1. Is it true? Always? Supporting
Reasoning-and-Proof Focused
Collaboration among Teachers
2013 NCTM Annual Conference
Denver, Colorado
Nicole Miller Rigelman * Portland State University
rigelman@pdx.edu
http://goo.gl/ys4Qd

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3. Session Overview
In this session we will:
Unpack what it is meant by reasoning-and-proving
and consider what it looks like to prompt such
thinking.
Explore tools to support teacher collaboration with
selecting tasks, planning, and observing/examining
classroom practice and student thinking with an eye
on supporting students with developing convincing
arguments.

4. Why focus on reasoning and proof in
your PLC?
Growing consensus in the community that it should
be “a natural, ongoing part of classroom discussions,
no matter what topic is being studied,” (NCTM, 2000,
p. 342).
What gets in the way?
It is difficult for teachers and student.
We have a limited vision of what counts as reasoningand-proving, particularly at the elementary grades.

5. Reasoning and Proof
Instructional programs [preK-12] should enable students to
develop and evaluate mathematical arguments and proofs.
- NCTM, 2000, p. 56
[In grades 3-5] mathematical reasoning develops in
classrooms where students are encouraged to put forth
their own ideas for examination. Teachers and students
should be open to questions, reactions, and elaborations
from others in the classroom. Students need to explain and
justify their thinking and learn how to detect fallacies and
critique others thinking.
- NCTM, 2000, p. 188

6. Reasoning and Proof
…both plausible and flawed arguments that are offered by
students create an opportunity for discussion. As students
move through the grades, they should compare their ideas
with others’ ideas, which may cause them to modify,
consolidate, or strengthen their arguments or reasoning.
Classrooms in which students are encouraged to present
their thinking, and in which everyone contributes by
evaluating one another’s thinking, provide rich environments
for learning mathematical reasoning.
- NCTM, 2000, p. 58

7. Reasoning and Proof
In the domain of Number and Operation…
Computation strategy. Purposeful manipulations that
may be chosen for specific problems, may not have a
fixed order, and may be aimed at converting one
problem into another.
Computation algorithm. A set of predefined steps
applicable to a class of problems that gives the
correct result in every case when the steps are carried
out correctly.
- K-5 Progressions, Number and Operations in Base Ten, 2011, p. 3

8. Standards 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 for Mathematics, 2010, pp. 6-7

9. Standards 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 for Mathematics, 2010, pp. 6-7

10. Mathematics Task Framework
Cognitive Demand
Design
Set-Up
Implementation
Student
Learning
Stein et al, 1998

11. Mathematics Task Framework
Mathematical task
as represented in
curricular/
instructional
materials
Mathematical task
as set up by the
teacher in the
classroom
*Task features
*Cognitive
demands
Factors
influencing setup
*Teachers’ goals
*Teachers’ knowledge
of subject matter
*Teachers’ knowledge
of students
Mathematical task as
implemented by
students in the
classroom
*Enactment of task
features
*Cognitive processing
Factors
influencing students’
implementation
*Classroom norms
*Teachers’ instructional
dispositions
*Students’ learning
dispositions
Students’
Learning
outcomes
Relationships among
various task-related
variables and students’
learning outcomes.
Henningsen and Stein,
1997, p. 528; Stein and
Smith, 1998, p. 270

12. Visiting Grandma
Jamie’s family visited their grandmother
who lives 634 miles from their house. On
the first day they drove 319 miles. How
many miles did they have left to drive the
second day?
From Investigations in Number, Data, and Space,
Russell & Economopoulos, 2008

13. Task Analysis Guide
Henningsen and Stein, 1997, p. 528; Stein and
Smith, 1998, p. 270
Memorization
•
•
•
involve either reproducing previously learned facts, rules,
formulas, or definitions or committing facts, rules,
formulas or definitions to memory.
cannot be solved using procedures because a procedure
does not exist or because a time frame in which the task is
being completed is too short to use a procedure.
are not ambiguous. Such tasks involve exact reproduction
of previously seen material, and what is to be reproduced
is clearly and directly stated.
Procedures with Connections
•
•
•
•
Procedures without Connections
•
•
•
•
•
are algorithmic. Use of the procedure either is specifically
called for or is evident from prior instruction, experience, or
placement of the task.
require limited cognitive demand for successful completion.
Little ambiguity exists about what needs to be done and
how to do it.
have no connection to the concepts or meaning that
underlie the procedure being used.
are focused on producing correct answers instead of on
developing mathematical understanding.
require no explanations or explanations that focus sole on
describing the procedure that was used.
focus students’ attention on the use of procedures for the purpose of
developing deeper levels of understanding of mathematical concepts
and ideas.
suggest explicitly or implicitly pathways to follow that are broad
general procedures that have close connections to underlying
concepts.
usually are represented in multiple ways, such as visual diagrams,
manipulatives, symbols, and problem situations. Making connections
among multiple representations helps develop meaning.
require some degree of cognitive effort. Although general procedures
may be followed, they cannot be followed mindlessly. Students need
to engage with conceptual ideas that underlie the procedures to
complete the task successfully and that develop understanding.
Doing Mathematics
•
•
•
•
•
require complex and non-algorithmic thinking -- a predictable, wellrehearsed approach or pathway is not explicitly suggested by the
task, task instructions, or a worked-out example.
require students to explore and understand the nature of
mathematical concepts, processes, or relationships.
demand self-monitoring or self-regulation of one’s own cognitive
processes.
require students to access relevant knowledge and experiences and
make appropriate use them in working through the task.
require students to analyze the task and actively examine task
constraints that may limit possible solution strategies and solutions.

14. Task Analysis Guide
Henningsen and Stein, 1997, p. 528; Stein and
Smith, 1998, p. 270
Procedures with Connections
• focus students’ attention on the use of procedures for the purpose
of developing deeper levels of understanding of mathematical
concepts and ideas.
• suggest explicitly or implicitly pathways to follow that are broad
general procedures that have close connections to underlying
concepts.
• usually are represented in multiple ways, such as visual diagrams,
manipulatives, symbols, and problem situations. Making
connections among multiple representations helps develop
meaning.
• require some degree of cognitive effort. Although general
procedures may be followed, they cannot be followed mindlessly.
Students need to engage with conceptual ideas that underlie the
procedures to complete the task successfully and that develop
understanding.

15. The Five+ Practices
0. Setting Goals and Selecting the task
1. Anticipating (e.g., Fernandez & Yoshida, 2004; Schoenfeld, 1998)
2. Monitoring (e.g., Hodge & Cobb, 2003; Nelson, 2001; Shifter, 2001)
3. Selecting (Lampert, 2001; Stigler & Hiebert, 1999)
4. Sequencing (Schoenfeld, 1998)
5. Connecting (e.g., Ball, 2001; Brendehur & Frykholm, 2000)

21. Orchestrating Productive Mathematical Discourse
Chart for Monitoring, Selecting, Sequencing, and Connecting Student Thinking
Strategy
A.
B.
C.
D.
Work of Specific Students
Sequence
Compare

22. Student Justifications
Students’ justifications can be separated into three
broad classes:
appeal to authority;
justification by example;
generalizable arguments.
Carpenter, Franke, Levi, 2003, p. 87

23. Discourse Types
Type 1 – Answering, Stating, or Sharing
Type 2 – Explaining
Type 3 – Questioning or Challenging
Type 4 – Relating, Predicting, or Conjecturing
Type 5 – Justifying or Generalizing
- Weaver, Dick, & Rigelman, 2005

24. Discourse Types
Type 1 – Answering, Stating, or Sharing A student
gives a short right or wrong answer to a direct
question or makes a simple statement or shares work
that does not involve an explanation of how or why.
Type 2 – Explaining
Type 3 – Questioning or Challenging
Type 4 – Relating, Predicting, or Conjecturing
Type 5 – Justifying or Generalizing
- Weaver, Dick, & Rigelman, 2005

25. Discourse Types
Type 1 – Answering, Stating, or Sharing
Type 2 – Explaining A student explains a
mathematical idea or procedure by describing how or
what he or she did but does not explain why.
Type 3 – Questioning or Challenging
Type 4 – Relating, Predicting, or Conjecturing
Type 5 – Justifying or Generalizing
- Weaver, Dick, & Rigelman, 2005

26. Discourse Types
Type 1 – Answering, Stating, or Sharing
Type 2 – Explaining
Type 3 – Questioning or Challenging A student asks a
question to clarify his or her understanding of a
mathematical idea or procedure or makes a
statement or asks a question in a way that challenges
the validity of an idea or procedure.
Type 4 – Relating, Predicting, or Conjecturing
Type 5 – Justifying or Generalizing
- Weaver, Dick, & Rigelman, 2005

27. Discourse Types
Type 1 – Answering, Stating, or Sharing
Type 2 – Explaining
Type 3 – Questioning or Challenging
Type 4 – Relating, Predicting, or Conjecturing A
student makes a statement indicating that he or she
has made a connection or sees a relationship to some
prior knowledge or experience or makes a prediction
or a conjecture based on an understanding of the
mathematics behind the problem.
Type 5 – Justifying or Generalizing
- Weaver, Dick, & Rigelman, 2005

28. Discourse Types
Type 1 – Answering, Stating, or Sharing
Type 2 – Explaining
Type 3 – Questioning or Challenging
Type 4 – Relating, Predicting, or Conjecturing
Type 5 – Justifying or Generalizing A student
provides a justification for the validity of a
mathematical idea or procedure or makes a
statement that is evidence of a shift from a specific
example to the general case.
- Weaver, Dick, & Rigelman, 2005

29. Discourse Analysis Tool
Discourse Types
Type 1 – Answering, Stating, or
Sharing A student gives a short right
or wrong answer to a direct question or
makes a simple statement or shares
work that does not involve an
explanation of how or why.
Predicted
Actual
Type 2 – Explaining A student
explains a mathematical idea or
procedure by describing how or what
he or she did but does not explain why.
Type 3 – Questioning or Challenging
A student asks a question to clarify his
or her understanding of a mathematical
idea or procedure or makes a statement
or asks a question in a way that
challenges the validity of an idea or
procedure.
Type 4 – Relating, Predicting, or
Conjecturing A student makes a
statement indicating that he or she has
made a connection or sees a
relationship to some prior knowledge
or experience or makes a prediction or
a conjecture based on an understanding
of the mathematics behind the
problem.
Type 5 – Justifying or Generalizing
A student provides a justification for
the validity of a mathematical idea or
procedure or makes a statement that is
evidence of a shift from a specific
example to the general case.
For more information about the Student Discourse Observation Protocol see, Weaver, D. & Dick, T. (September 2006). Assessing the Quantity and Quality of Student Discourse in Mathematics
Classrooms, Year 1 Results. Paper presented at Math Science Partnership Evaluation Summit II, Minneapolis, MN. Available at: http://ormath.mspnet.org/index.cfm/14122.

30. - Boaler &
Humphreys,
p. 37, 2005

31. Criteria for Representation-Based Proof
The meaning of the operation(s) involved is
represented in diagrams, manipulatives, or story
contexts.
The representation can accommodate a class of
instances (for example, all whole numbers).
The conclusion of the claim follows from the
structure of the representation.
- Schifter, Bastable, & Russell, 2008

32. Rate this presentation on the
conference app.
www.nctm.org/confapp
Download available presentation
handouts from the Online Planner!
www.nctm.org/planner
Join the conversation! Tweet us
using the hashtag #NCTMDenver