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Small Group and Whole Class Collaboration about Fractional Relationships Using Rod Models

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Title

Small Group and Whole Class Collaboration about Fractional Relationships Using Rod Models

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(type = doi)

doi:10.7282/T3QR5094

Abstract

This analytic is designed to show various forms of collaboration, which includes co-construction, integration, and modification, through small group and whole class discussions. Further, this analytic delves into how these three aspects of collaboration contribute to student’s understandings of the problems at hand. According to Elizabeth Uptegrove in her article “Shared communication in building mathematical ideas: A longitudinal study” (2015), “students build knowledge collaboratively by listening and reacting to each other’s ideas”, and this is demonstrated through each of the three aspects that make up collaboration (p. 108). Further, “as individual students’ ideas [are] adopted and elaborated by other students in the group, shared understandings emerge”, which is evident throughout this analytic (Uptegrove, 2015, p. 111).

Throughout these events, there are three different aspects of collaboration that will be discussed. One aspect is co-construction, which is where “dialogue occurs in a back and forth nature… until the argument is built” (Mueller, Yankelewitz, & Maher, 2012, p. 378). The second aspect is integration, which is where “a student’s argument is strengthened using arguments from their peers” (Mueller et al., 2012, p. 378). Lastly the third aspect, modification, occurs “as students attempt to correct a peer or assist him/her in making sense of a model or argument that was originally expressed in an unclear or incorrect way” (Mueller et al., 2012, p. 378).

In Event 1, Researcher Maher tells students to consider a train of one yellow and one light green rod put together as having the number name “two” and asks them to determine what a red rod should be called. Through small group discussion with Jacquelyn, Brian modifies his solution by agreeing with Jacquelyn’s claim that the red rod should be named “one half”. In Event 2, Audra is called upon to share the solution that she and Sarah co-constructed with the rest of the class. She states that they believe that the red rod should be called “one and one fourth” if the train is considered to be “two”, and the red rod should be called “one fourth” if the train is considered to be “one”. In Event 3, Researcher Maher asks Audra to explain the solutions that she and Sarah had offered. Upon completing her explanation, Erik states that he agrees with what Audra said about calling the red rod “one fourth” if the train of one yellow and one light green rod is referred to as “one”, and adds that “if the brown and the yellow and green they’re equal and they’re both called one, and four of the reds equal up to one, therefore that they’d have to be fourths, because there are four parts, they’re fourths”. This addition to the explanation can be viewed as Erik and Audra engaging in integration of their arguments.

In Event 4, students agree with the argument that was co-constructed by Audra, Sarah, and Erik, but disagree with the solution of calling the red rod “one and one fourth” when the yellow and light green train is considered to be “two”. Brian C. and Jacquelyn are then asked to come up and share their answer, which is when they claim that the red rod should be called “one half” in this situation. However, when asked for an explanation, they respond that they cannot remember their argument. In Event 5, David is asked to come up to the front of the class and explain why the red rod should be called “one half” if the yellow and light green train is called “two”. David then modifies his explanation when asked to clarify his solution by explaining that two red rods make up “one” since individually each red rod is half of the yellow and light green train. Thus, if the students take another red rod away from the half (the two red rods), leaving one (red rod), this demonstrates how the red rods would be named “one half”. Jacquelyn agrees with this modified explanation.

In Event 6, Maher introduces the “Candy Bar Problem” to students by providing them with the following scenario: “How many of you like chocolate? [Hands are raised]. Pretty much everybody. Right. Um, you know, we were talking about sharing things that people like and I was talking to, um, Tom earlier and I was talking to Amy. And, um, I said you know, Mrs. H. was nice enough to bring some candy because I thought we would look at some nice fraction problems and I said, well if we share these. So Tom said, I want one-half a candy bar and Amy said, I want one-half a candy bar. So I said okay, you each can have a half and I gave them each a half and they were so angry with me. They looked at me and said, well Tom was happy, but Amy said to me I don’t really like what you just did. That didn’t seem really fair. Now, how could that be? It seemed fair to me. I gave one-half to Tom and I gave one-half to Amy. Didn’t I do the right thing?” After posing this problem to students, Researcher Maher asks students to consider why Researcher Martino thought the sharing was unfair.

In Event 7, students are able to determine that Researcher Martino thought the sharing was because the halves were of candy bars of unequal size, that is, each candy bar was represented with a unit of different length. Thus, it is important for students to keep in mind that the unit must be fixed in a comparison of fraction parts. Researcher Maher then asks the students to consider which fraction is larger, one half or one third, by using Cuisenaire rods to build a model.

In Event 8, when given the problem of determining which fraction is larger, one half or one third, Erik and Alan collaborate on how to find a rod that they can partition into halves and thirds. They were unable to find a rod from the set of ten rods that could be “divided into anything”, so Alan suggests that they make their own rod using a train of rods to help them solve the problem.

In Event 9, Laura and Jessica co-construct an argument that designates a train consisting of one orange and one red rod as the unit “one”. They use the dark green rod to represent “one half” and a purple (pink) rod to represent “one third”. Using their train model, they claim that one half is bigger than one third by one red rod. This is when the researcher asks the class what number name they should give the red rod.

In Event 10, to answer Researcher Maher’s question of what the red rod should be called when a train of one orange and one red rod is considered to be “one”, Alan claims that the red rod should be named “one sixth” because “three reds make a dark green, and if there are three dark greens to make the orange and the red rod, then it would take six red rods to make the orange and the red rod”. Thus, through the use of Laura and Jessica’s model, Alan is able to determine what the red rod should be called. By using Laura and Jessica’s model, Alan engages in integration to make his claim.

In Event 11, using the class’s claim that one half is larger than one third, David designs a balance as a new representation using Cuisenaire rods to demonstrate that one half is indeed larger than one third. David conjectures that if he takes one light green rod off of the balance that he created at the same time that he takes two red rods off, the balance will fall to the side that has one light green rod left. It is interesting to note that David is able to switch rod representations from attending to the lengths of rods by comparing their sizes to attending to their weights using a balance beam representation. Through this representation, it can be inferred that David understands that if one half is indeed larger than one third, one half should also be heavier than one third. David checks this conjecture and determines that his prediction is correct.

In Event 12, Researcher Maher asks students what they would need to put on the other side of the balance that David has created to keep it from tipping when they remove two red rods and the green rod. David claims that he should use a white rod, which he justifies by demonstrating that a train of one red rod and one white rod is equal in length to one light green rod. At the end of the event, students are asked to consider what they have learned through collaboration with one another, as well as consider David’s model, over the weekend to be able to come back and discuss their thoughts.

References

Uptegrove, E. B. (2015). Shared communication in building mathematical ideas: A longitudinal study. The Journal of Mathematical Behavior, 40, 106-130. doi:10.1016/j.jmathb.2015.02.001

Mueller, M., Yankelewitz, D., & Maher, C. (2011). A framework for analyzing the collaborative construction of arguments and its interplay with agency. Educational Studies in Mathematics, 80(3), 369-387. doi:10.1007/s10649-011-9354-x

Video

Robert B. Davis Institute for Learning (1993). Reviewing rod relationships and the candy bar problem, Clip 2 of 6: What is the number name for red when the yellow and light green rod is two? Brian and Jacquelyn. [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3XP73MB

Robert B. Davis Institute for Learning (1993). Reviewing rod relationships and the candy bar problem, Clip 3 of 6: What is the number name for red when the yellow and light green rod is two? A whole class discussion. [video]. Retrieved from http://dx.doi.org/doi:10.7282/T32F7M3N

Robert B. Davis Institute for Learning (1993). Reviewing rod relationships and the candy bar problem, Clip 4 of 6: Switching units, candy bar metaphor. [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3668BTZ

Robert B. Davis Institute for Learning (1993). Reviewing rod relationships and the candy bar problem, Clip 5 of 6: Comparing one half and one third, part 1. [video]. Retrieved from http://dx.doi.org/doi:10.7282/T39W0D52

Robert B. Davis Institute for Learning (1993). Reviewing rod relationships and the candy bar problem, Clip 6 of 6: Comparing 1/2 and 1/3, David’s balance beam. [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3R49NZ4

Throughout these events, there are three different aspects of collaboration that will be discussed. One aspect is co-construction, which is where “dialogue occurs in a back and forth nature… until the argument is built” (Mueller, Yankelewitz, & Maher, 2012, p. 378). The second aspect is integration, which is where “a student’s argument is strengthened using arguments from their peers” (Mueller et al., 2012, p. 378). Lastly the third aspect, modification, occurs “as students attempt to correct a peer or assist him/her in making sense of a model or argument that was originally expressed in an unclear or incorrect way” (Mueller et al., 2012, p. 378).

In Event 1, Researcher Maher tells students to consider a train of one yellow and one light green rod put together as having the number name “two” and asks them to determine what a red rod should be called. Through small group discussion with Jacquelyn, Brian modifies his solution by agreeing with Jacquelyn’s claim that the red rod should be named “one half”. In Event 2, Audra is called upon to share the solution that she and Sarah co-constructed with the rest of the class. She states that they believe that the red rod should be called “one and one fourth” if the train is considered to be “two”, and the red rod should be called “one fourth” if the train is considered to be “one”. In Event 3, Researcher Maher asks Audra to explain the solutions that she and Sarah had offered. Upon completing her explanation, Erik states that he agrees with what Audra said about calling the red rod “one fourth” if the train of one yellow and one light green rod is referred to as “one”, and adds that “if the brown and the yellow and green they’re equal and they’re both called one, and four of the reds equal up to one, therefore that they’d have to be fourths, because there are four parts, they’re fourths”. This addition to the explanation can be viewed as Erik and Audra engaging in integration of their arguments.

In Event 4, students agree with the argument that was co-constructed by Audra, Sarah, and Erik, but disagree with the solution of calling the red rod “one and one fourth” when the yellow and light green train is considered to be “two”. Brian C. and Jacquelyn are then asked to come up and share their answer, which is when they claim that the red rod should be called “one half” in this situation. However, when asked for an explanation, they respond that they cannot remember their argument. In Event 5, David is asked to come up to the front of the class and explain why the red rod should be called “one half” if the yellow and light green train is called “two”. David then modifies his explanation when asked to clarify his solution by explaining that two red rods make up “one” since individually each red rod is half of the yellow and light green train. Thus, if the students take another red rod away from the half (the two red rods), leaving one (red rod), this demonstrates how the red rods would be named “one half”. Jacquelyn agrees with this modified explanation.

In Event 6, Maher introduces the “Candy Bar Problem” to students by providing them with the following scenario: “How many of you like chocolate? [Hands are raised]. Pretty much everybody. Right. Um, you know, we were talking about sharing things that people like and I was talking to, um, Tom earlier and I was talking to Amy. And, um, I said you know, Mrs. H. was nice enough to bring some candy because I thought we would look at some nice fraction problems and I said, well if we share these. So Tom said, I want one-half a candy bar and Amy said, I want one-half a candy bar. So I said okay, you each can have a half and I gave them each a half and they were so angry with me. They looked at me and said, well Tom was happy, but Amy said to me I don’t really like what you just did. That didn’t seem really fair. Now, how could that be? It seemed fair to me. I gave one-half to Tom and I gave one-half to Amy. Didn’t I do the right thing?” After posing this problem to students, Researcher Maher asks students to consider why Researcher Martino thought the sharing was unfair.

In Event 7, students are able to determine that Researcher Martino thought the sharing was because the halves were of candy bars of unequal size, that is, each candy bar was represented with a unit of different length. Thus, it is important for students to keep in mind that the unit must be fixed in a comparison of fraction parts. Researcher Maher then asks the students to consider which fraction is larger, one half or one third, by using Cuisenaire rods to build a model.

In Event 8, when given the problem of determining which fraction is larger, one half or one third, Erik and Alan collaborate on how to find a rod that they can partition into halves and thirds. They were unable to find a rod from the set of ten rods that could be “divided into anything”, so Alan suggests that they make their own rod using a train of rods to help them solve the problem.

In Event 9, Laura and Jessica co-construct an argument that designates a train consisting of one orange and one red rod as the unit “one”. They use the dark green rod to represent “one half” and a purple (pink) rod to represent “one third”. Using their train model, they claim that one half is bigger than one third by one red rod. This is when the researcher asks the class what number name they should give the red rod.

In Event 10, to answer Researcher Maher’s question of what the red rod should be called when a train of one orange and one red rod is considered to be “one”, Alan claims that the red rod should be named “one sixth” because “three reds make a dark green, and if there are three dark greens to make the orange and the red rod, then it would take six red rods to make the orange and the red rod”. Thus, through the use of Laura and Jessica’s model, Alan is able to determine what the red rod should be called. By using Laura and Jessica’s model, Alan engages in integration to make his claim.

In Event 11, using the class’s claim that one half is larger than one third, David designs a balance as a new representation using Cuisenaire rods to demonstrate that one half is indeed larger than one third. David conjectures that if he takes one light green rod off of the balance that he created at the same time that he takes two red rods off, the balance will fall to the side that has one light green rod left. It is interesting to note that David is able to switch rod representations from attending to the lengths of rods by comparing their sizes to attending to their weights using a balance beam representation. Through this representation, it can be inferred that David understands that if one half is indeed larger than one third, one half should also be heavier than one third. David checks this conjecture and determines that his prediction is correct.

In Event 12, Researcher Maher asks students what they would need to put on the other side of the balance that David has created to keep it from tipping when they remove two red rods and the green rod. David claims that he should use a white rod, which he justifies by demonstrating that a train of one red rod and one white rod is equal in length to one light green rod. At the end of the event, students are asked to consider what they have learned through collaboration with one another, as well as consider David’s model, over the weekend to be able to come back and discuss their thoughts.

References

Uptegrove, E. B. (2015). Shared communication in building mathematical ideas: A longitudinal study. The Journal of Mathematical Behavior, 40, 106-130. doi:10.1016/j.jmathb.2015.02.001

Mueller, M., Yankelewitz, D., & Maher, C. (2011). A framework for analyzing the collaborative construction of arguments and its interplay with agency. Educational Studies in Mathematics, 80(3), 369-387. doi:10.1007/s10649-011-9354-x

Video

Robert B. Davis Institute for Learning (1993). Reviewing rod relationships and the candy bar problem, Clip 2 of 6: What is the number name for red when the yellow and light green rod is two? Brian and Jacquelyn. [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3XP73MB

Robert B. Davis Institute for Learning (1993). Reviewing rod relationships and the candy bar problem, Clip 3 of 6: What is the number name for red when the yellow and light green rod is two? A whole class discussion. [video]. Retrieved from http://dx.doi.org/doi:10.7282/T32F7M3N

Robert B. Davis Institute for Learning (1993). Reviewing rod relationships and the candy bar problem, Clip 4 of 6: Switching units, candy bar metaphor. [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3668BTZ

Robert B. Davis Institute for Learning (1993). Reviewing rod relationships and the candy bar problem, Clip 5 of 6: Comparing one half and one third, part 1. [video]. Retrieved from http://dx.doi.org/doi:10.7282/T39W0D52

Robert B. Davis Institute for Learning (1993). Reviewing rod relationships and the candy bar problem, Clip 6 of 6: Comparing 1/2 and 1/3, David’s balance beam. [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3R49NZ4

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