As a math circle leader or math teacher, there are some basic pedagogical principles that must be observed. First, watch this 4-min video about my heuristic to problem-solving: The 3-Step Problem-Solving Cycle. The three steps are understanding, strategizing, and implementing. These transcend the realm of mathematics.

1. Understand – The first and most important step to solving any problem is to understand it. As mentors, we must understand this before we can be of any actual help in the discovery process of those under our care. Lead with questions. Therefore, use Socratic questioning when students are having difficulty solving a problem. Here are some examples:
   • Can you explain the problem in your own words?
   • Is there any word or concept you do not understand?
   • Do you understand the units in which measurements are given?
   • What are we looking for?
   • What is the unknown?
   • Is all of this information relevant?
2. Strategize – Avoid offering a strategy to solve a problem if students don’t understand it first. Strategies might naturally arise once they understand the problem. Once you are convinced that students understand the problem, give them some time to process the information and refine their understanding. If they still seem stuck, then vaguely suggest a general strategy and let them experiment. Employ Socratic questioning throughout.
   • Can you draw a picture?
   • Do you think a chart will help you?
   • Can you think of a problem that is similar but easier to solve?
   • Do you see a pattern?
3. Implement – Once students have selected a strategy, encourage them to implement it. Also encourage them to check their reasoning as they move along. Most importantly, when students are engaged playing with ideas or implementing a strategy, step back and observe.
   • Why don’t get your hands dirty?
   • Can you justify every step or is there any that you feel unsure about?
   • Have you solved the problem?
   • Can you restate the original problem and make sure all conditions are satisfied?
4. Repeat – Once students have gone through this cycle once, encourage them to go back to it until they solve the problem. Use Socratic question to entice them.
   • What else do you understand now that you didn’t understand when you implemented your n^th strategy?
   • Can you refine your strategy?
   • Do you want to try a different approach?
   • Are we making progress?

About the author: You may contact Hector Rosario at hr111@caa.columbia.edu.