Problem solving in 6 steps

Developing mathematical problem solving for the second stage of primary school

Product: Book

Trim size in cm: 21x29,7

Pages: 416

ISBN: 978-88-6137-436-2

Publication date: 01/05/2009

Suitable for: Primary 2nd level (ages 8-10), Lower secondary 1st level (ages 10-11)


A wise, thoughtful owl and a quick, cunning fox guide children through a complete programme for developing problem solving skills. The former with its reassuring manner will give instructions, whilst the latter will encourage metacognitive reflection in order to activate the self-evaluative processes needed for the stable acquisition of good problem solving skills. Created as a practical application of the SPM test (Test on mathematical problem solving abilities) and based on the contents of ministerial objectives for the second stage of primary school, the book is divided into two parts. The first part contains problems which test the entire flow of components commonly identified as the fundamental phases of the problem solving process (comprehension, representation, categorisation, solution plan, implementation, self-evaluation). In the second part the same competences are analysed in-depth with activities to be carried out individually or in groups. The various sections of the book do not necessarily have to be followed in order, but can be used flexibly according to the child’s proficiency and difficulties. This structure aims to test the dynamically interdependent relationship between the various components.

«Numerical and logical scientific cognition development programmes» edited by Daniela Lucangeli
The collection aims to translate findings on numerical and logical scientific cognition into practical programmes. In particular it refers to psychological and psychopedagogical research on intelligent learning processes. The strategies developed are presented in both paper-based and digital formats in order to guarantee flexibility and motivation.  The objective is to facilitate the development of basic cognitive abilities in childhood and adolescence with relation to scientific thought.

The CD-ROM is also available.

Presentation of the programme
The solution flow:
- Problems with the 4 operations;
- Problems with fractions;
- Problems with decimals;
- Problems with measurements;
- Problems with gross weight, net weight and tare weight;
- Problems with buying and selling
In-depth activities on individual areas:
- Comprehension;
- Representation;
- Categorisation;
- Solution plan;
- Implementation;
- Self-evaluation
The rulebook of the skilful problem solver

The “Numerical and logical scientific cognition development programmes" collection

Edited by Daniela Lucangeli – Università di Padova

This collection comprises a series of books which offer didactic programmes on calculation, geometry and problem solving, departing from traditional methods. Translating the latest research findings on numerical and logical scientific cognition into practical teaching strategies, the various books aim to build up cognitive processes in order to develop innate numerical intelligence.

What is numerical intelligence? We talk to Daniela Lucangeli

When do we develop numerical intelligence?
At only a few days old, a child recognises quantity way before being able to name it with words. Various scientific studies have demonstrated how a new-born baby in his mother’s arms discriminates 1. When dad arrives he discriminates 1 different from 1 and when the nurse arrives 1 different from 1 different from 1. Not only this, but within these 3 the baby is able to recognise greater than, less than and equal. Recognising quantity – which is the basis for counting – is in fact an innate skill, which has developed over thousand of years of evolution of our species.

If numerical intelligence is innate, why do we have difficulty with numbers?
Just like the fact that we don’t learn to speak unless somebody teaches us, we don’t learn to develop a competence if we don’t practice it in the right developmental stage. Generally, children practice the mechanisms of quantity only when they start school…too late! To give you an idea it’s like getting our children to practice talking from 4 years onwards. This is why in order to develop numerical intelligence and build competences on numbers and quantity it is essential that we act in the first five years of a child’s life.

What is the role of parents and teachers in developing numerical intelligence?
Through games, adults facilitate and speed up the processes of maturing and optimising learning. Like a spoon, which mixing coffee and sugar, facilitates and speeds up the transformation process and enables you to drink a tasty beverage in a short time.




Scientific research has demonstrated that the ability to understand and work on quantitative aspects in real life and to distinguish number and estimate it is a potential which is innate in children. These processes, however, should not be left, as often happens, to develop naturally; they require educational strategies and interventions designed to develop them.

Numerical intelligence in infancy Ages 18 to 36 months  
Numerical intelligence – volume 3 ages 8 to 11 years  
Numerical intelligence – volume 4 ages 11 to 14 years  
Numbers and space Visual spatial tools for counting, first calculations and times tables  
Dyscalculia test (Full Kit: book + CD-ROM) For the evaluation of calculus abilities and disabilities  
Dyscalculia trainer (Full Kit: book + CD-ROM) Consolidation activities and improvement of calculus skills difficulties  


An original and creative approach to arithmetic operations (and not only!) which are often an important milestone for youths of all ages. These books offers a series of strategies, which are part of a lengthy research project dedicated to eastern methodologies, in particular Vedic mathematics, and they are accompanied by observations and illustrations which provide a link between far eastern and western didactics. The manual, which contains a selection of techniques and strategies, comes complete with three books of in-depth explanations, full of examples and exercises for children and youths.

Calculation Strategies From vedic mathematics to numerical cognition  
Calculation Strategies- Vol. 1 Learn your times tables using your fingers  
Calculation Strategies – Vol. 2 Multiplications
Calculation Strategies – Vol. 3 Additions and subtractions  


Developing geometrical competences A complete programme in geometry, divided into two books (from ages 6 to 11 and from ages 11 to 14), which starts with the building of geometric elements, the main plane figures and related formulas for perimeters and areas and goes on to solids, to theorems and to analytical geometry, in a simple way which can be readily used by every student, on the edge of reasoning.

Developing geometrical competences- vol. 1 Cognitive and metacognitive abilities in building geometric knowledge for ages 6 to 11  
Developing geometrical competences - Vol.2 Cognitive and metacognitive skills in building geometric cognition for ages 11 to 14  
Let’s learn how to solve geometry problems Developing geometric problem solving for the second stage of primary school and for lower secondary school  

Geometry with paper An original and exciting practical programme based on folding paper in order to develop and build geometric competences in children at primary school and the beginning of lower secondary school, by stimulating their curiosity and creativity.

Geometry with paper – Vol. 1 Fold to unfold – Recognizing plane shapes  
Geometry with paper – Vol. 2 Fold to unfold – Fundamental Geometric Elements  
Geometry with paper – Vol. 3 Fold to unfold – Triangles and quadrilaterals  


Problem solving in 6 steps A wise, thoughtful owl and a quick, cunning fox guide children through the second stage of primary school in a complete programme for developing problem solving abilities. The owl with its reassuring manner will give instructions, whilst the fox will encourage metacognitive reflection in order to activate the self-evaluative processes needed for the stable acquisition of good problem solving skills.


Magically minded 1 A collection of fun recreational mathematical activities — to be carried out under the guidance of the «magician-teacher» — which stimulate pupil’s reasoning and curiosity.