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• Metacognition – necessities and possibilities in teaching and learning mathematics

  69-87

  Johann Sjuts

  Views:

  287

  This article focuses on the design of mathematics lessons as well as on the research in mathematics didactics from the perspective that metacognition is necessary and possible.
  Humans are able to self-reflect on their thoughts and actions. They are able to make themselves the subject of their thoughts and reflections. In particular, it is possible to become aware of one’s own cognition, which means the way in which one thinks about something, and thus regulate and control it. This is what the term metacognition, thinking about one’s own thinking, stands for.
  Human thinking tends to biases and faults. Both are often caused by fast thinking. Certain biases occur in mathematical thinking. Overall, this makes it necessary to think slow and to reflect on one’s own thinking in a targeted manner.
  The cognitive processes of thinking, learning and understanding in mathematics become more effective and successful when they are supplemented and extended by metacognitive processes. However, it depends on a specific design of the mathematics lessons and the corresponding tasks in mathematics.

  Subject Classification: 97C30, 97C70, 97D40, 97D50, 97D70

• Darstellungen und Vorstellungen und ihre Bedeutung für eine wirksame Metakognition beim Problemlösen und Begründen

  195-220

  Johann Sjuts

  Views:

  206

  Metacognition has one of the highest effect sizes concerning successful learning. However metacognitive activities during task solving and problem solving are not directly obvious. But they can appear by writing someone's thoughts down. The following analysis, which focusses on the level of argumentation as well as on the way of derivation, shows that the quality of representation is an essential condition for the possibility of metacognition.

• Is it possible to develop some elements of metacognition in a Mathematics classroom environment?

  123-132

  Márton Kiss

  Eszter Kónya

  Views:

  312

  In an earlier exploratory survey, we investigated the metacognitive activities of 9th grade students, and found that they have only limited experience in the “looking back” phase of the problem solving process. This paper presents the results of a teaching experiment focusing on ninth-grade students’ metacognitive activities in the process of solving several open-ended geometry problems. We conclude that promoting students’ metacognitive abilities makes their problem solving process more effective.

  Subject Classification: 97D50, 97G40

• Should we draw, or should we work with numbers? Investigating proportional reasoning among 5th to 7th graders

  1-28

  Ildikó Bereczki

  Csaba Csíkos

  Views:

  85

  Proportional reasoning is an essential component of our everyday life and our mathematics studies. The rate of development in this area varies between age groups. In order to find out the level of students in Grades 5–7, we developed an online test. We consider it important to emphasize and support the use of visual representations in this subject, and therefore the tasks of the test on the eDia (Csapó & Molnár, 2019) interface have three types of input and output.We distinguish between ratios represented visually in the form of discrete quantities, ratios represented visually in the form of continuous quantities and ratios represented by text or numbers. Our study aimed to explore the differences between task types. Results indicate a representation-dependent developmental shift: in Grades 5–6, students perform best on tasks involving visual discrete quantities, whereas in Grade 7, performance increases markedly on text-text tasks. This suggests that visual representations function as an early scaffold, while later instruction strengthens symbolic processing.

  Subject Classification: Primary: 97C30; Secondary: 97D40, 97D60

• A constructive and metacognitive teaching path at university level on the Principle of Mathematical Induction: focus on the students' behaviours, productions and awareness

  133-161

  Agnese Ilaria Telloni

  Nicolina Antonia Malara

  Views:

  367

  We present the main results about a teaching/learning path for engineering university students devoted to the Principle of Mathematical Induction (PMI). The path, of constructive and metacognitive type, is aimed at fostering an aware and meaningful learning of PMI and it is based on providing students with a range of explorations and conjecturing activities, after which the formulation of the statement of the PMI is devolved to the students themselves, organized in working groups. A specific focus is put on the quantification in the statement of PMI to bring students to a deep understanding and a mature view of PMI as a convincing method of proof. The results show the effectiveness of the metacognitive reflections on each phase of the path for what concerns a) students' handling of structural complexity of the PMI, b) students' conceptualization of quantification as a key element for the reification of the proving process by PMI; c) students' perception of the PMI as a convincing method of proof.

  Subject Classification: 97B40, 97C70