# Introduction: Introduction: Children often have a hard time understanding objects used as symbols. One reason is that they must think of the object both.

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Introduction: Introduction: Children often have a hard time understanding objects used as symbols. One reason is that they must think of the object both as an object in its own right, and as a symbol for something else. This ability is called dual representation. In teaching math, manipulatives are often used, but dual representation may pose challenges. If children cannot hold a dual representation of the object, they may have difficulty connecting the manipulatives to the ideas they represent. It is possible that the way children view a manipulative--as a toy or a mathematical representation--may influence the way they learn. We wanted to see if how we introduced the manipulatives could affect children’s views of manipulatives and their learning of the concept of mathematical equivalence. Manipulative: We used buckets and beanbags as the mathematical manipulative. Buckets represent two sides of equation Beanbags represent numbers Method: We introduced the manipulative (buckets and beanbags) in one of four ways. as toys to play with as math tools to solve equations as both toys and math tools as items (allowing them to form their own representation) Research questions: Does highlighting the manipulative’s representational use as a math tool help math learning: At posttest? Conceptually? At transfer test? Does highlighting the manipulative’s use as a toy hinder math learning? Participants: 59 children, 7-9 years old Pretest: Four equivalence problems (3 + 4 = 5 + _) Intervention: We introduced the buckets and beanbags in one of four ways. Participants then used them independently until three minutes had passed. Neither: Experimenter places 3 beanbags into each bucket while counting Toy Only: Experimenter tosses 3 beanbags into each bucket while counting Math Only: Experimenter places 3 beanbags into each bucket while counting Both: Experimenter tosses 3 beanbags into each bucket while counting and places 3 beanbags into each bucket while counting Lesson: The child was given a lesson using the manipulative to show how the equal sign is used in equations. Posttest: Four equivalence problems without the buckets and beanbags Conceptual understanding of equivalence: Statement sort Equal sign definition Transfer test: Four equivalence problems 3 + _ = 5 + 2 5 + 2 = 9 - _ Posttest/Conceptual understanding: Highlighting the manipulative’s mathematical function led to better learning. The math only condition appeared to be the most beneficial. As long as the mathematical use was highlighted, emphasizing the toy function did not help or hinder learning. Transfer test: Highlighting the manipulative’s function as both a math tool and toy resulted in the highest transfer performance. Participants were able to pull knowledge from two introductions instead of one. Focusing on both functions may have led to more flexible problem solving at transfer. Since most participants came in thinking of buckets and beanbags as toys, acknowledging the manipulative’s function as a toy may have decreased confusion. Implications: Educators should be thoughtful in the ways they introduce and use mathematical manipulatives. Introducing manipulatives, as toys to make math more fun, may not be the best way to help children learn. Limitations: Our sample size was the greatest limitation. Difficultly finding non-equivalent students old enough to participate Time frame for data collection was relatively small Future questions: With a larger sample size, might there be differences in learning between boys and girls? Are unfamiliar manipulatives more effective than familiar ones? We included participants who demonstrated little or no understanding at pretest (N=30). Participants were not evenly distributed across conditions. How does introducing manipulatives as a math tool vs. toy influence math learning? Posttest: Participants in the math conditions scored higher on posttest than those in the non-math conditions. The toy only condition showed the least learning at posttest. Conceptual understanding: Participants in the math only condition displayed higher conceptual knowledge of equivalence than those in the non-math conditions. Transfer test: Participants in the both- representations condition had the highest average score. For statistical analyses, additional data is needed. IntroductionDiscussionMethodsResults Representation and Math Learning with Manipulatives Kristjana Hrovat a, Martha W. Alibali b, Andrea Marquardt Donovan b a Madison West High School, b University of Wisconsin-Madison NeitherToyMath Both MATH TOYTOY

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