Cultivating procedural flexibility in school mathematics through collaborative learning

Good mathematical skills are not just about finding correct answers – they are also about being able to solve problems using different methods and choosing the most appropriate one for each situation. This is called procedural flexibility. For example, you can solve 334 – 296 with traditional column subtraction, but a more flexible approach would be to cleverly add four to both numbers first and calculate 338 – 300. Learning this kind of flexibility in school mathematics is a step towards creative problem solving needed in the future, where the key is to consider the advantages and disadvantages of different alternatives.

I studied procedural flexibility and its development from four perspectives: 1) I compared the flexibility of 791 Finnish, Swedish, and Spanish middle and high school students in equation solving, 2) I analyzed what types of mathematical tasks are good for learning, 3) I observed the role of collaboration when seventh graders were learning flexibility, and 4) I interviewed their teacher about her experiences.

The results show that using flexible solution strategies was quite rare among students from all three countries. Spanish students were more accurate, but Finnish and Swedish students were more flexible in their solutions. The "Reversed Equation Solving" task proved to be an effective way to help students find different solution methods and work together compared to the more typical equation solving lesson. An important finding was also that the teacher's strong belief in the importance of individual guidance hindered the implementation of collaborative learning. Strong attachment to one way can prevent learning different approaches – whether it is solution methods or teaching methods. Therefore, it is important for us to be aware of what we have become attached to.

In summary, I argue that mathematical tasks that encourage choices and multiple solution methods, combined with carefully guided collaborative learning, provide good opportunities for developing procedural flexibility. Learning to compare the strengths and weaknesses of different solution methods in school builds a strong foundation for solving future problems.