In this article we examine the relation between variation theory and Maria Montessori’s didactic theory. Montessori believed that training and sharpening of the child’s senses are crucial for their continued learning; she therefore developed specific sensorial materials to be used in Montessori preschools for such a purpose. As noted by interpreters of Montessori education, a key principle in this material, as well as in variation theory, is the use of variation and invariance. However, in this article, lessons in two different areas than the training of the senses are analysed from a variation-theoretical perspective on learning; these lessons originate from Montessori’s own writings and from extracts from Montessori training courses. The result shows that a systematic use of variation and invariance can be seen as a more fundamental part of Montessori’s didactic theory and is not only applied in the sensorial training. The article will offer theoretical concepts useful when explaining why lessons in various areas should be presented in the way they are described.

Montessori education is spread all over the world and the number of schools is constantly increasing (

However, a key principle in the application of the pedagogy, which has been noted by interpreters at a more theoretical level in recent years, is the use of variation and invariance (or contrast and sameness) within the training of the senses practised in Montessori preschools (e.g.

In the ordinary schools of today, teachers often give what are called ‘object lessons’ in which the child has to enumerate the various qualities of a given object: for example, its colour, form, texture, etc. But the number of different objects in the world is infinite, while the qualities they possess are limited. These qualities are therefore like the letters of the alphabet which can make up an indefinite numbers of words. If we present the children with objects exhibiting each of these qualities separately, this is like giving them an alphabet for their explorations, a key to the doors of knowledge. Anyone who has beheld not only the qualities of things classified in an orderly way, but also the gradations of each, is able to ready everything that their environment and the world of nature contains. (

Montessori’s idea (

Montessori (

The same plants surround the botanist and the ordinary wayfarer, but the botanist sees in every plant those qualities which are classified in his mind and assigns to each plant its own place in the natural orders, giving it its exact name. It is this capacity for recognizing a plant in a complex order of classification which distinguishes the botanist from the ordinary gardener, and it is

The aim of this article, however, is to explore, analyze and report on the validity of variation and invariance in other areas (and consequently other materials) than the training of the senses. The question we raise is whether the application of variation and invariance is valid in other areas as well and could therefore be seen as a fundamental idea in Montessori’s view of learning that has not been noted so far. If so, a variation-theoretical perspective on learning could be seen as an important part of Montessori’s didactic theory in general, thereby offering one answer to the question why lessons should be presented in the way described.

In the next section, we will initially describe some key concepts in variation theory. This section is followed by a description of the way in which teaching in Montessori education is implemented within two chosen areas at an elementary level. These descriptions are followed by analyses of the ways in which each description is related to a variation-theoretical perspective on learning. The article ends with a discussion of the results and their practical implications.

According to Marton (

One way is to make the object of learning (that which is to be learned) your own, to discern the important aspects of the content of learning and the relations between them. The other way is to learn what to do and say in order to meet the demands imposed upon the learner by the teacher or the test. (p. 14)

If the latter kind of learning is stressed, less of the first kind might happen. Hence the teacher should above all create conditions which will allow the students to acquire the necessary aspects of the object of learning and the relationships between them. In that case, students will learn how to do things by seeing how things are related to each other, rather than just learn a certain order as told by the teacher. This is significant for a variation-theoretical perspective on learning which indicates that “mastering an educational objective amounts to discerning and taking into consideration its necessary aspects” (ibid., p. 23). Thus in a variation-theoretical perspective learning is seen as “learning to see” (ibid., p. 36). According to Montessori (

The child then has not only developed in himself

When learning is seen as “learning to see”, it follows that someone has learnt something when he/she is aware of other or more aspects of a phenomenon than before (

According to variation theory, the learner has to be aware of the difference between at least two features in order to discern them. Marton (

Once the learner has found the meaning by contrast, he/she has to generalize the aspect which has previously been separated. If the aspect, for instance, is colour, generalization is achieved by keeping the colour invariant but varying other aspects such as form and size. The aim of generalization is not to find out what different aspects have in common; rather, it is to find out how different aspects vary. If the aspect is colour, the conclusion we will draw through generalization will therefore be something like: “so this can be red, and this and this”, rather than “they are all red”. As Marton (

The final step is to let the learner experience simultaneous variation in all relevant aspects. In variation theory, this pattern of variation is called fusion: “it defines the relation between two (or more) aspects by means of their simultaneous variation” (

Initially, we stated that Montessori, as in variation theory, emphasized that the child will develop their ability to “see” in the work with the sensorial materials in preschools by using patterns of variation and invariance. We will now look into the ways in which certain other areas are dealt with according to Montessori at an elementary level and whether it can be assumed that Montessori designed the materials and the teaching with such a purpose in other areas as well. We have chosen to look into one specific area in teaching arithmetic and one in teaching geometry. We decided to choose these areas as they are either described in detail in Montessori’s literature or in oral presentations within Montessori training.

When the teaching of numbers is introduced in Montessori education, teachers use a material called Number Rods, shown in Figure

The Number Rods. Photo by Eva-Maria T. Ahlqiuist.

In Montessori’s description (

When the rods have been placed in order of gradation, we teach the child the numbers: one, two, three, etc., by touching the rods in succession from the first up to ten. Then, to help him gain a clear idea of number, we proceed to the recognition of separate rods by means of the customary lesson in three periods. We lay the three first rods in front of the child, and pointing to them or taking them in the hand in turn, in order to show them to him we say: “This is

When the children have worked with the rods for some time, the teacher will introduce the Sandpaper Numbers, which consists of a box with cards on which the numbers from one to nine are cut out in sandpaper. Montessori (

Montessori also writes that another exercise associated with the child’s work with the boxes is to put all the Sandpaper Numbers on the table and place the corresponding numbers of cubes, counters and the like below (ibid.).

The Counting Boxes. Photo by Eva-Maria T. Ahlqiuist.

The didactic material used for teaching the first arithmetical operations is the same one as used for numeration, the Number Rods. Montessori (

The first exercise consists in trying to put the shorter pieces together in such a way as to form tens /…/ In this way we make four rods equal to ten. There remains the five, but turning this upon its head (in the long sense), it passes from one end of the ten to the other, and thus makes clear the fact that two times five makes ten.

These exercises are repeated and little by little the child is taught the more technical language; nine plus one equals ten, eight plus two equals ten, seven plus three equals ten, six plus four equals ten, and for the five, which remains, two times five equals ten. At last, if he can write, we teach the signs

Initially, we can note that the material presented above, in itself, isolates the quality “number” by its design. When the numbers 1, 2, 3… are introduced, it is only the numbers that vary. Other qualities in the material are identical. Furthermore, “one” is introduced in contrast to “two” and “three” and so on.

Another important aspect when it comes to the design of the lessons is the order in which these lessons are given. Looking at the sequences of the lessons, it seems clear that the purpose of such sequences is to make it possible for the child to initially find out the meaning of numbers by contrast and then, later, generalize the aspect which has previously been separated. This, for example, is done by working with different objects such as counters, cubes and the like. which the child matches with the Sandpaper numbers or the right compartment in the Counting boxes.

The importance of contrast is also evident when arithmetical operations are introduced with the Number Rods. In Montessori’s description of how this should be done, it is noticeable that addition is introduced in contrast to subtraction and that multiplication is introduced in contrast to division. The contrast between addition and subtraction, for example, is made by first putting rods together and then, later on, taking them apart. In this way it is possible for the child to “see” the relationship between, for example, 3 + 2 = 5 and 5 – 2 = 3. When Montessori links addition and subtraction together in this way, the relationship when it comes to what can be seen as parts and wholes is stressed, which may make addition easier to grasp since it is introduced in contrast to subtraction.

When comparing the work with the Number Rods and the Counting boxes, it might seem at first sight as if the children in their work with Counting boxes repeat the same exercise as with the rods. However, we have to look at the way the Number Rods and the Counting boxes are designed. If we say that the number that each rod corresponds to can be seen as “solid”, we then have to say that the pegs in the Counting boxes can be described as “loose”. This corresponds to two critical aspects, the ordinal and cardinal property of numbers, which the child has to “see” in order to grasp the rules of arithmetic. Ordinal property means that each number refers to a place in an order (1st, 2nd, 3rd…). Cardinal property refers instead to the “manyness” of things (one book, two books…). Both aspects can be noticed in the way the work with the Number Rods and Counting boxes is designed, but each material stresses different aspects. When the children are working with the rods, they grab “the manyness”, or as Montessori (1934) describes it, “one united whole”, that the rod in itself represents in their hands, even if they will also be able to identify the ordinal property when, for example, counting each section of the rod. The same can be said about the work with the Counting boxes, but in this case the ordinal property is more prominent when counting each peg than in the work with the Number Rods, even if the main aim of the work is to match each compartment with the right number of objects.

What can be seen as an additional critical aspect when handling the Number Rods as described above is that numbers are wholes that can be divided into parts. This may be noticed by the child in the work with arithmetical operations. When a child, for example, tries to put rods together in such a way that they form tens, this will illustrate that wholes can be divided into parts. In this example, the work done by the child illustrates that ten can be split into nine and one and that they are parts of the whole ten and so forth.

Geometry is presented in preschool by providing children with sensorial experiences and presenting the names of the different geometrical objects. Montessori argues:

Observation of form cannot be unsuitable at this age; the plane of the table at which the child sits to eat his soup is probably a rectangle; the plate which contains the meat he likes is a circle; and we certainly do not consider that the child is too immature to look at the table and the plate. (

The geometry material in preschools consists of blue Geometric solids containing objects of ten different shapes, a Geometry Cabinet with thirty-six plane figures and Triangle boxes used to construct polygons. These materials are also utilized in elementary education. This is, in fact, something that is fundamental in the Montessori curriculum: materials from preschool build the basis for further studies at higher levels. “They [the materials] form a long sequential chain of learning: each material can be placed within a hierarchy in which the simplest one forms the basis for the next. Nothing is left to chance in this sequence, everything is provided…” (

At an elementary level, there are more materials than mentioned above. Here, though, we will focus on the work with the Geometry Cabinet and how it is used to make it possible for the children to deepen their knowledge of triangles. The study of geometry in elementary classes is a work of experimentation and discoveries. Here we present extracts from the introductory notes to geometry from the AMI course in Bergamo:

Montessori’s psycho-geometry reveals the essential place that geometry holds in human development, both historically and now, in the educational system. Psycho-geometry seeks to show the geometry inherent in life: organic and inorganic nature.

For example, inorganically: crystals, snow-flakes and organically: formation of flowers, molecules etc. Further, we look at the supra-nature, the work of humans in constructive architecturally and in other designs. Similarly, it can be seen that geometry is based upon the observable order of our world. Geometry, therefore, cannot be seen only in the abstract. One can study geometry by studying the historical evolution of humans and also by observing carefully the world in which we pass our daily lives./…/ Geometry, γɛω; geo- “earth”, μɛτρία -metron “measurement”, the measurement of the Earth on which we live. This implies the relationship between humanity and the objects of our Earth, as well as knowledge of the relationship between these objects themselves. We study fundamental elementary Euclidian geometry./…/Our [the Montessori] geometry is made up of a) plane geometry, the study of the properties and relations of plane figures, and b) solid geometry, the study of figures in space, figures whose plane sections are the figures we have already studied in plane geometry.

In this article we will focus on the work with the Geometry Cabinet and how it is presented so as to expand the children’s knowledge of the different shapes. Here, we will concentrate on different types of triangles. At the elementary level, the geometry lessons, when adequate, will relate to the history of the subject area, and the etymology of words will be identified for each new concept the children meet. The study of triangles shown below will focus on the triangle examined by its side and by its angles and the work on uniting the sides and the angles.

The Geometry Cabinet consists of six drawers, each containing six wooden squares with geometric plane figures in the same colour

The presentation tray. Photo by Eva-Maria T. Ahlqiuist.

Each figure in the cabinet has a small handle in the centre, making it possible to lift up the figure when taking it out of the frame. The first drawer, shown in Figure

The second drawer has six rectangles, all with the same height, ten centimetres, and increasing from five centimetres in length to ten centimetres (the last one representing a square). The third drawer has six circles where the diameter increases from five to ten centimetres. The fourth drawer has regular polygons from a pentagon to a decagon and the fifth drawer has other quadrilaterals, such as an irregular quadrilateral,

The first drawer. Photo by Eva-Maria T. Ahlqiuist.

The children should be familiar with the name triangle and the etymological origin and be asked to pick out the triangle among other polygons from the cabinet and identify triangles by going out in nature or visiting the city.

The next step is to examine the angles of the triangles placed on the bottom of the drawer, starting with the right-angled triangle, with the right angle as a base letting one of the legs follow the base and the other pointing upwards. The children compare this right angle with the angle between the floor and the wall in the classroom. The teacher tells the children the name of the angle. The next triangle explored is the scalene. The teacher asks the child to compare the scalene angle with the right angle in order to discover the difference. The children will then be asked to compare the obtuse angle with the acute angle by letting the child touch both of them. Then the teacher asks the children to examine all three angles of the acute-angled triangle, discovering that all angles are acute. The same procedure is done with the right-angled and the obtuse-angled triangle.

The third step is to unite the sides and angles. The teacher asks the child to write labels with the names of the sides and labels with the names of the angles of all six triangles. Each triangle will have two labels. Then the children are asked to tear off the word triangle from the labels and then unite the words of the angles (for instance, acute-angled) and the words of the sides (for instance, scalene). Finally adding the labels on which the word triangle is written (here exemplified by the acute-angled scalene triangle). There is then a discussion about the equilateral triangle: Should the triangle be called equilateral triangle or “equiangular” triangle? The children are asked to look for the name commonly used and will choose the name equilateral. The labels are rewritten on an undivided label for each triangle.

The children now order the triangles by constructing a coordinate system with two axes. On one of the axes the children put the word Angles written on a label, and below three labels with the names of the angles. On the other axes, the children put the word Sides, and below the names relating to the sides. The coordinate system will in this way have nine spaces, and the child is asked to put the triangles in their right positions. When this is done, there will be three empty spaces. The children now have to explore if there are triangles missing which could be placed in the coordinate system. By constructing triangles with help from The Box of Sticks

The Box of Sticks. Photo by Eva-Maria T. Ahlqiuist.

Montessori argues that the child has to have embodied experiences in order to distinguish different shapes and she criticizes the traditional way of teaching as it does not help the child to recognize and remember the shapes.

The teacher draws a triangle on the blackboard and then erases it; it was a momentary vision represented as an abstraction; those children have never held a concrete triangle in their hands; they have to remember, by an effort, a contour around which abstract geometrical calculations will presently gather thickly; such figure will never achieve anything within them; it will not be felt, combined with others, it will never be an inspiration. (Montessori, 1917, p. 270)

Montessori education combines movement and language. This is an essential feature of Montessori’s didactic concept since manipulating an object facilitates the possibility to isolate the quality of the object in question. When starting by examining the different triangles, the fundamental condition is that the child already knows what characterizes a triangle. This was done with the presentation tray, where the triangle was initially contrasted with the square and the circle. What varies is the shape since the colour is invariant. In accordance with variation theory, the foundation of meaning here is the difference in shape. If instead the teacher had picked out three triangles of different colours, one blue, one red and one green, and told the child that all of them are triangles, the child would have had difficulty in grasping what a triangle is because there were no alternatives to a triangle. And even if there had been different geometrical shapes, but all of different sizes and in different bright colours, it would, according to variation theory, have been problematic for the child to focus on the essential aspect. As Feez (

When the child is able to identify the sides of triangles and knows what characterizes their angles, the two qualities are united in one and the same triangle. This act can be seen as what Marton (

The activities within the areas described above are the result of Montessori’s empirical research on how children learn. As shown in the analyses, the use of variation and invariance is to the fore in those activities. However, the latter is not made explicit by Montessori in her literature except for the sensorial training described in our introduction. In Montessori’s (e.g.

Our analyses show that the theory behind Montessori’s didactic material, due to the design of the material and how the lessons should be given, is supported by variation theory, and we reveal that Montessori has clearly searched for and identified what in variation theory is referred to as critical aspects. Montessori’s (

What Montessori implies by replicating lessons like the one described here is that the critical aspects must be identified by the teacher if the necessary conditions for learning are to be created. This is in accordance with Marton (

As the use of variation and invariation is not always clearly expressed in Montessori’s literature, even if the material and the sequences of lessons are described in detail, we believe that this article will have an impact on Montessori education. We also believe that it can contribute to variation theory with the idea that not merely seeing helps children to make knowledge their own. The fact that children are given the possibility to discover critical aspects by grasping them must be regarded as crucial. As Montessori (1934/2011) says, activities “involve the hand that moves, the eye that recognizes and the mind that judges” (p. 11). Viewing the body and the mind as interwoven (

With regard to didactics we refer to the basic questions:

Traditional here refers to a way of teaching in which the children have few opportunities to make experiences of their own. Rather, what is to be taught is mainly “transmitted” to the child by the teacher (

Extracts from personal notes by Ahlquist from the AMI, Associazione International Montessori course, 1981–1982.

Here we concentrate on just one section of the study of triangles. The Montessori material in geometry consists of other materials, such as the Constructive Triangle Boxes, the Box of Stars, the Metal Insets and the Yellow Area Material.

Montessori suggested that the geometric figures should all be blue and the bottom of each tray should be yellow. Some manufacturers of the material made the geometric figures red and the bottom of the tray white.

The last of the seven types of triangles, the obtuse-angled scalene triangle, is found in the sixth drawer.

In American English, it represents a trapezium.

In American English, it represents a trapezoid.

This is a special kind of trapezium as there are two pairs of sides of equal length or all four sides of equal length but none of the sides are parallel. The drawer could also contain a boomerang, depending on the manufacturer.

Some manufacturers include a third quatrefoil (an epicycloid). In those cases, the drawer contains six curvilinear figures.

Also known as the Reuleaux triangle.

Examples of such work are given in Ahlquist, Gustafsson & Gynther (

The Box of Sticks contains sticks from one unit to ten units, each unit in a different colour. Every stick has a hole in each end in order to be able to unify them with each other when constructing geometrical shapes. There are also neutral sticks with units from one to ten but of different lengths compared with the coloured sticks, as they represent irrational numbers. The material also consists of a set square, which is used to identify the angle as a right angle.

This manuscript has been peer-reviewed externally and the process was anonymous. The final decision was made by the Associate Editor Christina Gustafsson.