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The Importance of Weaving Alternative Approaches

Research supports that concepts are developed over a long period of time and through many different approaches. Convincing others that mathematics is more than a correct answer can be challenging! Careful descriptions of suggested activities will help you demonstrate and communicate this vision.

Activities are suggested throughout on these basic approaches:

The choices of activities are planned to weave together to build understanding of each of the skills.  The more approaches the greater the depth.


Once man learned to record the number of countable items, he found a need for whole numbers. Numeration is the way in which man represents his numbers. Numerals are the marks we make in order to preserve, record and share our numerical ideas. Children need to know that there are many ways to write numbers. They need to know and understand our base ten system of numeration, and they should realize that this system probably resulted from the fact that man has ten fingers. As they attempt to devise their own methods of recording number, children develop and appreciation for the preciseness of our decimal system.  This appreciation is further enhanced as children study number basis, other number systems, and various ways of recording number facts.

Numbers are extended to fractions, decimals and irrationals. Our study must include attempts to determine if these "new" numbers behave in the same manner as the counting numbers.

It is through examination of models, manipulation of objects that represent number, playing number games, stacking and unstacking, that we discover the algorithms or ways for computing numbers. It is possible to memorize answers provided by others, but this often leads to boredom and produces the kind of learning which does not remain with us. If the rules are discovered by the child, they are internalized and become personal, and therefore permanent.


Patterns exist in every dimension of learning. Recognizing and using them becomes a powerful tool for learning. There is no area in which the study of patterns is as fundamental as in mathematics. Developing the power to record, analyze, and generalize patterns depends on the understanding of relationships and constructs mathematical knowledge that goes beyond simple computation.

Patterns can be linear like jumps on the number line or two dimensional like "growing shapes" and three-dimensional as growing solids and multidimensional as illustrated with fractals on high speed computers.


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