No abstract

In our contribution we focus on the possibility to use an educational tool called Concept Cartoons in future primary school teachers' education, as an instrument for observing how future primary school teachers reason about fractions. In the introduction section we present Concept Cartoons, and also the primary school level of the fractions topic. In the first part of the research we analyse data obtained when future primary school teachers were solving a problem in the Concept Cartoon form. The task which we adapted to this form belongs to primary school mathematics, it focuses on the concept of a fraction per se (on the parts-and-whole decision and on comparison of two pre-partitioned models with diverse wholes). Using Concept Cartoons, we can observe which statements about the issue our respondents consider as correct, and which kinds of reasoning they use in their justifications. In the second part of the research we analyse problems that the respondents themselves posed in the Concept Cartoon form, with particular focus on tasks devoted to fractions.

In our contribution we focus on the possibility to develop open approach to mathematics in future primary school teachers during a university course on mathematics conducted in inquiry-based manner. In the first part of the research we analyse data obtained in the beginning and in the end of the mathematics course with respect to two main aspects related to open approach to mathematics: searching for all solutions of a task, and acceptance of different forms of notation of a given solution. Data analysis revealed in the participants three different shifts towards open approach to mathematics, and showed that after the active participation in the course each of the participants improved at least in one of the monitored aspects, and that none of the participants got worse in any of the aspects. In the second part of the research we analyse problems posed by participants several months after the end of the course, again with respect to the two aspects related to open approach to mathematics. As a special diagnostic instrument in our research we use an educational tool called Concept Cartoons.

Tento text je shrnutím našich dosavadních poznatků o uplatňování myšlenek badatelsky orientovaného vyučování matematice. Po nezbytném vymezení některých pojmů v něm představujeme, jak je chápáno badatelsky orientované vyučování matematice. Využíváme podněty přicházející z didaktiky přírodovědných předmětů a navazujeme na související podněty a poznatky z didaktiky matematiky. Ukazujeme, (a) jak existující didaktické teorie a podoby školní praxe v matematickém vzdělávání korespondují s badatelsky orientovaným vyučováním; (b) co může být zdrojem matematického bádání ve škole; (c) čím jsou charakteristické matematické úlohy, které mohou vést k badatelským aktivitám žáků. V závěru uvádíme náměty několika úloh podněcujících badatelsky orientované vyučování matematice.

The contribution investigates opportunities that an educational tool called Concept Cartoons can offer in future teachers' education, namely in comparison with word problems in standard written tests. The referred empirical study was conducted in two separated consecutive stages, with two groups of future primary school teachers (the first one from the Czech Republic, and the second one from Slovakia). The participants of the first stage solved four word problems (T1, T2, T3, T4) with increasing difficulty within the written test, and a problem with a similar structure and difficulty as T3 but in the Concept Cartoon form. The second stage of the study served as a complementary stage, its participants solved only the word problem T3 and the Concept Cartoon. In both stages, the comparison of results and solution procedures revealed many participants who mastered the word problem(s) but displayed a fundamental misconception when working with the Concept Cartoon. Two thirds of the participants presented noncorresponding responses to these two corresponding tasks: they solved one of them correctly and the other one incorrectly. All of the problems in the study were based on the part-whole interpretation of fractions, the revealed misconception consisted of incorrect determination of the whole.