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

Presented at the 1996 American Psychological Science Conference in San Francisco, California.

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 kindergartners are asked to solve simple addition problems, they often 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. The previous studies have focused mainly on accuracy with little or no examination of strategy use. Thus, no clear picture has emerged on the effect of manipulatives or how they influence strategy use. The present study investigated the effect of manipulatives on accuracy and children's use of strategies while solving addition problems.

The subjects were 27 kindergartners (M_age= 6.4 years) in public schools in Birmingham, Alabama. At the beginning of the study, all children were equivalent on basic number knowledge and accuracy for simple addition problems. Children were randomly split into two groups: 13 who had manipulatives available (forty small plastic bears) to solve problems and 14 who did not. Children were tested individually and given no instruction on strategy use or addition. Two sessions per week for 2 weeks were conducted in February (block one) and May (block two) of the children's kindergarten year. Each session consisted of 12 addition problems: 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, 12 categories of strategy use were scored with reliability greater than .90. There were 7 strategies which involved representation (including 4 finger counting strategies and 3 manipulative counting strategies), 4 strategies which involved no representation [i.e., counting aloud, retrieval ("I knew it")], and 1 guessing/no response strategy (see Table 1). [Note: Children who had manipulatives available could use all 12 of the categories, while children who did not have manipulatives were not able to use the 4 manipulative counting strategies.]

Overall, children with manipulatives available achieved significantly higher accuracy [F(1,25)=4.25, p<.05; 79% manipulatives available (MA) v. 69% with no manipulatives (NM)]; however there was a block x manipulative interaction [F(1,25)=4.47, p<.04] with the two groups only different at block one (68% v. 52%, MA v. NM, respectively). By block two, accuracy of the two groups was remarkably similar (89% v. 86%, MA v. NM).

A main effect of problem type was also found [F(2,50)=37.57, p<.001] in which accuracy decreased as problem type became more difficult (i.e., small addend to challenge). A three-way interaction of block x type x manipulative was also significant [F(2,50)=3.35, p<.04] indicating that when manipulatives were available, children's accuracy in block one was higher in magnitude on all problem types than when manipulatives were not available; however, by block two, children's accuracy was again remarkably similar (97%, 89%, 81% for small addend, large addend, and challenge problems for MA group v. 96%, 83%, 79% for NM group).

Eight of the 11 scorable strategies (excluding guessing/no response) had a significant main effect of block and/or type; of those, 5 had a significant block x type interaction. A block effect shows that strategy use either decreased or increased as children moved from block one to block two. For example, children who counted from one with fingers or bears were more likely to do so in block one, while children who counted from one of the addends (as a shortcut) or used retrieval were more likely to do so in block two. Strategies utilizing representation were used more frequently in block one, and there was a concomitant increase in the use of strategies not requiring representation ( i.e., retrieval) in block two. A type effect shows that whether a problem was small addend, large addend or challenge influenced the frequency with which a particular strategy was used. In general, for more sophisticated strategies, the frequency of strategy use increased as problem difficulty increased, while for less sophisticated strategies, the frequency of strategy use decreased. However, retrieval (a very sophisticated way to solve a problem) was the most frequently used strategy and followed a somewhat different pattern, being used most often on small addend problems (38%) but also used 25% of the time on both large addend and challenge problems.

Results from this study showed that the presence of manipulatives influenced accuracy, but only in block one; by the end of the kindergarten year, children with or without manipulatives were highly accurate in solving addition problems. The type of problem and whether it occurred in block one or block two seemed to have a strong influence on strategy use, whereas the presence of manipulatives did not.

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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."