MatCos 3.0

In this chapter, the authors present MatCos 3.0 software aimed at the primary school. First, the pedagogical-didactic and training bases on which the construction of the software is based will be exposed, albeit briefly. All the buttons on the user interface and all the axiomatic and genetic commands available are presented. The method used is a direct presentation aimed at technical use. Some pedagogical-educational observations will be given. Numerous examples and screenshots will be given. The commands will not be displayed in strictly alphabetical order, but a more logical and consequential line has been preferred. At the end of the chapter, the alphabetical list of all the commands available in the programming environment will be shown.

Computer-Based Mathematics Education

In this chapter, the authors mention, briefly, the attempts made from the 1970s to today to insert modern technologies in the teaching/learning of mathematics. They start with the first pocket calculators in the 1970s, which had a rapid technological diffusion that still exists. They focus on the impact that digital electronic technology has had on teaching/learning math. They will not follow a strictly chronological order, preferring to dwell on what, in their opinion, are the fundamental stages. So, the advent of the PC and programming languages—Logo, Basic, Pascal—CAI programs, DGS software, CAS. They conclude with their MatCos Project, after mentioning the new coding languages, including Scratch.

Epistemological Notes on Mathematics

This chapter is an attempt to show how mathematical thought has changed in the last two centuries. In fact, with the discovery of the so-called non-Euclidean Geometries, mathematical thinking changed profoundly. With the negation of the postulate for “antonomasia,” that is the uniqueness of the parallel for Euclid, and the construction of a geometric theory equally valid on the logical and coherence plane, called non-Euclidean geometry, the meaning of the word “postulate” or “axiom” changes radically. The axioms of a theory do not necessarily have to be dictated by real evidence. On this basis the constructions of arithmetic and geometry are built. The axiomatic-deductive method becomes the mathematical method. It will also highlight the constant link between mathematics and the reality that surrounds us, which tends to make itself explicit through an artificial, abstract language and with clear and certain grammatical rules. Finally, you will notice the connection with the existing technology, that is the new electronic and digital technology.

A Pedagogical Experiment in the Italian School

This chapter will summarize the extensive and multi-year experimentation carried out in many Italian secondary and primary schools, of the pedagogical-didactic proposal developed in the previous chapters. The teaching/learning of computer-based mathematics, as a programming tool, with software appropriate to the context, named MatCos, is given. In particular, the organization of the experimentation will be described, and the various phases will be illustrated. The results with the relative evaluation method will also be considered. Finally, works prepared independently by some participating students will be presented and commented. The opinion of some experimental teachers and school managers will close the chapter.

Laboratory Activities in Primary School Teaching-Learning Sequences (TLS)

In this chapter some practical activities to do in primary school classes are shown, and these concern classic themes of arithmetic and geometry but also more recent topics in statistics and probability. Naturally, all of them are based on pedagogical-didactic aspects noted in the preceding chapters. In particular, MatCos 3.0 environment is used. A complete TLS, from which the new methodology based on the MatCos programming environment emerges, will be presented. Finally, the simulation software package, DAF, is presented to illustrate the concept and related operations of the fractions.

A Statistical-Probabilistic Path

This chapter is borderline compared to the previous ones, since the topics are not really mathematical. In fact, the objectives and learning content of statistics and probability are different from those of mathematics. It also follows a methodological diversity of teaching. All this will be clarified below. However, the main focus of the goal, that is the possible contribution that the computer with the correct programming can give to the learning of the discipline, is no less evident and will be clearly delineated. In particular, graphical representations of statistical data will be widely examined. A TLS will also be schematized on the introduction of the concept of probability, in which the use of the computer with the MatCos 3.X environment will be very useful.

Activities in the Secondary School

With the same methodology of the previous chapter, in this chapter there is an outline of vertical path about geometric topics typical of the secondary school. Of course, the algorithm and computational aspect in the MatCos 3.X environment are more developed with respect to classical arguments, surely of interest. In particular, the presentation of conics in both the Euclidean and Cartesian plan is emphasized, based on construction algorithms by points, which can be easily implemented in the MatCos 3.X programming environment. Even solid geometry, or in three dimensions, will be characterized by effective construction algorithms of the solid figures presented. Some of these algorithms are general in nature.

Methodologies for Learning and/or Teaching

In this chapter, there is an introduction of the problem of methodologies and strategies for learning and/or teaching mathematics in primary and secondary school. The first methods are the symbolic-reconstructive and perceptive-motor. For the authors, the problem about contemporary teaching is characterized by the relationship between mathematics and electronic technology. Finally, a new didactic approach will be proposed. It is called “from concrete/virtual-concrete to abstract” and it consists in the introduction of a new computer-based phase, called graphic-numerical, in a good traditional didactic path.

Activities in the Secondary School

In this chapter there is the presentation of a vertical path on the main topics of arithmetic-algebra-infinitesimal calculus and numerical methods, which are an object of study in the secondary school. Naturally, the attention will be focused on the “virtual” phase, that is the applications with the computer and the MatCos 3.X environment, both as graphical-numerical experimentation, of intuitive support to the understanding of the concepts, that as a necessary moment for the actual calculation in the applications. It presents a TLS based on a real problem, from which the whole presented methodology shines through: from problem solving, to mathematical and numerical modeling, to the formulation of the solving algorithm and its implementation in the MatCos 3.X environment.

MatCos 3.X

The authors present a sketch of the Matcos 3.X environment. In particular, they present the axiomatic and genetic commands. There are genetic commands for the following: plane and solid geometry, arithmetic and classical algebra, calculus, and numerical methods, with an outline of real functions with two real variables, descriptive statistics, elements of combinatorics, probability, and linear algebra. Each command will be illustrated with an example. There are also comments and suggestions on the pedagogical and didactic use of some commands. The commands will not be displayed in strictly alphabetical order, but a more logical and consequential line has been preferred. At the end of the chapter, the alphabetical list of all the commands available in the programming environment will be shown.