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Conceptual mis(understandings) of fractions: From area models to multiple embodiments

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Abstract

Area-model representations seem to have been dominant in the teaching and learning of fractions, especially in primary school mathematics curricula. In this study, we investigated 40 fifth grade children’s understandings of the unit fractions, \( \frac{1}{2} \), \( \frac{1}{3} \) and \( \frac{1}{4} \), represented through a variety of different models. Analyses of pre-teaching test and interview data revealed that although the participants were adept at partitioning regional models, they did not cope well with questions for which unit fractions were embodied in non-area-model scenarios. Analyses of post-teaching test and interview data indicated that after their participation in an instructional intervention designed according to Dienes’ (1960) dynamic principle, the students’ performances on tests improved significantly, and their conceptual understandings of unit fractions developed to the point where they could provide reasonable explanations of how they arrived at solutions. Analysis of retention data, gathered more than 3 months after the teaching intervention, showed that the students’ newly found understandings had, in most cases, been retained.

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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.

Appendix

Appendix

Protocol for interviews

  1. 1.

    “I am going to say a word and, as soon as I say it, I want you to say something, or draw something, or do something do the first thing that comes into your head after I say the word. The word is … one-third.”

  2. 2.

    [Give the pupil a piece of string, about 15 in long, and say…] “Here is a piece of string. Fold it so that you show me…[three successive requests are to be made…] ‘(a)…one-half of it…,’ (b) ‘one-third of it,’ and (c) ‘…one-fourth of it’.”

  3. 3.

    [Show the pupil a picture of a circle (radius 6″) and say…] “Here is a picture of a circle, and a pencil. Color in…[three successive requests are to be made…] (a) ‘…one-half of it,’ (b) ‘…one-third of it,’ and (c) ‘… one-fourth of it’.”

  4. 4.

    [Show the pupil a picture of an equilateral triangle with corners labeled A, B, C, and say] “Here is a picture of a triangle with A at the bottom left corner, and C at the bottom right corner. Suppose you started at A [point to A] and began to walk around the triangle like this [indicate from A to B to C and back to A]. Point to where you would be when you are [three successive requests are to be made] (a) one-half of the way around the triangle (b) one-third of the way around the triangle and (c) one-fourth of the way around the triangle.”

  5. 5.

    [Show the child 12 identical blocks arranged in a 4 by 3 grid, and say] (a) “Give me one-half of these blocks,” (b) “[g]ive me one-third of these blocks,” and (c) “[g]ive me one-fourth of these blocks.”

  6. 6.

    [Show the child a glass full of water and four other identical glasses that are empty. Say, “This one is full of water. By pouring, show me…] (a) “exactly one-half of the water,” (b) “exactly one-third of the water,” and (c) “exactly one-fourth of the water.”

  7. 7.

    [Show the child a line with 0 and 1 on it. At the left end “0” is marked, and at the right end “1” is marked. Say] (a) “Show me where you think one-half is on the number line,” (b) “[s]how me where you think one-third is on the number line,” and (c) “[s]how me where you think one-fourth is on the number line.”

  8. 8.

    [Say] “Suppose you were with two friends, and the three of you wanted to share 12 blocks equally, so that each of you got the same number of blocks. How many blocks would each get? [Place three drawings of people separately, and give the interviewee the 12 blocks.].”

  9. 9.

    [Show the interviewee a drawing of a circle, with five 60° sectors colored, and one not colored. Say] “What fraction of the circle is not colored?”

  10. 10.

    [Say] “Suppose a girl runs around a triangle track, starting at A [point to A] and running in this direction [indicate that the person runs from A in a counter-clockwise direction]. Point to where the girl would be when she was (a) one-half of the way around the triangle (b) one-third of the way around the triangle and (c) ‘one-fourth of the way around the triangle’.”

  11. 11.

    [This is the same as the last interview task, only this time, we have a square rather than a triangle. The A is the bottom left corner, B is at the top left corner, C is at the top right corner, and D is at the bottom right corner. Arrows on the square show a path from A though B through C, through D, and then back to A. Say] “This is like two earlier problems we did only this time we have a square. Suppose you started at A [point to A] and began to walk around the square like this [indicate A to B to C to D, and then back to A]. Point to where you would be when you were (a) one-half of the way around the square” (b) one-third of the way around the square” and (c) one-fourth of the way around the square.”

  12. 12.

    “I am going to give you a product to work out in your head. You must not write anything down. Just work out the answer in your head as quickly as you can, and then tell me the answer. Here is the product: “What is the value of three times one-third?’ [After an answer is given, ask the interviewee to explain how he/she thought about the problem.]”

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Zhang, X., Clements, M.A. & Ellerton, N.F. Conceptual mis(understandings) of fractions: From area models to multiple embodiments. Math Ed Res J 27, 233–261 (2015). https://doi.org/10.1007/s13394-014-0133-8

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  • DOI: https://doi.org/10.1007/s13394-014-0133-8

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