Lisa A. Grupe, Lisa F. Huffman, & Norman W. Bray
University of Alabama at Birmingham

Presented at the 1996 Conference on Human Development in Birmingham, Alabama.

Mailing address: Department of Psychology and Civitan International Research Center, SC 313, University of Alabama at Birmingham, Birmingham, AL 35294. Phone: (205) 934-0657, FAX: (205) 975-6330. Send Internet email to: bray@cis.uab.edu

When children as young as kindergarten age are asked to solve simple addition problems, they externally represent the addends by counting on their fingers or by using manipulatives (blocks, etc.) (Baroody, 1992; Bray, Huffman, Ward & Hawk, 1994; Siegler & Jenkins, 1989). Manipulatives have been widely used in early elementary mathematics education because they provide a concrete, external representation of otherwise abstract numbers, thereby possibly improving performance. Conflicting findings about the use of manipulatives have been demonstrated in many studies (Fennema,1972b; Moody, Abell & Bausell, 1971). In a literature review of 15 studies on elementary math instruction comparing groups with and without manipulatives, Fennema (1972a) found that four reported significant differences favoring the use of manipulatives while one did not. Seven reported no significant differences and three reported mixed results.

Since the focus of previous studies has been on accuracy, no clear picture has emerged on how the presence of manipulatives may influence strategy use. The present microgenetic study (short-term longitudinal design with trial-by-trial observation of strategies) investigated the effect of manipulatives on accuracy and children's use of strategies while solving addition problems.

The subjects were 12 children (M_age= 6.4 years) enrolled in kindergarten programs of public schools in Birmingham, Alabama. At the beginning of the study, all 12 children were equivalent on basic number knowledge and accuracy for simple addition problems. The 12 children were randomly split into two groups of 6, those who had manipulatives available (forty small plastic bears) to solve problems and those who did not. Children were tested individually and given no instruction on strategy use or addition. There were two sessions per week for 12 weeks for a total of 24 sessions. Analyses for this study only compared results from the first four sessions (block one) to the last four sessions (block two). Each session consisted of 12 addition problems. There were six small addend problems (both addends 5), three large addend problems (one addend 5 and one between 6 and 9), and three challenge problems (one addend > 10, the other < 5).

Problems appeared on a computer monitor while the experimenter read the problem aloud ("What is 3 + 5"?). After each answer, the child was asked how s/he arrived at the answer. If no counting strategy was observed, the strategy was categorized based on the child's report. Using videotapes of the sessions, 19 categories of strategy use were scored with reliability greater than .90. There were nine counting strategies (including finger counting or counting aloud), eight of which had corresponding manipulative strategies (no corresponding strategy for counting with no fingers) with two remaining strategies being defined as retrieval ("I knew it") or guessing [see Table 1].

Overall, children with and without manipulatives did not differ in accuracy of problem solving (74% correct for both groups). There was a block x type interaction [F(2,20)=10.21, p<.001] with accuracy increasing most dramatically between blocks one and two for large addend (54% to 87%) and challenge problems (39% to 83%) and less dramatically on small addend problems (87% to 95%).

Of the 19 possible strategies, only 11 could be used if manipulatives were not available (8 strategies explicitly involved counting with manipulatives). Children with manipulatives used from 4 to 13 strategies, while children without manipulatives used from 2 to 9. Across blocks, 8 of the 12 children reduced the number of strategies used, always moving toward more advanced strategies.

Interestingly, the six children in the manipulatives group did not use manipulatives frequently (9% of all problems), and all manipulative use occurred in block one. All twelve children, however, used counting strategies (48% of all problems). Retrieval was used by 11 of the 12 children (34% of all problems), while guessing was used by 9 of 12 children (9% of all problems). For 11 of the 12 children, retrieval (the most advanced strategy) increased while use of counting strategies concomitantly decreased.

Results from this study showed that the presence of manipulatives did not facilitate accuracy, as children in both groups remarkably showed the same level of accuracy. Accuracy was related to the type of problem solved with the highest percentage correct demonstrated on small addend problems. However, by the end of the study, children were very accurate on all problem types: small addend, large addend, and challenge problems (95%, 87%, 83%, respectively). While children in both groups used a variety of strategies, manipulatives did not strongly influence the type of strategies used when they were provided, accounting only for 9% of strategy use on all problems.

This study does not provide support for the view that manipulatives provide extra support for young children learning to solve simple addition problems.

Baroody, A. J. (1992). The development of kindergartners' mental addition strategies. Learning and Individual Differences, 4, 215-235.

Bray, N. B., Huffman, L. F., Ward, J. L., & Hawk, T. M. (1994). A microgenetic study of addition strategies in young children with and without mental retardation. Paper presented at Gatlinburg Conference on Research and Theory in Mental Retardation and Developmental Disabilities, Gatlinburg, TN.

Fennema, E. (1972a). Models and mathematics. Arithmetic Teacher, 19, 635-640.

Fennema, E. (1972b). The relative effectiveness of a symbolic and a concrete model in learning a selected mathematical principle. Journal for Research in Math Education, 3, 233-238.

Moody, W. B., Abell, R., & Bausell, R. B. (1971). The effect of activity-oriented instruction upon original learning, transfer, and retention. Journal for Research in Math Education, 2, 207-212.

Siegler, R. S., & Jenkins, E. (1989). How children discover new strategies. Hillsdale, NJ: Erlbaum.

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STRATEGY	DESCRIPTION OF STRATEGY FOR "HOW MUCH IS 3 + 5?"
Counting Strategies
Sum	Put up 3 fingers, count "1,2,3". Put up 5 fingers, count "1,2,3,4,5". Begin counting again at 1, "1,2,3,4,5,6,7,8".
Verify One Addend	Verify one addend on fingers and then continue counting other addend without verifying. Verify "1,2,3", then continue on "4,5,6,7,8"
Hold Up Fingers as Unit	Hold up fingers as a unit first, then count fingers that are held up. Hold up three fingers, then hold up five, then count "1,2,3,4,5,6,7,8"
Successive Count	Count fingers successively by holding them up one-by-one while counting. As they extend fingers one by one, "1,2,3,4,5,6,7,8"
Representation Drop Out	Begin successive count from one, but as counting continues, child stops using fingers (drops out representation component).
Count from First Addend	Say "3,4,5,6,7,8" or "4,5,6,7,8", perhaps while putting up one finger for each count.
Min	Count from larger addend by saying, "5,6,7,8" or "6,7,8", perhaps while putting up one finger for each count.
Recognition	Put up 3 fingers, put up 5 fingers, say "8" without counting.
Count Without Fingers	Count from one without using fingers.
Corresponding Manipulative Strategies
Sum	Count out 3 bears, count "1,2,3". Count out 5 bears, count "1,2,3,4,5". Begin counting again at 1, "1,2,3,4,5,6,7,8".
Verify One Addend	Verify one addend with bears and then continue counting other addend without verifying. Verify "1,2,3", then continue on "4,5,6,7,8"
Count Out Bears as Unit	Pick up bears as a unit to represent each addend, then begin counting bears from one. Grab up three bears all at once, then grab up five bears at once, then count out five more, "1,2,3,4,5,6,7,8"
Successive Count	Count bears successively by laying them out one-by one while counting. As they lay out bears one by one, "1,2,3,4,5,6,7,8"
Representation Drop Out	Begin successive count from one with bears, but as counting continues, child stops using bears (drops representation component).
Count from First Addend	Say "3,4,5,6,7,8" or "4,5,6,7,8", while laying out a bear for each count.
Min	Count from larger addend by saying, "5,6,7,8" or "6,7,8", while laying out one bear for each count.
Recognition	Lay out 3 bears, lay out 5 bears, say "8" without counting.
Miscellaneous Strategies
Retrieval	Child describes solving the problem by saying, "I knew it".
Guessing	Child describes solving the problem by saying, "I guessed."