A descriptive phase model of problem-solving processes

Benjamin Rott,Birte Specht,Christine Knipping

doi:10.1007/s11858-021-01244-3

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.

Highlights

IntroductionProblem solving (PS)—in the sense of working on nonroutine tasks for which the solver knows no previously learned scheme or algorithm designed to solve them (cf. Schoenfeld, 1985, 1992b)—is an important aspect of doing mathematics (Halmos, 1980) as well as learning and teaching mathematics (Liljedahl et al 2016)
Problem solving (PS)—in the sense of working on nonroutine tasks for which the solver knows no previously learned scheme or algorithm designed to solve them—is an important aspect of doing mathematics (Halmos, 1980) as well as learning and teaching mathematics (Liljedahl et al 2016)
The goal of this paper was to present a descriptive model of PS processes, that is, a model suited to the description and analyses of empirically observed PS processes

Summary

Introduction

Problem solving (PS)—in the sense of working on nonroutine tasks for which the solver knows no previously learned scheme or algorithm designed to solve them (cf. Schoenfeld, 1985, 1992b)—is an important aspect of doing mathematics (Halmos, 1980) as well as learning and teaching mathematics (Liljedahl et al 2016). Almost all of the existing models are normative, which means they represent idealised processes. They characterise PS processes according to distinct phases, in a predetermined sequence, which is why they are sometimes called ‘prescriptive’ instead of normative. Normative models are generally used as a pedagogical tool to guide students’ PS processes and to help them to become better problem solvers. Real PS processes look different; they contain errors, detours, and cycles, and they do not follow a predetermined sequence. Actual processes like these are not considered in normative models.

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