Montessori and Mathematics
Dr. Montessori stressed that our first and foremost goal is to assist the development of the child, which entails supporting each child's progress along the path of self-construction. She believed that one section of that path was mathematical, and she worked to bring mathematics to children in such a way that they could understand it, and so that they could appreciate and enjoy it.
The mathematics introduced in the Montessori elementary environment includes arithmetic, geometry, and an introduction to algebra. For purposes of the adult's organization, arithmetic and an introduction to algebra are given as one sequence and geometry is given as another sequence.
Arithmetic: Numerical aspect of mathematics, or the science/art of computing. In the Montessori elementary the children study the four operations, fractions, decimals, numbers in different bases, measurement, etc.
Geometry: Originally a study of geodesy - Aspects of the earth (size, shape, weight of the earth). Today, a study of points, lines, planes, figures, solids, and an examination of their properties. In the Montessori elementary environment, the children are introduced to Euclidian geometry.
Algebra: A method for reasoning about numbers by employing symbols (usually letters of the alphabet) to represent them, and signs to represent their relationships. Algebra allows us to generalize and to symbolize. In the Montessori elementary, the children learn that algebra can also be used to solve problems, and algebraic expressions can be represented graphically.
Montessori intended that these three strands be introduced to the child simultaneously, and in a way that allowed them to reinforce and support one another:
Mathematics and the Second Plane Children
Experiences given to the child in the second plane of development continue to be based in the senses but now sensorial exploration alone is insufficient to support the development of second plane children. The material of the elementary is sensorial, able to be manipulated. However, the work that takes place is now not just activity of the hand. It is also activity of the mind.
The materials invite not just further concrete steps with the hand, but the taking of further conceptual steps with the mind.
Second plane children now utilize their growing power to abstract, and they explore using their imaginations. This new level of work allows the children to identify a variety of relationships.
After each presentation, the children are generally introduced to some form of application or "follow up" work. This provides the necessary repetition for acquisition of the information or skill involved.
The Reasoning Mind
The absorbent mind of the first plane fades away as the child enters the second plane, and in the reasoning mind gains prominence. Second plane children must be left free to reach their own understandings via that store of facts built in the Children’s House, and the relationships discovered now through the use of the reasoning mind.
We must take care to support the children's explorations: We must treat their conclusions with respect, both for the conclusion itself, and for the intellectual process that allowed them to reach the conclusion in the first place:
"If this, then this." - "Furthermore, that is also true." - "This problem is just like that one."
These are constructions that reveal to us the development of a system of logic in the child's mind. We observe the child spontaneously developing syllogisms as they work on a problem, or construct an argument or defense:
A = B B = C Therefore, A = C
There is a great power of the intellect at the second plane. This great power will fade away as the second plane draws to a close. It is important, therefore, to provide the children with maximum opportunity to exercise this capacity while it operates.
Second plane children responds best to challenging work that appears "difficult". Children of the second plane are drawn to the extremes and to the unusual. They tend therefore to construct extensive problems.
This characteristic provides a stimulating and exciting way to repeat, and to make independent discoveries.
The children should be shown how to construct their own problems. They will come up with problems that they can solve and they will create problems that they cannot solve, and therefore they come to ask for help. Learning occurs in both circumstances.
Adults should wait until help is needed or requested. Do not under-estimate the ability of second plane children and the remarkable results that emerge from their independent effort. Real learning may come from the making of errors, from righting those errors, and then by moving closer towards success.
This is the time when the children need and are capable of completing, an enormous amount of work. If they are not given work to do, to construct themselves as their nature demands, then the result is that they will assert themselves in less desirable ways. The evidence will be various less desirable and less acceptable activities.
Mathematics and History
From time to time, Maria Montessori referred to the "Mathematical Mind" of the human being.
Our human tendencies to abstract, imagine and then to make exact leads to this conclusion that we have this Mathematical Mind. History demonstrates that we symbolize what we discover. Human beings are able to generalize from particulars, and can then apply their discoveries to many new particular situations.
We use mathematics every day in a practical sense as we count our money, or as we attempt to fit various objects into certain spaces. Mathematics helps us to keep records of dates and amounts and measurements. These are activities that we can trace back to the earliest civilizations!
History brings mathematics to life. It allows children to glimpse some of the very real uses to which mathematics has been put, and it connects the children to the many fascinating individuals who have been instrumental in developing our mathematical knowledge. This in turn connects mathematics to society, and it also invites the children to contribute in their own time.
Elementary children are able to apply mathematics to their own daily lives, and to see it in action in the world and society in which they are immersed: "What are proportions of ingredients in the recipe for the dessert will I need to make enough for 100 people, rather than myself. Is there a formula that I can create that will work for 1 person, 10 people, 100 or 1000?"; "What is the shape of this leaf, or this wing, or this galaxy?"
In the history of our numbers and of mathematics and mathematicians the children see that people of the past have each made a contribution that led to the present state of our knowledge. Without them, we would be in another, less advanced place. They also see that these heroes of the past were simple people like themselves, inviting them to strive to make their own contributions in the future.