Project Math Access

Teaching Mathematical Concepts

Basic Number Facts and Operations

Collaborative and Inclusive Strategies

The Personal Perspective of Abraham Nemeth

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:

- Shape (square, circle, triangle, rectangle)
- Size (large, small, big, little)
- Weight (heavy, light)
- Length (short, long)
- Width (wide, narrow, thick, thin)
- Height (tall, short)

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:

- 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;
- matching, grouping and categorizing according to specific criteria; and
- classifying according to a variety of dimensions.

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

- Begin working on simple discrimination and matching with objects that are familiar to the child and that occur naturally in his or her world (e.g., shoes, toothbrush, squeeze toys, blocks, etc.), then move on to noting and analyzing specific attributes (e.g., shape, size); later, those specific attributes can be applied to naturally occurring objects in the environment (e.g., circle shape of a plate).
- Provide numerous opportunities for the child to handle and explore objects, note their critical features or attributes of shape, size, position in space, length, etc.
- Provide many opportunities for the child to match objects, and build groupings or sets of objects on the basis of specific attributes.
- Follow a logical or Piagetian sequence with regard to matching, grouping or categorizing, and later classifying: start with a single criteria or attribute by which to discriminate or group (e.g., shape/circle), change to a different criteria (e.g., small/large), progress to two attributes simultaneously (e.g., small circle), add additional attributes (e.g., small thin circle), and finally discriminate according to attributes NOT present (e.g., item that is not round, not small).

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.

- Involve children in daily living activities around the home or classroom. For example, helping to put silverware away in a divided tray with a sample in each section provides practice in matching, sorting and categorizing; helping to sort different sizes of towels or different items of clothing provides additional practice with these concepts.
- Give children numerous opportunities to use everyday items for matching and categorizing: eating utensils, grooming tools, foods, and toys for function; shoes and shoelaces for matching by size or length.
- To work on seriation, have children arrange boots belonging to family or class members from smallest to largest size; boots could also be arranged by height.
- The same type of activity could be carried out with other personal items such as belts of different lengths, books of different thicknesses, milk cartons of different sizes, or later with Unifix towers or Cuisenaire blocks. Students should not only identify the "extremes" of a series (e.g., longest or shortest), but also the "next shorter".
- Having family members or class members line up according to height can also help to facilitate understanding of seriation.
- Provide chances for children to work with the concept of conservation: give them a ball of clay and let them divide it into smaller amounts as they wish, and then combine the smaller shapes to demonstrate the constancy of amount.
- Using a sorting tray, place a variety of small items (buttons, paper clips, keys) in the larger section; to categorize, place one of each type of item in each of the smaller sections of the tray and have the child match and sort the remaining items; to classify, have the child form his or her own groups without providing a model. This activity could also be done using attribute blocks.
- Have children fold stiff fabric and paper to make different shapes. Squares can be folded to make triangles or smaller squares. Later, origami can be used to facilitate understanding of geometry.
- Children can explore shapes and size by building with Legos and Unifix blocks; they can also work with conservation by making a variety of different groupings from a given number of blocks.
- Have children copy simple shapes on geoboards; later they can make their own shapes based on names or clues such as "four corners", etc.
- Provide children with opportunities to explore and compare the three-dimensional shapes from Essential Geometric Forms which can be gotten from the American Printing House for the Blind.
- Have children walk, hop, run, jump through an obstacle course made from large shapes on frames, available from several children's catalogs, or arranged from items in the natural environment (e.g., jump 3 times in the circle, hop through the square, step in and out of the triangle).
- Use shapes, sizes, orders, patterns, planes, and eventually numbers in the real life environment (classroom, home) to teach concepts (e.g., compare the size of books to each other and to the size of tables, use corners of rooms to demonstrate angles, etc.).
- To practice positional ordering, have a student line up the rest of the children in a group, and then identify each as first, second, third, . . . last. Also have the student identify which child is before or after a particular individual, which one is next, etc. Children can also do the same activity by arranging toy cars or other manipulatives.
- Make a mathematical "pattern block" to enable students to build shapes and patterns with manipulatives that stay in place. To make the pattern block, drill ten or twelve evenly spaced holes into a long block (22 inches x 3 inches) such as the ones found in kindergarten block centers. Hammer thin wooden dowels or glue pieces of thick stranded wire into the holes, leaving about 2 inches; protruding up out of the block.
- Use magnet boards or felt boards for children to match shapes, size, position, order, and patterns; later, children can match numbers or form simple number statements to accompany the arrangement of manipulatives.

Assemble a collection of small objects that slide easily over the dowels or wires (e.g., beads of various sizes/shapes, washers, straws, plastic Unifix cubes, large paperclips, uncooked pasta, small pretzels). Students slide objects over the dowels in a left to right sequence to make a pattern (cube, cube, pretzel, cube, cube, pretzel, etc). The teacher can also start a pattern and have the student finish it. This device can also be used to teach ordinal number positions such as first, second, next, last.

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.