Number series and computer

This article describes the discovery and the subsequent proof of a hypothesis concerning harmonic series. The whole situation happened directly in the course of the process of a maths lesson. The formulation of the hypothesis was supported by the computer. The hypothesis concerns an interesting connection between harmonic series and the Euler\u27s number . Let denote the ’th partial sum of the harmonic series. Let’s notice the sums 1, 4, 11, . . ., where the partial sum reaches 1, 2, 3, . . . for the first time. Let’s mark the relevant indexes 1, 4, 11, . . . as 1, 2, 3, . . .. So is the index of such a partial sum for which the following is true: P−1 \u3c n, P \u3e n. The hypothesis lim Pn+1/Pn = e has been proved, in an elementary way, in the article

The Strategy the Use of False Assumption and Word Problem Solving

The paper describes one problem solving strategy – the Use of false assumption. The objective of the paper is to show, in accordance with Phylogenesis and Ontogenesis Theory, that it is worthwhile to reiterate the process of development of the concept of a variable and thus provide to pupils one of the ways helping them to eliminate usual difficulties when solving word problems using linear equations, namely construction of the equations. The paper presents the outcomes of a study conducted on three lower secondary schools in the Czech Republic with 147 14–15-year-old pupils. Pupils from the experimental group were, unlike pupils from the control group, taught the strategy the Use of false assumption before being taught the topic Solving word problems. The tool for the study was a test of four problems that was sat by all the involved pupils three weeks after finishing the topic “Solving word problems” and whose results were evaluated statistically. The experiment confirmed the research hypothesis that the introduction of the strategy the Use of false assumption into 8th grade mathematics lessons (14–15-year-old pupils) helps pupils construct equations more successfully when solving word problems

IMPACT OF HEURISTIC STRATEGIES ON PUPILS’ ATTITUDES TO PROBLEM SOLVING

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. 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 (Novotná 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 (Novotná et al., 2014); we focus on the dangers it bears when it is used by pupils. The conclusion from (Novotná 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

Relations between Scientific Reasoning and Culture of Problem Solving

The article reports the results of a study, the main aim of which was to find out correlations among the three components of the Culture of problem solving (reading comprehension, creativity and ability to use the existing knowledge) and six dimensions of Scientific reasoning (conservation of matter and volume, proportional reasoning, control of variables, probability reasoning, correlation reasoning and hypothetical-deductive reasoning). Further, we present the correlations among individual components of the Culture of problem-solving and individual dimensions of Scientific reasoning with pupils’ school performance in mathematics and physics. We conducted our survey among 23 pupils aged between 14–15 years in the Ústí nad Labem Region. The results have shown that one component of the Culture of problem-solving – the ability to use the existing knowledge – strongly correlates with three dimensions of the Scientific reasoning structure: proportional reasoning, control of variables and probability reasoning. However, no correlation was proved between the creativity and the dimensions of Scientific reasoning. We have found out also that the indicators of the Culture of problem-solving and the Scientific reasoning largely do not correlate with school performance either in mathematics or in physics

PROBLEM SOLVING IN SCHOOL MATHEMATICS BASED ON HEURISTIC STRATEGIES

The paper describes one of the ways of developing pupils’ creative approach to problem solving. The described experiment is a part of a longitudinal research focusing on improvement of culture of problem solving by pupils. It deals with solving of problems using the following heuristic strategies: Analogy, Guess – check – revise, Systematic experimentation, Problem reformulation, Solution drawing, Way back and Use of graphs of functions. Most attention is paid to the question whether short-term work, in this case only over the period of three months, can result in improvement of pupils’ abilities to solve problems whose solving algorithms are easily accessible. It also answers the question which strategies pupils will prefer and with what results. The experiment shows that even short-term work can bear positive results as far as pupils’ approach to problem solving is concerned

Pupils’ School Performance and Their Cognitive Abilities to Solve Problems

The paper describes the results of a study whose main aim was to find the correlation between a pupil’s school grades in Czech language (native), mathematics and physics and pupils' cognitive predispositions to problem solving in science and mathematics diagnosed by the Lawson Classroom Test of Scientific Reasoning and the Culture of Problem Solving test. The total of 180 pupils from the Czech Republic aged 14–15 took part in this study. The results show that pupils with better grades in the monitored subjects achieve better results in both tests. It also turns out that there are generally statistically insignificant differences between the results of pupils assessed by grades 1 or 2, and between the results of pupils assessed by grades 3 or 4. Pupils’ performance in the two tests might help to strengthen the objectivization of grading at school. They might also help to identify the indicators important for the development of problem-solving skills. The research specifically points at the need of developing algebraic thinking, conception of infinity, spatial imagination, geometric imagination in the plane, proportional reasoning and the abilities of the control of variables