Why problem solving?

Students entering a university-level mathematics program often have a particular image of what mathematics is, shaped by their experiences in secondary school. Typically, this image is biased toward calculational skills—such as solving equations and manipulating formulas. Much of the practice they encountered in high school focused on training these skills.

However, at university, students will increasingly find themselves solving exercises for which there is no clear a priori solutions method. This is what we call mathematical problem solving, after Polya in his classical text “How to solve it” ([2]).

Problem solving asks for much more than just knowledge of calculational routines. Important aspects (see also Schoenfeld [3]) include:

• Heuristics. If there is no clear recipe or solution method, how do you start? Experienced problem solvers use a set of diagnostic tools—called heuristics—to gain insight into a problem. For example: examine special cases, solve a simpler version of the problem, work backwards, or organize the information using a diagram or visual aid. First-year students typically have only a limited repertoire of such heuristics.
• Control. Problem solving is often a messy process. There is usually more than one path to a solution. Which do you choose? How long do you pursue a particular approach? How and when do you decide to abandon a strategy and try something else? These issues of control become increasingly important as problems grow in complexity. Inexperienced problem solvers often stick with an unproductive idea for too long.
• Perception of mathematics. The way students approach mathematical problems is heavily influenced by their idea of what mathematics is. First-year students often attempt to solve problems primarily through calculation or solving equations. However, other—often simpler—approaches may be more effective.

With our Problem-Solving sessions, we aimed to broaden students’ horizons: to show them that mathematics involves more than just solving equations and manipulating formulas, to help them develop heuristics, and to support them in gaining control over the problem-solving process. We selected problems accordingly: they should be relatively easy for students to begin exploring and to obtain partial results, but challenging to solve completely. A list of this year’s problems can be found at the end.

 

The setup

Another aspect students need to get used to is that problem solving takes time. Most students were quite good at mathematics in secondary school and are used to spending no more than a few minutes on an exercise. In our Problem-Solving sessions, however, students typically needed much more time to work through the problems. So how to keep them engaged?

We found inspiration in Liljedahl’s book “Building Thinking Classrooms”, in which the author gives many suggestions on how to set up classes to maximize student activity. We adopted the following design principles:

• Very short problem descriptions, preferably given orally;
• Students are randomly assigned to groups of three;
• Each group works standing at a whiteboard, using only one pen;
• Sessions end with a central discussion of the problem, along with each group’s results or conjectures;
• Students then sit down, write their solutions individually, and hand in their work;
• The teacher provides feedback on the work (but no grade);
• At the next session, the teacher discusses the students’ solutions, common mistakes, and observations.

The presence was mandatory. There was no final test or grade for this part of the course, only a pass/fail for participation and meeting some basic requirements for the hand-in solutions. We deliberately chose not to assign grades in order to create a low-pressure environment where students could engage without fear of making mistakes.

Our experiences

Positive experiences:

• The chosen setup works extremely well to keep students active. Even after an hour working, groups had to be stopped by the teacher.
• The sessions were highly effective in promoting bonding and building social cohesion. Since we ran the course in the first quarter of the first year, this was a very welcome side effect.
• The use of heuristics emerged organically; although we did not explicitly train students to use them, many independently adopted strategies such as examining special cases and creating diagrams or visual representations.
• Students’ control over the problem-solving process appeared to improve. Initially, they were hesitant to abandon a chosen solution method—even if it had led nowhere for 15 minutes. Over time, however, they became more willing to try alternative approaches.

Challenges:

• Precise formulation: Students found it very difficult to make precise statements and organize their arguments clearly. Even when they had the right ideas to solve a problem, their written work often contained ambiguous or unclear reasoning.
• Lack of transfer: While students were highly active during the Problem-Solving sessions, this engagement did not carry over to other courses. In particular, their behavior in regular exercise classes remained similar to that of students in previous years.
• “What did I learn?”: Most students enjoyed the problem-solve sessions, but it was not so clear to them what they learned. This is not surprising: it is hard to quantify or measure the improvement of heuristics and control.

 

Conclusion and outlook

The Problem-Solving sessions were a very positive experience for both teachers and students. The format has a lot of potential, and we believe there is still much more to be gained from it. In the coming years, we plan to place greater emphasis on helping students learn to formulate precise and unambiguous statements. Additionally, we are planning to conduct research on the specific learning outcomes.

If you have questions about our setup, feel free to contact us at B.vandendries@tudelft.nl. If you’re planning to try similar Problem-Solving sessions yourself—or if you’ve already implemented something similar—let us know. We’re very interested in hearing your ideas and experiences!

 

References

[1]. Liljedahl, P. (2021). Building Thinking Classrooms in Mathematics. Corwin Mathematics

[2]. Polya, G. (1945). How to solve it. Princeton University Press.

[3]. Schoenfeld, A.H. (1985). Mathematical Problem Solving. Academic Press, Inc.

 

The problems

We used the following problems:

1. Consider the following two equations: \(3x^3 + 5x^2 – 9x + 2 = 0 \text{ en } 2y^3 – 9y^2 + 5y + 3 = 0 \).
   Note that the coefficients are in opposite order. What is the relation between the solutions of these two equations?
2. How many zeros does \(1000!\) end with?
3. Consider sums of at least two consecutive positive integers. Which values can such sums attain? Which values are not possible?
4. Write 24 as sum of positive numbers, then take the product of these numbers. What is the highest possible product that you can get?
5. Suppose you cut a pancake with 9 straight cuts. What is the largest number of pieces you can get?
6. Fix a circle, and consider a triangle with vertices on that circle. How to choose the vertices to maxime the area of the triangle? What if you replace the circle by a regular octagon?
7. Consider the sequence \(1, 2, \ldots, 9\). Cross out two numbers and put the absolute value of the difference at the end of the sequence. Repeat until one number is left. What number(s) can this be?
8. Find all functions \( f:\mathbb{N} \to \mathbb{N} \) such that \( f(f(n)) = 10n \) for all \( n \in \mathbb{N} \). Repeat, but now with the condition \( f(f(n)) = n + 1 \) for all \( n \in \mathbb{N} \).