How to set up the teaching of mathematics?

**Purposes that educational paths for teaching mathematics should have**

(**1**)
A *programming* activity is needed which disarticulates and re-aggregates the themes (listed in the school programmes) into* didactic itineraries *which grasp the interactions between the various themes, exploit their reciprocal motivations and opportunities for technical exercise and consolidation, achieve economies and synergies by merging or integrating different mathematical topics
(the use of coordinates involves activities with numbers, operations, formulas and offers possibilities for alternative and more effective introductions of many geometric concepts, the use of the notion of function allows to simplify and connect various concepts, … not to mention the possibilities offered by use of means of calculation).

(**2**)
Themes such as statistics, probability, the use of means of calculation often create problems for teachers as their development must necessarily pass through* mathematization *activities
(every statistical or probabilistic problem involves the modeling of a random "phenomenon", representing an object or a mathematical situation with software involves its understanding and translation into a new formal language, …).

(**3**)
The nature of mathematics and its models (characteristics of mathematical models compared to models organized in other disciplines, internal organization of the discipline, role of definitions and arguments in mathematics, …) can be gradually understood through the construction of a complex network of cultural and experiential references.

**How to set up learning paths**

(**4**)
To achieve the interweaving between internal reflections and the use of mathematical models, it is necessary to organize teaching into** broad-based educational itineraries**,

(**5**)
As priority* objectives *of mathematics teaching we can assume (with obvious differences between the various school levels) the following:

) a |
to make pupils aware of the role and nature of mathematical models |

)b |
to make them reach a certain level of ability in applying, elaborating and comparing mathematical models
(through activities related to the way mathematics is done and used today:
delegation to calculators and computers of the more mechanical aspects, ability to orient oneself, to choose the appropriate mathematical models, to consult manuals, ... more than knowing how to make "mechanical" calculations and remember "recipes") |

)c |
to make them aware of the interactions (today and in history) of mathematics with the "rest", |

)d | to make the school perceived as a place of cultural formation, |

)e | to contribute to education to read, write, organize, doubt, … |

Objectives** d**

They are essential for conceptually interacting with pupils (as research on

** I**)
to prevent the knowledge developed by the school from being understood as an ad hoc culture (to be "retained" only superficially and temporarily), to be able to access the factors that are at the origin of pupils' conceptual difficulties, to transform their "needs" cultural in "interests", …

** II**)
to enable pupils: to understand definitions, arguments, texts of problems; to organize and communicate reasoning; …

**Teaching materials and verification**

(**6**)
As far as* teaching materials *and other* methodological aspects* are concerned:

** •** it is appropriate that the educational itineraries are organized in

rather than scattered exercises with restricted contexts, broader problematic situations are to be preferred, in which the mathematization activity is more significant (**•** several mathematical tools involved,
**•** more in-depth examination of the relationship between situation and mathematical models, **•** references to models of other disciplines,
**•** choice of situations which, in a culturally significant way for the pupils, "vehicle" and constitute "prototypes" for the mathematical concepts involved;

** •**
the

**•** parts to be *read*, **•** invitations to *discussions*,
**•** *questions* that require articulated answers (oral or written) in the natural language,
**•** more traditional mathematical questions,
**•** questions that require operational activities of another kind (see below) and
**•** more open-ended questions, which they involve organizational aspects (organising the worksheet, deciding which issue to address first, where to go and look for certain information, …), **•** questions to be addressed collectively and **•** questions to be addressed individually;

** •**
this teaching organization should make possible a dynamic verification of the pupils (having a more reliable idea of how individual pupils learn during the year and of the overall performance of the class);

** •** dynamic verification is particularly important if there are no "immediate" productivity objectives in mechanical calculation or in the reproduction of definitions and proofs, but the aim is:

**•** to develop the* mental organization *of concepts,

**•** to* bring out and compare or contradict the ideas, prejudices, distorted knowledge *of the students,

**•** to pay attention to the* ambiguities/confusions *that the* different semantics of common language and mathematical languages *can give rise to, … ; with an approach of this kind, which aims at deeper and more general acquisitions, the verification must be carried out over a longer period of time;

** •**
as regards the

** •** the worksheets should include many "traditionally" absent activities:

**•** *use of *calculators, ruler, square, protractor, graph and squared paper, …; **•** mental calculation activity; **•** observation, description, analysis of phenomena present in* daily life*;

in this way it is possible both **•** to activate the pupils more
(through moments of more operational/concrete work it is possible to restore to study and conceptually activate students not involved in traditional teaching), **•** and propose extracurricular exercise activities on what has been studied (and, indirecctely, to involve* families*, making the cultural nature of the proposed work perceived: families, for various factors, are often hostile to "non-traditional" approaches);

** •** the worksheets should be accessible online or made available to pupils or families in a format usable by all (pdf or html), so that they can also be used in classrooms equipped with a

** •** the teaching units should be accompanied by a

** •** we observe, then, that

** •** alongside the importance of the dynamic verification, it is necessary to underline the opportunity to stimulate the pupils to face verification tests independently,

(**7**) Furthermore, it is very important for a teacher* to discuss with colleagues*, debating* both live and online*, especially for those who use the same teaching material, not only to check the progress of the work, but also because the comparison between colleagues on the difficulties encountered and on the ways in which they were faced, on how the class responded to the proposed stimuli, … is useful for reflecting on one's own way of teaching (these aspects are often neglected in the collegial planning of schools), and for breaking the shell of individual schools.