Thursday, October 21, 2010

A child's mathematical mind - Part 2

The language area of the classroom also includes many activities that introduce important concepts in preparation for later mathematical experiences. "Just as the form of a language is given by its alphabetical sounds and by the rules for arranging its words, so the form of man's mind, the warp into which can be woven all the riches of perception and imagination, is fundamentally a matter of order." (The Absorbent Mind (1988) page 169).

One example of this is the Insets for Design. The child is reminded of the shapes in the geometric cabinet as she traces the shapes of the inset frames. Deep concentration is needed to fill in the designs with lines from left to right which also helps her to develop the precision needed for so many future tasks. As she develops her skill with the insets, her imagination is called into play and her designs become more and more intricate and abstract.

In the Cultural area, children are introduced to botany, zoology, scientific experiments, geography, history, music and art. This gives the child sensori-motor experiences based on the practical life activities, teaching the child to compare, grade and calculate.

Finally, we come to the mathematical area in a Montessori classroom. The mathematical exercises give children the satisfaction of learning by discovery. Basic arithmetic operations are learned as the child separates, combines, shares, counts and compares the mathematical materials. The specially designed math materials are presented in such a way that a natural progression takes place.

By working with quantity, then symbol, and eventually a combination of both, the child develops a true understanding of the numbers from 0 – 10 and a real reference for what a number means. This gives her a sound foundation for later work with the decimal system materials and in mathematical operations like addition, subtraction, multiplication and division. After working with the mathematics materials through touch and manipulative activity, the child discovers the physical nature of numbers and how, when combined, they make new numbers. Work in mathematics is profoundly satisfying to children because mathematical work corresponds with the way their minds naturally work.

This mathematical mind is that part of the mind which is orderly, observant, precise and imaginative.  Early preparation of the mathematical mind is achieved through use of manipulative materials from every area in the classroom.  These materials are carefully designed to present abstract concepts in concrete form and are presented in a specific way.  Through repeated work with the materials, the child is preparing herself for a smooth transition to more abstract thinking.  "There is a predisposition to exactness and detail, and it may be directed to detail of quantity. Arithmetic is a sort of abstraction and, therefore, brings this exactness to the abstract level.  The child, starting from the material, passes to the abstract number and thence to the more abstract stage of algebra, and he works with exactness in all three fields, material, abstract and algebraic, fascinated to be able to realise the play of the units (Education for a New World (1989) page 10).

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.

Sunday, October 3, 2010

Off to Newcastle Island

     Last Friday, Diana (the elementary teacher & Principal) and I took the elementary children on the first outing of the year.  This is something that Diana and the children do every year and it is an important bonding experience for them.  Each year, our oldest children leave, children join us from other schools, and a number of younger children move up from the Primary classes.  This means the group dynamics change each year and Diana has found that the trip to Newcastle Island and the elementary sleepover, are vitally important events for the group. 

     (It should be noted that parents are not invited to join this expedition for very specific reasons.  This trip is about independence and relationship.  Having one's parent along takes independence away from a child and changes her role within the group.)

      As Montessori teachers, Diana and I are trained to observe the children, only stepping in when the children need us.  This is crucially important for the bonding the children experienced on this trip.  Sitting quietly on a log, we watched as the children scattered, formed little groups, scattered again, formed different groups, scattered once more and finally all came together in a spontaneous group task.  We did not initiate any games nor did we issue rules and regulations.  All expectations had been discussed in the classroom.

     Anyway, enough of that.  Here are some photos of the day.

There was a mix up with the boat we were supposed to catch so we just waited for the small island ferry.  The children didn't mind because there is a great playground nearby the dock.

Finally, the little ferry showed up and we were on our way.

The island has been designated as a Provincial Park.  Considering it is right in the city harbour, it is a great place for picnics, camping, and day hikes.  Once off the ferry, we visited the washroom, filled up water bottles, investigated the buildings and then were on our way.  We walked for about 30 minutes.

Rather, I should say that we meandered along the path stopping for the occasional photo opp.

Some of the children needed encouragement and convincing to keep going  because it really wasn't that long a hike.

Others needed a little help from Diana to help resolve misunderstandings. 

With a chorus of 'Finally!' we reached Kanaka Bay.  The children spread out to eat their lunches but soon abandonded the food to the allure of the beach.

The children broke into small groups to investigate tidal pools,

inspect sea creatures,

and hunt for sea glass.

A couple of the boys began to repair an old dock that had been tossed onto the shore during a storm.  The plan, overheard by the teachers, was to get it seaworthy and float it over to Snake Island.  We silently wished them well.

The water was very cold - it is October, after all - but that didn't stop most of the children from going in. There were logs to be moved!

Teamwork became natural as various 'tasks' were discovered and attempted.  Like the ebb and flow of the tide, the children formed groups then broke apart to join other groups until.....

they all joined together in an effort to get that dock, now a Pirate ship, into the water!

Unfortunately (wink), our time was up before the craft was seaworthy and with promises to come back someday to finish, the children packed up and began the hike/run back to the ferry.

When we stopped for a drink of water, I was reminded that the beauty of this corner of the world never fails to take my breath away.

We made it! Tired but happy children piled onto the little ferry......

and made their way back across the harbour.

What a wonderful day!