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The child, at the Primary level, is introduced to mathematic concepts by using concrete, hands-on materials. Math concepts are accompanied in the practical life and sensorial areas by activities that stress the development of matching (one-to-one correspondence), sorting, and classifying. In the math area these concepts are enhanced by activities, which concretely present the concept of 1-to-10 and its numerical expression. Sandpaper numerals assist in the incorporation of this information and the appropriate language is added when the foundation has been established. In addition, students are introduced to the decimal system through hands-on materials called the Golden Beads. For example, the bead materials represent quantities such as 4,562 by using 4 cubes (containing one thousand beads each), 5 squares (one hundred beads each), 6 ten bars (ten beads each) and 2 units (single beads). The role of zero is illustrated with written numerals that are used with these materials. Concrete experiences with the mathematical operations may also be presented. Through the use of chains that represent the squares and cubes of the numbers 1 through 10, the children are introduced to skip counting with concrete materials. These materials increase and complexity and give the child a concrete foundation, which will help her better understand and remember abstract mathematic principles. All of these activities help to provide significant underpinnings for their later work in math.
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The Montessori math materials form the core of the math curriculum in the Lower Elementary classrooms. With these materials, the students are first introduced to math concepts to build an in-depth understanding of basic principles. Place value, linear counting and skip counting are reviewed; the functions of addition, subtraction, multiplication and division are re-presented with concrete examples using more abstract materials that encourage an understanding of fundamental process. As the children continue working with these materials, they begin to "discover" the abstract principles of mathematics for themselves, and they gain a new confidence in their abilities. In addition to activities that stress process, there are others that build toward the memorization of tables. The foundations for work with fractions are also established at this level.
An integral part of the math curriculum is the integration of math into other curricular areas. Real life problems are solved and practical applications are stressed so that students may come to understand the relevance of mathematics in everyday lives. Additional concepts such as time, money, graphing, measuring and estimating are also introduced.
The study of geometry that was begun in the Primary classroom on a sensorial level is expanded to incorporate the study of lines, angles and triangles through the use of hands-on materials. The exploration of spatial relationships provides the setting for learning the vocabulary of geometry.
Upper Elementary students continue to see new or expanded concepts such as common and decimal fractions, factoring, and the squaring and cubing of numbers through the use of materials. The materials also provide a means for the analysis of mathematical functions and the comparison of the relationship between mathematical operations. Students of this age are also introduced to the use of math texts or other abstract applications as a tool for reinforcements and solidification of previous work.
The use of math is incorporated into other work within the classroom whether calculating the distances on a timeline, determining how many bacteria will be replicated in a specified amount of time, or managing the money from lunchtime pizza sales. At this stage students also begin creating their own word problems and continue to solve those of others. They also incorporate estimation, averaging, and various forms of graphing.
Geometry studies review previous concepts and expand into the study of polygons and their analysis. Through the use of precisely designed materials, the students also explore the perimeter and area of plane figures and the volume of solids.
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