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doi:10.7282/T3CZ38Z5

There is much focus on the environment in which students undertake problem solving. When properly established, a collaborative setting can be a key factor in the growth and development of students’ ability to reason and construct arguments. Mathematically proficient students develop conjectures and construct arguments and in collaborative settings, propose these ideas to others and respond to the reasoning of others (NGACBP & CCSSO, 2010). Students should be able to differentiate between reasoning that is plausible and logical and reasoning that is incorrect. Critiquing the reasoning of others is an important skill for proficient mathematical students to develop and is a natural process in a collaborative setting when working towards solving a problem. Students should not only critique the reasoning of other’s, but students should also absorb and reflect upon feedback that is offered from their peers. In some cases, peer criticism may allow a learner to recognize and correct a misconception.


In this narrative, we focus on whole class debate and follow a fourth grade class as they unwrap ideas and make discoveries about comparing fractions using Cuisenaire rods. This narrative follows the class in debate after they explored a series of questions involving comparing fractions. Students explored the question, is 2/10 the same as 1/5, and which is greater ½ or 1/3 and by how much. The students are given an opportunity to compare and critique each other’s models. Through this critique and discussion, students determine that different models can be used to represent the same fractions, hence developing a deeper understanding of the topics of fraction equivalence. In whole class debate, students are able to compare reasoning strategies as to why it is true that 2/10 is the same as 1/5. In the first event, we observe a student Brian support this argument by comparing 1/10 as half of 1/5, while two other students, Meredith and Erik decide 2/10 is 1/5 by assigning a rod to 2/10 and 1/5 and showing it takes two tenths to equal the length of one fifth.
In their partner explorations, students also developed some false conjectures. By allowing students to share their ideas and discuss as a whole class, students are able to unearth and correct misconceptions. Many students are able to determine that ½ is larger than 1/3; however, we observe two students, Audra and Jessica develop a misconception in the second event, when asked how much larger ½ is. Audra and Jessica find a rod that represents the length, but name this rod 1/3 because they compare the length to 1/2, rather than refer back to the original whole. Two other students also develop a misunderstanding, by stating that ½ is “one” bigger than 1/3. In the final two events, we watch different students unearth these misconceptions. Through the whole class debate, students correct their misunderstandings and other students develop a stronger and more meaningful understanding of concepts with comparing fractions by critiquing arguments of others and receiving feedback on their own arguments.

Bibliography:

Maher, C. A. and Martino, A. M. (1996). The development of the idea of mathematical proof: a 5-year case study. Journal for Research in Mathematics Education, 27, 2, 194- 218.

National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards Initiative: About the standards. Retrieved from http://www.corestandards.org/about-the-standards

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