Because children utilize a myriad of different strategies to problem solve, the intention of my manipulative is for children to have an additional tool that they can use to help them work through equal share problems. My manipulative will serve as a visual representation of groups and how they can distribute whole, half, third, and quarter pieces to groups. Since some children have non-anticipatory coordination between groups and shares, my manipulative will be a way for children to utilize the pieces to see the distribution of shares to each group. They will be able to move pieces around to see the dynamics of how choosing different pieces will affect the groups and the shares. Additionally, equal sharing problems “allow children to create and manipulate fractional amounts and to reflect on the relationship between the numerator and denominator” (Empson, 2011, p. xxv) so children will be able to utilize my manipulative to begin gaining the concept of numerator and denominator which will demonstrate their understanding of parts, wholes and parts of wholes. Even without a specific word problem to work through, children will have the opportunity to tinker with the board and the pieces to find the relationships between the pieces and the groups.
I decided to make an 8×8 grid where children can use pieces to “fill in” or represent equal share problems. The grid is slightly raised and each square is hollow to allow for pieces to be inserted into each square. I decided that after the product (which is white) is printed, I will color in each “box” of four squares with a color. For example, the first “box” of four squares will be black, the second “box” of four squares will be pink, the third “box” of four squares will be black and so on so that children will have 16 groups (each delineated by being either black or pink). From there, children will have a variety of pieces from which to choose. One “whole” piece represents “one whole,” one half piece represents “one half of a whole,” one third piece represents “one third of a whole,” and lastly one quarter piece represents “one quarter of a whole.” Each of the pieces will be sized so that children can see the differences between the pieces to demonstrate larger and smaller units (ie: the one third piece will be bigger than the one quarter piece) and how they relate to one another. They will learn how parts combine to equal one whole.
Because children solve problems utilizing different strategies and models of representation, this manipulative will be one tool that students can decide to use while working through equal share problems. As the teacher, it will be my responsibility to properly launch the tasks and ask pointed questions where appropriate. However, I want students to work through their problems utilizing the modes and methods that are most helpful to them. Children will be encouraged to work together to discuss rationales for their work and to learn from one another. This manipulative will allow for great conversations as students work to discuss why they decided to use the tool, how they used it and what they learned from using it.
Since “equal sharing problems where the answer is greater than 1 are easier for young children” (Empson, 2011, p. 10), this manipulative will allow students to work through such problems. Problems will be designed to include a total number of items to evenly distribute. Additionally, “solving and discussing equal sharing problems prepares children to learn from a variety of other problems involving fractions and decimals” (Empson, 2011, p. xxi), therefore this tool will help to prepare children for future concepts. The manipulative (and other representations) will provide insight as to how the child interprets and thinks about equal sharing. The manipulative will also allow students the opportunity to visualize their thoughts and to make sense of their processes; the tool will be another way that students can communicate their ideas and provide rationale as to how they arrive at their solutions. Similarly, since equal sharing problems build off multiplication and division (partitive more specifically), I will present this manipulative after the concepts of multiplication and division have been introduced and taught.
The success of the manipulative will mean that children can make sense or are working toward understanding what it means to equally distribute shares. The success of the manipulative will mean that children have or are working toward understanding the difference between a whole, half, third, and quarter and how those shares when manipulated can equal one whole. The success of this manipulative will mean that children have the ability to utilize different mediums to work though problems and will be able to explain their thought processes based off their work on the manipulative.
If my design failed to work, I would garner information from children and teachers as to why they did not like working with the product. In efforts to improve the manipulative, I would want to know if the design, the aesthetics and/or the functionality were the reason why they preferred to utilize other methods over my manipulative, and I would ask for suggestions for improvement.
For testing, I can imagine presenting children and teachers with this manipulative coupled with some equal share problems. The reason why I want to test this product with teachers as well is because I believe teachers need to be comfortable utilizing the manipulative prior to presenting this tool as an option for students. I can imagine observing to determine what types of tools they use to problem solve as well as the strategies they use to solve said problems. I can imagine wanting to determine if my vision for the use of the manipulative is how children and teachers actually utilize it. I also want to determine if the users thought any part of the design was confusing, and if they had any suggestions on how to improve the manipulative. I want to know if they prefer utilizing the manipulative over other methods and the rationale for their decision. I want to determine if children and teachers solely utilize the manipulative, another method or a combination of both. I can imagine asking questions as to any ideas to better improve the product. Ideally, after the test and receiving feedback from the children and teachers, I would make any necessary changes and have the children and teachers utilize the new model for their feedback on the enhanced product.