1. Introduction
1.1. Despite mathematical permeation into our daily lives, many people fear math and believe that they are not “good” at it because of the way it is approached in schools.
2. Definition: What is Mathematics?
2.1. “The systematic study of number, shape, form, structure, relations, motion and change, and other concepts, represented by precisely defined abstractions; and the development and application of procedures for reasoning with those concepts”
2.2. Broadly defined as discovering patterns and relationships between entities.
3. The Mathematical Mind
3.1. A term borrowed from Pascal to describe the type of power our brain has to recognize patterns surrounding us and then to organize, classify and quantify them into our life experience
4. Mathematical mind and the developmental powers of the first plane
4.1. In combination with the Absorbent Mind, the Mathematical Mind of the child is actively storing information from the environment to be sorted later
4.2. In combination with the Human Tendencies for Exploration, Order, Exactness and Precision support the child to discover patterns and relationships amongst things and integrate them into what they have already learned
4.3. In combination with the child’s Sensitive Periods for Order and the Refinement of Sensory Perceptions is a great force driving the child to discover how to sort and group new discoveries spontaneously engaging the Mathematical Mind to classify and categorize the information. Filling up the mind like the shelves of a library
4.3.1. The result of the interwoven threads of this learning is the ability to form abstractions
5. History of mathematics
5.1. 30,000 years ago there is evidence of keeping track of seasons and other natural cycles
5.2. 4000 BCE as commerce grew in ancient Sumeria, so did the need for record keeping. The first mathematical systems had a base of 60 as opposed to the modern base 10
5.3. 3000 BCE Babylonians begin to use a true place-value system and the base 10 evolves
5.4. 400 CE India invents the use of 0 as a place holder
5.5. New tech means new math and we are developing new calculations and methods with great frequency. New discoveries are occurring rapidly and shaping our ever changing now
6. Why offer math to the young child?
6.1. Dr. Montessori observed that the children possessed a curious, seeking Mathematical Mind and strove to meet that need with her son Mario in the development of the math materials for the classroom
6.2. It was widely held that young children could not possibly be taught math as their brains were not ready for the concepts
6.3. Mathematics lead to the formation of abstraction and was the impetus for their inclusion in the casa
7. How math is offered in the Casa
7.1. Math begins as a very Sensorial experience in the Casa with tangible materials and a great deal of indirect preparation
7.1.1. “Math exists only in the realm of the mind, in the world of pure abstraction:
7.1.2. Traditional schools begin with the abstraction and confusing representation of quantity leaving the child to flounder and flop their way through early math experiences
7.1.3. In the Casa, we form the base of abstraction through offering the child a tangible representation of what the quantity is. The child will literally wrap their hands around the quantity and be able to fully abstract what it is to mean “five”
7.2. General Principles in Math:
7.2.1. We introduce new concepts, one at a time, with concrete, sensorial materials
7.2.2. The learning is progressive, with each concept building on ideas previously encountered, and then adding in a little bit more
7.2.3. Three period lessons are used to teach new vocabulary
7.2.4. Each child works at their own pace, taking as much time as they need to build their own abstractions
7.2.5. The child learns through exploration and discovery
7.3. Timing and Pacing
7.3.1. We wait to offer the first lessons in math until the child has reached a sufficient maturity to grasp the abstractions offered, usually around age 4
7.3.2. We must wait and watch patiently until the child has shown us that they have prepared themselves to be psychologically ready for the math works
7.3.3. It is crucial not to rush the child through any of the math areas. They must fully interact with the concrete materials before moving on, but we must also be ready to offer the next thing so the child does not get stuck in a slump
8. Indirect Preparations for mathematics
8.1. Practical Life
8.1.1. Logical sequencing, cause and effect, spatial awareness/relations, physics and gravity through pressure, measure and calculate quantities, and psychological preparations for future math work
8.2. Sensorial
8.2.1. Experiencing mathematically exact materials, matching, grading, geometry, prepares the mind for one to one correspondence, experience with the base 10 system, comparisons, accuracy and precision, and the friendliness with error
8.3. Language
8.3.1. Daily encounters with mathematical language. Numerical concepts through song
9. The organization of the Math Materials
9.1. General pattern of activity
9.1.1. Quantity in isolation
9.1.1.1. Concrete, sensorial representation
9.1.2. Symbol in isolation
9.1.2.1. Written symbols
9.1.3. Association o quantity and symbol
9.1.3.1. Bringing the two together
9.1.4. Repetitions, with variationst
9.1.4.1. Using the symbols and the concrete to practice the new concept in a variety of ways
9.1.5. The “test”
9.1.5.1. Gentle assessment so that the child and the guide can both assess whether the child has gotten it
10. Overview of the Groups
10.1. Group 1
10.1.1. The Numbers One to Ten
10.1.1.1. First experience with numerals and their quantities using concrete sensorial materials in the number rods followed with the sandpaper numbers
10.2. Group 2
10.2.1. The Decimal System
10.2.1.1. Children love and are fascinated by really big numbers so this group works with the concept of place value and manipulating numbers into the thousands
10.3. Group 3
10.3.1. Continuation of Counting
10.3.1.1. Exploration of number sequences greater than 10
10.4. Group 4
10.4.1. Exploration and Memorization of Essential Combinationss
10.4.1.1. Offered a variety of ways to explore the essential combinations for the four basic functions
10.5. Group 5
10.5.1. Passage to Abstraction
10.5.1.1. All the knowledge and experience gained thus far converges. At some point the child will gain the realization that their memory is faster than using the materials and will move into pure abstraction
10.6. Group 6
10.6.1. Fractions
10.6.1.1. Quantities less than one through manipulation of concrete sensorial materials