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Manipulatives are concrete objects meant to help children represent and understand abstract mathematical ideas (McNeil & Jarvin, 2007). Several researchers believe that the theory of embodied cognition can explain the mechanisms through which manipulatives are beneficial for learning. According to this perspective, it is through perceptual and physical interactions with the environment that various cognitive processes are developed and extended. As such, the sensorimotor information that concrete objects provide can guide the development of mathematical concepts (Barsalou, 2008; Nathan, 2008; Pouw et al., 2014).
With respect to mathematics instruction, the type of learning environment and instructional guidance might also play a role in how children interact with manipulatives, which in turn would affect their learning (Martin & Schwartz, 2005). Direct instruction, for example, could discourage children from exploring manipulatives, thus preventing the discovery of new information (see workshop: Bonawitz et al., 2018). The present study examined differences in the way children physically interacted with manipulatives as a function of instructional environment. We explored how children touched and manipulated concrete objects in three instructional contexts that differed in the degree of explicit instruction they were given on how to use the objects to solve mathematical problems.
Sixty-three first-graders met individually with a researcher. The objective of the session was for children to assign a quantitative referent of 2 to a novel manipulative called a “polly” (Figure 1). Children were asked to represent specific quantities with pollies. Ten cubes, for example, were represented by 5 pollies. Children were randomly assigned to one of three instructional conditions: (1) Direct Instruction (DI), where children were explicitly told that the referent for each polly was 2; (2) Guided Exploration (GE), where the researcher first let the children explore and decide on a referent for each polly before constraining the task with prompts that would lead children to “discover” that the referent was 2; (3) Control, where children were permitted to attribute the referent of their choice to the pollies with no feedback.
The analysis focused on how children interacted with the pollies during instruction. There were no condition differences in the proportion of time children touched and handled the pollies. Children in the GE condition, however, were more likely to interact with specific parts of the manipulatives than children in the other conditions, chi-square(2) = 18.22, p < .001 (Figure 2). Some children in the GE condition touched and counted two of the extremities of each polly, “one-two,” to show that the referent was 2. In contrast, most children in the DI condition did not attend to the pollies’ physical features and only touched them to keep track of their counts.
In sum, it appears that when children are given the opportunity to explore and make sense of a problem with manipulatives, they more readily take advantage of the physicality of the objects to interpret their meanings. We are currently analyzing additional data to assess the extent to which these observed qualitative differences impacted the children’s use of the pollies to solve novel problems.