a. What makes a good challenge ?

A good challenge should

  • be easy to get started on, drawing on likely knowledge and past experience of the student 
  • arouse curiosity – successful challenges often build on apparent inconsistencies or start from seemingly unintuitive premises
  • offer an engaging environment in which the student is invited (and indeed, seduced) to make conjectures 
  • invite extensions to more general and/or analogous situations
  • not be perceived as a question to which there is a single correct answer


Possible opening sentences for challenges include

  • Why do you think…?
  • Which of these explanations make more sense to you and why?
  • If  ______  is true, then how come  _______ ?
  • How do you think this situation does or does not resemble  ________ ?
  • What do you think might be true about this  ________ ?
  • What do you think this is a case of?


In each of the challenges on this site I have tried to pose questions that adhere to these guidelines. 

I hope these guidelines will serve as inspiration to teachers to pose new sorts of questions to their students and to encourage both teachers and their students to formulate and explore conjectures of their own.



 a short list of quotes about the importance of good questions -

Judge a man by his questions rather than by his answers.”   Voltaire

Too often we give our children answers to remember rather than problems to solve.”  Roger Lewin 

The formulation of a problem is often more essential than its solution, which may be merely a matter of skill.....To raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks real advances.”  Albert Einstein

Perplexity is the beginning of knowledge. “  Kahlil Gabran

Philosophy may be defined as the art of asking the right question…awareness of the problem outlives all solutions...  The answers are questions in disguise, every new answer giving rise to new questions.”  Abraham J. Heschel

Why and How are words so important that they cannot be too often used.”  Napoleon


Dan Finkel in Seattle has a wonderful TED talk that is very much in the spirit of this page and this website.

Judah L. Schwartz

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mathMINDhabits by Judah L. Schwartz is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.