Contributions from: Marie Pupo and Tina Newman.
Abstract
There is considerable evidence that many intellectually disabled children have difficulty learning the basic operations of arithmetic. Our research has indicated that some such children are unable to learn the most basic operation, addition, without the use of concrete referents such as blocks, tokens or fingers. Many of these children are ashamed of their reliance on objects or fingers for addition and would rather guess incorrectly than show their dependence on such concrete referents. We have been studying the potential of a dot-notation approach to the teaching of addition. When adding, the student initially learns to count the positions of dots embedded in numerals from 1 to 9. With practice, the child learns to count the positions of the dots so that the actual dots can be removed. This approach can be adapted to the four operations and allows special needs children to appear to do arithmetic like everyone else which should make them feel more at ease and be more accepted when being mainstreamed into regular classes.
Much attention has been placed on integrating children with disabilities into the regular school system. To this end, knowing how to add is useful when attempting to adapt to the demands of the regular classroom. Several studies have indicated that many intellectually disabled children learn to add by following the same sequence of addition strategies that are used by their non-disabled peers (Cayho, Gunn, & Siegal, 1991; Hanrahan, Rapagna, & Poth, 1993; Irwin, 1991; Newman, 1994; Pupo, 1994).
Three basic addition strategies have been identified. The first of these has been described as a "count-all" strategy and is normally employed with the aid of concrete referents such as blocks or fingers. Using this strategy a child will typically solve an addition problem such as 2+3 by first counting out two blocks, then counting out three blocks and finally counting all 5 blocks. Children eventually develop the more sophisticated strategy of "counting-on". Here the child states the cardinal value of one of the addends and counts the other addend. For example, when adding 2+3 the child will say "two, pause, three, four, five". Some children notice that there is less to count if they begin with the larger addend, even when it is the second addend. Eventually the sums of various number combinations are committed to long term memory which is considered to be the third strategy.
Our research indicates that many children with intellectual disabilities spend a long time using the count-all strategy. Indeed, it appears that some such children will not progress beyond this basic strategy (Hanrahan, Poth, & Rapagna, 1993; Hanrahan & Newman, 1996). The count-all strategy requires the child to use concrete objects or fingers for addition and this becomes difficult for the child when number pairs with sums larger that 10 are added. However, it is possible to adapt the count-all strategy to the advantage of the intellectually disabled child. This involves a process of placing a dot(s) on the numbers 1 to 9. For example, the number 1 has a dot embedded at the top of the number while the number 2 has a dot embedded at the beginning and end of the number. To add 1 + 2 the child need only to count the dots. Using this system, the child is first taught to count numbers with the dots clearly marked at fixed positions on the numbers. In time the child memorizes the positions of the dots and the dots are removed from the numbers. The child then counts the dot positions rather than the actual dots. The dot-notation system was first developed by Kramer & Krug (1973) for the purpose of instructing children with intellectual disabilities in basic arithmetic. The system was revised by Bullock, Pierce & McClelland (1989) who have developed a curriculum based on dot-notation suitable for all children. Their program, which focuses on the 4 operations, is called Touch Math.
Utility of the Dot-notation Approach
There has been little research done on the usefulness of the dot-notation approach with intellectually disabled children; however, what is available is promising. Kokaska (1975) suggested that the method was appropriate for children with intellectual disabilities. Scott (1993) used an adaptation of the Touch Math program (Bullock, 1991) to teach addition and subtraction to a small group of mildly intellectually disabled students. Pupo (1994) had similar success when teaching addition to a group of 3 moderately intellectually disabled children ranging in age from 11 to 12 using an adaptation of the Touch Math method. At the beginning of her study all children were able to add single digit addends up to sums of 10 using concrete objects, fingers or tally marks. With one-to-one instruction on dot-notation, the subjects learned to add all possible combinations of single digit numbers up to the sum of 18. Newman (1994) also investigated the utility of the dot-notation approach with a sample of four Trisomy-21 Down Syndrome children ranging in age from 8 to 15 years. Her subjects were able to count objects to 10 and to recognize numbers to 10. She also used an adaption of the Touch Math program in an attempt to teach her subjects to add number pairs with addenda from 1 to 5. She reported that all subjects were able to learn the basics of the dot-notation system and learned to add the required number pairs. However, there was considerable variation in the time it took subjects to master the addition problems, the older subject meeting the objectives of the experiment in two weeks while the younger subjects required almost three months to complete the program.
Advantages of the Dot-notation Approach for Inclusion
The dot-notation approach seems particularly well suited for use by intellectually disabled children who are using the count-all strategy for addition. Such children must use concrete referents when adding. Usually these referents are fingers which makes the addition of number pairs with sums greater than 10 difficult or objects such as blocks, coins or tokens which are cumbersome even when adding small numbers. With the dot-notation approach, the child counts memorized dot positions which permits the addition of single digit pairs of numbers to 18 or single digit series of numbers as high as the child can count. Moreover, because this can be accomplished without the use of obvious concrete referents, the child appears to be doing arithmetic mentally like everyone else. Our research has indicated that it is very important for intellectually disabled children to appear like their non-disabled peers. We have observed numerous examples of intellectually disabled children preferring to guess when adding numbers rather than to count using their fingers or concrete objects (Hanrahan, Rapagna, & Poth, 1993; Poth. 1994). Pupo (1994) observed that 2 of her 3 subjects attempted to avoid tapping the numerals or crossing out the top numeral which are required steps when learning to add using the Touch Math method. Rather, they insisted on bypassing these steps and counting the dot configurations mentally. The dot-notation approach, then, allows intellectually disabled children to use the count-all strategy discretely.
For children who appear to be locked into the count-all stage of addition, the dot-notation approach may lead to a more mature understanding of addition than would be the case if the child were only required to memorize number pairs. Using the dot-notation approach the child is constantly reminded that the number 6 for example, represents 6 things. The dot-notation method permits activities such as partitioning of sets and the coordination of pointing and oral counting which are activities recommended for the understanding of meaningful rather than rote counting (Cayho, Gunn, & Siegal 1991; Gelman & Cohen, 1988; Yarmish, 1988). As well, if a previously memorized sum is forgotten, the child can easily regenerate the sum using this approach.
The dot-notation approach may be particularly well suited to children with Down syndrome as there is some evidence that these children have stronger visual than auditory skills (Hodapp, 1999). Dot-notation also lends itself to multisensory instruction which has proven to be effective with intellectually disabled children. Intellectually disabled children who quickly learn the basics of dot-notation may be able to learn double digit addition, subtraction and even multiplication and division. As the system can be easily applied to regular classroom materials, teachers have the option of using commercially available materials or developing their own instructional materials. In our opinion, the dot-notation approach is a promising addition to the repertoire of methods available to the teacher of intellectually disabled children.
References
Bullock, J.K. (1991). Touch Math addition kit (4th ed.). Colorado Springs: Innovative Learning Concepts, Inc.
Bullock, J.K., Pierce, S., & McClelland, L. (1989). Touch Math. Colorado Springs, Colorado: Innovative Learning Concepts, Inc.
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