Manipulatives And Strategy Use
Lisa A. Grupe
Jodie Malmberg
Tanya Shunnarah
Laura Barnes
Lisa F. Huffman
Norman W. Bray, Ph.D.
University of Alabama at Birmingham
Presented at the 1998 Conference on Human Development in Mobile, Alabama.
Mailing address: Department of Psychology and Civitan International Research Center, SC 313, University of Alabama at Birmingham, Birmingham, AL 35294. Phone: (205) 934-9768, 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 on their fingers or with manipulatives (blocks, etc.) (Bray, Huffman, Ward & Hawk, 1994; Siegler & Jenkins, 1989). Manipulatives are widely used in elementary mathematics education because they provide a concrete, external representation of otherwise abstract numbers. However, a review of 15 studies on elementary math instruction comparing groups with and without manipulatives showed mixed results (Fennema, 1972a). These studies focused mainly on accuracy with little or no examination of 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 48 kindergartners. A pretest determined all children were equivalent on basic number knowledge and accuracy for simple addition problems. Twenty-seven of the 48 kindergartners participated in an experimental condition where they were randomly split into two groups: 13 who had manipulatives (40 small plastic bears) available to solve problems and 14 who did not. Children were tested individually twice per week for 12 weeks in the spring of their kindergarten year and received no instruction on strategy use or addition. Each session consisted of 12 addition problems: six small addend, three large addend, and three "challenge" problems (one addend > 10, the other < 5). The remaining 21 children participated in a control condition of the exact same design, except they only attended the first two sessions (in February) and the last two sessions (in May); ten children had manipulatives available while 11 did not.
Problems appeared on a computer monitor while the experimenter read them aloud ("How much is 3 + 5"?). After each answer, the child was asked how s/he arrived at the answer. Eleven categories of strategy use (see Table 1) were scored from videotape.
Overall, accuracy of children in the control condition was not significantly different from children in the experimental group (71% vs. 76%, respectively). By group, children without manipulatives in both conditions were remarkably similar in their accuracy (73% control vs. 72% experimental). However, children with manipulatives in the control condition demonstrated a significantly lower accuracy than their counterparts in the experimental condition (68% vs. 80%). Closer examination revealed that the major change came during the last two sessions (75% control vs. 89% experimental).
That children with manipulatives who participated in all 24 sessions achieved higher accuracy than children with manipulatives who only participated in four sessions suggests a practice effect in the manipulative condition only. Perhaps children who attended only four sessions never achieved proficiency with the manipulatives. In fact, while most children in the control condition continued to use manipulatives in the last sessions, children in the experimental condition stopped using the bears about halfway through the 24 sessions. In reality, then, the children in the experimental condition demonstrated strategy use more like that of children without manipulatives, and their accuracy resembled that of children without manipulatives. Other preliminary findings on strategy use show that though children in the control condition appear to choose more efficient strategies as time passes, they do not do so to the same degree as children in the experimental condition. It seems that the additional 20 sessions give the children opportunity to progress more fully through the strategy continuum presented in Table 1, suggesting a possible practice effect for strategy use. Further analyses on these issues is ongoing and will be fully included in the poster presentation.
| Strategy Continuum | Description of Strategy for "How much is 3 + 5?" |
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| 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. |
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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. |
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Miscellaneous Strategies | |
| Retrieval | Child describes solving the problem by saying, "I knew it". |
| Guessing | Child describes solving the problem by saying, "I guessed." |