Project Math Access

Teaching Mathematical Concepts

Basic Number Facts and Operations

Collaborative and Inclusive Strategies

The Personal Perspective of Abraham Nemeth

In addition to the concepts discussed in the section on basic concepts, understanding the one-to-one correspondence of object to object is also necessary before the child can carry out meaningful counting and higher calculations.

Children can find many opportunities in their daily life to experience one-to-one correspondence. They can place one sock inside one shoe or one shoe on one foot; they can get one napkin or snack for each member of the family or class; they can place one lid on each of several containers; they can place pieces in one-piece puzzles.

Once children understand these relationships, they can link one number with one object and then count with understanding. “Rote memorization of a set of numbers is meaningless” (Moore, 1973, p. 67) and counting is a skill which should not be stressed until the child has shown understanding of basic classification, conservation, seriation and set comparison at both the quality level (attributes of objects) and the quantity level (general amounts in groups or sets).

When students are ready to develop the skill of counting, they can benefit from learning several counting strategies to increase their accuracy and efficiency. Students sometimes develop one or more such strategies on their own, but it is to their benefit to provide training in this area. As with any concepts or skills, it is important to start working with real objects and manipulatives and to continue providing these as learning aids.

Objects to be counted are often found in one of several types of arrays: linear, circular, rectangular, or random. The following steps can be helpful for young children in identifying the counting situation, organizing it, and keeping track of their progress as they count the items in the array.

- Scanning-The child moves his hands across the top of each item in the array to be counted, in order to obtain information about the objects and the general field over which they are spread. The child could also pick up and examine items and replace them in the tray.
- Organizing-If items are randomly displayed, the child can move all items to one side in preparation for counting. If items are already arranged in a linear fashion, the child can locate the first item in the series and scan to confirm the arrangement.
- Partitioning-The child can count individual items and move counted items to a separate area on the tray. The child could also pick up items one at a time, give them a name, and replace them apart from those yet to be counted. The child could also individually touch each item to be counted with one hand, giving each a numeral name, while the other hand keeps track of the next item to be counted.

When teaching counting skills, these suggestions might help:

- Pair word problems with calculations at the earliest levels, even if it involves only an easy oral “story” problem to go with sets being counted.
- As soon as possible, tie the use of manipulatives and oral counting and number statements to the representation of these numbers on paper with the brailler and on the abacus. Use manipulatives alongside these tools during transition to the brailler and the abacus.
- Provide the student with notes on basic number concepts; these can be kept simple with examples to illustrate. A small flip chart such as those available in teacher stores could be labeled in braille.
- Modify a braille meter stick by covering it with clear braillon; re-label it with 0 in the middle, 0-50 going to the right, and negative numbers moving to the left. A similar modification could be made to a raised line ruler.

Activities for teaching counting

- Have the child compare/match/sort groups of objects into sets; then have him or her identify the number of items in each set, expressing them by name and by some pattern (e.g., clapping or ringing a bell the same number of times as the number in the set).
- Use counting songs and fingerplays to practice counting forward, backward, by twos, by fives, by tens, etc.
- Using the brailler, have the student count spaces to get to the bell, starting from different points along the line; the student can also depress full cells to correspond with a particular number.
- Have the child count objects aloud as he or she individually drops them into containers; start by dropping one item at a time, then two at a time, and so on.
- Keeping track of game scores can be a motivating and relevant way of applying counting skills. For example, the child can count the number of points earned by individuals in a card game, or in a ball game.
- Record specific directions on tape for the student’s independent practice. For example, using a tray with dividers, the student could place a certain number of items in the first section, a different number of items in the second section, and so on. Students could also be directed to place a card with the correct Nemeth Code symbol in each of the sections to correspond with the number of items.
- Have a “counting scavenger hunt.” Tell the child the location of several containers of objects (depending on the student’s memory, he or she could be given one location at a time, or several at once). The child must go to the locations, obtain the container of objects, count the number in the container, and then order the containers in correct number sequence. The student can then count up all the items for a grand total.
- Use a “talking tablet” device with overlays containing rows of tactual dots and shapes; program the device to speak the number of the shapes sequentially from left to right or top to bottom. The child touches shapes in the correct sequence and receives reinforcement as to the number place in sequence. The child must confirm the correct sequence of numbering (good for a student who has limited fine motor ability).
- The development of an autobiographical timeline (in cooperation with a student’s family) requires the student to actually plot significant events sequentially. This provides concrete reinforcement of number line concepts and the value and sequence of numbers, in a personally relevant and interesting format.
- Students can play a game called “Guess My Number” (Petreshene, 1985), to reinforce and practice the concepts of “greater than” and “less than.” Braille a numeral between 1 and 100 on a piece of paper without informing the student of the number. Ask the student to discover the secret numeral by asking “greater than” and “less than” questions, keeping track of the answers by recording them in braille. For example, if the teacher has chosen the number 19, the student might ask if the number is greater than 10; he or she would then record, “>10.” The next question might be whether the number is 20; he or she would record “<20.” Interim guidance can be provided if necessary; for example, the student could be told that now he or she knows the number is somewhere between 10 and 20. Roles can be switched, with the teacher guessing the student’s number. The score is kept by entering a tally for each guess; the person with the fewest tallies (guessing the secret numbers in the fewest attempts) wins the game.

These types of numeracy readiness activities can be used to lay a solid foundation for a greater understanding of mathematics.

Moore, M. (1973). Development of number concept in blind children. *Education of the Visually Handicapped*, V, 65-71.

Petreshene, S. S. (1985). *Mind joggers! 5 to 15 minute activities that make kids think*. West Nyack, NY: The Center for Applied Research in Education, Inc.