Abstract
The paper is a sequel to the article (Novotna et al., 2014), where the authors present the results of a 4-month experiment whose main aim was to change pupils’ culture of problem solving by using heuristic strategies suitable for problem solving in mathematics education. (Novotna et al., 2014) focused on strategies Analogy, Guess – check – revise, Systematic experimentation, Problem reformulation, Solution drawing, Working backwards and Use of graphs of functions. This paper focuses on two other heuristic strategies convenient for improvement of pupils’ culture of problem solving: Introduction of an auxiliary element and Omitting a condition. In the first part, the strategies Guess – Check – Revise, Working backwards, Introduction of an auxiliary element and Omitting a condition are characterized in detail and illustrated by examples of their use in order to capture their characteristics. In the second part we focus on the newly introduced strategies and analyse work with them in lessons using the tools from (Novotna et al., 2014). The analysis of results of the experiment indicates that, unlike in case of the strategy Introduction of an auxiliary element, successful use of the strategy Omitting a condition requires longer teacher’s work with the pupils. The following analysis works with the strategy Systematic experimentation, which seemed to be the easiest to master in (Novotna et al., 2014); we focus on the dangers it bears when it is used by pupils. The conclusion from (Novotna et al., 2014), which showed that if pupils are introduced to an environment that supports their creativity, their attitude towards problem solving changes in a positive way already after the period of four months, is confirmed.
Highlights
It is a fact generally agreed on by teachers, mathematics educators and researchers in the field of mathematics education that problem solving is the cornerstone of mathematics taught at all school levels
The paper is a sequel to the article (Novotná et al, 2014), where the authors present the results of a 4-month experiment whose main aim was to change pupils’ culture of problem solving by using heuristic strategies suitable for problem solving in mathematics education. (Novotná et al, 2014) focused on strategies Analogy, Guess – check – revise, Systematic experimentation, Problem reformulation, Solution drawing, Working backwards and Use of graphs of functions
The following analysis works with the strategy Systematic experimentation, which seemed to be the easiest to master in (Novotná et al, 2014); we focus on the dangers it bears when it is used by pupils
Summary
It is a fact generally agreed on by teachers, mathematics educators and researchers in the field of mathematics education that problem solving is the cornerstone of mathematics taught at all school levels. In accordance with (Polya, 2004) we define an auxiliary element as an object that is not explicitly present in the problem assignment and that we introduce into the problem, hoping it will help us see the solution more . Polya calls this strategy Auxiliary elements and understands it in a broader context than as we present it because we focus exclusively on school problems. In case of algebraic problems we usually add a convenient number to both sides of the equation or we introduce a new unknown As this element is not explicitly present in the problem assignment and we introduce it into it, we call this type of solution the strategy of Introduction of an auxiliary element. Let us illustrate the described strategy by two problems
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