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Fourth Graders Analyses of Equivalence: 1/5 or 2/10?

## Descriptive

TypeOfResource
Text
TitleInfo
Title
Fourth Graders Analyses of Equivalence: 1/5 or 2/10?
Identifier (type = doi)
doi:10.7282/T3WW7KFN
Abstract
Maher and Martino (1996) contend that when students are given sufficient time to work on a problem and are given the opportunity to discuss their solutions with their peers, they often express differences of opinion. These conflicts are not always resolved immediately; sometimes they are pushed off to a later class session, possibly being deferred over an extended period of time. This allows students to think about the problem and upon revisiting the identical or similar problem at a later date, students are able to build on their ideas and solidify their understanding. As noted by Francisco and Maher (2005), revisiting a concept in a related problem, “helps students build rich and durable forms of mathematical understanding of mathematical concepts” (p. 371).

This analytic highlights students’ foray into the world of fraction equivalence and shows how revisiting a task helps students build strong understanding of a fundamental mathematical concept. Their journey began in an earlier session where students showed uncertainty regarding whether or not two fractions are equivalent. This analytic portrays two subsequent sessions in which students revisit the concept of fraction equivalence and ultimately come to a clear consensus of the mathematical truth of equivalent fractions. The researcher, Carolyn Maher, then introduces the fraction notation used to demonstrate equivalence.

In an earlier session, the students had been asked to give a number name to two white rods if the orange rod was given the number name one. Mark and Andrew had offered the solution of one fifth and Meredith had countered that with justification for the solution of two tenths. The class did not reconcile the two explanations and the researcher Amy Martino therefore left the discussion for a later time. This analytic begins with a session in which the researcher, Carolyn Maher, continues the discussion and asks the students what they remembers about the problem “Is 1/5 = 2/10?” In revisiting this problem, Meredith builds on the work of a previous session and shows that the white rods are called tenths and that the red rods are called one fifth. She then demonstrates the equivalence of two white rods and one red rod and concludes that one fifth equals two tenths. The students once again revisit and strengthen their understanding of this concept in a later session when working on the task “Which is larger, one half or two thirds, and by how much.” Many students come up with models to show the solution as one sixth. Meredith demonstrates that the solution can be two twelfths as well, enabling students to further solidify their understanding of fraction equivalence. The researcher then records the students’ ideas in mathematical notation to show that the two solutions are equivalent.

References
Francisco, J. M., and Maher, C. A. (2005). Conditions for promoting reasoning in problem solving: Insights from a longitudinal study. The Journal of Mathematical Behavior, 24(3), 361-372.

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), 194–214.
Genre (authority = RULIB)
Effective teaching
Genre (authority = RULIB)
Student model building
Genre (authority = RULIB)
Reasoning
Genre (authority = RULIB)
Representation
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TitleInfo
Title
RBDIL Analytics
Identifier (type = local)
rucore00000002136
Name (id = NAME-1); (type = personal)
NamePart (type = family)
Gerstein
NamePart (type = given)
Miriam
DisplayForm
Miriam Gerstein
Role
RoleTerm (authority = RULIB)
Creator
Affiliation
Rutgers University
Name (id = NAME-2); (type = personal)
NamePart (type = family)
Maher
NamePart (type = given)
Carolyn
DisplayForm
Carolyn Maher
Role
RoleTerm (authority = RULIB)
Publisher
Affiliation
Rutgers University
OriginInfo
DateCreated (qualifier = exact)
2014-07-30T14:29:09-0400
OriginInfo
DateOther (qualifier = exact); (type = modified)
2015-06-16T16:50:49-0400
OriginInfo
DateOther (qualifier = exact); (type = published)
2015-06-16T17:06:21-0400
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## Rights

RightsDeclaration (AUTHORITY = RULIB); (ID = rulibRdec0007)
The author owns the copyright to this work.
Status
Availability
Status
Open
Reason
Note
RightsHolder
Name (TYPE = personal); (ID = R-NAME_0001)
Miriam Gerstein
Role
Author
ContactInformation
ContactInformationDate
2014-08-21T13:39:22-0400
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## Technical

ContentModel
Analytic
PreservationLevel
full
Generation
born digital source
MimeType (TYPE = file)
application/xml
FileSize (UNIT = bytes)
14201
CreatingApplication
CreatingApplicationName
RUanalytic
CreatingApplicationVersion
2.0
CreatingApplicationDateCreated
2015-06-16T17:06:21-0400
Note
Generated using the RUanalytic tool export function.
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