Our presentation, which got published as a scientific paper, focuses on the pedagogical use of Cuisenaire method, one of the most imaginative and creative tools for structuring numbers and mathematical thinking through images and supervisory training material. “Building Numbers with rods” method, describes the potential for creating simplicity within complexity and enables a way of working through observations and senses’ mixture that engages the playful and creative powers of the users.
We would like to express our sincere feelings of gratitude to Mr. Antonis Panagiotopoulos, Emeritus Professor in the Department of Computer Science of the University of Piraeus, as he is the inspirer to implement successfully this method in several primary schools in Greece, with extraordinary results.
1. The Method
The “Building Numbers with Rods” method, constitutes a program of mathematical education from 4 to 9 year olds. The program aims at instructing children into reaching and comprehending mathematical thought through, its theoretical and practical part. The importance of mathematical education is determing even at younger ages, as children learn to understand mathematical terms better, or are able to come to conclusions through passive observance.
The recommended program philosophy is exclusively based on the creation of a learning environment through the use of manual means, so that the child can get used to combining learning with its natural tendency to play and interaction. It is worth mentioning the fact that the theoretical foundation of their being introduced to mathematical education, started from Comenius and Pestallozi, who recognized the necessity and value of observance for empirical teaching. Piaget was the first to claim that perception develops with age and that children do not have the required mental maturity to perceive abstract mathematical concepts through words and symbols from a very young age.
To realize any action and assessment though, it is necessary that the right educational material be found, so that it can accurately render the meaning. At the same time, in order that a teaching material can constitute an effective and sustaining means of teaching and learning Mathematics, educational planning should take place in advance.
The Cuisenaire method teaching material which was initially developed in 1920 by Belgian Mathematician and nursery school teacher Georges Cuisenaire, is composed by rods of ten different colors and sizes ranging from 1 cm to 10 cm, where each color corresponds with a certain size and consequently a certain number. Therefore, each size and color, clearly leads to the corresponding ideogram enthusiastically forming, this way, the geometrical ratios of the great ancient Greek mathematicians and rendering us able to realize them and form the “abstract concept of the mathematical symbol (Figure 1).
2. Getting acquainted with Cuisenaire Method
All the rods are divided into 65 families according to their colors. Namely, they are length models which children handle for composing and comparing length models technological development is an indisputable fact, education can do nothing but include it in its future dimensions (Figure 2).
Taking notice of the aforementioned number families, we should take the following things into account concerning the first numbers:
• Number 1 with the white color, builds all the rest numbers as such, it cannot be included into the category of first numbers.
• Number 7 with the black color, is the ONLY first number within the first ten numbers which has no multiples within them. The choice of number 7 for the week is not a random choice as it is the next number after number one which can generate numbers within the first ten numbers.
• The multiples of number 2 are numbers 4 and 8, that is why they all belong to the red family.
• The multiples of number 3 are numbers 6 and 9, that is why they all belong to the azure family.
• The multiple of number 5 is number 10, that is why they both belong to the yellow family.
The colors divided into families represent a different first number, linking children’s observation abilities with mental arithmetic. The lengths of the rods, facilitate children into forming metal images in parallel with gradually perceiving the concept of the size using the senses of sight and touch. Consequently, they are already in the position of mentally linking their senses with the images produced in their brains. Finally, the combination of colors and measurements activates children’s all five senses, exercising at the same time their mental abilities and abstractive thought.
We see that the cornerstone of the method consists of three parts: observation, activation of all senses and self-activity of the learner. Let us not forget however that the experiential teaching, turned in the manner mentioned in the previous section, can lead to spectacular results if run through the frame of New Technologies and Communications Networks. Already at a very young age, children are familiar the per interact, play and perceive through online applications and programming environments. Therefore , it becomes urgent and necessary, our direct actuation in that direction , so that children become familiar with Cuisenaire method, through both of said material, and through the development of appropriate training for online applications, an act which are committed in their direct implement, either with paper and pencil, or using Web 1.0 or Web 2.0 tool.
The systematic occupation with the aforementioned stuff facilitates children’s observation, self – initiated action, creativity, verification and comprehension skills thus leading them to mathematical thought. Generally speaking, the systematic occupation with activities which utilize this stuff, facilitates children into being accustomed to the relations between sizes, colors and numbers as symbols. It is very important for children to understand and develop, through a group of selected activities, the potential relations between the first ten numbers and the first one hundred numbers accordingly. These potential relations, which are stabilized and fully assimilated through systematic use, constitute the cornerstone of any arithmetic operation.
Therefore, the Cuisenaire stuff facilitates children into doing plenty of preparatory exercises so that they can fully understand the meaning and value of each number as well as the meaning of all four arithmetic operations before starting to occupy with mental calculations and the symbols representing numbers. Aiming at further facilitation, the rods are accompanied with a track, a wooden rectangular framework with two grooves where students can put the rods while in the space between there is the numbers line (Figure 3).
The child using the rods on the track has the chance of gradually structuring the numbers tracing various arithmetic combinations (Figures 4, 5).
As we can easily understand, the premise behind this educational material is based on the creation of enriched learning environments through the use of teaching materials, tools and means. The use of teaching stuff and tools is considered to gain a prevalent role in the formation of the appropriate conditions so that children’s actions and perception skills can be studied with reference to mathematical concepts. The teaching stuff as well as their qualities on their own do not lead to the elaboration of children’s mathematical thought. In contrast, it is believed that children’s physical actions relating to them and their interaction between them along with their reasoning concerning these actions contribute to shaping and reshaping of children’s mental figures and functions.
Teaching Maths using this material renders dealing with maths a pleasant, appealing as well as a life – like procedure, which facilities the student in particular into being led from this concrete stuff to the mathematical concepts represented. Thus a bridge is being built in a stable way gapping the child’s work on this particular mathematical stuff with its abstractive thought relating to symbols and numbers. In this way the child itself is being stimulated to develop what is called “atypical learning”, that is representations leading to the exploration of perception skills along with the future evolution of concept.
As it is easily understood, children can learn about and explore each number in its essence and sheer entirety on their own. They can check and verify their calculations by themselves, so they themselves can correct their errors.
In this way, though creative imagination and the respective mental calculations, children gradually take further steps towards comprehending the concept of amounts and the relations concerning the numbers and their operations. After all, according to Galperin’s “The psychology of acting”, the aim of the mental procedure is the formation of mental images by means of using the right materials on the right way. Therefore, through the right use of the teaching material on the one hand and the cultivation of mathematical thought and activation of all four senses on the other, children learn to perceive the number as a set of three qualities, those of color, size and ideogram developing, this way, the skills of analyzing and composing numbers though various combinations.
As Cobb points out the tangible representations (material) in combination with the pursued mental representation that is indispensable to students, renders the detection of all the mathematical relations though the use of the right stuff and tools an absolutely achievable target. Consequently, in a natural way, children’s thought smoothly advances from the handling of mere objects to the stable mathematical reasoning. However, it is in our immediate plans, implementation and promotion of Cuisenaire method through interactive online applications for further engagement of children and simultaneous familiarization with new technologies.
Through the following link through you can download the correspondent scientific publication of our proposal:
It was a wonderful educational experience, fully organized, targeted and ontinuity… Our gratitude at all the teachers and conference coordinators!!!
Dear fellow teachers, we invite you to sign up for free, to our website teachers.arnos.gr and it we will be our pleasure to keep you always informed about our next steps of action and creativity.
¨Do not forget that our thoughts are the ones to create boundaries… and those boundaries are captured from our inaction“.
Krokos P. Yannis: Mathematician – Civil Engineer
Mantsis Kosmas: Mathematician
Tsilivis K. Vasilis: Mathematician