In this field, the name of Caleb Gattegno is frequently associated with a certain number of tools that he either invented or promoted: Geoboards, Cuisenaire rods, Nicolet's silent geometry films (see Maths Films), the use of the set of fingers as a foundation for elementary mathematics, and specially constructed tables of numeration.

Though they continue to be of importance, they are not, however, the essence of his work.

Mathematics as a reference

With his background in mathematics, Gattegno observed for more than forty years how mathematical thinking is constructed.

For Gattegno, mathematics is something any of us can do, since it involves reformulating mental activity in mathematical terms: an ability that all human beings possess. He called that process "mathematization."

Thus, the essential role of the mathematics teacher is to lead their students to discover the mathematician within themselves.

A constructive and personal approach

Mathematical situations are proposed to learners who, invited to participate actively, become aware, little by little, of the relationships that structure the situations, and at the same time understand better the dynamics of their own mental functionings.

Centered on a given, clear and tangible problem, the situations presented for students' exploration lead each student to use their varied mental faculties, to formulate hypotheses and to test them, to develop personal strategies and to arrive at precise and profound understandings.

Through those situations, from a perception of what they contain and from actions performed on them, mental representations are created and generalizations made, first verbalized and finally represented symbolically, in classic mathematical notation.

As the learners explore a mathematical situation, teachers observe them and lead the class, always ready to modify their pedagogical actions, based on what the learners are doing, what they are saying, their hesitations and errors, their understandings and misunderstandings, their attitudes and various reactions.

Thanks to such constant, formative assessment and by paying attention to each learner, teachers can accompany their students and guide them step-by-step towards new insights and increased know-how.