larger problem
- What can you do to solve the subproblems you find?
- What can you learn from the unknown or the given data?
- If you drop some of the conditions, how does that effect the unknown(s) in the problem?
- What patterns, insights, or questions can you create from the data in the problem?
- What other data might be useful for determining the unknown(s)?
- How can you change the unknown or the data or both to produce new unknowns or data that are nearer to each other?
- Have you used all the information you can?
- Have you used all the data?
- Have you used the entire condition?
- Have you taken into account all essential notations involved in the problem?
- What types of example(s) can you generate to give you insights into how to solve the problem?
STEP 3: CARRY OUT THE PLAN
- Carry out your plan of the solution.
- Can you see clearly that the step(s) are correct?
- How might you prove that the step(s) are correct?
- What types of example(s) can you generate to check that your results work?
STEP 4: LOOK BACK
- How can you check the results?
- How can you check the argument?
- How might you derive the results differently?
- Can you see the result(s) at a glance?
- How might you use the result or method for some other problems?
- How can you make sense of your results?
- How would you teach your result to someone who has no knowledge of this problem?
- Describe your work in abuelita (simple, highly intuitive) language?
- Describe your work in nerdy (formal, technical) language?
- How can you revise your notation to improve understanding for others?
For those of you that would like to download, distribute, or print this list, please see below:
Community Challenge
- I make a awe-inspiring assertion in this post. Specifically, I am claiming that if you get good at asking yourself the questions listed above, you can learn to solve almost any problem you can imagine. And this skill is not only limited to mathematical exploration. What comes up for you when you think about my claim here? What benefit might you derive from developing your ability to solve hard problems?
- Next time you are in STEM class that requires you to solve problems, crack open this list and go to work. Then, write about your experience. How did these questions impact your progress? What did you learn from using this resource?
- Too often, educational systems and classroom rules get in the way of learning. This is particularly true when it comes to learning how to solve problems. Great problem solving doesn’t happen on a timer. In fact, problem solving is often a slow, deliberate process that culminates into solutions only after you’ve spent a long time thinking about the problem. But our education systems put so many rules and time restrictions on your learning. How would your life in school be different if you where not constrained by time limits and if you could use any resource you have at your finger tips to solve problems?
- What if your teachers were not required to give you grades. Instead, what if your teacher(s) acted as a coach and mentor by giving you guidance as you solved problems that related to your career and academic interests? What policy changes would we need to make as a society to realized such a reality inside your classroom(s) and in our institutions? What can you do to work towards such a system?
Jeff Anderson 