Benedetto Di Paola

Discrete Mathematics in Lower School Grades? Situation and Possibilities in Italy

This paper presents an overview of the Italian situation in teaching discrete mathematics in primary and middle school, taking into account the national teaching guidelines and their connection with the subject. We describe research conducted with about 150 teachers, interviewed in a preliminary questionnaire. The data collected shows, for all teaching grades, interest in having more discrete mathematics in the school curriculum even if there are some difficulties in teaching it and in inserting it in the usual mathematical activities at school, mostly related to teachers’ knowledge and self-confidence about the subject. We also discuss results and future plans for a continuing research project in the field. We describe in the conclusion a design research project involving teachers in the activity-designing process, aimed at bringing new mathematical knowledge and competences to students.

Analysing the Conceptions on Modelling of Engineering Undergraduate Students: A Case Study Using Cluster Analysis

The problem of taking a set of data and separating it into subgroups where the elements of each subgroup are more similar to each other than they are to elements not in the subgroup has been extensively studied through the statistical method of Cluster Analysis . This method can be conveniently used to separate students into groups that can be recognized and characterized by common traits in their answers, without any prior knowledge of what form those groups would take (unsupervised classification). In the last years many studies examined the consistency of students’ answers in a variety of situations. Some of these papers have tried to develop more detailed models of the consistency of students’ reasoning, or to subdivide a sample of students into intellectually similar subgroups by using Cluster Analysis techniques. In this paper we start from a description of the data coding needed in Cluster Analysis, in order to discuss the meanings and the limits of the interpretation of quantitative results. Then a method commonly used in Cluster Analysis is described and the variables and parameters involved are outlined and criticized. Section 3 deals with the application of this method to the analysis of data from an open-ended questionnaire administered to a sample of university students, and discusses the quantitative results. Finally, some considerations about the relevance of this method in Physics Education Research are drawn.

The Chinese Written Language as Tool for a Possible Historical and Epistemological Reflections on the Mathematics and the Impact of Teaching/Learning of Mathematics

Strong and undisputed vector of the Chinese cultural tradition has been (till now), as told in the preceding chapters, its peculiar system of fixation grafemica.

The Experimental Epistemology as a Tool to Observe and Preview Teaching/Learning Phenomena

In previous chapters, we discussed some general considerations of epistemological nature, and in this chapter, we introduce two groups of investigations of experimental behaviors in the years and in the spirit of the experimental epistemology of mathematics. The first group refer to an experiment conducted with Palermo and Nanchino students of Superior Secondary School on the comparison of the schemes of reasoning regarding the logical–linguistic paradoxes in this two cultures. The analysis of two cases is introduced, then one of which had already been analyzed for about a year in 1984/85.

Strategy and Tactics in the Chinese and European Culture: Chess and Weich’i

In previous chapters we observed that hypothetic-deductive reasoning may represent in western culture a fundamental reference for “strategic” choices. It is concerned with maths and also with other cultural contexts including economy. What for Chinese Culture?

The Meta-rules between Natural Language and History of Mathematics

Before going on to outline teaching, we need a short historical–epistemological process that can draw criticism, quite basic, the key stages of the long journey and suffered from the trend on Algebra as the Arithmetic. We believe the combination of arithmetic-algebra as significant for the relationship between mathematical thinking. Considerations should be made parallel to the geometric area in which we mention in the discussion of the chapter but that remains an issue open to possible future theoretical and experimental investigations.

Common Sense and Fuzzy Logic

As mentioned in Chapter 1, it is our opinion we that believe different types of logic can lead to different patterns of reasoning. Examples are the use of the logic of a bivalent and fuzzy set of argumentation in stages used in mathematical contexts and beyond. This observation may be significant, in our opinion, problem addressed and on the integration of different cultural references. With this conviction, we believe useful in discussing, albeit briefly, some significant aspects of common sense and fuzzy logic. This chapter presents considerations within epistemological nature validated by experimental investigations conducted in the classroom in Palermo. It remains open to the interpretation of the results obtained from a comparison of cognitive styles between Chinese and Italian.

Rhythm and Natural Language in the Chinese and European Culture

Appendix 2 of Chapter 6 has been taken into account the classification of the arts in the two main cultures. In Chinese culture, the music was made in the first place: “Music, discipline related to hearing and also to sophisticated processes that regulate vibrations. It is the practice of the inner qi. It concerns also with embodiment referred to the musical rhythms. Today we know that it is related to natural language and numbers learning”. In European culture, the music is part of the quadrivium of the arts and is considered one of the sciences together with arithmetic, geometry and astronomy.