Problem solving for fostering students’ analytical thinking: from lessons as an undergraduate to secondary school application
In the last few years, Inspectie van het Onderwijs has reported a decline in the analytical thinking level of secondary school students in the Netherlands. This decline impacts their mathematical level and understanding, further impairing their later studies. In 2023, under the guidance of Nelly Litvak, I introduced a problem solving method based on Polya’s method in a K-8 class to study whether their analytical thinking could be fostered by problem solving and reflection upon their own work.
Back in September of 2018, when I had just started my BSc Applied Mathematics at the University of Twente, I had the great fortune of following the course Linear Structures I, at the time given by Nelly Litvak. This was for me the very first time encountering such a theoretical subject and, like many other students, I initially struggled. However, I soon was able to overcome my issues within the subject thanks to Nelly Litvak’s problem sets, which the students had to solve according to Polya’s method with one small variation: much more emphasis was placed on reflecting on the work carried out (Litvak & Weedage, 2023). This implementation, as I would later on learn, was thanks to Eric Mazur (Mazur, 2016).
This particular problem solving method left a great impression on me, guiding much of my efforts when solving exercises in later subjects and helping me develop analytical thinking skills and so I wondered if it could do the same for much younger students, potentially setting up for success by shaping their understanding of what mathematics is beyond simple calculational skills in order to better prepare them for university.
Similarly to what Nelly Litvak did in her course, I introduced problem sets to be solved according to a problem solving method. This problem solving method was based on Polya’s (Polya, 2014) with a few key differences: students did not have to come up with a plan and reflection was used as a focus point. In particular, students were handed problem sets to solve within a week according to the following three steps:
- Preparation: students must read through the problem and describe in own words what is asked of them. Students must also give a summary of the data available to them, providing a visualisation if possible. In addition, students must indicate which pieces of information are necessary to find the final answer.
- Calculations: students must perform the calculations or steps of a proof in an orderly manner by writing every step out clearly and by placing each new step on a new line. The lines must then also be numbered and accompanied by a short description of what the students are doing in that step.
- Explanation and Reflection: students must explain each step of their calculations in full sentences using the same numbering as in step 2.
Setting up a plan is not included as a step since it is my experience and that of Nelly Litvak that students find this step particularly challenging, to the point of giving up on the method and therefore rendering the method unproductive for their own development and the fostering of analytical skills.
Figure 1: Scheme of the complete method
Students were required to bring their written out solutions to the problem sets to class, where the reflection step of the method would take place. During these in-class sessions, students were divided into groups to compare and discuss their solutions, providing one another with feedback and discovering different solution methods. This was so as to give every student the opportunity to actively talk about mathematics.
The group discussion was followed by a discussion of the solution with the teacher, who wrote out the answer based on input given by students. Finally, students were handed a reflection sheet, on which to reflect upon their work. This sheet required them to write out what had gone wrong in their solution, what type of mistake they had made, but also what had gone well, letting students reflect on all aspects of their work and, therefore, making them more aware of their own skills as well as what did and didn’t work throughout their process.
Although this study spanned across a limited amount of time, both the attitude and skills of students involved improved significantly, leading to a better atmosphere in class during mathematics lessons and to students approaching mathematics with more insight and interest. This, in turn, lead to better outcomes when working on homework and, finally, to more complete and explained in full sentences answers during the final test to the chapter during which this research took place.
If you’re interested in the complete set up and findings of this study, you can read more about it here: Lanting, L. S. (2025). Fostering the Development of Young Students’ Analytical Thinking by use of a Problem-solving Method. Journal of Research in Science, Mathematics and Technology Education, 8(SI), 377-401. https://doi.org/10.31756/jrsmte.4117SI
References
Litvak, N., & Weedage, L. (2023). Do we teach what we preach? https://research.utwente.nl/en/publications/do-we-teach-what-we-preach
Mazur, E. (2016). Private communication with prof.dr. N.V. Litvak (Nelly).
Polya, G. (2014). How to Solve It: A New Aspect of Mathematical Method. Princeton University Press.