Introduction
Using concrete materials to teach mathematics is a long-established pedagogical strategy (Brownell, 1928). Based on theories claiming that children need concrete referents to develop abstract mathematics concepts (Piaget, 1966), and supported by research showing qualified advantages from using concrete materials (Sowell, 1989), educators have advocated using manipulatives for instruction (Burns, 1996). Other researchers and educators have argued that these materials are not automatically helpful (Ball, 1992; Resnick & Omanson, 1987). In general, research on manipulatives has focused on whether they were generally good or bad for instruction. We doubt that hands-on materials are beneficial or harmful for learning mathematics overall. Instead, it is likely that the effects of materials are more qualified and specific. In this paper, we report on work examining how using different manipulatives affects student learning in the domain of fraction addition. We found that working with particular materials helps students scaffold their learning in new situations.
The resurgent controversy surrounding the use of manipulatives makes this timely research. Following the upswell of support for manipulatives in the past two decades, new mathematics standards and textbook adoptions in some states (e.g., CA), emphasize concrete models less and algorithms more. Research on how manipulatives affect math learning could help teachers make decisions about their use in instruction.
There are multiple perspectives concerning how manipulatives help students learn mathematics, though little evidence firmly supporting any one view (Chao et al., 2000). One idea is that exposure to multiple representations leads to better understanding of underlying mathematical principles (Moreno & Mayer, 1999). This view implies that using many different manipulatives to teach a mathematics concept is the best instructional strategy. Another hypothesis is that it is the symbolic aspect of manipulatives that conveys any benefit they have (Uttal et al., 1997). From this perspective, useful manipulatives have structures that mirror the semiotic systems they are meant to represent, such that each action on a manipulative corresponds to a semiotic action, one-to-one. Another view is that external resources primarily help problem solvers keep track of problem elements without wasting internal memory resources (Cary & Carlson, 1999).
Our view is slightly different. We are looking at manipulatives as quantities that students structure through their activity to scaffold their own learning. Students build partial structures capitalizing on quantitative properties of the manipulatives, and in an iterative process, these partial structures can support the emergence of new structures students had not anticipated at the outset. For example, students who need to make 1/3 of 9 tiles might start grouping the tiles into piles. Given the partial structure of the groups, they might try to make the groups the same size, and given groups of the same size, they might further recognize that they can make three groups, notice 3 is one of the numbers in the problem, and so on. Ideally, given proper success and appropriate manipulatives, students may move to a new physical situation and manipulate it to scaffold their learning as well.
We will report on two studies of upper-elementary students using manipulatives to learn fraction addition. In these studies, students solved fraction addition problems with different types of materials and kinds of training. We hypothesized that learning with active structuring materials -- ones that required them to structure their environment to support learning --would prepare students better for a new situation than passive structuring materials that blocked this practice. In the context of the part-whole relationships of fractions, individual square tiles are active structuring while fraction pies are passive structuring.
Pies provide the whole as part of their spatial layout, whereas with tiles, students need to structure a whole from the pieces. We found that when students learned with the active structuring materials, they were prepared to work with new materials better than students who learned with the passive structuring materials.
Methods
Study 1
Over three days of guided discovery, students learned fraction addition with one of two materials, tiles or pies (the learning phase), and then tried new materials (the transfer phase). We hypothesized that learning with the active structuring material, tiles, would help students learn to solve problems with new materials better than learning with the passive structuring material, pies.
Sixteen students from three fifth-grade classes at the same school participated. We chose students from the middle of the score range on a fraction test and randomly assigned them to the tiles or pies condition. Students in the two groups had similar math achievement scores.
In the learning phase of the study, half of the students used tiles and half used pies. Two of the transfer materials, bars and beans (See Figure 2), were the same for all students. The third transfer material differed depending on the material used in learning. Students who learned with tiles used pies and vice-versa.
In the learning phase, students worked individually with an interviewer for 20 minutes on each of three days to learn to solve problems that were presented symbolically (e.g., "what is 1/2 + 1/4?"). We taught a counting method that asked them to build numerators and denominators (See Figure 3). Six problem levels progressed from same denominator fractions and a sum less than one to different denominator fractions in which the denominators were not multiples of each other and a sum greater than one. Different students learned to different levels over the days.
A subsequent transfer phase on each day lasted 10 minutes and tested students without feedback on problems at the level they reached during the learning phase. Students first tried to solve problems mentally and then with the other materials.
This study employed a microgenetic design with the between-subjects factor of learning material (tiles or pies), and within-subject variables of day and transfer material. One dependent measure was highest problem level reached in the learning phase each day. Measures during transfer included accuracy and fraction model. A response was coded as accurate if the student said the correct answer and built a workable model of the problem with the materials. We created four categories of these models (See Figure 4). In the Part Model, students model only parts, they do not represent the whole. In the Separate Model, students represent the right numbers for the numerator and the denominator, but not the relation between the two. In the Individual Model, students accurately model single fractions, but not the relation between two fractions. Finally, in the Integrated Model, students represent both fractions and their relationship.
Results
Students in both conditions learned to do fraction addition problems with their primary material. The groups did not perform significantly differently as measured by highest problem level reached in the learning phase (All reported results are significant at p < .05). In the transfer phase, students who learned with tiles were more accurate at using new materials to do fraction addition than the students who learned with pies (See Figure 5). These students could not do these problems without resources - they were actively using the materials. Students who learned with tiles also made more advanced models than the pie students in the transfer phase. The tile group made mostly integrated models with the new materials, whereas the pie students made mostly part and separate models (See Figure 6).
Both groups were taught to make the same integrated models with the counting method, but the tile group made more of those integrated models in transfer. Even though the pies group counted out the right number of pieces for the whole repeatedly in training, we think they relied on the visible whole because when they did problems in the transfer phase with materials that did not show wholes, they could not represent it. They made part models. The tiles do not have a spatially built in whole. Presumably, the students using them learned to make a whole.
Our favorite explanation for this result is that the tile group actively structured wholes. Alternatively, the counting method may have hurt the pies students. To rule out this possibility, in a second study we taught the pie group with a method that capitalized on the whole represented with the pies, the spatial method. The tiles students still learned the counting method. We expected that if active structuring helps students learn, the tiles students would again do better than the pie students when using new materials.
Study 2
We taught students to do fraction addition problems with either tiles or pies using methods tailored to each material. The tiles group learned with the counting method. The pies group learned with the spatial method (See Figure 7).
Two fourth-grade classes participated in this study. We split each class in half and switched rooms so that half of each class was in a classroom learning with pies and half was in another classroom learning with tiles simultaneously. Two experimenters taught the lessons. For three days, the students worked as a whole class on same and different denominator fraction addition problems with manipulatives and recorded their results on paper. On the fourth day we interviewed each child individually (32 students participated in the interviews). The posttest interview asked students to solve fraction problems with the material that they learned with and the material the other group learned with. We compared the groups′ accuracy and models.
Results
When working with their learning material, the pie students were more accurate (See Figure 8) and made more advanced models (See Figure 9) than the tile students. But, when asked to try a new material, the students who learned with tiles were more accurate and made more advanced models of fractions. The spatial method did seem to help the pies students work with pies, but did not help them structure their learning in new situations. Even though the tiles students looked worse at learning, they performed better at transfer. These results support the idea that materials that help children scaffold their ability to structure important mathematical elements of the situation are more helpful than materials that pre-scaffold learning by building in those features.
Conclusions
When students used tiles to learn, they did not just learn to do the problems well with tiles. They learned to structure their activities in adaptive ways to learn in new situations. Learning with the pies helped students do fraction problems but did not help them prepare for new situations.
We do not want students to use manipulatives to do math forever. Students in these studies had not yet reached the point where they could solve these problems reliably without resources. The ways that we as educators can help students use concrete experiences to develop understanding and move toward mastery are not clear. These studies suggest that manipulatives may be helpful to students because they prepare them to learn in new situations. Structuring your environment may play an important part in learning – you are not just learning to do a certain problem but learning to make an environment conducive to solving kinds of problems. We need to examine further how this structuring shapes the development of mathematical concepts over time.
References
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Uttal, D. H., Scudder, K. V., & DeLoache, J. S. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. Journal of Applied Developmental Psychology, 18, 37-54.
Paper presented at the American Educational Research Association Conference in New Orleans, LA, April 2002.
Last modified: April 7, 2002