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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.
!"#$"%&"'&%(%)&%&*"+,+-%.&%&*,%$)/*01!!
! ! !"#$%!&'$()*+!*&!%#,!',(%$+-.,!/,.*0!12!2#$3,34!!
!!!!!5$6'$!2$72!%#$%!8!*&!%#,!',(%$+-.,!12!2#$3,39!:*!7*6!%#1+;!2#,!12!(*'',(%4!!!!!:,&,+3!7*6'!$+20,'9!!!!
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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)!
!!!!!!!!!!!!
! !
!$#,&'$*-(".,#+//(0"#($*,(1/+--!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#!$%&'!()*'+&,-!&'!.%/0'*1!&-!+*/2'!,3!4')56/,).'78!9,)!20:!,/!20:!-,+!5*!05;*!+,!
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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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