A descriptive phase model of problem-solving processes

Abstract

Complementary to existing normative models, in this paper we suggest a descriptive phase model of problem solving. Real, not ideal, problem-solving processes contain errors, detours, and cycles, and they do not follow a predetermined sequence, as is presumed in normative models. To represent and emphasize the non-linearity of empirical processes, a descriptive model seemed essential. The juxtaposition of models from the literature and our empirical analyses enabled us to generate such a descriptive model of problem-solving processes. For the generation of our model, we reflected on the following questions: (1) Which elements of existing models for problem-solving processes can be used for a descriptive model? (2) Can the model be used to describe and discriminate different types of processes? Our descriptive model allows one not only to capture the idiosyncratic sequencing of real problem-solving processes, but simultaneously to compare different processes, by means of accumulation. In particular, our model allows discrimination between problem-solving and routine processes. Also, successful and unsuccessful problem-solving processes as well as processes in paper-and-pencil versus dynamic-geometry environments can be characterised and compared with our model.

Knowledge for teaching mathematical problem-solving with technology: An exploratory study of a mathematics teacher’s proficiency

“…These results are in line with our previous findings obtained with middle grade students (Carreira & Jacinto, 2019;Jacinto & Carreira, 2017a and with other studies that report on the non-linearity of problemsolving processes (Carlson & Bloom, 2005, Rott et al, 2021. In fact, these results not only contribute to contradict a view of mathematical problem-solving activity as a straightforward progression from the givens to the goals, as they support the claim that it develops around "iterative cycles of expressing, testing, and revising current ways of thinking" (Lesh & Zawojewski, 2007, p. 772), also when technology plays a significant role in the activity.…”

“…A few research teams have been adding evidence on the problem-solving strategies and ways of reasoning developed by students and teachers by means of technological tools (e.g., Jacinto & Carreira, 2017aKoyuncu et al, 2014;Santos-Trigo, 2019;Santos-Trigo & Reyes-Martínez, 2019). General studies on mathematical problem-solving expertise have proposed idealised prescriptive models and, more recently, descriptive models anchored in empirical data (e.g., Rott et al, 2021). Actual mathematical problem-solving seldom occurs in a linear, straightforward way; instead, as shown with young or adult problem solvers, the process takes place in a cyclic way (Carlson & Bloom, 2005).…”

“…Actual mathematical problem-solving seldom occurs in a linear, straightforward way; instead, as shown with young or adult problem solvers, the process takes place in a cyclic way (Carlson & Bloom, 2005). This is also the case when problem solvers make use of digital tools in carrying out the process (Carreira & Jacinto, 2019;Jacinto & Carreira, 2021;Rott et al, 2021). However, the role of technology in triggering this cyclic activity and pushing the solver to move forward is still insufficiently understood.…”

Pólya’s influence on (my) research

“…Another part of my PhD research was developing an empirical process model for describing and analyzing problem-solving processes. Apart from analyzing students' processes, I did an extensive literature review on models of the problemsolving process (Rott, 2014;Rott et al, 2021). There are a lot of (slightly) different frameworks by different authors developed for different purposes.…”

The Effect of Numerical Literacy Activities in Problem Based Learning Environment toward Mathematical Reasoning Ability of Elementary School Students

“…Problem Based Learning (PBL) model is one alternative that can be applied in learning mathematics in the context of developing students' problem-solving abilities (AK Amin et al, 2021, Rott et al, 2021. The PBL syntax is orienting students to problems, organising students, guiding the investigation process, presenting student work and analysing and evaluating the problem-solving process (Kelly et al, 2016, Basadur et al, 2014, Meister et al, 2018.…”