Mathematics is the basis of all practical things and the imagination. A child needs to understand mathematics in order to adapt and orient himself to his own time and place. Dr. Montessori borrowed the term ‘Mathematical Mind’ from Pascal, the French Philosopher and Mathematician. Pascal said that a man’s mind is mathematical by nature. He believed that knowledge and progress come from accurate observation. A mathematical mind is a combination of abilities:
· Curiosity leads to investigation.
· Interest in the world which is an affective quality that desires accurate observation especially in calculation and measuring (interest in an emotion). It’s an emotional state that begins at birth, and if nurtured, interest in the world grows with the person.
· Cognitive ability to reason and make judgements (cognition).
· Ability to make order out of chaos (remember us mentioning this in our community meeting about Sensorial?!)
· Language to help organize and classify data.
· Ability to move physically and mentally using repetition, exactitude and precision. Mathematics shows us how to achieve harmony.
· Ability to abstract from the above and the development of the imagination constructs.
As you might begin to notice, all preceding and intertwining activities the child experiences in the Children’s House is preparation for mathematics. Think of the one-to-one correspondence involved in setting the table each day for lunch. Recall the discrimination, recognition of similarities and differences, comparison and construction of series and finding relationships to understand exact terminology the child experiences in the Exercises for the Education of the Senses. The child has had innumerable opportunities to build a concrete understanding of the world around her which will aid in her ability to abstract.
The Learning Pattern for Mathematical Concepts
For Each new concept we follow the same 5 steps:
1. Sensorial, concrete experience associated with language
2. Written signs associated with language
3. Bringing together the concrete, sign and language
5. Test (a material for the child and adult to see what she knows)
An early example of the number progression is as follows: The exercises with the number rods piggy-back the Red Rods in Sensorial. They offer the same dimensions with a new mathematical concept for the child to discover. The child will discover the idea of quantity 1-10. Each quantity is represented as a whole with its own name. The Sandpaper Numbers represent the written symbol of the name, are presented separately and lend muscle memory for number writing. The number rods and cards bring quantities, symbols and language together. The child will later use the materials to prepare herself indirectly for addition, subtraction and multiplication. After work with number rods the child will work with quantities as separate entities with spindle boxes, cards and counters and memory game of numbers.
The Golden Bead Material shows the different categories of the Decimal System; units, tens, hundreds and thousands. The child sees, handles and uses language to gain a sensorial experience of the relationship between these different categories. The child will soon learn to read quantities conventionally through Formation of Numbers (i.e., one thousand, two hundred and twenty four; opposed to one thousands, two hundreds, two tens and four units). Next the child will experience all four mathematical operations, addition, subtraction, multiplication and division simultaneously. The lessons are presented in groups, are full of movement and directed by mathematical laws. It is the process of the operations that are important during this stage; not the answer. Once the child has had much practice with an operation, he will practice through an independent exercise called Stamp Game which is a step toward abstraction. The essence of carrying form one category to another is emphasized with Dot Game. The test of this area is in Word Problems for each operation.
Just like most of Montessori’s mathematic activities, counting, also is not thought to a child as an abstract exercise. After a child learns the decimal system, they learn the teen numbers through a combination of gold bars and units. The child gets the language of the numbers from 11-19 while later they can visually see the association of the numeric symbols and beads. This activity is called Teen Boards and the same concept applies to the next activity which is Ten Boards. These two activities give the child the opportunity to sensory see the amount (using beads) while they learn the name and symbolic form of numbers from 11-99.
Linear counting exercises come after when the child works with the golden chains. In the 100 and 1000 chain, the child becomes comfortable with the sequence of numbers from 1-100 or 1-1000. The counting, then becomes mechanical and with repetition, the child can count automatically from 1-1000.
The Chain Cabinet contains the square and cube of each number from 1-10. After the child works with the Gold Chains and gets comfortable with the sequence of counting by 10, the child gets introduced the Long chains which are the cubes. For instance when a child works with the cube chain of 6, he learns how to count the chain while skipping 6 units at a time.
Gradually the child realizes the different quantities in square and cube of each number. This method of counting also is an indirect preparation for Multiplication. When a child counts the square chain of 5, he literally sees by counting of the strain of 5 units 2 times, makes 10.
In Children’s House, “memorization” is embedded in many of Math activities. From Snake Game to Strip or Work Charts and Bead frames; this concept applies to all the mathematic operations. Although the materials themselves are designed to give the child a tangible experience but throughout the repetition, the child realizes that for example does not need to put the strips of 3 and 5 together in order to make 8. For instance, the Work Charts, either for addition, subtraction, multiplication or division, designed in a way to display series of operations with different ways for the child to discover the answers. While the following charts eliminates more information from the chart, makes it more challenging for the child to find the answers. At last the Chart is only given the necessary elements of the operations with no answers and here is where the child has to refer to his memory to remember the answer.
However, there is memorization towards the completion of these activities, the fact that the child can rely on his memory to find the answers is very satisfying and fun for him. The memorizations are not based on abstract concepts thus the child can indicate what he has been sensory exposed to earlier in order to analyze an equation.
Passage to Abstraction
Up to now, the sequence of introducing mathematics to the child has been the same: introduction to the quantity, introduction to the symbol and then the association of the two together. In the Passage to Abstraction, the child moves towards working with new materials, such as the small bead frame, that rely on his work with the four operations of the decimal system and the memorization work through the charts.
With this knowledge, the child begins to take an equation, calculate it abstractly and have the material confirm his calculations. He records his findings on notation paper that imitates the traditional way of documenting his work. This reinforces that there are no more than 9 in any category, the use of zero as a place holder for a category, and the importance of organization of the numbers on the sheet to keep each category of numbers in their correct column.
We then introduce the child to the wooden hierarchical materials in it’s concrete form. This allows the child to calculate numbers into the ten thousand, hundred thousand, and the millions. Once this is done, the large bead frame is introduced as another tool to work with addition, subtraction and multiplication and the racks and tubes which works with division.
When the child has a solid understanding of whole numbers, the fractions frames and the four mathematical operations with fractions are introduced in it’s most simplistic form. Throughout my training and parent meetings, I often hear “If only we were taught about fractions this way.”
Time: In the Children’s House we introduce time based vocabulary to the children. We discuss night and day; darkness and light; yesterday, today, tomorrow; season; and names of the days and months. The clock is also introduced to give the child the concept of how we measure the time of the day.
Science: In the Montessori Environment, science is presented to the older child in the Children’s House. They will better understand the concepts behind the science activity. Such activities include: Sinking and Floating, living and non-living, light and shadows, magnification, magnets, and prism for light refraction. We also introduce basic temperature and weather observations.
A Peek Inside our Children's House Classrooms
We can hardly believe that it's already Spring! The children have been absolutely thriving and we continue to celebrate their growth with them each day. They are respectful, courteous, cohesive and eager to learn from one another. We have had a few new students join us this month and it's heartwarming to observe the more seasoned children welcome them with open arms and hearts. If you happen to notice new parents at drop off or pick, please extend a friendly hand and smile!
The children were busy harvesting and exploring new recipes with the fruits and vegetables in our garden this month. We also prepped and planted new seeds and transplanted a few vegetable plants. The seedlings are already sprouting, it is true magic!
Thank you for your support and donations for the charity event. Our project and basket exceeded our hopes and dreams and your contributions make a difference for the children in our community!
Believing in the children as hope for tomorrow,
Ms. Lauren and Ms. Yadira
We spent a wonderful day planting!! The children were excited to get their hands dirty. However, the joke was on me when I was the one with the most dirt on my clothes. All the children had a wonder laugh!! The day was beautiful as we planted flowers and vegetables. Our tomatoes are still producing and the children are enjoying them at snack time.
A big thank you to all your support and donations for the upcoming auction. Our classroom art project, "When you wish upon a star" looks amazing. The children all had a great time creating their stars. The photo book that accompanies the art piece is so beautiful. It features the children creating the art piece. Our classroom basket, "Family Outings" is full of incredible activities for the family to do throughout the year. Hope to see you all at the auction.
Ms. Johna and Miss Eugenia
March has been a fantastic month as we welcomed spring and talked about Nowruz, the Persian New year, starting on the first day of spring. We have been enjoying discussing this month’s musician, “Chopin”. The children showed great interest in learning about a musician’s childhood and culture. We also got to play few pieces of Chopin’s work during our daily work time.Gardening had played a huge rule in our schedule in the past few days. The children had a chance to plant flowers, seeds and different vegetable plants in our garden. The vegetable plants were chosen purposefully to serve our food preparation activities.
The auction class project has also been a highlight of our events this month. We were proud to be part of Creo’s third Charity event and delighted to present our Copper Framed Mirror, embossed by the children at Bobcats.
Thank you to all who attended Creo’s Charity event, all the contributions, donations and to all the volunteers who made that magical night happen.