Introduction to Mathematics
Mathematics is the study of relationships as formulated by the abstract mind. Certain logical patterns emerge and provide avenues of learning.
The origins of mathematics began long ago, when humans bound together, and with a unity of communal effort they invented new ways to relate to the environment and meet their fundamental needs. Humans’ earliest efforts came in the measurement of the earth. The advent of agriculture brought this earth measurement into the accumulated science of geometry. Concrete numbers emerged out of this new science of the relationships of shapes in space. The power of concrete numeration inspired man to invent arithmetic, with operations of numbers and with symbols. This new arithmetic came in handy as commerce and trade became a part of human culture.
With a deeper understanding of the laws of arithmetic, algebra was created. As early as 1650 BC an unknown writer set out to create a handbook of everyday mathematics that would be useful for merchants for purposes of business, paying taxes and measuring things. This is our first known record of algebra. Algebra is the science of treating the properties of numbers by means of general symbols. The use of the variable came into being with this new avenue of mathematics.
Higher levels of geometry were also attained by the early Greeks as they expressed number relationships in geometrical form. The human mind, free to contemplate, went beyond the physical world and took relationships of form and numbers to a science called analytical geometry.
Set theory is also a branch of mathematics which studies the interrelationships of sets of objects. Calculus looks at the infinitesimally small and the relationships of the unseen. Finally, statistics is the study of frequency and probabilities.
A Human Tendency
Dr. Montessori realized that the work of the child in ordering perceptions differed greatly than other creatures because only the human had the power to order and abstract. Unless man could imagine and make abstractions, he would not be intelligent; or his intelligence would be like that of higher animals, that is to say, it would be rigid and restricted to some particular form of behavior, and this would prevent its expansion.
The mind of the human makes inferences extracted from the physical world around them but these impressions generalize in the mind then go beyond physical reality. The process of the intellect creates abstractions based on inference. This action is the mathematical mind at work, creating concepts beyond reality and ordering them into relationships. These perceptions are the true work of mathematics, to stimulate the imagination to create order from abstract thought. The communal wealth of each human’s ideas in the ordering of relationships of abstract thought create the science of mathematics.
This formal thinking that we call mathematics is not the only form it takes. A great unconscious power is at work in the developing child which aids him in adapting to his world. The child must order his own impressions into relationships to create the construction of his personality. This informal mathematics is the work of the unconscious mind in classifying, categorizing perceptual impressions. What Dr. Montessori discovered was that young children have a mind that is mathematical in its approach to ordering perceptions. She called this aspect of the human the mathematical mind.
The Role of Language
The outward manifestation of this inner work is the child’s ability to classify his experiences with language. The myriad of impressions absorbed by the child are only that, unless he can sort and classify these with a special abstract tool that only man was given, the tools of language. These then are the final abstractions of real experience. This allows the child to share in the communal process of accumulated knowledge, for without language man would have no base for passing on his learning. The child utilizes the inner powers of imagination and abstraction to put the final ordering on perception and materialize it with language. These two powers of the mind (imagination and abstraction), which go beyond the simple perception of things actually present, play a mutual part in the construction of the mind’s content. … For words, if they are to be utilized and enrich the language , must be capable of taking their place in the ground work of sounds and grammatical order. And what happens in the construction of language happens in the construction of the mind.
The Role of Imagination
The power of the mind to imagine takes man beyond the real experience and provides the ability to abstract and change experience to meet his needs. With imagination man can take the orderly and exact perceptions in his mind as a starting point and utilize them to create something new. This ability to interrelate isolated perceptions and manipulate them into new constructs is the true power of the mathematical mind.
The true basis of knowledge is the real experience which the mind configures with the imagination. The imagination uses real experience to alter impressions to develop possibilities beyond reality.
The Role of Reasoning
The power of reasoning utilizes abstractions through imagination and generalizes them, allowing the mind to go beyond simple conclusions to complex relationships. Reasoning and imagination work together, for imagination without it is idle fantasy unconnected to the real world. The exercise of intelligence, reasoning within sharply defined limits, and distinguishing one thing from another, prepares a cement for imaginative constructions; because these are the more beautiful the more closely they are united in form, and the more logical they are in the association of individual images. The fancy that exaggerates and invents coarsely does not put the child on the right road. 3 Reasoning draws upon the mathematical mind’s ability to order perceptions and utilizes the imagination to take the impressions beyond the real experience. Yet the key aspect of the reasoning mind is the ability to ground this imaginative configuration by relating it to other real experience, distinguishing it from fantasy, and giving it the substance of intelligence.
- Number itself cannot be defined and understand
- The concept of number grows from experience with real objects but eventually they become abstract ideas.
- It is one of the most abstract concepts that the human mind has encountered. No physical aspects of objects can ever suggest the idea of number. The ability to count, to compute, and to use numerical relationships are among the most significant among human achievements.
- The concept of number is not the contribution of a single individual but is the product of a gradual, social evolution. The number system which has been created over thousands of years is an abstract invention. It began with the realization of one and then more than one. It is marvelous to see the readiness of the child’s understanding of this same concept.
- Arithmetic deals with shape, space, numbers, and their relationships and attributes by the use of numbers and symbols. It is a study of the science of pattern and includes patterns of all kinds, such as numerical patterns, abstract patterns, patterns of shape and motion.
- Little children are naturally attracted to the science of number. Mathematics, like language, is the product of the human intellect.
- The concepts covered in the Primary class are numeration, the decimal system, computation, the arithmetic tables, whole numbers, fractions, and positive numbers.
- Arithmetic is the science of computing using positive real numbers. It is specifically the process of addition, subtraction, multiplication and division. The materials of the Primary Montessori classroom also present sensorial experiences in geometry and algebra.
Mathematics in the Children’s House
The mathematical mind plays a key role in the preparation of the environment in the Children’s House. Experiences are structured to appeal to the child’s natural urges to explore with his perceptions the immediate world of the here and now. Sensorial activities provide the opportunity to hone discrimination and appeal to the mathematical mind in their orderly isolation of sense experiences. This builds impressions which provide the grist for language to develop and express the abstractions of the physical properties the child has generalized from his experiences. As the child masters language he is able to manipulate symbols, and discovers the process of writing. This leads him to early understandings of the mechanics of reading.
In his mathematical exposure he comes on the quantities and symbols of base ten, and learns the operations of these. He explores the decimal system and the regular progressions of math facts necessary to accomplish arithmetic.
In all this work in the Children’s House there is a benefit for the mathematical mind. From the sequential activities of practical life, the seriated and graded activities of sensorial, through the exploration of the mechanics of reading and writing, to the early work in geometry and arithmetic, the Children’s House experience lays a mathematical foundation for the elementary child.
The key to this benefit lies in what Dr. Montessori called “materialized abstractions”. Her concrete materials presented relationships for higher thinking later in life by creating impressions for abstraction. She saw that the child used his hands as the tools of the mind, and provided for this natural activity in a systematic way. From spontaneous individual work the child gains what formal education could never teach later, the physical manipulation of abstract concepts.
- No longer the perceptual explorer, the elementary child still needs to explore with his hands and his senses the concepts of mathematics.
- He explores with a reasoning mind and discovers mathematical implications.
- The powers of imagination and the vigor of the age allow the elementary student to engage in “great works”, where he integrates concepts on a wider scale then ever before.
- He discovers that abstractions are shared imaginations growing from human history.
- His perceptual experiences with Montessori materials build the basis for more materialized abstractions in the elementary. He learns to draw conclusions from simple to complex relationships.
- These processes in the elementary help the child to infer, abstract, relate and recognize theorems.
- Through repetition and variety the child internalizes concepts and develops a growing ability to represent them in abstract work.
Montessori stresses the importance of manipulating materials to discover answers, rather than merely memorizing math techniques. How much richer to arrange colored strips on a board to see the sum of two addends, instead of reading a flash card! It is fundamentally different to use your hands to compare fractional pieces, instead of just learning rules for comparing numbers. Who among us really understood why it worked to invert the second number when dividing fractions? Our children do!
There are several reasons that Montessori math materials promote optimal psychological development. First, multi-sensory learning allows students to use various parts of their brains to learn. Hands-on manipulations encourage active, discovery learning. Second, materials encourage children to work together, and these collaborations involve lots of discussion and rationalization. Third, Montessori students learn to strive for accuracy because the materials provide feedback. They see their math work as puzzles to be solved, instead of assignments to be corrected. Finally, Montessori math materials are elegantly designed to use geometric relationships to show algebraic ones.
Montessori math constitutes superb brain-based learning. Robert Sylvester (A Biological Brain in a Cultural Classroom) states, “It is not the number of neurons itself that determines our mental characteristics; it is how they are connected;” and Eric Jensen (Teaching with the Brain in Mind), “The key to getting smarter is growing more synaptic connections between brain cells and not losing existing connections…The single best way to grow a better brain is through challenging problem solving.” Michael explained that unused connections are pruned, so the neurological rule is, “Use it or lose it.” All of this suggests that the best way to learn is to engage all the lobes of cerebrum, activating visual and auditory memory, along with controlled movement and problem-solving. Activities that cross hemispheres force interactions of logic and creativity.
Parents need to “trust the benefits of Montessori mathematics and support the school, the teachers and your child.” Parents need to be dissuaded from “helping” their children by teaching number tricks or doing lots of drills, because those short cuts can short circuit deeper learning. We ask that parents avoid judging their child’s mathematics progress only by standardized test scores or the ability to answer correctly when quizzed. We advocate for our children’s mathematical “heart and soul.”