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doi:10.7282/t3-k1tp-fb53

Students find challenges in interpreting and working with fractions. Many concepts, such as equivalent fractions, difference between fractions, and sums of fractions do not seem to make sense to them, and these topics are approached with a lot of difficulty and abstractness. This is in part due to the abstract introduction and development of concepts in fractions (teaching without models and teaching aids, rules without meaning, etc.). For many students, fractions (as a topic) are not looked at from a wider perspective of understanding beyond chalk-board examples and rules.
Argumentation is often “image dependent” in a way that students fold back through image making and image having as they justify their claims during argumentation. The students’ Cuisenaire rod models (both their verbal descriptions of the models and the physical models) become an essential referent in their argumentation (Krummheuer,1995). Although students do not use formal mathematical language, to express their ideas, their reasoning often takes the form of valid arguments. Given the opportunity to express ideas and provide backing for these ideas by building models, students are engaged in productive argumentation that enhances conceptual understanding of fraction ideas. Analysing fraction relationships using physical models, can become essential in interpreting students’ reasoning. (Maher & Van Ness, 2018). In the video clips used in this analytic, Alan and Erik form conjectures and (using their models, confirm their conjectures with evidence, turning them into claims providing justification by use of arguments.
Argumentation includes actions, tools, notations, drawings, models, numerical data, as well as verbal statements. Models often play a key role in the building of students’ arguments, often providing the warrants and backing that make the argument more sophisticated (Douek, 1999; Krummheuer, 1995; Yackel, 2002). Sometimes models are the starting point for students’ argumentation, enabling them to construct a convincing verbal argument. At other times, students use their models to explain or support their verbal arguments, providing more convincing evidence. When students make sensible arguments, they often rely on their models. (Maher & Van Ness, 2018).
Mathematical ideas can be interconnected and extended as learners work together to make sense of each other’s ideas and to build convincing arguments for the solutions of problems (Davis,1992; Maher, Powell, & Uptegrove, 2010; Maher & Yankelewitz, 2017).
Interactions with peers have been shown to provide opportunities for students to reconsider their existing thoughts and beliefs. In some cases, this reconsideration leads to new knowledge constructions (De Lisi & Golbeck, 1999). In combination with argumentation, students are encouraged to explore mathematical ideas while developing into mathematical thinkers and sense makers. In the process of engaging in argumentations, students explain and justify their ideas, often refining their reasoning, having an opportunity to make connections between and among mathematical concepts (Whitenack & Yackel, 2002). Through argumentation, students can construct and refine their understanding of mathematical ideas and build together a shared understanding (Lampert, 2003). In a natural way, even young children in justifying their solutions to problems produce reasoning that is “proof like” (Maher & Martino, 1996).
This analytic focuses mainly on the collaborative argumentation and reasoning used by Alan and Erik, two fourth-grade students. The students express their understanding of fraction ideas in building solutions to different tasks. By building and using Cuisenaire rod models and through argumentation and collaboration, Alan and Erik explain, present, and support their arguments, convincing other students and the researchers about their answers. Engagement and collaboration of students in small groups are key to effective learning. In the process of engaging in argumentations, students explain and justify their ideas, often refining their reasoning, having an opportunity to make connections between and among mathematical topics (Whitenack & Yackel, 2002). Therefore, when concepts are effectively introduced to students, they will be better prepared to build their own understanding through forms of reasoning, such as case building, reasoning by contradiction and induction (Maher & Yankelewitz, 2017).
Promoting children’s building of conceptual understanding of mathematical ideas is essential before requiring that they apply algorithms and procedures that often rely on rote memorisation with rules that do not make sense. This type of meaningless learning can result in confusion as in the case of “Benny” (Erlwanger,1973) or for rules to be forgotten. If the goal for mathematics instruction is for students to be engaged and confident in their ability to build new knowledge, opportunities must be provided for students to explore, conjecture, build models, and reason.

Problem tasks
Which is larger and by how much?
½ or ¼. (Events 2 and 3)
¾ or 2/3. (Events 4 through7).

Video Clip references
Comparing fractions, a whole class debate, Clip 5 of 5: The difference is one sixth [video]. Retrieved from https://doi.org/doi:10.7282/T32R3Q8Q

Continuing to explore fraction comparisons, Clip 7 of 7: Alan and Erik compare two third and three fourths https://doi.org/doi:10.7282/T39G5KCK

Discovering equivalent fractions and introducing fraction notation, Clip 3 of 5: Alan and Erik compare one half and two thirds https://doi.org/doi:10.7282/T34Q7SKC


Citations
1. Alston, A. & Van Ness, C. (2017). Switching the unit. In Maher, C. A. & Yankelewitz, D. (Eds.). Children’s reasoning while building fraction ideas (pp. xiii-xvi). Rotterdam & Boston, MA: Sense Publishers (pp. 65-80).
2. Douek, N. (1999). Argumentation and conceptualization in context: A case study on sun shadows in primary school. Educational Studies in Mathematics 39(1-3) 89-110.
3. Erlwanger, S. H. (1973/2004). Bennyʼs conception of rules and answers in IPI Mathematics. In T. P. Carpenter, J.A. Dossey, & J. L. Koehler (Eds.), Classics in mathematics education research (pp. 48-58). Reston, VA: NCTM.
4. Krummheuer, G. (1995). The ethnography of argumentation: NJ: Erlbaum.
5. Lampert, M. (2003). Teaching problems and the problems of teaching. New Haven: Yale University Press.
6. Maher , C. A. (2005). How students structure their investigations and learn mathematics: Insights from a long-term study. The Journal of Mathematical Behavior. 24(1) 1-14
7. Maher, C. A and Yankelewitz, D. (Eds). Children’s reasoning while building fractions ideas; Rotterdam & Boston, MA: Sense Publishers.
8. Pirie, S., & Kieran, T. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics, 26, 165-190.
9. Van Ness, C., & Maher, C. A (in press). Analysis of the argumentation of nine-year-olds engaged in discourse about comparing fraction models. The Journal of Mathematical Behavior.
10. Whitenack, J., & Yackel, E. (2002). Making mathematical arguments in the primary grades: The importance of explaining and justifying ideas. Teaching Children Mathematics. 8(9) 524-527.

Effective teaching

Student collaboration

Student model building

Reasoning

Representation

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