Sunday, October 17, 2010

A child's mathematical mind - Part 1

From the very first moments of life, a child is bombarded by impressions of the world around her. At first, the child only absorbs information but gradually begins to form an understanding of relationship and patterns. It is not enough, however, for the child's mind just to receive and record impressions. There must be some sort of order imposed upon them for the child to use the information intelligently. For example, a new born child quickly learns to recognize the smell of her mother's milk. Later on she will learn to recognize her mother's form, the sound of her voice and the comfort of her arms. Eventually, the child will order and arrange these impressions and attach language to them. The impressions become "Mommy".

This impulse to produce order out of disorder is called the "mathematical mind". Dr. Montessori borrowed this phrase from the works of Blaise Pascal, a French philosopher and mathematician, who said "man's mind was mathematical by nature and that knowledge and progress came from accurate observation." (The Absorbent Mind (1988) page 169). For the purpose of this post, however, it is important to distinguish between the words arithmetic and mathematics.

As defined by the Oxford dictionary, arithmetic is "the science of numbers" while mathematics is defined as the "abstract science of space, number and quantity" and mathematical is "rigorously precise". Therefore, when discussing the "mathematical mind", one is not necessarily speaking only of numbers and calculations but is also taking into consideration the concepts of order, precision, imagination and abstraction.

Observation, representation and investigation of patterns and relationships are also involved in mathematics. "Mathematics is often defined as the science of space and number…but a more apt definition is that mathematics is the science of patterns. The mathematician seeks patterns in number, in space, in science, in computers, and in imagination. Mathematical theories explain the relations among patterns." (The science of patterns (1988) pages 611-616). To summarize, the spontaneous ability to organize, classify and quantify patterns and relationships within the context of daily experiences is what Dr. Montessori calls the mathematical mind.

Now here comes the good stuff! Dr. Montessori recognized that the characteristics of a mathematical mind (order, observation, precision and imagination) can be sharpened with careful nurturing. This nurturing is accomplished in the Montessori environment practically from the first day a child enters the classroom.

When a young child of three enters the prepared environment of a Montessori classroom, she is introduced to the practical life activities. As she becomes absorbed with the work she is laying a foundation for future learning. For example, the activity of opening and closing boxes compels the child to pay close attention to the size and shape of the lids.  Close observation may lead her to imagine the boxes fitting inside one another or whether the lids are interchangeable. She requires precise movement in order to correctly replace the lids on the boxes and, by completing the cycle of work and returning the activity to the shelf, she is responding to her internal sense of order. The skills and self-discipline learned in the simplest of practical life activities prepare the child for the precise manipulation required later when working with the mathematical materials. They instil the concentration, organization, precision and order required in mathematics.

Once the child has developed the ability to concentrate and focus on her work, she will be introduced to the sensorial materials. Through repeated work with these materials the child absorbs the properties of the physical objects in the environment. These impressions are then abstracted and related to the environment through their application. Aspects of number, shape and space are isolated in the various materials and the child is led to focus on essential qualities and variations in qualities and quantities rather then merely upon the objects themselves. As E.M.Standing explains, (in Maria Montessori: Her Life and Works) "Examine any one of the sensorial materials from this point of view and you will see that it consists of a series of objects that have to be arranged in accordance with some principle of order latent within them."

The sensorial dimensional materials make a huge contribution to the ordering of the child's sense perceptions.  These materials encourage observation and enhance awareness of differences in dimension.  The pink tower, for example, consists of 10 cubes that increase in size by increments of one centimeter squared.  This provides a visual and muscular perception of dimension and prepared the child for mathematices.  The fact that there are ten cubes is a preparation for the decimal system.  The child forms a sensory impression of the precise size of one cubic centimeter as well as the size of a litre.  It is also a preparation for later work with cube roots in the bead cabinet.

Part 2 to follow.


Leptir said...

Thank you! Great post :-)

k-barker said...

Amazing stuff. I can't wait for part 2!

Rose said...

I loved this post. It is amazing what a child is learning through the simplest of acts.