By manipulating the differently colored triangles to create new triangles of different types, the child gains tactile preparation for later geometry. 5) Fraction boards.:

Understanding fractions not as numbers but as spatially and sensorially meaningful 'partial' objects is reinforced through this activity.

6) Binomial and trinomial cubes:

Doubling and tripling as physical entities through manipulation of cubes first used to teach simple numbers helps build upon previous sequential learning of tactile concepts.

A g) Montessori materials for concept and symbols 1 to 10:

1) Number rods:

Red rods' which vary only in length, from one to ten centimeters in high, indicate variation in length and how numbers ascend in value from one to ten in a visually meaningful and observable fashion.

2) Sandpaper numerals:

Matching smooth and rough surfaces teaches discrimination and 'pairing' while tracing sandpaper numbers teaches how to write the numerals and reinforces student physical coordination.

3) Spindle box with introduction to "zero": The concept of zero, an abstract concept, is rendered visually and physically understandable in this activity.

4) Cards and counters:

Using cards and counters allows the teacher to ask what or who is missing. It empowers the child to find the necessary card, even if it is out of sequence. Counters reinforce notions of sequencing, which cards allow the child to visualize and then see when something is out of sequence in the line of cards.

5) Secret numbers and memory and number games:

Number games makes memorizing numbers fun and gives a child a sense of excitement of the unknown of a particular solution, and reinforces the need to remember numbers sequentially and nonsequentially in a way that gives numbers meaning.

6) Writing numerals:

Writing numerals, beyond their importance in math, helps students understand how concepts and written symbols relate to concepts like ascending or descending amounts, and again teaches coordination through manipulation of writing instruments.

Section III

List ways in which a mathematically prepared environment meets the child's developmental needs. Give reasons based on Montessori philosophy and developmentally-appropriate practice for your choices.

'prepared' environment:

Reinforces sensory concepts

Teaches concepts of identity...

Tasks must be fun and engaging to generate such concentration, and be relevant to children's lives.
Children need to eye- hand coordination to do math and to write word problems, so manipulating tactile numbers is important.

The scientific method and word problems, all require independent thinking and problem solving skills as well as the ability to write (Shu-Chen, Jenny Yen, 1995-1999, Math album: introduction).

Students from ages 0-3 first are prepared to learn math by observing, touching feeling, and understanding the 'tenses' of their lives in terms of the structure of their day.

Students from ages 3-5 learn more abstract concepts through concrete engagement like units of measurement

Then, when developmentally capable from years 5-7 the students began to understand mathematical relationships and units relate to previously observed ideas such as length, weight, volume. They must reach the neurological stage in which they are capable of understanding how objects retain their integrity, even when they change shape and form.

Section IV

List the correct sequence of the presentation of the Montessori materials for the direct teaching of mathematical concepts, indicating the approximate age of introduction.

In traditional instruction, teachers lecture, and correct student errors when students are responsible for replicating the material.

Montessori philosophy believes that "Children need to see, without hurry or pressure, how numbers change and grow and relate to each other," first by acting on objects, then modifying and transforming the mathematical objects, then understanding the process of this transformation (Shu-Chen, Jenny Yen, 1995-1999, Sensory Motor Index: Introduction).

Works referenced

Shu-Chen, Jenny Yen (1995-1999) Math album.

Retrieved January 11, 2009 at http://www.ux1.eiu.edu/~cfsjy/mts/math/_link.htm

Shu-Chen, Jenny Yen (1995-1999) Sensorial motor development index.

Retrieved January 11, 2009 at http://www.ux1.eiu.edu/~cfsjy/mts/sensor/_link.htm