Mathematics Primary School OILER School

Numbers, operations, mental arithmetic

PITAGORAS is a didactic tool that focuses on mental calculation. We believe that developing the ability and consequently the habit of mental calculation not only helps whenever you need to calculate something, even in written form, but in general helps acquire the sense of number. Mental calculation is therefore useful both in didactic environments, for example to check the result if the calculation was performed in another way or to estimate an order of magnitude, and in everyday life, on all occasions when quantitative reasoning is required. PITAGORAS uses the completion of a “pyramid” as its main technique: we will explain how this works in detail later, anticipating for now that each brick contains a number and that you go from one level to the upper one by performing an operation. In certain contexts, this structure can be funnier and richer than usual computation, as we will see later. In particular, it implies the idea of inverse operation and can be useful as an implicit introduction to the concept of an equation.
cards
blue expansion cards: tens and ones
yellow expansion cards: multiplication tables

Data Sheet

CLASSES: 1 to 5.

Skills:
The pupil:

  • calculates confidently, in writing and mentally, with natural numbers;
  • solves simple problems in every content area, keeping track of both the solving process and the results;
  • describes the process followed and recognises solution strategies different from their own.

METHODS:

Starting from activities on addition and multiplication, the pathway aims to develop number sense and promote mental calculation. Several studies stress the importance of mental calculation not only for understanding arithmetic, but also for moving from an additive to a multiplicative scheme, which leads to a better conceptualisation of number. Mental calculation requires a certain mental agility: one has to be able to change strategy along the way, to accept mistakes as part of the process and to focus on the “here and now”. This suggests the value of classroom settings where pupils calculate and describe to one another how they did their calculations. In this process the teacher plays a key role: highlighting connections and properties, valuing the different approaches with clear language and offering tools for recording strategies together.

There are at least four good reasons to teach mental calculation (Threlfall, 2002): it is the kind of calculation most used in everyday life; it develops number sense; it strengthens problem-solving skills; and it provides a solid basis for learning written calculation. Pupils should also get used to calculating without writing anything down, because the strategies used on paper differ from those of mental calculation.

This pathway was designed by Luigi Bernardi and Giorgia Damiano.

No activities yet: more are coming soon!

PITAGORAS is a didactic tool that focuses on mental calculation. We believe that developing the ability and consequently the habit of mental calculation not only helps whenever you need to calculate something, even in written form, but in general helps acquire the sense of number. Mental calculation is therefore useful both in didactic environments, for example to check the result if the calculation was performed in another way or to estimate an order of magnitude, and in everyday life, on all occasions when quantitative reasoning is required. PITAGORAS uses the completion of a “pyramid” as its main technique: we will explain how this works in detail later, anticipating for now that each brick contains a number and that you go from one level to the upper one by performing an operation. In certain contexts, this structure can be funnier and richer than usual computation, as we will see later. In particular, it implies the idea of inverse operation and can be useful as an implicit introduction to the concept of an equation.
cards
blue expansion cards: tens and ones
yellow expansion cards: multiplication tables

No activities yet: more are coming soon!

Data Sheet

CLASSES: 1 to 5.

Skills:
The pupil:

  • calculates confidently, in writing and mentally, with natural numbers;
  • solves simple problems in every content area, keeping track of both the solving process and the results;
  • describes the process followed and recognises solution strategies different from their own.

METHODS:

Starting from activities on addition and multiplication, the pathway aims to develop number sense and promote mental calculation. Several studies stress the importance of mental calculation not only for understanding arithmetic, but also for moving from an additive to a multiplicative scheme, which leads to a better conceptualisation of number. Mental calculation requires a certain mental agility: one has to be able to change strategy along the way, to accept mistakes as part of the process and to focus on the “here and now”. This suggests the value of classroom settings where pupils calculate and describe to one another how they did their calculations. In this process the teacher plays a key role: highlighting connections and properties, valuing the different approaches with clear language and offering tools for recording strategies together.

There are at least four good reasons to teach mental calculation (Threlfall, 2002): it is the kind of calculation most used in everyday life; it develops number sense; it strengthens problem-solving skills; and it provides a solid basis for learning written calculation. Pupils should also get used to calculating without writing anything down, because the strategies used on paper differ from those of mental calculation.

This pathway was designed by Luigi Bernardi and Giorgia Damiano.

Pitagora of Samos

The stories about the life of Pythagoras are a mix of legend and reality. Nothing is certain, because he left no writings, but a good deal of information has come down to us through his disciples and some writings of Aristotle, Herodotus and Plato. So we know that he lived in the 6th century BC, that he was born in Greece but travelled widely in Asia Minor and Egypt (it is said that it was Thales who introduced him to the mathematical knowledge of the East), before reaching Italy, at Croton, where he founded a School: a scientific, philosophical and religious community whose disciples followed certain moral rules and pledged not to divulge the philosophical and mathematical knowledge they had learned from their master Pythagoras.
Everyone knows the Pythagorean theorem, but in fact the Pythagorean doctrine was based not on geometry but on arithmetic. The other disciplines that made up mathematics, namely music, geometry and astronomy, were founded on arithmetic. For Pythagoras, number was the essence of all things. Knowing the properties of whole numbers was part of a mystical path, and special representations, such as triangular or pyramidal numbers, took on great importance.
It seems that the Pythagorean school was thrown into crisis by the discovery of irrational numbers, or rather by the fact that the ratio between the diagonal and the side of a square cannot be expressed as a ratio between whole numbers. But that is another story...