University of Zadar

Erkki Pehkonen

In this paper, the task “Number Triangle” stands for problem fields. The problem field in question is planned by the author to be used in the Finnish comprehensive school. With its aid, our purpose is to show how open teaching acts in practice. The most important in these problem tasks is the way they are presented in teaching: A problem field should be offered to pupils little by little. And the continuation in the problem field depends always on pupils’ answers. The answers of problems are not given here, since they are not as important as pupils’ independent solving of problems. How far the teacher proceeds with the problem field in question depends on answers given by pupils.

Bernd Zimmermann

Teaching of problem solving as well as research in problem solving are subjected to many different constraints and hidden assumptions. Many of the constraints for teaching hold as well for research in these areas. The detection and the analysis of such issues are carried out on the background of concrete elucidating examples (“proofs of existence”). We conclude with some theses, related to such constraints as “As well teachers as researchers should be much more aware on their respective belief-system” and “There should be more analysis and reflection about possible constraints and limits on own studies on problem-solving and metacognition.” Some own reflections on possible constraints of this study are presented as well.

András Ambrus, Dániel Katona

There is still a deep gap between the theories of the didactics of mathematics and mathematics teaching practice worldwide. In our article, we analyse our trial to reach practicing mathematics teachers and summarize their opinion about some basic issues of teaching mathematics problem-solving from the point of view of cognitive load theory, what is a quite new topic in mathematics didactics society. We asked on the one hand, teachers from a small town in Hungary, and on the other hand, expert teachers and four young teachers from elite schools in the capital. The four young teachers have also started their PhD studies in mathematics education, besides school teaching. The opinions of the two groups of teachers reflect different attitudes towards teaching problem-solving, but in both cases relevant and important perspectives of the Hungarian school reality. The base of our study was a talk and an article of the first author, related to the role of human memory in learning and teaching mathematical problem-solving. We have been interested in how classroom teachers can take into consideration some results of the cognitive load theory, e.g. the split-attention effect and schema automation in their teaching practice, as well as in their attitudes to the use of worked examples and distributed practice. We analyse the results mostly from the perspective of desirable developments in in-service teacher training in Hungary.

Günter Graumann

Problem orientation as a part of mathematics education is required by mathematicians already for a long time and since the beginning of ProMath meetings the situation in school has been improved a little bit, but it is still practiced not sufficiently. Resistors among other enhancements are special beliefs about mathematics education and insufficient knowledge about problem orientation. This is the reason I offered again a seminar for teacher students about problem orientation in mathematics education in summer 2016. In the following first I will briefly discuss different type of problems and then as the main part report and discuss results of a written survey on knowledge to problem orientation at the beginning of the seminar as well as personal comments the students have made during the seminar sessions and in protocols on the seminar sessions.

Ana Kuzle

Problem solving in Germany has roots in mathematics and psychology but it found its way to schools and classrooms, especially through German Kultusministerkonferenz, which represents all government departments of education. For the problem solving standard to get implemented in schools, a large scale dissemination through continuous professional development is very much needed, as the current mathematics teachers are not qualified to do so. As a consequence, one organ in Germany focuses on setting up courses for teacher educators who can “multiply” what they have learned and set up their own professional development courses for teachers. However, before attaining to this work, it is crucial to have an understanding what conceptions about teaching problem solving in mathematics classroom mathematics teacher educators hold. In this research report, I focus on mathematics teacher educators’ conceptions about problem solving standard and their effects regarding a large-scale dissemination.

Lars Burman

This article is a complementary article to three earlier published articles (Burman & Wallin, 2014; Burman, 2014; Burman, 2016) about the use of problem sequences in mathematics instruction in the grades seven to nine in Finland. The pupils work with problems using the same strategy in different contexts, or with only one problem where they gradually proceed towards the solution. In both cases, the problems are solved in steps under the guidance of the teacher. The article focuses on the considerations designer of a problem sequence has, as the design of the sequence is accomplished. In general, the pupils are supposed to be provided with the possibility to think creatively, to work mostly in groups but also individually, and to be inspired by tasks related to real-world situations.

Tatjana Hodnik Čadež, Vida Manfreda Kolar

This paper presents a discussion of the problem-solving approaches of primary and secondary school pupils in relation to the following issues: developing strategies, communicating, and receiving guidance. Guiding is the role of the teacher who should be sensitive enough to support pupils’ thinking, when necessary, but not direct it. A group of pupils (35 pupils between 10 and 19 years old) were given a geometrical problem that required them to define the number of parts created when a single plane was divided by straight lines. Each pupil tackled the problem individually, while prospective teachers from the Faculty of Education observed and guided them. After analysing the prospective teachers’ research reports on guiding pupils through the problem we came to the following conclusions: all the pupils needed guiding in order to make progress in problem solving towards general rule, most of the pupils need to learn about heuristics more systematically, prospective teachers got better inside view on thinking process for problem solving of different age groups of pupils. From the success at problem solving point of view we observed the following: until presented with a problem that required a geometrical approach, the differences among the age groups in terms of successful problem solving were not that noteworthy, the difference among age groups was observed in examples of more complex problem solving where a shift towards an arithmetical approach was needed.

Ljerka Jukić Matić

Problem solving in schools begins with mathematics teachers. The degree to which mathematics teachers are prepared to teach for, about and through problem solving influences on their implementation of problem solving in school. We conducted a small scale study where we examined the effect of implementation of heuristic strategies and Polya’s steps in mathematics method course. We assessed pre-service teachers’ knowledge and attitudes about them as problem solvers before and after the course. Moreover we assessed their beliefs of problem solving in school mathematics. Those beliefs were assessed in two occasions: right after the course and after finished teaching practice. Although students’ knowledge on problem solving was improved, the results of students’ beliefs show that it is important that pre-service teachers, and consequently in-service teachers, are constantly reminded on the positive effect of constructivist and inquiry-based approach on teaching mathematics.

Irena Mišurac, Maja Cindrić

There is a strong link between teaching activities in teaching mathematics and students’ outcomes. Activities that teachers and students conduct in mathematics are encouraged to specific mathematical competence of students. In the present research, we wanted to establish to what extent the Croatian class teachers know the guidelines of teaching mathematics and their awareness of the importance of performing activities that encourage contemporary mathematical processes. The goal of the research was to establish which activities teachers carry out when teaching mathematics in order to foresee the competences to be developed in their pupils. We have done our research on a sample questionnaire of 400 class teachers that teach mathematics 4 classes a week. To determine which activities were conducted by teachers with students in class mathematics and how often, we defined 26 activities for teachers to determine the intensity of their use on the Likert scale from 1 (never) to 5 (almost always). We selected 15 activities typical of modern teaching of mathematics and 11 activities typical of traditional teaching, which we offered in mixed order in the survey. In like manner, we worked out 26 competences (15 competences emphasized by contemporary teaching of mathematics and 11 emphasized on traditional teaching), while teachers marked the number of competences they considered to be important for pupils. In order to test the theoretical assumption on the difference in access to teachers who work in a modern or traditional way, we conducted a process of factor analysis. The factor analysis clearly distinguished the two groups of activities and two groups of competences, and as expected the way to the variables that saturate the first factor consists of contemporary activities/competences and variables that saturate the second factor consists of traditional activities/competences. This confirms our theoretical setting of modern and traditional approach to teaching mathematics. We noticed that most teachers carried out traditional activities more frequently than the contemporary ones, but that most of them evaluated contemporary competences with better scores than traditional ones.