Mid Sweden University

Begrepspförståelse är en central del av elevers lärande i matematik, som innebör en djupare förståelse för bland annat begrepps egenskaper och hur olika matematiska begrepp hänger samman, och som hjälper dem lösa matematiska problem. Genom att arbeta med utmanande problem, som kräver kreativa lösningar, främjar elevernas förmåga att tänka kritiskt och reflektera över matematiska begrepp, vilket i sin tur stärker deras långsiktiga lärande.

One way to stimulate conceptual considerations is through mathematical problem solving, which requires students to construct new solution methods, potentially including new representations. Hence, this study focuses on students’ use of and shift between representations during problem solving, specifically regarding the fundamental and often challenging concept of rational numbers. Swedish elementary school students were observed when tackling mathematical problems in pairs. The findings revealed that the students’ considerations concerned several conceptual properties of rational numbers through the use of different representations. However, students tended to use the representations presented in the problem, favouring the most familiar representation rather than constructing their own. As a result, they may have overlooked opportunities to explore a wider range of representations which could have deepened their understanding of the concept. This suggests a potential opportunity to design problems that require students to transfer between representations and grapple with unfamiliar ones.

Problemlösning är en fundamental del av matematisk verksamhet, men hur ska problem utformas om syftet är att utveckla elevers begreppsförståelse? Hör har flera olika problem testats och reviderats för att skapa begreppsliga utmaningar.

A mathematical problem that involves conceptual challenges, may offer students opportunities to show and develop their conceptual understanding. However, to meet specific conceptual learning goals, a deliberate design of the problem is important. In the study, a single problem is used in different versions to investigate what conceptual understanding of the concept perimeter is visualized in students’ solutions. The result shows that the chosen problem offers reasonable conceptual challenges and thus is suitable for visualizing and possibly developing students’ conceptions. Three conceptual aspects are identified and suggested to be used to support further discussions on task design: properties and definitions of concepts, the use of non-typical examples, and the relations between processes and concepts.

Mathematical problem solving has proven to be valuable for students’ learning. Yet, problem solving is often referred to and discussed without distinguishing between creative and conceptual aspects. The purpose of this study is to advance understanding of mathematical problem solving through the analysis of students’ work with mathematical problems in terms of the creative and conceptual challenges they encounter. It is proposed that the characteristics of creative challenges can be used to obtain a more detailed description of the problem-solving process. Furthermore, the characteristics of conceptual challenges can provide a basis for discussion of what conceptual understanding may entail. Moreover, the analytic framework developed in this study is proven to be of assistance in the visualization of challenges and may be useful in future efforts to investigate challenges and students’ problem solving in mathematics.

Problem solving has proven valuable for students’ learning. This presentation includes results from three studies where students’ opportunities to engage in mathematical problem solving have been studied. The results show that textbook tasks to a great extent are of routine character rather than mathematical problems. Furthermore, students mainly work on the easier sets of tasks, including even fewer mathematical problems. Thus, students’ opportunities to engage in mathematical problem solving are limited. In an ongoing project an analytic framework is developed to better understand some of the important components of mathematical problem solving and possibly also be of support in the design of mathematical problems. 

A selection of secondary school mathematics textbooks from twelve countries on five continents was analysed to better understand the support they might be in teaching and learning mathematical problem solving. Over 5700 tasks were compared to the information provided earlier in each textbook to determine whether each task could be solved by mimicking available templates or whether a solution had to be constructed without guidance from the textbook.There were similarities between the twelve textbooks in the sense that most tasks could be solved using a template as guidance. A significantly lower proportion of the tasks required a solution to be constructed. This was especially striking in the initial sets of tasks.Textbook descriptions indicating problem solving did not guaranteethat a task solution had to be constructed without the support of anavailable template.