Seeing Squares: The Art of Question Generation

A first draft at assessing problem-solving with Further Mathematicians.

  1. Find a problem, written in 1950 for university applicants to Stanford University1
  2. Work on it with Tom, and then work on it a bit more alone.2
  3. Distil the question to its bare bones3

    Too vague? Just right?
  4. Students work, at first alone, and then in front of 30 visiting teachers, on the problem.
    Name-labels on backs so observers can write about students without disturbing them. 
    A confused student?
    First: attack problem alone

    Aftermath: discuss problem-solving with a wide variety of teachers.
  5. After noticing how hard the students found the task of generating questions from the prompts, I asked PHD students to do it too…
  6. Students analyse all questions that were asked, and start to analyse what makes a good question. Could you guess which were written by professionals and which by students?IMG_9157

We came up with the following rules for question generation:

Do Don’t
  1. Ask “How” questions
  2. Have an end goal in mind
  3. Look at interaction between two variables
  4. Make links to other topics
  5. Examine the scenario from multiple viewpoints
  6. Ensure your question is comprehensible to a stranger
  1. Be trivial
  2. Be vague
  3. Ask questions with a yes/no answer
  4. Make assumptions in your questions – include them as sub-questions instead

Examples of good questions:

  • What are the conditions required to create a 45 degree angle from the lines going through a square’s vertices? What would the collection of possible “45 degree points” look like?
  • Suppose the squares have side length 1 – what is the largest possible perimeter for the triangles formed from the straight lines and the square (with the 90 degree angle still assumed)

Examples of poor questions:

  • Is it drawn to scale?
  • How can I link this to trigonometry?
  • Are those squares?

7. After all of this warm-up, when the students saw the original question they were remarkably quick at finding the answer (once they had parsed the word-heavy question). Great recall of circle theorems.  I made a toy on Geogebra to illustrate the problem

Exuberant student work

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