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Enriching student concept images: Teaching and learning fractions through a multiple-embodiment approach

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Abstract

This study investigated how fifth-grade children’s concept images of the unit fractions represented by the symbols \( \frac{1}{2} \), \( \frac{1}{3} \), and \( \frac{1}{4} \) changed as a result of their participation in an instructional intervention based on multiple embodiments of fraction concepts. The participants’ concept images were examined through pre- and post-teaching written questions and pre- and post-teaching one-to-one verbal interview questions. Results showed that at the pre-teaching stage, the student concept images of unit fractions were very narrow and mainly linked to area models. However, after the instructional intervention, the fifth graders were able to select and apply a variety of models in response to unit fraction tasks, and their concept images of unit fractions were enriched and linked to capacity, perimeter, linear and discrete models, as well as to area models. Their performances on tests had improved, and their conceptual understandings of unit fractions had developed.

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Acknowledgments

The authors would like to thank Mr. X and his students for their enthusiastic participation in this study. We would also thank Dr. Jeffrey Barrett and Dr. Jennifer Tobias for their support and constructive advice.

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Correspondence to Xiaofen Zhang.

Appendices

Appendix 1

The list of all category A responses (by grade 5 students) at the pre-teaching stage

  1. 1.

    “Erin and Jonathan have eaten \( \frac{4}{8} \) of a cherry pie. Then Levi and Jessica eat \( \frac{1}{8} \) of the pie. Finally Keegan and Hayley eat \( \frac{2}{8} \) of the pie. How much pie is left?”

  2. 2.

    “Mark ordered a pizza for his friends and him. They ordered pineapple on it. Mark ate \( \frac{1}{4} \) of the pizza. His 3 friends ate the rest. Mark and his friends were stuffed after they ate”.

  3. 3.

    “Once upon a time there were three kids named Billy, Joe, and Timmy. It was Timmy’s birthday and his Mom bought him a cake. The kids all fought over who would get the most birthday cake. Then Billy figured out that they could each have one third of the cake. And they lived happily ever after”.

  4. 4.

    “You could write about \( \frac{1}{6} \) is not needed any more or there is no circle to go with \( \frac{1}{6} \) so she had to go with the square”.

  5. 5.

    \( \frac{1}{6} \) is equivalent to \( \frac{2}{12} \) or \( \frac{4}{24} \) or \( \frac{8}{48} \) or \( \frac{16}{96} \) or \( \frac{32}{194} \) or…” [The accompanying pictures are shown in Fig. 13.]

    Fig. 13
    figure 13

    A story for \( \frac{1}{6} \)

    Responses regarded as belonging to category A (continued)

  6. 6.

    “Sally had a pizza that had 4 equal slices. She wanted to share the pizza with 2 of her friends. All of them ate 1 slice. How much pizza was left?” [The accompanying picture is shown in Fig. 14(a).]

    Fig. 14
    figure 14

    Pictures that students drew to illustrate their stories for \( \frac{1}{4} \) (ac)

  7. 7.

    “There once was a number called \( \frac{1}{4} \). He had a friend named zero. Zero didn’t like \( \frac{1}{4} \) of him because it was covered in freckles and the rest of him wasn’t. But, \( \frac{1}{4} \) was like zero’s \( \frac{1}{4} \) of his face without the freckles. So zero asked \( \frac{1}{4} \) to be the last part of his face. \( \frac{1}{4} \) said he’d do it. Now, \( \frac{1}{4} \) won’t ever get lonely and will always be living on his friend’s face”. [The accompanying picture can be seen in Fig. 14(b).]

  8. 8.

    “Sam drank \( \frac{1}{5} \) amount of water how much water would be left [?] \( \frac{1}{4} \) amount of water would be left” [see Fig. 14(c)].

Appendix 2

The list of all category C responses (by grade 5 students) at the pre-teaching stage

  1. 1.

    “Mary needed about \( \frac{1}{4} \) of [a glass of] water but she got a whole glass of water So Tom subtracted \( \frac{3}{4} \) and he got \( \frac{1}{4} \) of [a glass of] water”.

  2. 2.

    “Nathan and Ashton went to buy 9 apples. It cost five dollars. When they came back to the Danty, Ashton ate \( \frac{1}{3} \) of the apples Nathan ate \( \frac{1}{3} \) of the apples, so together they ate \( \frac{2}{3} \) of the apples. Koletin sneaked over there and stole \( \frac{1}{3} \) of the apples so now all the apples are gone”.

  3. 3.

    “I would tell her: You walk downstairs and get a treat. You eat \( \frac{1}{6} \) of the treat. You leave it out on a plate for the next day. You see your dog eat \( \frac{1}{6} \) of your treat. Then you eat another \( \frac{1}{6} \) of your treat. You eat the rest of your treat and then you walk back upstairs to watch TV”.

  4. 4.

    “One sixth can be changed into different numbers. You can add subtract multiply and divide to it. You can change it to \( \frac{2}{12} \) or \( \frac{1}{6} \) or 0. You can do anything”.

  5. 5.

    “Once long ago \( \frac{1}{4} \) was a 0, because he wasn’t born yet of course. There were already 3 family members in his family. Then when he was born he had become the fourth person in their family. That is when he got his name: \( \frac{1}{4} \)”.

  6. 6.

    “You could make a story about a boy that is making a tree house and he needs a \( \frac{1}{6} \) size so he goes to the hardware store”.

  7. 7.

    “Once there was a man. However he really was \( \frac{1}{4} \) of a man. He would eat \( \frac{1}{4} \) of everything he even had \( \frac{1}{4} \) of a house. One day he went to jail for robbing \( \frac{1}{4} \) of a bank. ”

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Zhang, X., Clements, M.A.(. & Ellerton, N.F. Enriching student concept images: Teaching and learning fractions through a multiple-embodiment approach. Math Ed Res J 27, 201–231 (2015). https://doi.org/10.1007/s13394-014-0137-4

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