The Toolbox: Objects and Tools for Doing Mathematics

The Toolbox: Objects and Tools for Doing Mathematics

Alessio Drivet (Teacher of Applied Mathematics, Italy)
DOI: 10.4018/978-1-4666-2122-0.ch068


This chapter concerns the use of concrete objects and tools connected with mathematics. Currently, over a hundred items have already been provided by the author, and others are in preparation. The topics are part of what is called “education and recreational use of games and simulations,” and so the objects also come in a technological version. Part of the materials were used in an educational experience with the aim of stimulating students to identify objects and situations that speak to us of mathematics and to teach them how to retrieve and organize information, promoting non-traditional learning of the discipline. The most interesting aspect is the use of a very artisanal approach, starting from objects that students can, in part, found in everyday life.
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In this article I will examine the possibility of using iconic representations of objects to introduce or explain parts of mathematics in a concrete mode. There are many different artifacts, some with obvious disciplinary connotations (Abacus, Dice, Geoboard, Napier's Bones, Tangram, etc.), and others which are just a starting point to explore more or less usual mathematical themes (Bicycles, Glasses, Potatoes, Spaghetti Measures, T-Shirts, etc).

Somebody might define artifacts as learning objects. In my opinion, this is an understatement. The more structured attempt with direct involvement on the students’ part carried out in 2009-2010 is related to a fourth year Italian class in a secondary school (work was continued the following year);then using the tools of cloud computing, each student collaborated to prepare work sheets about some mathematical objects. This work can therefore, at least in part, be referred to socio-constructivist learning principles.

Of course one can ask whether this approach has yielded more satisfactory results than the traditionally established methods which, in general, refer to the sequence: textbook - lesson - homework – assessment (the dominant paradigm in the Italian school).For an assessment of the effects on training, validation tools should be prepared that are not normally possible to use in a local situation. Therefore, the only tool, purely qualitative, that I have employed is related to a positive evaluation of the curiosity and interest shown by the students.

The experience I will describe revolves around two key questions:

  • 1.

    Is it right to start from maths and then get to the “objects and instruments” or is it better to start from objects and tools to “do maths”?

It is really not correct to consider this choice as the basis of a paradigm, and so one should mix the two levels, sometimes proceeding from a mathematical question, sometimes reversing the process.

  • 2.

    Is it necessary to use the computer when teaching maths?

The answer, as often happens, is “it depends.” Those who play cards can read books on the subject, or use dedicated software, but they cannot satisfactorily replace the materiality of those colored rectangles called “cards.”

To the two questions I have tried to respond by building a toolbox, bringing together artifacts and digital versions, all layered on Sheets designed to summarize concepts, proposals and references. The mathematics teacher still has to get used to entering the classroom with a box full of tools and materials and with a computer full of virtual tools.

Of course, things are not so easy. The teacher must have a good knowledge of the discipline and learning models to build his/her own educational paradigm. In addition, technological updating, reading books and articles, collecting information on the Internet are necessary conditions for the development of “best practices” and they take time and commitment.

The context is a technical high school set in a suburb of Turin (an industrial city in the North of Italy), with students coming from working families with a pretty poor cultural background. The experience, performed at the Technical Commercial Institute Bertrand Russell of Turin, has involved 16 students in a class of accountant programmers (Mercury experimental course).

As part of the curriculum of a technical college, theirs was a course where applied mathematics had a certain weight. In fact, the program was very extensive and five hours of lessons were provided weekly with the use of the computer lab. This was in the past because the recent reform programs (2010) have unfortunately “resized” the hours.

That class was chosen because the year before it had demonstrated a good degree of collaboration and interest in overcoming the narrowness of curricular areas. In practice, I had already employed mathematical objects to introduce some of the topics and therefore the students were prepared for this approach.

The experience should be seen as part of a project that has been going on for many years, aimed at leading students to the understanding of mathematics as a tool and at allowing them to recognize and interpret the world which surrounds us. Mathematics is often viewed as an arid subject, made up of formulas, numbers, theorems that discourage students. But it may also be fascinating in so far as it allows us to model and better understand the world around us.

Key Terms in this Chapter

Mathematical Object: In the sense of “real object,” which refers to a mathematical “object/concept.”

Cloud Computing: Technology that allows both to process and store data through the use of distributed and virtualized network resources.

Didactics of Mathematics: A field of research that aims at studying the theory and methods of learning the discipline.

Educational Play: A way of teaching which provides learning through the use of laboratory activities, games, etc.

Learning Objects: A small instructional component that can be reused a number of times in different learning contexts.

Social Constructivism: A theory of knowledge that argues that humans generate knowledge and meaning from an interaction between their experiences and their ideas.

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