Abstract
This paper seeks to establish what kind of arguments pupils (aged 12–13) use and how they make their assumptions and generalizations. Our research also explored the same phenomenon in the case of graduate mathematics teachers studying for their masters’ degrees in our faculty at that time. The main focus was on algebraic reasoning, in particular pattern exploring and expressing regularities in numbers. In this paper, we introduce the necessary concepts and notations used in the study, briefly characterize the theoretical levels of cognitive development and terms from the Theory of Didactical Situations. We set out to answer three research questions. To collect the research data, we worked with a group of 32 pupils aged 12–13 and 19 university students (all prospective mathematics teachers in the first year of their master’s). We assigned them two flexible tasks to and asked them to explain their findings/formulas. Besides that, we collected additional (supportive) data using a short questionnaire. The supporting data concerned their opinions on the tasks and the explanations. The results and limited scope of the research indicated what should be changed in preparing future mathematics teachers. These changes could positively influence the pupils’ strategies of solving not only flexible tasks but also their ability to generalize.
Highlights
We often get used to generalizing in most situations in life
If teachers are unaware of its presence and are not in the habit of getting students to work at expressing their own generalizations, mathematical thinking is not taking place. (Mason, 1996, p. 65)
Solving strategies which we expected from predominantly numerical generalizators are: S1: the pupil uses the operation and procedures which can “fit” this problem; the solution lacks any evidence of noticing a mathematical structure, mostly counting the objects in the figures is used, the general formula is missing or is invalid
Summary
We often get used to generalizing in most situations in life. Not all our generalizations are correct, and some are based on one or two preconceptions Generalization is a part of mathematics education and mathematics literacy. Generalization is a heartbeat of mathematics and appears in many forms. If teachers are unaware of its presence and are not in the habit of getting students to work at expressing their own generalizations, mathematical thinking is not taking place. Mathematics is just a set of rules to memorize. In talking to the masters students we found out that presenting the rule and solving “problems” with its application is one of the favorite ways of teaching mathematics in the classroom
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