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,
itineraries referring to "real" situations in which to introduce, develop, … mathematical-type models, also making reference to concepts from other disciplines  (in general, the applications of mathematics are mediated by the models of other disciplines: mathematics does not contain all the tools for direct "physical contacts"),
and  (in particular starting from secondary school)  itineraries referring to explicitly mathematical themes  (reflections on the "mathematics" discipline: languages to describe the various types of models, generalizations, properties of models, relationships between different mathematical models, mathematical models that abstract properties of other mathematical models, …).

(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-e are to be pursued through cultural setting choices - see (4) - and methodological choices - see (6).
They are essential for conceptually interacting with pupils (as research on learning processes also highlights):

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 teaching units each consisting of several worksheets:

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 worksheets  (for the first classes made up of images and a few written requests, to be addressed in a non-formal way, intertwining, without apparent differentiation, different disciplinary skills;  for the successive classes, more articulated, until they become material used as a trace or documentation for the work carried out in class, and then used by the pupils for a review of what they have covered in school)  it is good that they present:

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 adaptation of the teaching material  (personal or shared worksheets  or  a reference book)  to different levels of preparation/ability and to different school situations,   the performances, the questions, … through which the work is articulated they should be graded and presented in a way that they can be tackled with different levels of abstraction,   and the spiral resumption of the topics in successive teaching units should make less difficult  (compared to "traditional" teaching forms, with a vision of internal mathematics, with no cultural scope)  for everyone to participate in the "thread of discourse" and in understanding the essential aspects of the mathematical concepts introduced;   optional paragraphs or parts may be foreseen, in order to allow the use of the material in classes with different levels of "users";

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 projector;   and, starting from the last years of primary school, they should provide ideas for computer activities  (to be carried out also in the classroom with the projector or, in some schools, in the computer classroom, and at home by the pupils, with free usable software, from the software standards for drawing, to software for writing, to software for increasingly advanced mathematical activities);

the teaching units should be accompanied by a store of exercises  (exercises to be used during or after - for revision or consolidation - carrying out the worksheets, for homework, for classwork, for consolidation activities aimed at some pupil, for activities to be proposed at the end and at the beginning of the school years, …, with the aim of both exploring the learning difficulties of the pupils and evaluating the effectiveness of their own teaching proposals).

we observe, then, that new technologies also involve other reflections:  pupils (both primary and secondary school and university) are more accustomed to using new technologies than teachers, but they use them in a superficial, often uncritical way;   new technologies are changing ever more rapidly, and the school runs the risk of becoming increasingly detached from the use that pupils make of them;   on the other hand, the software that companies that make computer products for schools or "experts" in information technology offer us is often unrelated to the cultural objectives that the teacher must set for himself;   it is necessary to build a do ut des educational relationship with the pupils, in which they update us on new technologies and we teach them to use them critically;   and the evolution of technologies must be taken into account:   innovations will be ever faster,  the use of "mobile phones" will gradually lead to new problems/prospects,  computers will be increasingly transportable and normal instruments of everyone's daily life,  computer labs will disappear, …

alongside the importance of the dynamic verification, it is necessary to underline the opportunity to stimulate the pupils to face verification tests independently, without the teacher's control, with the possibility of self-monitoring, of trying again, of challenging themselves, of managing their times, of proceed in several ways, by successive approximations, from a particular or from the general, putting together or rearranging pieces of things already done … (for this too the use of the computer is very useful, with appropriate indications from the teacher).

(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.