The following problem set is being given to our department before we talk about how we want to teach problem-solving.
For the following problems, please use our 5-step annotation:
(i) note the context,
(ii) underline the question,
(iii) circle useful information,
(iv) write a sentence frame for the answer,
(v) “build a bridge” from the useful information to the answer (ie, solve).
Problem 1: A 20.5 gallon fish tank is 4/5 full. How many more gallons will it take to fill the tank? [Kelemanik]
Problem 2: Imagine a triangle inside a rectangular box. How much of the box does the triangle take up? [Lockhart]
Problem 3: Are the following lists the same list, or not?
- Guess and check
- Make an orderly list
- Eliminate possibilities
- Use symmetry
- Consider special cases
- Use direct reasoning
- Solve an equation
- Look for a pattern
- Draw a picture
- Solve a simpler problem
- Use a model
- Work backward
- Use a formula
- Use your head
- Look for patterns
- Represent symbolically
- Check for plausibility
- Take things apart
- Change or simplify the problem
- Work backwards
- Re-examine the problem
- Change representations
- Make sense of problems and persevere in solving them
- Reason abstractly and quantitatively
- Construct viable arguments and critique the reasoning of others
- Model with mathematics
- Use appropriate tools strategically
- Attend to precision
- Look for and make use of structure
- Look for and express regularity in repeated reasoning
Problem 4: Describe in detail what went on in your mind to “build the bridge” in problems 1 and 2. (Extra credit: same for Problem 3.) (Extra extra credit: same for Problem 4.)
Problem 5: Describe the relationship between the bridge-building you did in previous problems, and the list(s) from Problem 3.
Problem 6: To what extent do you relate to the following?
Mathematicians frequently report that often one of the most helpful things they can do to solve a problem they’re stuck on is step away from it. Jacques Hadamard (1949) examined his own experiences and also talked to many of his colleagues to work out what the common structure of this experience was, and determined that there seems to be a fairly predictable sequence to it:
- Intensely focus on the problem, working through every permutation you can think of that’s likely to produce an answer.
- Walk away from the problem and think about something else.
- The magic genie in your head might eventually, and often unexpectedly, yell a possible insight into your awareness.
For instance, Henri Poincaré reported struggling to work on Fuchsian functions over the course of several weeks and then being forced to walk away from the proof he had been stuck on due to a planned vacation. One day he was stepping onto a bus with his mind certainly not on mathematics, and suddenly the key insight he needed to finish the proof appeared in his mind. It was as though a part of his mind had been secretly working on the problem and then brought the finished product into his awareness. [Valentine]
Problem 7. Watch Grace Kelemanik’s Ignite talk, which starts around 20:25. (The other talks are also terrific.)
Problem 8. What experiences will help children become more confident in, competent at, and excited about solving problems?
Problem 9. How can we give those experiences to kids in the confines of a school schedule and in the context of required content and skills?