PurposesEffective teaching; Lesson activity; Professional development activity; Student model building; Reasoning; Representation

DescriptionThe purpose of this RUanalytic is to illustrate instances of generic reasoning that occurred as fourth graders, participating in a longitudinal study on student reasoning, explored fraction comparison concepts. According to Yankelewitz (2009), generic reasoning occurs when a student reasons about the properties of a paradigmatic example that are representative of and can be applied to a larger class of objects in which it is contained and lends insight into a more general truth about that class. The consideration of the general application of these properties in turn verifies the claim made about the particular example (Rowland, 2002a, 2002b). According to Balacheff (1988), generic reasoning is a valid form of justification and is easily understood by students at all levels. Alibert and Thomas (1991) suggest that this generic form of reasoning is more intuitive than many other forms of proof. Events depicting the students generic reasoning are included, as well as episodes that show how this reasoning was supported. The data for this RUanalytic were collected on October 1, 1993 and October 6, 1993, during Sessions 6 and 8.

During the longitudinal study, the students began to explore fraction comparison during Session 3 on September 24, 1993. At first, they worked with partners to build models. During later sessions, students were encouraged to build multiple models to compare two fractions, and to share their thoughts with the class. During these sessions, students used various reasoning strategies as they built models to compare fractions and shared their thoughts verbally and in writing. This RUanalytic traces the students’ development of generic reasoning during these sessions. The episodes of generic reasoning come from Sessions 6 and 8.

Prior to Event 1 of this RUanalytic, on September 27, 1993, during Session 4, the students built models to compare 1/2 and 1/3. One student, Erik, used generic reasoning to explain why halves are bigger than thirds, "Because see, if you have one whole, and you want to divide it into halves, the halves have to be so big that you can only divide them into two parts. So, and if you wanted to divide it into thirds, they have to be big enough to divide into three parts. So if you only wanted to divide it into two parts, you have one whole, the whole has to be big enough to divide into two parts, two equal parts. So if you have two parts, two is less than three, but if you divide it into two parts, they have to be bigger than the thirds." Additionally he stated, "The thirds would be smaller because two parts of one, like a circle or something, you cut it into two parts, they’re going to have to be bigger because it’s two parts you’re cutting it into. But if you’re cutting it into three parts, the thirds are going to have to be bigger, I mean not bigger, smaller, because you’re cutting it into three parts, and three parts, is, the number three is larger than two but if you’re cutting something into two parts it’s going to have to be larger than three." Erik concluded his argument saying, “So technically, if you’re counting by numbers, the smaller number is the larger.” With this, Erik used the generic example of halves and thirds to make a generalized statement about fraction comparisons.

Erik’s partner, Alan, interjected, explaining recursively, "If you cut this [the unit] into thirds, it would have three equal parts. If you would cut it into halves it would have two equal parts and if you cut it into fourths it would have four equal parts. The fourths would be smaller than the thirds and the thirds smaller than the halves." Erik then continued, using the example of thirds and fourths to show that his rule held. He used generic reasoning to show that his generalization held for another example, and why that generalization was true. He said, "What I’m saying is, see if you divide it into thirds and fourths, four is a larger number than three, but three, you’re dividing it into, um, you’re dividing it into three parts, so instead of dividing it into four parts you cut it four times into fourths and then, and that would be much smaller than the, a third. And if you divide it- if you cut it only three times, it’d be bigger. So therefore, four may be bigger than three, but the smaller the number, the larger the piece. As a result of further questioning, Erik, continuing his reasoning using his generic example, stating that, "the smaller the number [a whole is divided into], the bigger the pieces [referring to the fractional parts].” It is interesting to note this generic reasoning came before the generic reasoning demonstrated by Alan and Erik in the events in this RUanalytic (see Event 6).

The events in this RUanalytic begin in Session 6. During Session 6, as shown in Event 1, students work on the task of comparing 1/2 and 1/4. After students come up with different models to show the comparison, Andrew offers a generic explanation to show that 1/2 is always larger than 1/4, no matter which model is built to represent the difference. He stated that all of the models that had been built and shared during the session “always had the room for one more fourth, and I think that because usually the fourths, or two of ‘em are equal up to the half, so then it would be a fourth.”

The remaining events in this RUanalytic are taken from Session 8, during which students continue to explore fraction comparison problems, and have the opportunity to record their solutions and accompanying justifications as they work with their partners. As students work to compose their justifications and record them on paper, they use generic reasoning to explain their ideas.

In Session 8, two of the instances of generic reasoning can be observed. One occurs as Brian and Michael write a justification for the difference between 1/2 and 3/4, as shown in Event 5. Events 2 through 4 illustrate the exploration of different models that lead to the instance of generic reasoning in Event 5.

The second instance of generic reasoning occurs in Event 6 when Alan explains to Researcher Maher that the difference between 3/4 and 1/2 is always 1/4, no matter which model is built. The beginning of the event describes the investigation of different models that took place prior to the instance of generic reasoning.

It is interesting to note that two of the instances of generic reasoning occur as students record their justifications in written form, possibly because writing a justification encourages students to try to record their ideas in a more general and abstract manner. For example, in Event 5 when Brian and Michael attempt to explain their solution to comparison of fraction problems, rather than explaining their model alone, they try to explain why their solution makes sense in mathematical terms and how it applies to any model built to compare the fractions. This pattern is important to note, as it suggests that encouraging students to record their justifications may foster the abstraction and generalization of mathematical arguments.

Also evident in this RUanalytic is that in every episode of generic reasoning, students explore and build a variety of models to show each comparison. Thus, the generic reasoning is a result of considering all of the models for each comparison task, rather than just one model.

For a more detailed view of students’ investigation of fraction comparison, viewers are encouraged to access, "Comparing fractions and evaluating models that represent solutions, Clip 8 of 8: Comparing one half and one fourth, multiple models," "Continuing to explore fraction comparisons, Clip 4 of 7: Michael and Brian find four models to compare one half and three fourth," and " Continuing to explore fraction comparisons, Clip 6 of 7: Erik and Alan compare one half and three fourths" at videomosaic.org.

References

Alibert, D. & Thomas, M. (1991). Research on mathematical proof. In D. Tall (Ed.), Advanced mathematical thinking, (pp. 215-230). Dordrecht, The Netherlands: Kluwer Academic Publishers.

Balacheff N. (1988) Aspects of proof in pupils’ practice of school mathematics. In: Pimm D. (ed.) Mathematics, Teachers and Children (pp.316-230). London: Hodder and Stoughton.

Rowland, T. (2002a). Generic proofs in number theory. In S. Campbell and R. Zazkis (Eds.) Learning and teaching number theory: Research in cognition and instruction. (pp. 157-184). Westport, CT: Ablex Publishing.

Rowland, T. (2002b). Proofs in number theory: History and heresy. In Proceedings of the Twenty-Sixth Annual Meeting of the International Group for the Psychology of Mathematics Education, (Vol. I, pp. 230-235). Norwich, England.

Yankelewitz, D. (2009). The development of mathematical reasoning in elementary school students’ exploration of fraction ideas. Unpublished doctoral dissertation, Rutgers, The State University of New Jersey.

During the longitudinal study, the students began to explore fraction comparison during Session 3 on September 24, 1993. At first, they worked with partners to build models. During later sessions, students were encouraged to build multiple models to compare two fractions, and to share their thoughts with the class. During these sessions, students used various reasoning strategies as they built models to compare fractions and shared their thoughts verbally and in writing. This RUanalytic traces the students’ development of generic reasoning during these sessions. The episodes of generic reasoning come from Sessions 6 and 8.

Prior to Event 1 of this RUanalytic, on September 27, 1993, during Session 4, the students built models to compare 1/2 and 1/3. One student, Erik, used generic reasoning to explain why halves are bigger than thirds, "Because see, if you have one whole, and you want to divide it into halves, the halves have to be so big that you can only divide them into two parts. So, and if you wanted to divide it into thirds, they have to be big enough to divide into three parts. So if you only wanted to divide it into two parts, you have one whole, the whole has to be big enough to divide into two parts, two equal parts. So if you have two parts, two is less than three, but if you divide it into two parts, they have to be bigger than the thirds." Additionally he stated, "The thirds would be smaller because two parts of one, like a circle or something, you cut it into two parts, they’re going to have to be bigger because it’s two parts you’re cutting it into. But if you’re cutting it into three parts, the thirds are going to have to be bigger, I mean not bigger, smaller, because you’re cutting it into three parts, and three parts, is, the number three is larger than two but if you’re cutting something into two parts it’s going to have to be larger than three." Erik concluded his argument saying, “So technically, if you’re counting by numbers, the smaller number is the larger.” With this, Erik used the generic example of halves and thirds to make a generalized statement about fraction comparisons.

Erik’s partner, Alan, interjected, explaining recursively, "If you cut this [the unit] into thirds, it would have three equal parts. If you would cut it into halves it would have two equal parts and if you cut it into fourths it would have four equal parts. The fourths would be smaller than the thirds and the thirds smaller than the halves." Erik then continued, using the example of thirds and fourths to show that his rule held. He used generic reasoning to show that his generalization held for another example, and why that generalization was true. He said, "What I’m saying is, see if you divide it into thirds and fourths, four is a larger number than three, but three, you’re dividing it into, um, you’re dividing it into three parts, so instead of dividing it into four parts you cut it four times into fourths and then, and that would be much smaller than the, a third. And if you divide it- if you cut it only three times, it’d be bigger. So therefore, four may be bigger than three, but the smaller the number, the larger the piece. As a result of further questioning, Erik, continuing his reasoning using his generic example, stating that, "the smaller the number [a whole is divided into], the bigger the pieces [referring to the fractional parts].” It is interesting to note this generic reasoning came before the generic reasoning demonstrated by Alan and Erik in the events in this RUanalytic (see Event 6).

The events in this RUanalytic begin in Session 6. During Session 6, as shown in Event 1, students work on the task of comparing 1/2 and 1/4. After students come up with different models to show the comparison, Andrew offers a generic explanation to show that 1/2 is always larger than 1/4, no matter which model is built to represent the difference. He stated that all of the models that had been built and shared during the session “always had the room for one more fourth, and I think that because usually the fourths, or two of ‘em are equal up to the half, so then it would be a fourth.”

The remaining events in this RUanalytic are taken from Session 8, during which students continue to explore fraction comparison problems, and have the opportunity to record their solutions and accompanying justifications as they work with their partners. As students work to compose their justifications and record them on paper, they use generic reasoning to explain their ideas.

In Session 8, two of the instances of generic reasoning can be observed. One occurs as Brian and Michael write a justification for the difference between 1/2 and 3/4, as shown in Event 5. Events 2 through 4 illustrate the exploration of different models that lead to the instance of generic reasoning in Event 5.

The second instance of generic reasoning occurs in Event 6 when Alan explains to Researcher Maher that the difference between 3/4 and 1/2 is always 1/4, no matter which model is built. The beginning of the event describes the investigation of different models that took place prior to the instance of generic reasoning.

It is interesting to note that two of the instances of generic reasoning occur as students record their justifications in written form, possibly because writing a justification encourages students to try to record their ideas in a more general and abstract manner. For example, in Event 5 when Brian and Michael attempt to explain their solution to comparison of fraction problems, rather than explaining their model alone, they try to explain why their solution makes sense in mathematical terms and how it applies to any model built to compare the fractions. This pattern is important to note, as it suggests that encouraging students to record their justifications may foster the abstraction and generalization of mathematical arguments.

Also evident in this RUanalytic is that in every episode of generic reasoning, students explore and build a variety of models to show each comparison. Thus, the generic reasoning is a result of considering all of the models for each comparison task, rather than just one model.

For a more detailed view of students’ investigation of fraction comparison, viewers are encouraged to access, "Comparing fractions and evaluating models that represent solutions, Clip 8 of 8: Comparing one half and one fourth, multiple models," "Continuing to explore fraction comparisons, Clip 4 of 7: Michael and Brian find four models to compare one half and three fourth," and " Continuing to explore fraction comparisons, Clip 6 of 7: Erik and Alan compare one half and three fourths" at videomosaic.org.

References

Alibert, D. & Thomas, M. (1991). Research on mathematical proof. In D. Tall (Ed.), Advanced mathematical thinking, (pp. 215-230). Dordrecht, The Netherlands: Kluwer Academic Publishers.

Balacheff N. (1988) Aspects of proof in pupils’ practice of school mathematics. In: Pimm D. (ed.) Mathematics, Teachers and Children (pp.316-230). London: Hodder and Stoughton.

Rowland, T. (2002a). Generic proofs in number theory. In S. Campbell and R. Zazkis (Eds.) Learning and teaching number theory: Research in cognition and instruction. (pp. 157-184). Westport, CT: Ablex Publishing.

Rowland, T. (2002b). Proofs in number theory: History and heresy. In Proceedings of the Twenty-Sixth Annual Meeting of the International Group for the Psychology of Mathematics Education, (Vol. I, pp. 230-235). Norwich, England.

Yankelewitz, D. (2009). The development of mathematical reasoning in elementary school students’ exploration of fraction ideas. Unpublished doctoral dissertation, Rutgers, The State University of New Jersey.

Created on2015-11-17T18:28:10-0500

Published on2016-02-04T20:22:49-0500

Persistent URLhttps://doi.org/doi:10.7282/T3417044