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Laura Says !th is Shaded Problem (Grade 4)!STUDENT WORK SAMPLE ARGUMENTATION RESOURCE PACKET

This packet was produced as part of the Bridging Math Practices Math-Science Partnership Grant (2014 -2015). The purpose of the packet is to help a) reveal what students can do with respect to generating an argument in response to mathematical questions, including the variety of their arguments; b) highlight features that should be considered when reviewing students’ arguments, and c) identify what counts as a quality argument in light of the review criteria. What is a mathematical argument? A mathematical argument is

a sequence of statements and reasons given with the aim of demonstrating that a claim is true or false. This links to the Connecticut Core Standards of Mathematical Practice #3, construct viable arguments and critique the reasoning of others, as well as other standards.

This resource packet is a product of work by participants in the UConn Bridging Math Practices Math-Science Partnership Grant, which included faculty and graduate students from the University of Connecticut’s Neag School of Education and Department of Mathematics,

and teachers and coaches from the Manchester Public Schools, Mansfield Public Schools, and Hartford Public Schools. This resource packet reflects significant contributions from An’drea Flynn, Christine Giaquinto, Kylie Hoke, Teresa Maturino, and Diane Ozmun. Many thanks for all their insights and contributions! For more information about the grant, or for additional argumentation-related

materials and resource, please see the project website: http://bridges.uconn.education.edu The Mathematics and Science Partnership (MSP) grant is a federal program funded under Title II, Part B, of the Elementary and

Secondary Education Act and administered by the U.S. Department of Education (ED).

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What is a high quality mathematical argument? A high quality mathematical argument is an argument that shows that a claim must be true. It leaves little room to question. The chain of logic leads the reader to conclude that the author’s claim is true. What are the characteristics of a high quality argument? A high quality argument can be described by the following components and criteria:

Criteria Description 1. A clearly stated claim

The claim is what is to be shown true or not true.

2. The necessary evidence to support the claim

Evidence can take the form of equations, tables, charts, diagrams, graphs, words, symbols, etc. It is one’s “work” which provides the information to show something is true/false.

3. The necessary warrants to connect the evidence to the claim

Warrants can take the form of definitions, theorems, logical inferences, agreed upon facts. Warrants explain how the evidence is relevant for the claim, and collectively they chain the evidence together to show the claim is true or false.

4. Language use and computations are at a sufficient level of precision and accuracy

The language used and computations must be at a sufficient level of precision or accuracy to support the argument. Language use needs to be precise enough to communicate the ideas with sufficient clarity.

These criteria are helpful for discussions. It is important not to lose sight of the “big picture” however, and that is whether the argument offered shows that the claim is (or is not) true. This is the goal and purpose of a mathematical argument. You will see in many of these packets that students can approach an argumentation prompt from many different perspectives. It matters less which mathematical tools they use, and matters more whether their chain of reasoning compels the result. !!!!!!!!!!!!!

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!!In this packet you will find !!!

"# A blank copy of the task: "Laura says 1/4 is shaded. Is she right?" and a description of the task implementation and/or other important considerations regarding student work samples included in this packet. !

$# A protocol that can help you and your colleagues discuss student work related to this task. The use of the protocol is optional.!%# Selected work samples on this task from 4th-grade students in classes of teacher participants in the UConn Bridging Math Practices

project to be used with the protocol. ! Work Samples Classification and Commentaries: the student work samples ordered by whether they seem to be high, adequate, or

low quality responses with respect to the criteria described on page 2 along with commentaries that support the classification. Among the samples are some that present a well-structured argument, but have important mathematical flaws, which prevent them from being classified as the highest quality. !

!!Important note: The teachers and project members that discussed these work samples were not always unanimous in their determinations of quality. Although we might even agree on what the student did do, did not do, and strengths of the argument, there were differences in how much “weight” people put on different strengths and weaknesses. Thus, two teachers might see the same things in the student work sample, but one might want to classify the argument as, say, adequate quality and the other as low quality. This points to the importance of professional discussions and talking through the work samples with colleagues. There is no one absolute answer to whether a student work sample is high, adequate or low. Rather, trying to do the categorization leads to important conversations and helps a group clarify strengths, weaknesses, and what we value. That said, the teams reviewing these work samples had focused on argumentation for a year and had some level of shared vision for this work which we think is helpful to share and is reflected in the commentaries. !!

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THE TASK

Source of the task: Adapted from Illustrative Mathematics https://www.illustrativemathematics.org/content-standards/tasks/881

!CONTEXT !This problem was given to 4th-grade students at one school. The purpose of this problem was to see how students combine visual representation of fractions and conceptual understanding of equivalent fractions to justify their answer to the question in the problem. We offer one set of commentaries here. Student expectations may vary based on the manner in which the task is administered (time of the year, classroom norms, and prior knowledge about creating arguments). The student work samples in this set represent only a selection of the whole classroom work. It was apparent from the whole classroom set that students had been exposed to the idea that rearranging shaded parts on a visual representation does not change the value of the fraction it represents. Student work contained a variety of explicit and implicit warrants, and at times no warrants were presented.

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Protocol Guided Sorting Activity: (33–40 mins) Bridging Math Practices Math-Science Partnership Grant #$%&!'()*)+),!-.&!+(/.*/0!1)(!*$/!'2(')&/!)1!(/3%/-%45!&*20/4*!-)(67!8*!%&!9)0%1%/0!1()9!*-)!)1!*$/!'(/3%)2&,:!'(/&/4*/0!'()*)+),&!%4!*$/!;.4+$/&*/(!2%,0%45!.!+)4&/4&2&!.()240!-$.*!+)24*&!.&!.!$%5$!S2.,%*:!.(529/4*7!!!0. Assign Roles

The Handler – places work samples in agreed-upon pile Facilitator – ensures space is made for all to contribute; supports finding consensus Time Keeper – keeps time and ensures group doesn’t exceed section time limits. Can prompt movement to next section even if full time is not used. All– share ideas and keep notes on own set of work samples

A: Setting the context for discussion (5 mins) #/.9!9/9>/(&!(/.0!.40!0)!*$/!'()>,/97!#/.9!9/9>/(&!0%&+2&&A!D$.*!-.&!*$/!W>%5!%0/.X!)1!*$/!*.&6F.&&/&&9/4*V!D$.*!(/&2,*!)(!+,.%9!4//0/0!Y2&*%1%+.*%)4V!!! B: Quick sort: Reviewing student work (15 mins) =)!.!01#-2'34"*')1!&*20/4*&?!-)(6!>:!*$/!0/5(//!)1!'()1%+%/4+:!T$%5$R!.0/S2.*/R!,)-U!0/9)4&*(.*/0!-%*$!'()3%0%45!.4!.(529/4*!)1!*$/!(/,/3.4*!+,.%9T&U7!#$/!Q.40,/(!',.+/&!.!+)':!)1!*$/!&*20/4*!-)(6!%4*)!*$/!.''()'(%.*/!'%,/!.&!.5(//0!2')4!>:!*$/!5()2'7!Z)2!9.:!%4%*%.,,:!4//0!.!W[)*!'!!!!!!!!

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!"#$%&'()*+#$(HIGH Quality (high quality mathematical argument)

ADEQUATE Quality (adequate mathematical argument)

LOW (low quality mathematical argument)

!!!!!!!!!!

! !

C: Strengths and areas for growth? (5 mins) Group member summarize key ideas from their Sorting Discussion regarding the strengths and areas for growth for individual samples, each group1 (High Quality, Adequate, Low) of samples, or the overall set with respect to the argumentation? (HIGH Quality (high quality mathematical argument)!

ADEQUATE Quality (adequate mathematical argument)!

LOW (low quality mathematical argument)!

!!!!!!!!!!!!

! !

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D: Reading ARP Commentaries (optional: 5-7 mins) As deemed useful, group members read the commentaries in the Argumentation Resource Packet to gain new perspectives on selected student work samples, their strengths and areas for growth, and what counts as a high quality argument. E: Reflection (5 mins) Each person shares The facilitator guides the group to take turns in sharing a reflection. Group may decide to reflect on the same question, or each share a take away. a. What did you learn about argumentation and how students engage argumentation from

looking at the work of these students? You might also consider aspects of task design. !b. Did you have any ah hah moments? c. What questions remain for you? What would you like to lean more about? d. What will you take away from this discussion back to your classroom? What ideas

might impact your planning or teaching? !!F: Reflection on Protocol Implementation (3 mins)!Facilitator guides a reflection on how the protocol process worked. Group members contribute ideas. Members make suggestions for modifications to future protocol as needed.!!

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