```Supporting Rigorous Mathematics
Teaching and Learning
Constructing an Argument and Critiquing the
Reasoning of Others
Tennessee Department of Education
Elementary School Mathematics
© 2013 UNIVERSITY OF PITTSBURGH
Mathematical Understandings
[In the TIMSS report the fact] that 89% of the U.S. lessons’
content received the lowest quality rating suggests a general
lack of attention among teachers to the ideas students
develop. Instead, U.S. lessons tended to focus on having
students do things and remember what they have done. Little
emphasis was placed on having students develop robust ideas
that could be generalized. The emergence of conversations
about goals of instruction – understandings we intend that
students develop – is an important catalyst for changing the
present situation.
Thompson and Saldanha (2003). Fractions and Multiplicative Reasoning. In Kilpatrick et al. (Eds.), Research companion
to the principles and standards for school mathematics, Reston: NCTM. P. 96.
In this module, we will analyze student reasoning to determine
attributes of student responses and then we will consider how
teachers can scaffold student reasoning.
2
Session Goals
Participants will learn about:
• elements of Mathematical Practice Standard 3;
• students’ mathematical reasoning that is clear,
faulty, or unclear;
• teachers’ questioning focused on mathematical
reasoning; and
• strategies for supporting writing.
© 2013 UNIVERSITY OF PITTSBURGH
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Overview of Activities
Participants will:
• analyze a video and discuss students’
mathematical reasoning that is clear, faulty, or
unclear;
• analyze student work to differentiate between
mathematical reasoning; and
• study strategies for supporting writing.
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Making Sense of Mathematical Practice
Standard 3
Study Mathematical Practice Standard 3: Construct a
viable argument and critique the reasoning of others,
and summarize the authors’ key messages.
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Common Core State Standards:
Mathematical Practice Standard 3
The Common Core State Standards recommend that students:
•
construct viable arguments and critique the reasoning of others;
•
use stated assumptions, definitions, and previously established
results in constructing arguments;
•
make conjectures and build a logical progression of statements to
explore the truth of their conjectures;
•
recognize and use counterexamples;
•
justify conclusions, communicate them to others, and respond to
the arguments of others;
•
reason inductively about data, making plausible arguments that
take into account the context from which the data arose; and
•
compare the effectiveness of two plausible arguments, distinguish
correct logic or reasoning from that which is flawed, and—if there
is a flaw in an argument—explain what it is.
Modified from the Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO
6
NCTM Focal Points:
Reasoning and sense making are of particular
importance, but historically “reasoning” has been limited
to very selected areas of the high school curriculum,
and sense making is in many instances not present at
all. However, an emphasis on student reasoning and
sense making can help students organize their
knowledge in ways that enhance the development of
number sense, algebraic fluency, functional
relationships, geometric reasoning, and statistical
thinking.
NCTM, 2008, Focus in High School Mathematics:
Reasoning and Sense Making
7
The Relationship Between Talk and Understanding
We come to an understanding in the course of communicating it.
That is to say, we set out by offering an understanding and that
understanding takes shape as we work on it to share it. And finally
we may arrive cooperatively at a joint understanding as we talk or
in some other way interact with someone else (p. 115).
This view is supported by Chin and Osborne‘s (2008) study. They
state that when students engage socially in talk activities about
shared ideas or problems, students must be given ample
opportunities for formulating their own ideas about science
concepts, for inferring relationships between and among these
concepts, and for combining them into an increasingly more
complex network of theoretical propositions. For Hand (2008), the
oral language component is heavily emphasized in the social
negotiated processes in which students exchange, challenge, and
debate arguments in order to reach a consensus.
(Chen, Ying Chih, 2011 Examining the integration of talk and writing for student knowledge construction through
argumentation.)
8
Determining Student Understanding
What will you need to see and hear to know that
students understand the concepts of a lesson?
Watch the video. Be prepared to say what students
know or do not know. Cite evidence from the lesson.
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Context for the Lesson
Teacher:
Rebecca Few
Principal:
Roseanne Barton
School:
John Pittard Elementary School
School District:
Murfreesboro, Tennessee
4
Date:
February 4, 2013
The teacher is a TN Common Core Coach. She has been working
on both the Mathematical Content and Mathematical Practice
Standards. She is interested in gaining a better understanding of
ways of encouraging classroom talk.
© 2013 UNIVERSITY OF PITTSBURGH
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1
4
Jolla has of a pizza.
Sarah has
Maria has
30
of a pizza.
100
3
of a pizza.
100
Tim’s pizza is shaded on the pizza. How much pizza is Tim’s
share?
Jake has
Juan has
3
of a pizza.
10
1
of a pizza.
5
1. Show each of the student’s amount of pizza.
2. Compare the students’ amounts of pizza. Explain with
words and use the >, <, or = symbols to show who has the
most pizza.
3. Explain with words and use the >, <, or = symbols to show
who has the least amount of pizza.
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The Pizza Task (continued)
Jolla’s Pizza
Tim’s Pizza
Juan’s Pizza
Sarah’s Pizza
Maria’s Pizza
Jake’s Pizza
© 2013 UNIVERSITY OF PITTSBURGH
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The CCSS for Mathematics: Grade 4
Number and Operations – Fractions
4.NF
Extend understanding of fraction equivalence and ordering.
4.NF.A.1
Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by
using visual fraction models, with attention to how the number and size
of the parts differ even though the two fractions themselves are the same
size. Use this principle to recognize and generate equivalent fractions.
4.NF.A.2
Compare two fractions with different numerators and different
denominators, e.g., by creating common denominators or numerators, or
by comparing to a benchmark fraction such as 1/2. Recognize that
comparisons are valid only when the two fractions refer to the same
whole. Record the results of comparisons with symbols >, =, or <, and
justify the conclusions, e.g., by using a visual fraction model.
Common Core State Standards, 2010, p. 30, NGA Center/CCSSO
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The CCSS for Mathematics: Grade 4
Number and Operations – Fractions
4.NF
Understand decimal notation for fractions, and compare decimal
fractions.
4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction
with
denominator 100, and use this technique to add two fractions with
respective denominators 10 and 100. For example, express 3/10 as
30/100, and add 3/10 + 4/100 = 34/100.
4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100. For
example, rewrite 0.62 as 62/100; describe a length as 0.62 meters;
locate 0.62 on a number line diagram.
4.NF.C.7
Compare two decimals to hundredths by reasoning about their size.
Recognize that comparisons are valid only when the two decimals
refer to the same whole. Record the results of comparisons with the
symbols >, =, or <, and justify the conclusions, e.g., by using a visual
model.
Common Core State Standards, 2010, p. 31, NGA Center/CCSSO
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Essential Understandings
Essential Understanding
Equal Size Pieces
A fraction describes the division of a whole or unit (region, set,
segment) into equal parts. A fraction is relative to the size of the whole
or unit.
Meaning of the Denominator
The larger the name of the denominator, the smaller the size of the
piece.
Use of Benchmarks
Comparison to known benchmark quantities can help students
determine the relative size of a fractional piece because the
benchmark quantity can clearly be seen as smaller or larger than the
piece. One significant benchmark quantity is one-half.
Equivalency
A fraction can be named in more than one way and the fractions will be
equivalent as long as the same portion of the set or area of the figure
is represented.
Creating Equivalent Fractions
When the denominator is multiplied or divided then the numerator is
automatically divided into the same number of pieces because it is a
subcomponent of the denominator.
© 2013 UNIVERSITY OF PITTSBURGH
CCSS
4.NF.C.7
4.NF.A.2
4.NF.C.7
4.NF.A.2
4.NF.C.7
4.NF.C.5
4.NF.A.1
15
The CCSS for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of
others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO
16
Determining Student Understanding
(Small Group Work)
What did students know and what is your evidence?
Where in the lesson do you need additional information
to know if students understood the mathematics or the
model?
Cite evidence from the lesson.
© 2013 UNIVERSITY OF PITTSBURGH
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Determining Student Understanding
(Whole Group Discussion)
In what ways did students make use of the third
Mathematical Practice Standard?
Let’s step back now and identify ways in which student
understanding shifted or changed during the lesson.
Did student understanding evolve over the course of the
lesson? If so, what ideas did you see changing over
time?
What do you think was causing the shifts?
© 2013 UNIVERSITY OF PITTSBURGH
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Common Core State Standards: Mathematical
Practice Standard 3
(Whole Group Discussion)
How many of the MP3 elements did we observe and if not, what are
we wondering about since this was just a short segment?
The Common Core State Standards recommend that students:
• construct viable arguments and critique the reasoning of others;
• use stated assumptions, definitions, and previously established results
in constructing arguments;
• make conjectures and build a logical progression of statements to
explore the truth of their conjectures;
• recognize and use counterexamples;
• justify conclusions, communicate them to others, and respond to the
arguments of others;
• reason inductively about data, making plausible arguments that take
into account the context from which the data arose; and
• compare the effectiveness of two plausible arguments, distinguish
correct logic or reasoning from that which is flawed, and—if there is a
flaw in an argument—explain what it is.
Modified from the Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO
19
Imagine Publicly Marking Student
Behavior
In addition to stressing the importance of effort, the
teachers were very clear about the particular ways of
working in which students needed to engage. D. Cohen
and Ball (2001) described ways of working that are
needed for learning as learning practices. For example,
the teachers would stop the students as they were
working and talking to point out valuable ways in which
they were working.
(Boaler, (2001) How a Detracked Math Approach Promoted Respect, Responsibility and High Achievement.)
20
Talk is NOT GOOD ENOUGH
Writing is NEEDED!
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The Writing Process
In the writing process, students begin to gather,
formulate, and organize old and new knowledge,
concepts, and strategies, to synthesize this information
as a new structure that becomes a part of their own
knowledge network.
Nahrgang & Petersen, 1998
When writing, students feel empowered as learners
because they learn to take charge of their learning by
increasing their access to and control of their thoughts.
Weissglass, Mumme, & Cronin, 1990
22
Talk Alone is NOT GOOD ENOUGH!
Several researchers have reported that students tend to process
information on a surface level when they only use talk as a learning
tool in the context of science education.
(Hogan, 1999; Kelly, Druker, & Chen, 1998; McNeill & Pimentel, 2010)
After examining all classroom discussions without writing support, they
concluded that persuasive interactions only occurred regularly in one
teacher’s classroom. In the other two classes, the students rarely
responded to their peers by using their claims, evidence, and
reasoning. Most of the time, students were simply seeking the correct
answers to respond to teachers’ or peers’ questions. Current research
also suggests that students have a great deal of difficulty revising
ideas through argumentative discourse alone.
(Berland & Reiser, 2011; D. Kuhn, Black, Keselman, & Kaplan, 2000)
Writing involves understanding the processes involved in producing
and evaluating thoughts rather than the processes involved in
translating thoughts into language.
(Galbraith, Waes, and Torrance (2007, p. 3).
(Chen, Ying Chih, 2011 Examining the integration of talk and writing for student knowledge construction through argumentation.)
23
The Importance of Writing
Yore and Treagust (2006) note that writing plays an
important role―to document ownership of these claims,
to reveal patterns of events and arguments, and to
connect and position claims within canonical science
(p.296). That is, the writing undertaken as a critical role
of the argumentative process requires students to build
connections between the elements of the argument
(question, claim, and evidence).
When students write, they reflect on their thinking and
come to a better understanding of what they know and
what gaps remain in their knowledge (Rivard, 1994).
(Chen, Ying Chih, 2011 Examining the integration of talk and writing for student knowledge construction through
argumentation.)
24
Writing Assists Teachers, TOO
Writing assists the teacher in thinking about the child as
learner. It is a glimpse of the child’s reality, allowing the
teacher to set up new situations for children to explain
and build their mathematics understanding.
Weissglass, Mumme, & Cronin, 1990
25
Analyzing Student Work
• Analyze the student work.
• Sort the work into two groups—work that shows
mathematical reasoning and work that does not show
sound mathematical reasoning.
What can be learned about student thinking in
each of these groups, the group showing reasoning
and the group that does not show sound reasoning?
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Student 1
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Student 2
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Student 3
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Student 4
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Student 5
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Student 6
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Student 7
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Essential Understandings
Essential Understanding
Equal Size Pieces
A fraction describes the division of a whole or unit (region, set, segment) into equal
parts. A fraction is relative to the size of the whole or unit.
Meaning of the Denominator
The larger the name of the denominator, the smaller the size of the piece.
Use of Benchmarks
Comparison to known benchmark quantities can help students determine the
relative size of a fractional piece because the benchmark quantity can clearly be
seen as smaller or larger than the piece. One significant benchmark quantity is onehalf.
Equivalency
A fraction can be named in more than one way and the fractions will be equivalent
as long as the same portion of the set or area of the figure is represented.
Creating Equivalent Fractions
When the denominator is multiplied or divided then the numerator is automatically
divided into the same number of pieces because it is a subcomponent of the
denominator.
© 2013 UNIVERSITY OF PITTSBURGH
34
Common Core State Standards:
Mathematical Practice Standard 3
The Common Core State Standards recommend that students:
• construct viable arguments and critique the reasoning of others;
• use stated assumptions, definitions, and previously established
results in constructing arguments;
• make conjectures and build a logical progression of statements
to explore the truth of their conjectures;
• recognize and use counterexamples;
• justify conclusions, communicate them to others, and respond to
the arguments of others;
• reason inductively about data, making plausible arguments that
take into account the context from which the data arose; and
• compare the effectiveness of two plausible arguments,
distinguish correct logic or reasoning from that which is flawed,
and—if there is a flaw in an argument—explain what it is.
Modified from the Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO
35
Two Forms of Writing
Consider the forms of writing below. What is the
purpose of each form of writing? How do they differ from
each other?
when solving a problem
• Writing about the meaning of a mathematical
concept/idea or relationships
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A Balance: Writing About Process
Versus Writing About Reasoning
Students and groups who seemed preoccupied with
“doing” typically did not do well compared with their
peers. Beneficial considerations tended to be
conceptual in nature, focusing on thinking about ways
to think about the situations (e.g., relationships among
“givens” or interpretations of “givens” or “goals” rather
than ways to get from “givens” to “goals”).
This conceptual versus procedural distinction was
especially important during the early stages of solution
attempts when students’ conceptual models were more
unstable.
Lesh & Zawojewski, 1983
37
Strategies for Supporting Writing
© 2013 UNIVERSITY OF PITTSBURGH
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Strategies for Supporting Writing
How might use of these processes or strategies assist
students in writing about mathematics? Record your
responses on the recording sheet in your participant handout.
Reflect on the potential benefit of using strategies to support
writing.
1. Make Time for the Think-Talk-Reflect-Write Process
2. The Use of Multiple Representations
3. Construct a Concept Web with Students
4. Co-Construct Criteria for Quality Math Work
5. Engage Students in Doing Quick Writes
6. Encourage Pattern Finding and Formulating and Testing
Conjectures
© 2013 UNIVERSITY OF PITTSBURGH
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1. Make Time for the Think-Talk-ReflectWrite Process
Think:
Work privately to prepare a written response to one of
the prompts, “What is division?”
Talk:
What is division? Keep a written record of the ideas
shared.
Reflect:
Reflect privately. Consider the ideas raised. How do
they connect with one another? Which ideas help you
understand the concept better?
Write:
Write an explanation for the question, “What is
division?” Think about what everyone in your group
said, and then use words, pictures, and examples to
explain what division means. Go ahead and write.
Hunker & Lauglin, 1996
40
2. Encourage the Use of Multiple
Representations of Mathematical Ideas
Pictures
Manipulative
Written
Models
Symbols
Real-world
Situations
Oral & Written
Language
Modified from Van De Walle,
2004, p. 30
41
3. Construct a Concept Web with
Students
• Analyze the concept web. Students developed the
concept web with the teacher over the course of
several months.
• How might developing and referencing a concept web
help students when they are asked to write about
mathematical ideas?
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3. Construct a Concept Web with
Students (continued)
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4. Co-Construct Criteria for Quality Math
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Students worked to solve
high-level tasks for several
weeks. The teacher asked
questions daily. Throughout
the week, the teacher
pressed students to do
quality work. After several
days of work, the teacher
showed the students a
quality piece of work, told
them that the work was
“quality work,” and asked
them to identify what the
characteristics of the work
were that made it quality
work. Together, they
generated this list of criteria.44
© 2013 UNIVERSITY OF PITTSBURGH
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5. Engage Students in Doing Quick
Writes
A Quick Write is a narrow prompt given to students after
they have studied a concept and should have gained
some understanding of the concept.
Some types of quick writes might include:
• compare concepts;
• use a strategy or compare strategies;
• reflect on a misconception; and
• write about a generalization.
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Brainstorming Quick Writes:
What are some Quick Writes that you can ask students
to respond to for your focus concept?
© 2013 UNIVERSITY OF PITTSBURGH
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6. Encourage Pattern Finding and
Formulating and Testing Conjectures
• How might students benefit from having their
conjectures recorded?
• What message are you sending when you honor and
record students’ conjectures?
• Why should we make it possible for students to
investigate their conjectures?
© 2013 UNIVERSITY OF PITTSBURGH
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Checking In: Construct Viable Arguments and
Critique the Reasoning of Others
How many of the MP3 elements did we observe and if not, what are
we wondering about since this was just a short segment?
The Common Core State Standards recommend that students:
• construct viable arguments and critique the reasoning of others;
• use stated assumptions, definitions, and previously established
results in constructing arguments;
• make conjectures and build a logical progression of statements
to explore the truth of their conjectures;
• recognize and use counterexamples;
• justify conclusions, communicate them to others, and respond
to the arguments of others;
• reason inductively about data, making plausible arguments that
take into account the context from which the data arose; and
• compare the effectiveness of two plausible arguments,
distinguish correct logic or reasoning from that which is flawed,
and—if there is a flaw in an argument—explain what it is.
Modified from the Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO
49
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