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Basic Concepts

For students to understand and work with formal mathematical concepts successfully, they must understand the concepts of classification, conservation, seriation, ordering and one-to-one correspondence. Students must first work with and understand these concepts on the basis of quality (e.g., attributes such as shape, size, weight) before moving on to their application to general quantity (e.g., attributes such as many, few, none) and then on to number (e.g., attributes such as "fiveness", 100=10x10, 4+1=1+4.

In order for students to develop their innate number sense, and a working knowledge of the above concepts, they must have a great variety of interactions with their environment, exploring and manipulating, comparing, arranging and rearranging real objects and sets of objects. Many of these types of interactions and experiences occur incidentally for sighted children, but the blind child is at great risk for missing valuable and relevant incidental information. Therefore, it is critical that teachers and parents provide both structured and informal opportunities to handle and explore, note likenesses and differences, match, group and classify, order, and experience other relationships with real objects to prepare them for understanding the same relationships with numbers.

One of the earliest concepts to be developed is that of classification. Classification involves discrimination, matching, and grouping or categorizing according to attributes and attribute values. A sampling of these attributes and attribute values at the quality level follows:

At the quantity level, these attributes would involve general number concepts (e.g., many, few, more, less, none), and later, more specific number values (e.g., sets of 2, sets of 10, sets of values greater than 2).

The development of classification concepts involves several sequential stages:

  1. discriminating between same and different (note: if a child has difficulty with the dichotomy of same/different, the dichotomy of same/not same may be more effective to begin with); attention should be called to the critical features of objects and their attributes;
  2. matching, grouping and categorizing according to specific criteria; and
  3. classifying according to a variety of dimensions.

To promote the development of classification concepts, the teachers can:

Another basic concept that children must understand is that of seriation, or ordering objects, then quantities, and eventually numbers, according to specific given criteria. As with the concept of classification, the child must begin working in this area with real objects on the basis of quality (e.g., ordering family members' shoes or belts according to attributes such as length). Only then will the child be able to apply the concept to quantity (e.g., ordering jars of coins or chains of keys–one having many, one having several, one having few and one having one or none), and later to number (e.g., ordering the numerals 2,10, 3, 5). The concepts of classification and seriation can be taught in conjunction with each other very effectively. For example, after the child can match and sort according to size, he or she can work on ordering from largest to smallest.

In addition to the understanding of the concepts of classification and seriation, the child must develop an understanding of conservation-knowing that a given amount remains the same though its appearance may change. Also, as with classification and seriation, the concept of conservation must be developed first with real objects (e.g., a bowl of cake mix is the same amount as when it is divided into 12 cup cakes). This must be understood before a child can be expected to understand the "partners" that make up numbers (10=5+5, 10=7+3, 10=6+4), units of measurement and money (a nickel is the same amount as five pennies), fractions (one whole is the same amount as two halves or four quarters) or the associative principle (7x3 equals the same as 3x7).

In addition to the concepts of classification, seriation, and conservation, children need to understand basic spatial and positional concepts. For example, the concepts of top, bottom, around, middle, center, corner, line, straight, curved, next to, beside, are very relevant to basic mathematical understanding. Later, concepts such as diagonal, parallel, perpendicular, intersecting, angles, and rotating will be relevant. Positional ordering concepts are also critical for sorting, for seriation, and for working with sets; these include concepts such as first, second, third, next, last, before, and after. However, these concepts require basic counting ability.

When teaching any of these basic concepts, it is important to start with real three dimensional objects, progressing to two dimensional shapes or diagrams and finally to more symbolic representations. It is also advantageous to have students develop the ability to express their discriminations in complete sentences (e.g., "These are the same because they are both square," or "This is the longest belt.") because doing so helps them to focus their attention on the concept rather than simply naming a descriptor.

Activities for teaching basic concepts

The above described activities can be used to good advantage in helping young severely visually disabled children to lay the groundwork for understanding the fundamental concepts underlying the study of mathematics.