Catherine Yeulet/iStockphoto.com
Discourse is the mathematical communication that occurs
in a classroom. Effective discourse happens when students
articulate their own ideas and seriously consider their
peers’ mathematical perspectives as a way to construct
mathematical understandings. Encouraging students to
construct their own mathematical understanding through
discourse is an effective way to teach mathematics,
especially since the role of the teacher has transformed
from being a transmitter of knowledge to one who
presents worthwhile and engaging mathematical tasks.
Professional Standards for Teaching Mathematics
(NCTM 1991) identifies Communication, with discourse
as a key component, as one of the six Standards for
teaching mathematics. The questions below may stimulate your thinking about this topic.
• How do you choose tasks and/or questions that
engage and challenge all students’ thinking in your
classroom? How do you ensure that these tasks
remain at this level?
• How do you encourage your students to listen carefully to one another’s ideas? To disagree? To question?
• How do you decide whether or not to pursue a mathematical idea? How do you decide when to give more
information or let students grapple with their ideas?
• How do you help students clarify and justify their ideas?
• How does discourse encourage reasoning and sense
making in your classroom?
• How do teacher-to-student conversations in your
classroom become student-to-student conversations?
How do you give more ownership of the classroom to
students?
The Editorial Panel of Mathematics Teaching in the
Middle School (MTMS) encourages readers to submit
manuscripts concerning discourse. We especially invite
responses from middle school classroom teachers who
are incorporating action research into their practice to
reflect on how discourse impacts students’ learning and
understanding.
Send submissions to this open-ended call for
manuscripts by accessing mtms.msubmit.net. On the
Keywords, Categories, Special Sections tab, select this
specific call from the list in the Department/Call section.
mathematics
Middle School
teaching in the
Call for Manuscripts
Discourse
Copyright © 2010 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Linear Functions
What is the slope of the line passing
Solving Quadratic
Equations
What is the solution set of the
System of Equations
through (-3, 2) and (6, 7)?
equation 6x2 -22x – 28?
Use elimination to solve: 2x + 5y = 7
3x + 6y = 3
Find the range for the data listed
Statistics
Statistics
Common Core Math I
25
20
37
41
44
45
25
19
13
2
34
63
Find the image of the polygon, given
Dilations
vertices: F(3, 4), G(6, 10), H(-3, 5)
after a dilation centered at the origin
with a scale factor of 2
Exponential Functions
Find the annual percent increase that is
modeled in: y = 1298(1.63)x
Find x. Round to the nearest tenth.
Trigonometric Ratios
x
24°
19
Statistics
Probability
Common Core Math II
Find the probability of x successes in n
trial for the given probability of
success p:
x = 4, n = 10, p = 0.2
Logarithmic Functions
Find the inverse of: y = log4 x
Trigonometric Identities
Simplify:
cos θ + sin θ tan θ
Sequences
Fundamental
Theorem of
Statistics
Algebra
Common Core Math III
Is the following sequence geometric?
If so identify the common ratio and
find the next 2 terms:
1, 2, 4, 8 . . .
Find all the zeros for:
y = 2x3 + x2 + 1
"
)
[p.1]
~
Handout 5.h
)
\
Math Talk Learning Community Rubric
Describing Levels and Components of a Math-Talk Learning Community
Source: Reprinted with permission from Journal for Research in Mathematics Education, copyright © 2004 by the National Council of Teachers of Mathematics. All rights reserved
Overview of Shift over Levels 0-3: The classroom community grows to support students acting in central or leading roles and shifts from a focus on answers to a
focus on mathematical think' .•• .::1.
A. Questioning
D. Responsibility
for learning
B. Explaining mathematical
C. Source of mathematical
thinking
ideas
Shift from teacher as
Students increasingly take responsibility for learning
Students increasingly explain and
Shift from teacher as the source of
questioner to students and
and evaluation of others and self. Math sense
articulate their math ideas.
all math ideas to students' ideas
teacher as questioners.
become the criterion for evaluation
also influencing direction of lesson.
A.
Questioning
Teacher is the only questioner.
Short frequent questions
function to keep students
listening and paying attention
to the teacher.
Students give short answers
and respond to the teacher
only. No student-to-student
math talk.
Level
her-d'
-~ - -- 0:
- - Trad'"
_. _. __ ._---_ .. -- --_ ..
- .... _---- dcl_._--_.__. -_._-ith _.brief
--- _ ... _--_.
-------Explaining mathematical
C. Source of mathematical
thlnklnq
ideas
No or minimal teacher elicitation of
Teacher is physically at the board
students thinking, strategies, or
usually chalk in hand, telling and
explanations; teacher expects answershowing students how to do math.
focused responses. Teacher may tell
answers.
No student thinking or strategyStudents respond to math
focused explanation of work. Only
presented by the teacher. They do
answers are given.
not offer their own math ideas.
B.
Level 1 : Teach _. - ~ .. .... -- _._-- -----,.d
...
B. Explaining mathematical
thlnklnq
Teacher questions begin to
Teacher probes student thinking
focus on student thinking and
somewhat. One or two strategies may
focus less on answers.
be elicited. Teacher may fill
Teacher begins to ask followexplanations herself.
up questions about student
methods and answers.
".
Teacher is still the only
auestioner.
As a student answers a
Students give information about their
question, other students listen
math thinking usually as it is probed by
passively or wait for their turn.
the teacher (minimal volunteering of
thoughts). They provide brief
descriptions of their thinkinq.
"
A.
<,
Questioning
~"-'"
-_
.. _._
h _. .- --_ ..
Source of mathematical
ideas
_.- . ~............
C.
Teacher is still the main source of
ideas, though she elicits some
student ideas. Teacher does some
probing to access student ideas.
Some students ideas are raised in
discussions, but are not explored.
,I
Session 5 - How Can Professional Development Enable Teachers to Improve Student Achievement?
.-,,,. Secondary Lenses on Learning: Team Leadership for Mathematics in Middle and High Schools
Corwin Press
Copyright 2009
....
D.
--_
d-
...
--_.
Responsibility
Teacher
directed
students'
showing
for learning
repeats student responses (originally
to her) for the class. Teacher responds to
answers by verifying the correct answer or
the correct answer method.
Students are passive listener; they attempt to
imitate the teacher and do not take responsibility
the learning of their peers or themselves.
..
-_ -_ ...
_.- .... th
_ .. - ... __ ..
h-talk...
D. Responsibility
'--"'--
for
-
for learning
Teacher begins to set up structures to facilitate
students listening to and helping other students.
The teacher alone gives feed back.
Students become more engaged by repeating
what other students say or by helping another
student at the teacher's request. This helping
mostly involves students showing how they solved
'a problem.
"'-,
)
.•
)
)
[p.2]
Level 2: Teacher modeling and helping students build new roles. Some co-teaching and co-learning begins as student-to-student talk increases. Teacher
de or back of th - . ....
B. ExplaininQ mathematical thinking
D. Responsibility
for learning
C. Source of mathematical
ideas
A. QuestioninQ
Teacher follows up on explanations
Teacher encourages students' responsibility
Teacher probes more deeply to learn
Teacher continues to ask
and builds on. them by asking
for understanding the mathematical ideas of
about student thinking and supports
probing questions and also asks
detailed descriptions from students.
students to compare and contrast
others. Teacher asks other students
more open questions. She also
them. Teacher is comfortable using
questions about student work and whether
facilitates student-to-student talk,
Teacher open to and elicits multiple
student errors as opportunities for
they agree or disagree and why.
strategies.
e.g., by asking students to be
learning.
prepared to ask questions about
other students' work.
Students usually give information as it is
Students exhibit confidence about
Students begin to listen to understand one
Students ask questions of one
probed by the teacher with some
their ideas and share their own
another. When the teacher requests, they
another's work on the board,
volunteering of thoughts. They begin to
thinking and strategies even if they
explain other students' ideas in their own
often at the prompting of the
are different from others. Student
words. Helping involves clarifying other
stake a position and articulate more
teacher. Students listen to one
ideas sometimes guide the direction
another so they do not repeat
information in response to probes. They
students' ideas for themselves and others.
explain steps in their thinking by
of the math lesson.
Students imitate and model teacher's probing
questions.
providing fuller descriptions and begin to
in pair work and in whole-class discussions.
defend their answers and methods.
Other students listen supportivelv,
1"""
•••••.••••.••••
J
_ .••••
~ ••• -
~-
••• ..,.
-
,-
_.
__
Level 3: Teacher as co-teacher and co-learner.
A. QuestioninQ
Teacher expects students to ask
one another questions about their
work. The teacher's questions
still may guide the discourse.
---------
Student-to-student talk is studentinitiated, not dependent on the
teacher. Students ask questions
and listen to responses. Many
questions are "Why?" questions
that require justification from the
person answering. Students
repeat their own or other's
questions until satisfied with
answers.
Teacher monitors all that occurs, still fully engaged. Teacher is ready to assist, but now in more peripheral
••• _. "~-'"
'.:':J • _..•.••_ ••
..- .... -'.... '_ ..
B. Explalnlnq mathematical thinklnq
C. Source of mathematical
ideas
D. Responsibility for laarnlnq
Teacher follows along closely to student
Teacher allows for interruptions from
The teacher expects students to be
students during her explanations; she
descriptions of their thinking,
responsible for co-evaluation of everyone's
lets students explain and "own" new
work and thinking. She supports students as
encouraging students to make their
explanations more compete; may ask
strategies. (Teacher is still engaged
they help one another sort out
and deciding what is important to
probing questions to make explanations
misconceptions.
She helps and/or follows up
continue exploring.) Teacher uses
more complete. Teacher stimulates
when needed.
student ideas and methods as the
students to think more deeply about
basis for lessons or mini-extensions.
sireieaies.
Students describe more complete
Students interject their ideas as the
Students listen to understand, then initiate
strategies; they defend and justify their
teacher or other students are
clarifying other students' work and ideas for
teaching, confident that their ideas
answers with little prompting from the
themselves and for others during whole-class
teacher. Students realize that they will
are valued. Students spontaneously
discussions as well as in small group and
be asked questions from other students
compare and contrast and build 011
pair work. Students assist each other in
when they finish, so they are motivated
ideas. Student ideas form part of the
understanding and correcting errors.
content of many math lessons.
and careful to be thorough. Other
students support with active listening.
Session 5 - How Can Professional Development Enable Teachers
Sf'condary Lenses on Learning: Team Leadership for Mathematics
>,:i:
I,
to Improve Student Achievement?
in Middle and High Schools
Corwin Press
Copyright 2009
Question
What should you not write there?
If you have a mapping, what do you put in the left circle and the right circle?
What do I write here?
Do we want to use the 3 twice?
What do we do now? For the mapping?
How do I draw the lines?
How can we tell from the mapping whether they repeat?
How can you see the repeating?
Do we have multiple lines?
Is this a function?
What does the Vertical Line Test do for us?
How do we use that?
Do you remember?
What does that tell us?
How do you know its not a function?
The next section stays flat. What would we say for that?
What words would you use?
Another one?
The next section?
What does this mean to square?
What's a negative times a negative going to be?
What's the domain again?
If they give you the x‐values, what do you think they want us to do?
How do they want us to write this?
How would you find that?
What is f(x) equal to?
What is g(x)?
What do they want us to do with those things?
On which days did Jack shop a Dollar Deals?
How could you tell?
Is there a pattern you saw?
Do you remember what we called that yesterday?
Why do we call that the identity function?
How much did he spend per item at Stuff Mart?
How do you know?
Where does Jack shop most often?
At Dollar Deals, what's the cost for 10 items?
What is the ordered pair?
Why is that? Which is what?
What does that ordered pair mean?
What can you tell me about this function?
Tell me what you know about this function?
What else can you tell me?
What do you mean by that?
Give me something else.
Who's dependent on who here?
Is that what that means?
How does it work?
What did she give you?
Why did she give you $6?
So that makes sense?
Did anybody figure our what goes up here?
Why does it have to be a 2?
Why did it have to be 2 times x?
What could you tell me in comparison?
What are some differeneces between Dollar Deals and Puggly Wuggly?
Which one of the slopes is going to be steeper?
Why?
Why would that make it steeper?
What can you tell me about that store?
What else can you tell me based on what we just said?
How would you be able to tell (which day the manager was out sick)?
What kind of pattern did those two days make?
How do you know it was a Dollar Deals function?
What points is that going to go through?
What can you tell me based on those two lines?
What does that help you see?
Which one is steeper?
How can we tell?
Why is red steeper?
What is continuous data?
Does everyone see the same thing I see?
How were they charged (Customer 2 and 3)?
How could you tell?
What does that mean to you?
What would be the function?
What would be the slope?
What is the slope or cost per item?
How does Fivealiscious compare to Dollar Deals and Puggly Wuggly?
Is it steeper?
Type
Bloom's Level
Yes/No
1‐2 Word Ans
Short Ans
Extended Ans
Rhetorical
Remembering
Understanding
Applying
Analysing
Evaluating
Creating
CLASSROOM DISCOURSE
Students
Are actively engaged in the lesson and
demonstrate their understanding of
mathematics in the way they think, talk
agree, and disagree by:
Posing and answering questions
Student to student
Student to teacher
Listening to the reasoning of others
Making conjectures
Presenting solutions
Exploring examples
Examining counter examples
Using a variety of tools
Using multiple representations
Making connections
Providing logical mathematical arguments
to defend answers
Teachers
Elicit, engage and challenge students
thinking by:
Selecting appropriate worthwhile
mathematical task
Posing questions that require use of higher
order thinking
Asking probing questions to initiate the
thinking process
Listening to students’ ideas
Requiring students to clarify and justify
Facilitating participation of all students
Acknowledging multiple pathways
Allowing appropriate wait time
DATA TOOL
 Comments
 Comments
Adapted from
Field Experience Guide Elementary And Middle School Mathematics Teaching Developmentally,
John A. Van de Walle
OMLI Classroom Observation Protocol
Instructions
About Mathematical Discourse
The OMLI Classroom Observation Protocol is a tool for documenting the quantity and quality of
mathematical discourse that transpires during mathematics lessons observed as part of the OMLI
project. For this research study, we are interested in documenting evidence of mathematical
discourse that engage students in thinking about mathematical concepts and procedures. Several
aspects of this definition require elaboration. First, the observation is looking for evidence of
mathematical thinking among students. The teacher may initiate the discourse and may be
involved in the discussion, but the student is the focus of the observation. The observer should
not document evidence of mathematical thinking on the part of the teacher if it does not engage
students. Second, the evidence must center around mathematical ideas or procedures.
Interactions around classroom logistics or management are not part of mathematical discourse.
Exhibit 1 provides examples of typical classroom activities that are and are not considered
mathematical discourse for the purposes of this study.
Exhibit 1—What Is and Is Not Student Mathematical Discourse
IS Considered Discourse
IS NOT Considered Discourse
A student asks, “I don’t understand how you got that
answer. Could you explain it again?”
The teacher provides an explanation of a mathematical
procedure to the class.
A student explains, “I first added 20 and 40 to get 60.
Then I subtracted 2 and added 3 to get 61.”
The teacher provides further explanation in response to a
student’s question.
A student explains, “I saw that 18 + 43 was the same as
(20 + 40) – 2 + 3.”
Two students discuss the scores of last week’s football
game.
Students write in their journals about their thinking to
solve a problem.
The teacher provides instructions to the class about an
activity they are about to engage in.
A student states, “I think I see a pattern. Each one goes up
by 3 more than the one before it.”
A student asks a question about nonmathematical
procedures related to an assignment such as when the
assignment is due, whether students need to show their
work, and the like .
Two students discuss whether a procedure suggested by a
student will work in all similar situations.
Students practice applying a procedure to solve problems
of a specific type (seat work).
A students challenges an algorithm posed by a student by
saying, “Yes, but how does it work with 37 x 98?”
The teacher provides a counter example to a method posed
by a student.
A student answers a question in response to the teacher.
Notation System for Classroom Discourse
This classroom observation protocol includes a notation system that enables observers to quickly
and accurately record evidence of student discourse. Notation involves recording the mode, type,
and the tools used by the students who are engaged in mathematical discourse in each lesson
August 2005
1
” RMC Research Corporation‹Portland, OR
OMLI Classroom Observation Protocol
observed. The follow section provides a detailed description of each aspect of the notation
system and outlines the method observers should use to record evidence of mathematical
discourse among students.
Mode of Discourse—Mathematical discourse—that is, the act of articulating mathematical ideas
or procedures—may take place in several modes. The observer should identify who the student is
addressing. Exhibit 2 provides the codes, definitions, and descriptions of the various modes that
are applicable in this study.
Exhibit 2—Modes of Mathematical Discourse
Code
Definition
Explanation
T
Student to Teacher
The student primarily addresses the teacher even though the entire class or
group hears the student’s comments.
S
Student to Student
The student addresses another student.
G
Student to Group or Class
The student addresses a small group of students or the entire class.
IR
Individual Reflection
The student documents his or her reflections about mathematics in writing.
Please note that the teacher to student and teacher to group or class modes, although common,
are not listed because they relate to the mathematical thinking of the teacher, not the student.
Types of Discourse—Effective mathematical discourse is an iterative process by which students
engage in a variety of types of discourse at different cognitive levels. Student questions lead to
explanations and justifications that may be challenged and subsequently defended, which might
in turn lead to the formation of new generalizations or conjectures, thereby initiating a new
cycle. Exhibit 3 describes the types of mathematical discourse the observer should document
during classroom observation.
Exhibit 3—Types of Mathematical Discourse
Code
Level
A
1
Answering
A student gives a short answer to a direct question from the teacher or another
student.
S
2
Making a
Statement or
Sharing
A student makes a simple statement or assertion, or shares his or her work with
others and the statement or sharing does not involve an explanation of how or why.
For example, a student reads what she wrote in her journal to the class.
E
3
Explaining
A student explains a mathematical idea or procedure by stating a description of
what he or she did, or how he or she solved a problem, but the explanation does
not provide any justification of the validity of the idea or procedure.
Q
4
Questioning
A student asks a question to clarify his or her understanding of a mathematical idea
or procedure.
C
5
Challenging
A student makes a statement or asks a question in a way that challenges the
validity of a mathematical idea or procedure. The statement may include a counter
example. A challenge requires someone else to reevaluate his or her thinking.
R
6
Relating
A student makes a statement indicating that he or she has made a connection or
sees a relationship to some prior knowledge or experience.
August 2005
Definition
Explanation
2
” RMC Research Corporation‹Portland, OR
OMLI Classroom Observation Protocol
Code
Level
Definition
Explanation
P
7
Predicting or
Conjecturing
A student makes a prediction or a conjecture based on their understanding of the
mathematics behind the problem. For example, a student may recognize a pattern
in a sequence of numbers or make a prediction about what might come next in the
sequence or state a hypothesis a mathematical property they observe in the
problem.
J
8
Justifying
A student provides a justification for the validity of a mathematical idea or
procedure by providing an explanation of the thinking that led him or her to the
idea or procedure. The justification may be in defense of the idea challenged by the
teacher or another student.
G
9
Generalizing
A student makes a statement that is evidence of a shift from a specific example to
the general case.
Tools for Discourse—Students may employ a variety of tools to help them communicate the
mathematical ideas or procedures. The tools they choose to use are important indicators of their
level of sophistication with respect to mathematics. Exhibit 4 describes some of the tools that
students are likely to use.
Exhibit 4—Tools for Mathematical Discourse
Code
Definition
Explanation
V
Verbal
A student communicates mathematical ideas or procedures verbally (orally).
A
Gesturing/Acting
A student makes gestures or other body movements to communicate
mathematical ideas or procedures.
W
Written
A student writes a narrative of mathematical ideas or procedures.
G
Graphs, Charts, Sketches
A student uses tables, graphs, charts, sketches, or other visual aids to depict
mathematical ideas or procedures.
M
Manipulative
A student uses physical objects to model mathematical ideas or procedures.
S
Symbolization
A student uses informal, nonmathematical notation to communicate
mathematical ideas or procedures.
N
Notation
A student uses standard (formal) mathematical notation to communicate
mathematical ideas or procedures.
C
Computers/Calculators
A student uses computers, calculators, the Internet, or other forms of
technology to communicate mathematical ideas or procedures.
O
Other
A student uses tools other that those described above.
Using the Notation—The observer will use the codes that appear in Exhibits 2 through 4 to
document the quantity and quality of the mathematical discourse that occurs among the students
in the classrooms observed. Exhibit 5 provides examples of observer’s notations of evidence of
mathematical discourse along with explanation of each set of notations.
August 2005
3
” RMC Research Corporation‹Portland, OR
OMLI Classroom Observation Protocol
Exhibit 5—Examples of Evidence Notation
Mode
Type
Tools
T
Q
V
A student verbally asked the teacher a question to clarify a mathematical idea or procedure
he or she did not understand.
G
E, J
V, A
A student addressed the class to give a verbal explanation of a mathematical idea or
procedure; the student used hand gestures and the explanation included justification of the
idea or procedure.
S
S
E, J
Q
G
V
A student presented a mathematical idea or procedure to another student using tables and
graphs. The second student asked questions to clarify his or her understanding of the idea or
procedure but did not challenge its validity.
G
G
V
A student shared with the class an observation that he or she made about a pattern in a
number sequence.
IR
E, J
W
Students individually reflected on a mathematical idea or procedure and wrote their thoughts
in their journals.
T
A
V
A student answers a question from the teacher with a correct answer.
S
S
V
A student reads what he wrote in his journal to another student.
G
J
M
A student used manipulatives to build a model to justify a mathematical idea or procedure
and presented the model to the class.
N
Explanation
Students did not engage in any discourse during the lesson episode observed.
S
S
VM
One student in a small group uses a wooded cube to point out (make a statement) that a cube
has 8 corners, 12 edges, and 6 flat surfaces.
G
E
V, G
A student drew a diagram on the board and explained to the class how he or she solved a
mathematics problem.
G
G
V
S
S
S
E, J,
C
J
G, N
N
G
A student verbally shared with the class a generalization or conjecture regarding a
mathematical idea or procedure.
Two students engaged in high-level dialogue over a single mathematical idea. The exchange
involved an explanation and justification by one student, a challenge to the validity by the
other student, followed by a defense of the idea by the first student. The students used
graphs and mathematical notation during the process. (The observer’s notations represent
several exchanges between the 2 students, but all of the exchanges were around a single idea
or procedure.)
Classroom Observations Procedures
Step 1: Schedule Observations
RMC Research staff drew a random sample of 25 participating schools for in-depth evaluation.
Within each school, teachers were randomly selected for periodic observation throughout the
duration of the project. Each graduate student observer was assigned approximately 16 to 18
teachers whom they will observe according to a schedule provided by RMC Research. If a
selected teacher teaches more than one mathematics class, the observer should consult the
teacher to select a class that would best typify the teacher’s practices. The observer should
observe the same class for each subsequent observation during the same school year.
August 2005
4
” RMC Research Corporation‹Portland, OR
OMLI Classroom Observation Protocol
RMC Research will send a letter to the teachers selected to participate in the observations
explaining their involvement and how and why they were selected and inviting them to
participate. Copies of the letters will also be sent to the school principals. The letter will include
a consent form that the teachers will sign and return if they choose to participate. Those teachers
who participate will receive $100 in 2 installments.
RMC Research will notify the appropriate observer once a teacher agrees to participate. At that
point the observer should follow up with a telephone call to schedule the exact date and time for
the observation. Observers must remember to schedule time for both the pre- and
postobservation interviews and the observation itself. Contact information for teachers is
available on the OMLI Professional Development Database (www.rmccorp.com/OMLI).
Step 2: Prepare for the Observation
Observers may find the following tips helpful when preparing for an observation:
ƒ
ƒ
ƒ
ƒ
ƒ
Make sure you have enough copies of the Discourse and Summary forms. You will need
one copy of the Classroom Observation Summary Form for each observation but will
likely need several copies of the Classroom Observation Discourse Form for each
observation.
Bring a tablet for taking notes, pencils and pens, and possibly a clipboard.
Be sure you know how to find the school. Observers may wish to ask for directions when
scheduling the observation or use an online map service such as MapQuest
(www.mapquest.com) to help find the school. The address of all participating schools
appears in the OMLI Professional Development Database.
Check on the availability of parking if you are visiting a high school. Observers may wish
to ask the teacher about parking when scheduling the observation.
Allow enough time to drive to the school, park, sign in at the main office, obtain a
visitor’s pass, and find your way to the teacher’s classroom.
Step 3: Conduct the Pre-observation Interview
The observer must gain information about the context of the lesson before it starts. Exhibit 6 lists
several questions that observers can use to learn about the context of the lessons. Observers may
elect to gather some of this information when scheduling the observation.
Exhibit 6—Suggested Pre-observation Interview Questions
1. What has this class been covering recently?
What unit are you working on?
What instructional materials are you using?
2. What do you anticipate doing with this class today/on the day of the observation?
What would you like the students to learn during this class?
3. Is there anything in particular that I should know about the students in this class?
August 2005
5
” RMC Research Corporation‹Portland, OR
OMLI Classroom Observation Protocol
The information gained through the preobservation interview will assist in the completion of the
lesson context portion of the Classroom Observation Summary Form. Observers should be sure
to express appreciation to the teachers for allowing the observation and should answer any
questions they have about confidentiality, the use of the data collected, the incentive, and so on.
If the teacher is using published materials, be sure to note the complete name of the materials,
publisher, chapter, section, and pages that relate to the lesson observed. If the teacher developed
the lesson, get a copy of the lesson plan and include it with your submission.
Step 4: Observe the Lesson
The observer must be as unobtrusive as possible during the lesson. Avoid distracting the students
by staying out of the spotlight as much as possible. Avoid interacting with the students in a way
that takes their attention away from the lesson. Definitely avoid the urge to help the students with
the activities or assignments.
Any lesson observed is likely to comprise distinct episodes and transitions between the episodes.
Episodes have a distinct beginning and end and usually focus on 1 or 2 instructional objectives.
The time during which students work in small groups to solve problems using manipulatives is a
distinct episode. A large group discussion that engages students in sharing a variety of
approaches to solving a problem followed by time for students to write in their journals is 2
episodes: the large group discussion is one episode and the journal time is another episode. Not
all episodes will present opportunities for mathematical discourse among students. For example,
a lesson may include materials cleanup. Such episodes do not require the observer to record
evidence of mathematical discourse because none is likely to occur.
Observers should collect data on each distinct episode that has an instructional focus. The
approach to data collection will change depending upon the type of episode that is observed.
Exhibit 7 provides guidelines for collecting data on each type of episode. Observers should use
the Classroom Observation Discourse Form to document evidence of mathematical discourse
and ensure that all information required is captured for each episode that occurs during the
lessons.
Exhibit 7—Episode Data Collection Guidelines
Episode Type
Data Collection Guidelines
Large group (All or most all
students)
Observe the entire group and record the evidence of mathematical discourse as it occurs.
Pairs or small groups
Randomly choose one of the pairs or small groups and observe the interaction among
the members of the selected group, recording evidence of mathematical discourse as it
occurs. If the group is off task, move to another group of the same size.
Individual
Circulate among the students and observe what they are working on. If students are
solving problems, it is unlikely any mathematical discourse will occur unless student
interaction is involved. If all students are writing in their journals, record a single
notation indicating as much (IR/E, J/W). If the teacher is circulating among the students
or working with individual students, follow the teacher and record evidence of
mathematical discourse on the part of the students.
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OMLI Classroom Observation Protocol
The Classroom Observation Discourse Form is intended for use during the observation to record
lesson episodes and the evidence of mathematical discourse that is observed during each episode.
Because a lesson may involve any number of distinct episodes, observers must have a supply of
blank Classroom Observation Discourse Forms readily available. Observers should indicate the
teacher’s name, the date of the observation, and page number at the top of each Classroom
Observation Discourse Form to ensure that the forms can easily be associated with the
corresponding Classroom Observation Summary Form. Exhibit 8 provides guidelines for
completing each column of the Classroom Observation Discourse Form.
Exhibit 8—Classroom Observation Discourse Form Field Definitions
Field
Explanation
Episode Type
Check the ONE column that best describes how students are grouped for the episode. A
change in the grouping is a good indicator that an episode has ended and a new one is about
to begin.
Start/End Times
Record the time of day that the episode starts and when it ends to the nearest minute. It is
very important that both of these times are recorded.
Students Observed
Record the number of students being observed during the episode.
Episode Description
Write a brief description of the episode, describing what students are doing.
Discourse Codes
Use these columns to record every incident of student mathematical discourse observed
during the episode using the specified notation system described earlier. Assign a mode,
type, and tools code to every incident.
Tally
For each incident of mathematical discourse that occurs, tally the number of times that it is
observed during an episode. Remember to tally the first case.
Episodes that have a management or logistics focus such as cleanup or roll call need not be
recorded. When one episode ends and another begins, draw a horizontal line across the
Classroom Obseration Discourse Form to indicate the transition between episodes. Be sure to
note the time each episode begins and ends. Use as many copies of the form as necessary to
document each episode that has an instructional focus. Gaps in segments of the lesson with
instructional focus should be indicated as a gap between the end time of one episode and the start
time of the next instructional episode.
Step 5: Conduct the Postobservation Interview
Conduct a brief postobservation interview with the teacher as soon after the classroom
observation as possible. Exhibit 9 lists questions that observers can use to obtain the information
needed to complete the Classroom Observation Summary Form and to assess the degree to which
the class observed represented a typical class taught by this teacher. Observers should express
appreciation for the opportunity to observe the class at the conclusion of the postobservation
interview.
Exhibit 9—Suggested Postobservation Interview Questions
1. Did this lesson turn out different from what you planned? If so, in what ways?
2. How typical was this lesson for the students?
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OMLI Classroom Observation Protocol
3. What do you think the students learned from this lesson, and what they still need to
learn?
4. What challenges did you confront in encouraging students to engage in the mathematical
discourse?
5. What do you plan to do in the next lesson with these students?
Step 6: Complete the Classroom Observation Summary Form
Observers should complete the Classroom Observation Summary Form as soon after each
observation and postobservation interview as possible. The form includes a Lesson Context
section and an Observation Summary section.
Lesson Context—Use this section of the form to document the lesson context. Be sure to
complete all items in this section. Exhibit 10 provides an explanation of each fields in this
section of the form.
Exhibit 10—Classroom Observation Summary Form Lesson Context Field Definitions
Field
Explanation
Observer
The first and last name of the person who conducted the classroom observation and
completed the form.
Date
The date the observation took place. Not the date the form was submitted.
Teacher
The first and last name of the teacher of the class that was observed.
School
The name of the school where the observation took place.
Grade(s)
The grade or grade range of the students in the class.
Course
The name of the course (e.g., Algebra I, Interactive Math, Grade 3 Math)
Unit/Topic
The name of the unit and topic the students were studying the day of the observation (e.g.,
percentage, polynomials, whole number multiplication)
Learning Objective
A brief statement that explicitly describes what the teacher intended the students to learn
from the lesson. This statement should not describe what students were intended to do, but
what they should have learned.
Instructional Materials
A specific reference to the instructional materials (including manipulatives) that were used
in the lesson. If the materials were printed, please record the title, publisher, chapter,
section, and page. If the lesson is teacher developed, get a copy of the lesson plans.
Math Class Began/Ended
The time of the day the class began and ended.
Students
The total number of students present during the observation. If the number of students
changed during the class period, the maximum number of students.
Percent Minority
An estimate of the percentage of the students present during the observation who were
ethnic minority (non-White).
Relationship to previous
and future lessons
A brief description of students had learned prior to the lesson observed and what the
teacher planned to address in future lessons. This description should place the lesson
observed in the overarching instructional.
Other comments
Other comments regarding the aspects of the lesson context not already addressed (e.g.,
the presence of an instructional aide, information about the classroom environment,
unexpected events that occurred such as a fire drill).
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OMLI Classroom Observation Protocol
Observation Summary—Use this section of the form to rate the overall lesson according to key
lesson characteristics. Base the ratings on the information gathered during the observation and
the interviews. Provide a rationale for extreme ratings and general impressions regarding the
lesson on the last page of the form (use the back side if necessary).
Step 7: Submit the Results
Observers are responsible for submitting the classroom observation results to RMC Research via
the OMLI Professional Development Database. The URL for the web site is:
http://www.rmccorp.com/OMLI
Passwords for access to the web site will be issued to each observer by RMC Research staff. The
observations forms can be found under the data collection menu.
Once the data have been submitted electronically, mail the original forms to:
Dave Weaver
RMC Research Corporation
522 SW Fifth Avenue, Suite 1407
Portland, OR 97204-2131
If you have any questions regarding classroom observations procedures or about submitting data,
feel free to contact Dave by phone at (503) 223-8248 or (800) 788-1887 or by e-mail at
dweaver@rmccorp.com.
References
Some of the items used in this protocol were adapted from instruments available from the
following sources:
Horizon Research, Inc. (2003). Local systemic change 2003–04 core evaluation data
collection manual. Chapel Hill, NC: Author.
Secada, W. & Byrd, L. (1993). Classroom observation scales: School-level reform in the
teaching of mathematics. Madison, WI: National Center for Research in
Mathematical Sciences Education.
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Classroom Observation Discourse Form
Evidence of Mathematical Discourse
Teacher: ______________________________________
August 2005
Date: ______________________
Discourse Codes
Start/End
Times
Students
Observed
Individual
Pairs/Small
Group
Large
Group
Episode Type
Page: _____
Episode Description
1
Mode
Type
Tools
Tally
” RMC Research Corporation‹Portland, OR
OMLI Classroom Observation Protocol
Classroom Observation Reference Sheet
Preobservation Interview Questions
MODES
1. What has this class been covering recently?
Code
a. What unit are you working on?
b. What instructional materials are you using?
2. What do you anticipate doing with this class
today/on the day of the observation?
a. What would you like the students to learn
during this class?
3. Is there anything in particular that I should
know about the students in this class?
NOTE: Get specific instructional materials
reference or a copy of the lesson plans.
Postobservation Interview Questions
1. Did this lesson turn out different from what
you planned? If so, in what ways?
Definition
T
Student to Teacher
S
Student to Student
G
Student to Group or
Class
I
Individual Reflection
TYPES
Code
Definition
A
Answering
S
Stating or Sharing
E
Explaining
Q
Questioning
C
Challenging
R
Relating
P
Predicting or
Conjecturing
J
Justifying
G
Generalizing
2. How typical was this lesson for the students?
3. What do you think the students learned from
this lesson, and what they still need to learn?
4. What challenges did you confront in
encouraging students to engage in the
mathematical discourse?
5. What do you plan to do in the next lesson with
these students?
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TOOLS
Code
Definition
V
Verbal
A
Gesturing/Acting
W
Written
G
Graphs, Charts, Sketches
M
Manipulative
S
Symbolization
N
Notation
C
Computers/Calculators
O
Other
” RMC Research Corporation‹Portland, OR
Classroom Observation Summary Form
Lesson Context
Observer: ____________________________________
Date: __________________________
Teacher: __________________________________ School: ____________________________
Grade(s): _____________________ Course: _______________________________________
Unit/Topic ____________________________________________________________________
Learning Objective ______________________________________________________________
______________________________________________________________
______________________________________________________________
Instructional Materials: ___________________________________________________________
___________________________________________________________
Math Class Began: _________________
Math Class Ended: ____________________
Students: ________________
Percent Minority: _____________ %
Relationship to previous and future lessons:
Other comments regarding the lesson context:
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Observation Summary
Assess this lesson based on your observation data and the information gathered during the preand postobservation interviews.
A. Representativeness—How typical was the lesson observed in comparison to other lessons
taught by this teacher?
Somewhat Typical
Mostly Typical
Very Typical
The teacher clearly made special
preparations for the observation.
The lesson was very contrived.
Student behavior seemed
rehearsed and the students were
clearly unaccustomed to the
instructional approach employed
in the lesson.
Many parts of the lesson seemed
contrived. Students seemed
uncomfortable and unfamiliar
with the instructional approach.
The teacher may have stated that
he or she tried to show you what
you wanted to see.
A few parts of the lesson seemed
contrived but for the most part
the students seemed comfortable
and familiar with the instructional
approach. The teacher might have
made a few modifications for the
observation.
This lesson was very typical of
the lessons normally conducted
by this teacher. The students
appeared very familiar with the
instructional approach. There was
no evidence the lesson was
contrived.
To a Great
Extent
Not at all Typical
Mostly
e
Some
d
Very Little
c
Not at All
b
1
The instructional objectives of the lesson were clear and the teacher was
able to clearly articulate what mathematical ideas and/or procedures the
students were expected to learn.
b
c
d
e
f
2
The lesson design provided opportunities for student discourse around
important concepts in mathematics.
b
c
d
e
f
3
The teacher appeared confident in his/her ability to teach mathematics.
b
c
d
e
f
4
The pace of the lesson was appropriate for the developmental level/needs
of the students and the purpose of the lesson.
b
c
d
e
f
5
The teacher’s questioning strategies for eliciting student thinking
promoted discourse around important concepts in mathematics.
b
c
d
e
f
6
The teacher was flexible and able to make adjustments to address student
needs or to take advantage of teachable moments.
b
c
d
e
f
7
The teacher’s classroom management style/strategies enhanced the quality
of the lesson.
b
c
d
e
f
8
The vast majority of the students were engaged in the lesson and remained
on task.
b
c
d
e
f
Rate the extent to which each of the following characteristics
was evident in the lesson observed.
B. Lesson Design and Implementation
C. Mathematical Discourse and Sensemaking
1
Student asked questions to clarify their understanding of mathematical
ideas or procedures.
b
c
d
e
f
2
Students explained mathematical ideas and/or procedures.
b
c
d
e
f
3
Students justified mathematical ideas and/or procedures.
b
c
d
e
f
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Not at All
Very Little
Some
Mostly
To a Great
Extent
4
Students thought critically about mathematical ideas and/or procedures
and in an appropriate manner challenged each other’s and their own ideas
that did not seem valid.
b
c
d
e
f
5
Students defended their mathematical ideas and/or procedures.
b
c
d
e
f
6
Students determine the correctness/sensibility of an idea and/or procedure
based on the reasoning presented.
b
c
d
e
f
7
Students shared their observations or predictions.
b
c
d
e
f
8
Students made generalizations, stated observations, or made conjectures
regarding mathematical ideas and procedures.
b
c
d
e
f
9
Students drew upon a variety of methods (verbal, visual, numerical,
algebraic, graphical, etc.) to represent and communicate their
mathematical ideas and/or procedures.
b
c
d
e
f
10
Students listened intently and actively to the ideas and/or procedures of
others for the purpose of understanding someone’s methods or reasoning.
b
c
d
e
f
Rate the extent to which each of the following characteristics
was evident in the lesson observed.
D. Task Implementation
1
Tasks focused on understanding of important and relevant mathematical
concepts, processes, and relationships.
b
c
d
e
f
2
Tasks stimulated complex, nonalgorithmic thinking.
b
c
d
e
f
3
Tasks successfully created mathematically productive disequilibrium
among students.
b
c
d
e
f
4
Tasks encouraged students to search for multiple solution strategies and to
recognize task constraints that may limit solution possibilities.
b
c
d
e
f
5
Tasks encouraged students to employ multiple representation and tools to
support their ideas and/or procedures.
b
c
d
e
f
6
Tasks encouraged students to think beyond the immediate problem and
make connections to other related mathematical concepts.
b
c
d
e
f
E. Classroom Culture
1
Active participation of all students was encouraged and valued.
b
c
d
e
f
2
The classroom climate was one of respect for the students’ ideas,
questions, and contributions.
b
c
d
e
f
3
Interactions reflected a productive working relationship among students.
b
c
d
e
f
4
Interactions reflected a collaborative working relationship between the
teacher and the students.
b
c
d
e
f
5
Wrong answers were viewed as worthwhile learning opportunities.
b
c
d
e
f
6
Students were willing to openly discuss their thinking and reasoning.
b
c
d
e
f
7
The classroom climate encouraged students to engage in mathematical
discourse.
b
c
d
e
f
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F. Overall Rating—For each section below, mark the choice that best describes your overall
summary of the lesson based on the observation.
1. Depth of Student Knowledge and Understanding—This scale measures the depth of the students’
mathematical knowledge as evidenced by the opportunities students had to produce new knowledge
by discovering relationships, justifying their hypotheses, and drawing conclusions.
c
Knowledge was very superficial. Mathematical concepts were treated trivially or presented as nonproblematic.
Students were involved in the coverage of information which they are to remember, but no attention was paid to the
underlying mathematical concepts. For example, students applied an algorithm for factoring binomials or used the
FOIL method of multiplication—in either case with no attention to the underlying concepts.
d
Knowledge was superficial or fragmented. Underlying or related mathematical concepts and ideas were mentioned
or covered, but only a superficial acquaintance with or trivialized understanding of these ideas was evident. For
example, a teacher might have explained why binomials are factored or why the FOIL method works, but the focus
remained on students mastering these procedures.
e
Knowledge was uneven; a deep understanding of some mathematics concepts was countered by a superficial
understanding of other concepts. At least one idea was presented in depth and its significance was grasped by some
students, but in general the focus was not sustained.
f
Knowledge was relatively deep because the students provide information, arguments, or reasoning that
demonstrate the complexity of one or more ideas. The teacher structured the lesson so that many (20% to
50%) students did at least one of the following: sustain a focus on a topic for a significant period of time;
demonstrate their understanding of the problematic nature of a mathematical concept; arrive at a reasoned,
supported conclusion with respect to a complex mathematical concept; or explain how they solved a
relatively complex problem. Many (20% to 50%) students clearly demonstrated understanding of the
complexity of at least one mathematical concept.
g
Knowledge was very deep. The teacher successfully structured the lesson so that almost all (90% to 100%)
students did at least one of the following: sustain a focus on a topic for a significant period of time;
demonstrate their understanding of the problematic nature of a mathematical concept; arrive at a reasoned,
supported conclusion with respect to a complex mathematical concept; or explain how they solved a complex
problem. Most (51% to 90%) students clearly demonstrated understanding of the complexity of more than
one mathematical concept.
2. Locus of Mathematical Authority—This scale determines the extent to which the lesson
supported a shared sense of authority for validating students’ mathematical reasoning.
c
Students relied on the teacher or textbook as the legitimate source of mathematical authority. Students accepted an
answer as correct only if the teacher said it was correct or if it was found in the textbook. If stuck on a problem,
students almost always asked the teacher for help.
d
Students relied on the teacher and some of their more capable peers (who were clearly recognized as being better at
math) as the legitimate sources of mathematical authority. The teacher often relied on the more capable students to
provide the right answers when pacing the lesson or to correct erroneous answers. As a result, other students often
relied on these students for correct solutions, verification of right answers, or help when stuck.
e
Many (20% to 50%) students shared mathematical authority among themselves. They tended to rely on the
soundness of their own arguments for verification of answers, but, they still looked to the teacher as the authority
for making final decisions. The teacher intervened with answers to speed things up when students seemed to be
getting bogged down in the details of an argument.
f
Most (51% to 90%) students shared in the mathematical authority of the class. Though the teacher intervened when
the students got bogged down, he or she did so with questions that focused the students’ attention or helped the
students see a contradiction that they were missing. The teacher often answered a question with a question, though
from time to time he or she provided the students with an answer.
g
Almost all (90% to 100%) of the students shared in the mathematical authority of the class. Students relied on the
soundness of their own arguments and reasoning. The teacher almost always answered a question with a question.
Many (20% to 50%) students left the class still arguing about one or more mathematical concepts.
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3. Social Support—This scale measures the extent to which the teacher supported the students
by conveying high expectations for all students.
c
Social support was negative. Negative teacher or student comments or behaviors were observed. The classroom
atmosphere was negative.
d
e
Social support was mixed. Both negative and positive teacher or student comments or behaviors were observed.
f
Social support from the teacher was clearly positive and there was some evidence of social support among students.
The teacher conveyed high expectations for all, promoted mutual respect, and encouraged the students try hard and
risk initial failure.
g
Social support was strong. The class was characterized by high expectations, challenging work, strong effort,
mutual respect, and assistance for all students. The teacher and the students demonstrated these attitudes by
soliciting contributions from all students, who were expected to put forth their best efforts. Broad participation was
an indication that low-achieving students received social support for learning.
Social support was neutral or mildly positive. The teacher expressed verbal approval of the students’ efforts. Such
support tended, however, to be directed to students who were already taking initiative in the class and tended not to
be directed to students who were reluctant participants or less articulate or skilled in mathematical concepts.
4. Student Engagement in Mathematics—This scale measures the extent to which students
engaged in the lesson (e.g., attentiveness, doing the assigned work, showing enthusiasm for
work by taking initiative to raise questions, contributing to group tasks, and helping peers).
c
Students were disruptive and disengaged. Students were frequently off task as evidenced by gross inattention or
serious disruptions by many (20% to 50%).
d
Students were passive and disengaged. Students appeared lethargic and were only occasionally on task. Many
(20% to 50%) students were either clearly off task or nominally on task but not trying very hard.
e
Students were sporadically or episodically engaged. Most (51% to 90%) students were engaged in class activities
some of the time, but this engagement was uneven, mildly enthusiastic, or dependent on frequent prodding from the
teacher.
f
Student engagement was widespread. Most (51% to 90%) students were on task pursuing the substance of the
lesson most of the time. Most (51% to 90%) students seemed to take the work seriously and try hard.
g
Students were seriously engaged. Almost all (90% to 100%) students were deeply engaged in pursuing the
substance of the lesson almost all (90% to 100%) of the time.
Rationale/General Impressions:
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COMMON CORE STATE STANDARDS FOR MATHEMATICS
Standards for Mathematical Practice
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should
seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding
importance in mathematics education. The first of these are the NCTM process standards of problem solving,
reasoning and proof, communication, representation, and connections. The second are the strands of mathematical
proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning , strategic
competence, conceptual understanding (comprehension of mathematical concepts, operations and relations),
procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately) and productive
disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in
diligence and one’s own efficacy).
1
Make sense of problems and persevere in solving them.
Mathematically proficient students:
• explain to themselves the meaning of a problem and looking for entry points to its solution.
• analyze givens, constraints, relationships, and goals.
• make conjectures about the form and meaning of the solution attempt.
• consider analogous problems, and try special cases and simpler forms of the original problem.
• monitor and evaluate their progress and change course if necessary.
• transform algebraic expressions or change the viewing window on their graphing calculator to get information.
• explain correspondences between equations, verbal descriptions, tables, and graphs.
• draw diagrams of important features and relationships, graph data, and search for regularity or trends.
• use concrete objects or pictures to help conceptualize and solve a problem.
• check their answers to problems using a different method.
• ask themselves, “Does this make sense?”
• understand the approaches of others to solving complex problems.
2. Reason abstractly and quantitatively.
Mathematically proficient students:
• make sense of quantities and their relationships in problem situations.
ü decontextualize (abstract a given situation and represent it symbolically and manipulate the representing
symbols as if they have a life of their own, without necessarily attending to their referents and
ü contextualize (pause as needed during the manipulation process in order to probe into the referents for the
symbols involved).
• use quantitative reasoning that entails creating a coherent representation of quantities, not just how to compute
them
• know and flexibly use different properties of operations and objects.
3 Construct viable arguments and critique the reasoning of others.
Mathematically proficient students:
• understand and 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.
• analyze situations by breaking them into cases
• recognize and use counterexamples.
• justify their 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
• compare the effectiveness of plausible arguments
• distinguish correct logic or reasoning from that which is flawed
ü elementary students construct arguments using objects, drawings, diagrams, and actions..
ü later students learn to determine domains to which an argument applies.
• listen or read the arguments of others, decide whether they make sense, and ask useful questions
4 Model with mathematics.
Mathematically proficient students:
• apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.
ü In early grades, this might be as simple as writing an addition equation to describe a situation. In middle
grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the
community.
ü By high school, a student might use geometry to solve a design problem or use a function to describe how
one quantity of interest depends on another.
• simplify a complicated situation, realizing that these may need revision later.
• identify important quantities in a practical situation
• map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas.
• analyze those relationships mathematically to draw conclusions.
• interpret their mathematical results in the context of the situation.
• reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
5 Use appropriate tools strategically.
Mathematically proficient students
• consider available tools when solving a mathematical problem.
• are familiar with tools appropriate for their grade or course to make sound decisions about when each of these
tools
• detect possible errors by using estimations and other mathematical knowledge.
• know that technology can enable them to visualize the results of varying assumptions, and explore
consequences.
• identify relevant mathematical resources and use them to pose or solve problems.
• use technological tools to explore and deepen their understanding of concepts.
6 Attend to precision.
Mathematically proficient students:
• try to communicate precisely to others.
•
use clear definitions in discussion with others and in their own reasoning.
• state the meaning of the symbols they choose, including using the equal sign consistently and appropriately.
• specify units of measure and label axes to clarify the correspondence with quantities in a problem.
• calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the
context.
ü In the elementary grades, students give carefully formulated explanations to each other.
ü In high school, students have learned to examine claims and make explicit use of definitions.
7 Look for and make use of structure.
Mathematically proficient students:
• look closely to discern a pattern or structure.
ü Young students might notice that three and seven more is the same amount as seven and three more.
ü Later, students will see 7 x 8 equals the well-remembered 7 x 5 + 7 x 3, in preparation for the distributive
property.
2
ü In the expression x + 9x + 14, older students can see the 14 as 2 x 7 and the 9 as 2 + 7.
• step back for an overview and can shift perspective.
• see complicated things, such as some algebraic expressions, as single objects or composed of several objects.
8 Look for and express regularity in repeated reasoning.
Mathematically proficient students:
• notice if calculations are repeated
• look both for general methods and for shortcuts.
• maintain oversight of the process, while attending to the details.
• continually evaluate the reasonableness of intermediate results.
How to Get Students Talking!
Generating Math Talk That Supports Math Learning
By Lisa Ann de Garcia
Due to the attention in the last few years on discourse and its importance to student learning, educators
nationwide are finding that they can help children become confident problem solvers by focusing on getting
them to talk and communicate in partnerships, small groups, whole groups, and in writing. In addition,
English Language Learners are flourishing as they experience focused opportunities for talking and trying on
new mathematical vocabulary.
So what exactly is discourse? What are the teaching practices associated with successfully establishing
an environment to support it, and as a result, to improve mathematical proficiency? How does one begin
to elicit meaningful talk during math lessons? As a profession, we share a vision about the role student
discourse has in the development of students’ mathematical understanding, but are often slow to bring the
students along. Children do not naturally engage in this level of talk.
This article addresses the above questions and concerns—and more. It opens with a look at discourse
through NCTM’s definition and its involvement with the Common Core State Standards. It then focuses on
literature available on discourse, specifically the book Classroom Discussions, and addresses five teaching
practices focused on the how to of getting students talking about mathematics. The article concludes with
journaling insights on discourse from a kindergarten and second-grade classroom. This article is by no
means an exhaustive list of discourse “to dos;” hopefully it will however get us all started in thinking about
and implementing best talk practices.
What is Discourse in the Mathematics Classroom?
NCTM’s Definition
The National Council of Teachers of Mathematics (NCTM) in their 1991 professional standards describes
discourse as ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged
and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the
nature of the learning environment.
A View Through The Common Core Lens
As much of the country begins to implement the new Common Core State Standards, it is important to
reflect on the role of discourse in these new standards. The Common Core was created based on five
process standards: communication, reasoning and proof (another form of communication), problem solving,
representation, and connections. Evidence of the importance of communication in learning mathematics is
found in the Common Core introduction in statements such as, “One hallmark of mathematical understanding
is the ability to justify . . . a student who can explain the rule understands the mathematics and may have a
better chance to succeed at a less familiar task . . .” (p. 4). In the grade-specific standards, the importance of
communication in learning mathematics is reflected in statements such as , “Students also use the meaning
of fractions, of multiplication and division, and the relationship between multiplication and division to
understand and explain why the procedures for multiplying and dividing fractions make sense” (p. 33).
These Common Core statements make it clear that conceptual understanding must be connected to the
procedures, and that one way to deepen conceptual understanding is through the communication students
have around concepts, strategies, and representations.
Learning from Literature on Discourse
One of the leading resources for discourse is Classroom Discussions: Using Math Talk to Help Students Learn
(Chapin, O’Connor, and Anderson 2009). This resource and others highlight five teaching practices associated
with improving the quality of discourse in the classroom.
Five Teaching Practices for Improving the Quality of Discourse in Mathematics
Classrooms
1)
2)
3)
4)
5)
Talk moves that engage students in discourse,
The art of questioning,
Using student thinking to propel discussions,
Setting up a supportive environment, and
Orchestrating the discourse.
Practice 1: Talk Moves That Engage Students in Discourse
For the first practice, the authors of Classroom Discussions propose five productive talk moves that can get
talk going in an otherwise silent classroom. The first is revoicing. An example would be, “So you are saying
that . . .” This revoicing allows the teacher to check in with a student about whether what the student said
was correctly heard and interpreted by the teacher or another student. A way to encourage students to
listen to their peers is through asking them to restate someone else’s reasoning, such as, “Can you repeat
what he just said in your own words?” Another way is to ask students to apply their own reasoning to
someone else’s using questions such as “What do you think about that?” and “Do you agree or disagree?
Why?” This helps prevent students from just thinking about what they want to share and focuses their
attention on what their classmates are saying. It also helps to strengthen the connections between ideas.
Simple questions such as, “Would someone like to add on?” are ways teachers can prompt for further
participation. This helps elicit more discussion when not many students are talking, especially when they
are not accustomed to explaining their thinking. Again it helps students to tune in to what others are saying
so that they are able to expand on someone else’s idea.
Perhaps the most valuable talk move suggested by Chapin, O’Connor, and Anderson is the use of wait
time. Often teachers are too quick to answer their own questions when no one chimes in. Children quickly
become accustomed to this. Waiting provides think time and sets the expectation that someone will indeed
respond and that the teacher will wait until someone does. Another important use for wait time is to provide
English Language Learners or anyone who needs extra time with an opportunity to process the question and
formulate an answer. One teacher reported that in his initial uses of wait time, one of his English Language
Learners was able to participate in class discussion for the first time.
Practice 2: The Art of Questioning
Questioning is another critical component in supporting students to engage in meaningful discussions. The
NCTM Standards outline roles questions have in the math classroom. The first role, helping students to work
together to make sense of mathematics, is addressed by the five talk moves discussed above. The second
role, helping students to rely more on themselves to determine whether something is mathematically correct,
can be supported by questions such as, “How did you reach that conclusion? Does that make sense? Can
you make a model and show that?” Questions such as, “Does that always work? Is that true for all cases?
Can you think of a counterexample? How could you prove that?” are designed to help students to learn to
reason mathematically. To help students to learn to conjecture, invent, and solve problems, the teacher
might ask, “What would happen if? Do you see a pattern? Can you predict the next one? What about the last
one?” Finally, teachers use questions to help students connect mathematics, its ideas and applications by
asking, “How does this relate to . . .? What ideas that we have learned were useful in solving this problem?”
Practice 3: Using Student Thinking to Propel Discussions
Because discussions help students to summarize and synthesize the mathematics they are learning, the use of
student thinking is a critical element of mathematical discourse. When teachers help students build on their
thinking through talk, misconceptions are made clearer to both teacher and student, and at the same time,
conceptual and procedural knowledge deepens. When doing so, the teacher must be an active listener so she
can make decisions that will facilitate that talk. She also needs to respond neutrally to errors, so that the students
can figure out misconceptions themselves. For example, the teacher can ask the whole class, “What do you think
about that?” when a student offers an incorrect strategy or can ask the rest of the class to prove whether or not
the strategy works. Through the conversation, the misconception becomes apparent to the class. This practice
results in an authentic discussion focused on the mathematics and not on the individual student. The teacher
also needs to be strategic about who shares during the discussion, since it is not a show-and-tell session, and
choose ideas, strategies, and representations in a purposeful way that enhances the quality of the discussion.
Practice 4: Setting Up a Supportive Environment
When setting up a discourse-rich environment and one that enhances student engagement, both the physical
and emotional environment must be considered. Teachers who have studied engagement find that it is very
effective if students face each other, either sitting in a circle or semi-circle on the floor or sitting in chairs
arranged in a circle. Teachers can sit with students as part of the circle to encourage peer-to-peer discussion.
If teachers are still having difficulty getting children to talk, they can remove themselves from the group and
stand outside the circle. As a result, students are left looking only at each other, which encourages them to
direct their comments to one another.
Careful consideration of the placement of visual aids and mathematically related vocabulary is important
in supporting the level of talk. If charts are not visually accessible when they need to be, they will likely not
be resourced by the students during whole group conversations. To increase the extent to which English
Language Learners participate in group discussions, having related vocabulary and sentence frames where
they can be easily accessed is critical.
For rich discussions, the emotional environment of the classroom must be safe and must be one where
students want to learn and think deeply about the mathematics. When these elements are not present, the
discussion stays at the surface level. Imagine a third grade classroom where the teacher introduces division
for the first time and is met with cheers. It can happen! It happens when the value is on learning, challenging
each other, and working together to solve problems as opposed to just getting the right answer. For more
on setting up a supportive classroom environment for discourse, see Chapter 8 of Classroom Discussions.
Practice 5: Orchestrating the Discourse
The teacher becomes not unlike a conductor as he supports students to deepen their understanding of
mathematics through a carefully orchestrated environment. In Orchestrating Discussions, Smith, Hughes,
Engle, and Stein outline the Five Practices Model, which gives teachers influence over what is likely to
happen in a discussion.
The Five Practices Model
The teacher’s role is to:
1) anticipate student responses to challenging mathematical tasks;
2) monitor students’ work on and engagement with the tasks;
3) select particular students to present their mathematical work;
4) sequence the student responses that will be displayed in specific order; and
5) connect different students’ responses and connect the responses to key mathematical ideas.
Even if the teacher is focused, he still needs to hold students accountable. Otherwise the discussion
will be unproductive. A lot of explicit teaching must go into how to engage in each level of discussion:
whole group, small group, and partnerships. In the younger grades, one will find teachers showing students
exactly what they should look like and sound like when discussing their thinking. Teachers may say things
like, “Today in math, we are going to practice turning and talking with our partner. When I say go, you are
going to turn like this and look at your partner. When I say stop, you are going to turn around and face me.
Let’s practice that right now.” Even older students need to be explicitly taught what to do and say. A teacher
might teach how a partnership functions by saying, “It sounds like you have an idea and you have an idea,
but what seems to be lacking is for you two to put your ideas together to come up with a solution. So, what
is your plan?”
One very effective method of holding students accountable is to let them know exactly what they should
be saying when they are talking in their partnerships or small groups. For example, “Today, when you are
talking to your partners and describing your solid shapes, I expect to hear you using the words faces, edges,
and vertices.” It is also supportive to let students know what they should be focusing on when someone
is sharing a strategy, so they have a lens for listening, which heightens the level of engagement. A teacher
might say, “When he is sharing his thinking, I want you to be thinking of how his way is similar or different
to your way.”
Students need to be aware of themselves as learners, and a great way to heighten this awareness is
through self-evaluation and goal setting. Sometimes the child is the last one to know that he is distracting
or not listening. Part of developing a safe culture is supporting students in being open with each other
regarding their strengths and weaknesses so they can improve their communication skills and behaviors.
It is wonderful to hear one child compliment another when she has participated for the first time or give
gentle correction when another has been dominating the conversation. This level of self-awareness happens
through consistent venues such as class meetings and tracking the progress of personal goals related to
participation in mathematical discussions. The more students open up about themselves as learners, the
deeper the relationships and, as a result, the deeper the trust.
Kindergarten
Teaching Points
Sept
Oct
Nov
Dec
Jan
Feb
Partnerships
Partner Talk Expectations
Problem solving possible partner problems, such as:
“What do you do if you both want to go first?”
“How do you talk to your partner if they are not sharing?”
Modeling language such as, “You can go first, or I can go first”
X
“Turn and Talk”, “hip to hip,” “knee to knee”
X
Demonstrating with a partner
Modeling with another student how to share
Showing Eye Contact
What Listening Looks Like
Teaching students to ask and answer a question on cue
Ex: “Turn and talk. First partner ask . . . second partner answer . . .”
Using partnerships to move towards whole group share of what they did together
Comparing their work with a partner
Ex: Asking partner, “How did you sort?” Partner answers, “I sorted by . . .”
Have partners share in front of the whole group
Introducing story problem procedures by saying the story a few times while
students listen, then having them repeat it with the teacher a few times, then turn
and tell their partner the story, then solve.
Holding class meetings to help a partnership problem solve something related to
working as partners
Formulating own question to ask their partner
X
much less
prompting
X
X
X
X
X
X
X
prompting prompting
refine
X
X
X
X
X
X
X
X
Whole Group Discussion
Comparing their work as a whole group
“Is what so and so did the same or different as what s/he did?”
Eye contact towards speaker
Can you tell me what so and so said? (revoicing)
“What do you notice about . . .”
( this promoted a lot of talk)
Learning to compare their work with others
Prompting, “Who is talking?” “What should you do?”
Turning and looking with just the heads and not entire bodies
Whole group physical behaviors
Very
Guided
X
X
X
X
X
X
X
Supporting Language and Vocabulary
Use Sentence Stems
“When you turn and talk to your partner, I want you to use the words . . .”
Model Language: “I say it, you say it.”
X
X
Responding, “I did it like so and so”
Language when comparing work: “same/different, because”
Use of co-created charts / prompting students to reference them
Vocabulary: agree/disagree
Teaching how to ask a question back & generate own spontaneous questions
Vocabulary: accurate / efficient
X
X
mimic with
a partner
X
X
X
X
X
X
X
X
prompting
X
X exposure
Table 1: Teaching points of a kindergarten teacher during the year
2nd Grade
Teaching Points
Sept
Whisper to your partner (during whole group)
“Did you and your partner agree or disagree?” (beginning listening and repeating
Tell me what your partner said
X
X
Oct
Nov
Dec
X
X
X
Partner
coaching really
paying off!!
Jan Feb
Partnerships
“You two don’t agree? Who is right?” Don’t just let it be,
but push-back on each other
“How can you figure that out?” “Can your partner help you with that?”
Students are pushing on each other and keeping each other accountable
X
X
Students are voicing
disagreement on
own respectfully
X
X
Coaching on how to wait for your partner to finish her turn. “Watch your partner.”
“Do you agree with how she took her turn?”
Model how to help telling with out telling answer. “You could say...you have a lot
of coins, do you think you could trade?”
Disagreeing and justifying
“Is the way he/she did it the same as how you did it?”
Providing list of questions students were to ask as partnership during games (race
to a stack with beans and cups)
Talk to your partner about ____’s way
Modeling how to ask partner to repeat and how to explain
Using sentence starters
Providing limited tools to promote discussion in small groups
Provide team activities where members have to decide how to solve
and which strategy to share
Table 2: Teaching points of a second grade teacher during the year for Partnerships
X
X
X
X
X
X
X
X
X
X
X
2nd Grade
Teaching Points
Sept Oct
Nov
Dec
Jan
Feb
Whole Group Discussion
Teach “quiet thumb”
Respect: No laughing, mistakes are learning opportunities
Good listening behaviors: No touching manipulatives, eye contact.
Physically adjusting eyes, heads, body
Begin number talks; collecting all answers without judgments
Choosing kids to explain
Ask questions to draw out solutions, such as, “How did you figure that out?” “How did you
count?” “Where did you start?” “Did you count like this or a different way?” Modeling if they
still cannot explain
Strengthening listening by asking another child to repeat/explain strategy of another student
Ask questions to hold students accountable for listening and deepening understanding such as
“Does that make sense?” “What do you think of what ____ said?” “Do you agree/disagree?”
“Any questions for____?” “Who can explain ____’s strategy?” “What should you say if you
didn’t understand, couldn’t hear, etc.?”
Chart and name strategies students use, such as: “Oh, you counted all, counted on, made a 10,
used doubles.” Chart as the students talk to make steps visible.
X
X
X
X
X
X
Referring to other kids’ ways as a way to celebrate students taking risks by trying a new way
“Is your strategy the same or different than _____’s strategy?” “Which strategy did _____
use?” (referring to the chart)
Teacher scripting children’s strategies on their papers and on the chart.
Highlighting students who try on another student’s strategies
Trying to get students to see that their peers are their teachers to foster reason for listening
more carefully
Getting students to try on another someone else’s strategy and acknowledging it with
students, such as “Oh, Marquis did it like Yosef did yesterday.”
Helping students learn how to articulate their thinking (e.g., “What did you do? Tools you
used? Where did you start?”) to be easier understood by others
Helping students to record their thinking. Model how to record each step
so the listeners can see what you did
Highlighting different ways of recording and different tools used in solving a problem
(“Let me show you another way to record” “When you put the blocks together,
how can you show that on paper?”)
Slowing down the person sharing between each step and ask class “Does that make sense?”
“Do you understand” “Who can explain that step” “Why do you think she did that?”
“Which ways are the same or kind of the same?” “Who’s might you try on?”
Having preselected student writing strategies to share
Discussing incorrect answers to see if kids will listen and respectfully agree and disagree
Allow time for the other person to react to partner during share out
Moving position from front of the room to promote explaining
Share partner’s strategy rather than your own
What do you think _______did next (heighten engagement)
Using document camera more for share out since students have become
more proficient with recording
X
Reminders
Reminders
X
X
X
X
X
X
X
X
X
X
X
Kids starting to
notice, “Oh, that
is how __ did it”
Reminders
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Table 3: Teaching points of a second grade teacher during the year for Whole Group Discussion
First Discourse Experience 3rd- 6th Grade
Teaching Points
Whole Group Discussion
Explain that we are having a conversation about what we built (model for problem given)
What do we do when someone is explaining his/her thinking?
*Listen (not just hearing, but thinking about what they said)
*Listening to compare to see if we thought the same thing the speaker did
*What does paying attention look like?
Don’t merely think what you are going to say next, rather respond back to the speaker - adding on or comparing
How do we talk like adults? - taking turns, not raising hands
Who would like to share? - opening it up to anyone (sometimes - other times choosing someone specific this depends on if the focus is on the act of sharing or a specific strategy.
When one person shares, ask some to restate
Teach students how to ask someone to speak up or to repeat themselves if they weren’t listening or if they couldn’t hear
“Could you please say that again, I wasn’t listening.”
Lots of turn and talk to partner with something specific to talk about
I have to listen so I can highlight a partnership and ask students to think about their thinking
Asking students to try on someone else’s way and explain what they did.
Asking lots of questions such as “Does their way make sense?”
**It is necessary to remind students often where their eyes need to be and to listen to what the speaker is saying.
Partner Talk
Generally on the first day I go around and listen and make sure that the partnerships are working together rather than side-by-side play and coach
accordingly
I will ask questions such as, “Do you know what he did?” “Can you explain it?”
Direct when necessary (if students are having trouble working together) by saying, “When we share out, I want you to explain what your partner
did.”
Note:
At the end of one lesson, the discourse is not beautiful, but if the teacher is explicit with expectations and how to engage in discourse. children
will talk, mostly to partner, as they are a little shy about the group at first. Students definitely engage in what the other students are thinking and
make sense of other strategies. I would expect to be emphasizing the above points repeatedly for the next couple of months.
Table 4: Teaching points that can be made on the first day in an upper grade classroom around discourse
Managing a classroom that makes students are responsible for their own learning means that the teacher
has to become accustomed to not doing all of the work for them. One of the hardest things for teachers is
to stop jumping in too soon and answering their own questions. Once a teacher I was working with told me
that if she wasn’t always doing the talking, she felt that she was not doing her job. Just because the students
are the ones who should be doing the thinking and talking doesn’t mean that the teacher does not play a
significant role. One of the biggest jobs of the teacher is that of decision maker. The NCTM Standards state
that teachers must decide what to pursue in depth, when and how to attach mathematical notation and
language to students’ ideas, when to provide information, when to clarify an issue, when to model, when to
lead, and when to let a student struggle with difficulty, and how to encourage each student to participate.
These decisions, so well-articulated by NCTM, are central to effective math teaching and remain crucial as
we move into the implementation of the Common Core State Standards for Mathematics.
A Look into Classrooms: Journaling About Discourse
Recently, a kindergarten and a second-grade teacher were invited to spend most of one school year
journaling exactly what they do to explicitly teach meaningful mathematical discourse. I also reflected on
what I do when I go into a 3rd – 6th grade classroom for the first time for a demonstration lesson and how I
start to get students to talk when they are not accustomed to it. This analysis was further broken down into
partnerships and whole group discussion. In the case of the kindergarten teacher, the explicit teaching she
did to support language and vocabulary was also noted. The following tables outline the teaching points
and what time of year each was a primary focus. For example, in kindergarten, the teacher worked on the
children turning and talking in September and October. In November, much less prompting was needed, and
after that it became a norm in the classroom culture.
Each group of students is unique and has different needs. The above insights are not meant to be a
checklist or recipe of how to facilitate deep mathematical discourse in your individual classroom, but they
can serve as a resource of the types of behaviors teachers need to explicitly teach and pay attention to when
trying to deepen the quality of talk. They can also serve as a reminder that it is best to teach behaviors in small
segments, especially with younger children. When teaching older children, unless they exhibit significant
social difficulties, it may be possible to focus on several different aspects of talk at once, but these behaviors
need to be reinforced on an ongoing basis. Once these behaviors become part of the classroom culture, it
is important to refine and deepen the talk by addressing specific needs of the individual group of students.
Carrying Discourse into the Individual Classroom
Mathematics educators nationwide agree that student engagement in meaningful mathematical discourse
has a positive effect on their mathematical understanding as they increase the connections between ideas
and representations. As we begin to implement the new Common Core State Standards, we need to not
only have a vision for what meaningful talk might look like, but also be equipped on how to get the talk
going. Teachers need to explicitly teach the social behaviors necessary in engaging in discourse on a whole
group, small group, and partnership level. Although there are common behaviors most teachers can initially
address, most behaviors are unique to the dynamics of an individual classroom.
Works Cited
Chapin, S. H., C. O’Connor, and N.C. Anderson. Classroom Discussions: Using Math Talk to Help Students
Learn, Grades K-6, Second Edition (Math Solutions, 2009)
Professional Standards for Teaching Mathematics (National Council of Teachers of Mathematics, 1991)
Smith, M. S., E. K. Hughes, R. A. Engle & M. K. Stein. Orchestrating Discussions, (Mathematics Teaching in
the Middle School, 14 (9). 548-556, 2009)
Recommended Reading List
Classroom Discussions: Using Math Talk to Help Students Learn, Grades K-6, Second Edition, S. H. Chapin,
C. O’Connor, and N.C. Anderson.
Classroom Discussions: Seeing Math Discourse in Action, Grades K-6, N.C. Anderson, S.H. Chapin, C.
O’Connor, ( Copyright © 2011 by Scholastic, Inc.)
Good Questions for Math Teaching: Why Ask Them and What to Ask, K-6, Peter Sullivan and Pat Lilburn
(Copyright © 2002 Math Solutions)
Good Questions for Math Teaching: Why Ask Them and What to Ask, Grades 5-8,
Lainie Schuster and Nancy C. Anderson (Copyright © 2005 Math Solutions)
Let’s
Talk
Promoting Mathematical
Discourse in the Classroom
Catherine C. Stein
A
s part of reform-based mathematics,
much discussion and research has
focused on the idea that mathematics should be taught in a way that
mirrors the nature of the discipline
(Lampert 1990)—that is, have students use mathematical discourse to make conjectures, talk, question, and agree or disagree about problems in order
to discover important mathematical concepts. In
fact, communication, of which student discourse
is a part, is so important that it is one of the Standards set forth in Principles and Standards for
School Mathematics (NCTM 2000).
The use of discourse in the mathematics classroom, however, can be difficult to implement and
manage. The same students participate in every
discussion while others contribute only when
called on, and even then their contributions are
sparse. Some students make comments that relate
to procedure but never reach the deeper-level
mathematical concepts. This article discusses what
research tells us about mathematics discourse in
the classroom and explores the ways in which
teachers establish the classroom community at the
beginning of the year, facilitate discussion, and
assess the quality of discourse.
Vol. 101, No. 4 • November 2007 | Mathematics Teacher 285
Copyright © 2007 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Train 1
Train 2
Train 3
Adapted from Phillips et al. (1991), pp. 49–50
Fig. 1 Hexagon perimeter train
SETTING THE STAGE
Student 2
How do I set up the classroom community to encourage
students to participate?
Solution: p = 4n + 2
Explanation: Each hexagon has at least four sides
on the outside of the train, so I multiplied four by
the number of hexagons (n). The hexagons on the
end have one extra side, so I added two for the two
sides on the end.
Teachers send messages about what is important
to them by the way they establish their classroom
community. Of course, accuracy is essential in
mathematics, but to encourage discourse, teachers
must show students that they value understanding
concepts rather than just getting the right answer.
Turner et al. (2003) grouped the messages that
teachers send into four categories: (1) messages
about tasks, learning, and expectations for students;
(2) relationships with the teacher; (3) relationships
among students; and (4) rules and management
structures. Teachers in classrooms supportive of
discourse showed enthusiasm for learning, set
expectations that all students would learn, and
established classroom relationships and management systems based on respect.
One could rightly argue that these principles for
establishing a community are true of classrooms in
general and are not specific to mathematics classrooms and mathematics classroom discourse. Yackel
and Cobb (1996), however, argue that establishing a
mathematical community also includes sociomathematical norms, the norms of the mathematics community. Although these norms may never be overtly
stated, through discussion the teacher and students
come to an understanding about what counts as
mathematical difference, sophistication, and explanation. Consider two students’ responses to a task
that asks them to find a rule for determining the
perimeter of any given hexagon train (see fig. 1).
Student 1
Solution: p = 4(n – 2) + 10
Explanation: The middle blocks of the train have
four sides out of six total sides that can be counted
in the perimeter. So n equals the number of blocks.
I took away the two blocks on the end since I’m
only counting the middle. Then I multiplied by four
to find the number of sides that can be counted for
the perimeter. The two end blocks each have five
sides showing, so I added ten.
286 Mathematics Teacher | Vol. 101, No. 4 • November 2007
Are the two solutions different mathematically?
Are the solutions efficient? Are the explanations
provided acceptable? The answers to these questions will be negotiated as the classroom community participates in discourse, but they will ultimately depend on the teacher. Teachers send both
explicit and hidden messages about what they value
in mathematics and what they expect of students.
FACILITATING DISCOURSE
My students understand the expectations and norms.
Now what do I do?
There is a misconception that the shift toward the
use of classroom discourse in teaching mathematics
means that the teacher simply presents the problem and then stands aside while students discuss
and solve it (Chazan and Ball 1995). The teacher’s
instructional role is perceived as “don’t tell the
answer.” This perception severely underrates the
complexity of the teacher’s role in classroom discourse (Chazan and Ball 1995). So what should
teachers do during discussions to increase participation and conceptual understanding? There are
two aspects of teacher discourse to be considered:
cognitive discourse and motivational discourse.
Cognitive discourse refers to what the teacher
says to promote conceptual understanding of the
mathematics itself. Kazemi and Stipek (1997)
found that some inquiry-based classrooms,
described as low-press, are still not effective in
facilitating student discourse because they focus
only on explanations of procedure and do not link
to a conceptual understanding of mathematics. In
the following example, a teacher and a student are
discussing the student’s solution to the Skeleton
Tower problem (see fig. 2).
Tower 1 Tower 2
Tower 3
Use the blocks to build the fourth tower in the
sequence. How many cubes did you use? How
many cubes would you need to build the fifth
tower? The 12th tower? The 20th tower? The
100th tower? Write a rule to help you find the
number of cubes for the nth tower.
Adapted from Stoker (2006)
Fig. 2 Skeleton Tower problem
Ms. D. Please explain how you found the rule for
the towers.
S. The center of each tower has the same number of
cubes as the tower number, so that equals n cubes.
Ms. D. Okay, then what?
S. There are four arms coming out from the center
in the shape of triangles.
Ms. D. Triangles?
S. Yeah, when you flip them over you get two rectangles. The height of the rectangle is the same as
the center, and the width is one less. So
2n(n – 1) + n gives you the number of cubes.
Ms. D. 2n(n – 1) + n. Does everyone agree? [“Yeahs”
heard from around the room.] Does everyone
understand how he got the answer? [More “yeahs”
from the class.] Okay, who else has a solution?
In contrast, in high-press classrooms, teachers
push students to link the strategies and procedures
used to the underlying concepts. The following
exchange begins in the same way as the previous
one. In this example, however, the teacher presses
the student for more information about his thinking.
S. The center of each tower has the same number of
cubes as the tower number, so that equals n cubes.
Ms. K. Okay, then what?
S. There are four arms coming out from the center
in the shape of triangles.
Ms. K. Can you explain what you mean by
triangles?
S. The cubes look like the shape of a triangle.
Ms. K. Let’s be sure everyone understands. Can you
show us one of the triangles on the model you
built of the fourth tower?
S. Sure. When you look at one of the arms coming
out from the center [pulls the cubes away from
the rest of the model], you have a piece with three
cubes on the bottom, two on the middle level,
and one on the top level. It looks like a triangle.
Ms. K. Okay, I see. Why are the triangles
important?
S. Because if I can figure out how many cubes are
in the triangles for each tower, I can add that
number to the center tower and figure out how
many cubes total. [The exchange continues as the
student continues explaining.]
In addition to helping students make connections,
teachers of high-press classrooms take better advantage of helping students learn from mistakes and
stress individual accountability so that all students
are engaged.
The issue of engagement necessitates the second
type of teacher discourse, motivational discourse.
Motivational discourse refers not only to praise
offered to students but also to supportive and nonsupportive statements teachers make that encourage or discourage participation in mathematics
classroom discussions. Students’ lack of participation in classroom discourse can be a result of selfhandicapping, failure avoidance, or a preference for
avoiding novelty (Turner et al. 2002). Sometimes
students who disagree remain silent rather than
express a mathematical argument (Lampert 1990).
Turner et al. (2002) found that when teachers used
supportive motivational discourse in addition to
pressing for conceptual understanding, the reported
levels of these behaviors decreased.
Supportive motivational discourse occurs when
teachers focus on learning through mistakes, collaboration, persistence, and positive affect (Turner
et al. 2003). Consider the following exchange in
which a student explains her solution to the teacher.
Ms. K. Explain to the class how you built the fourth
tower.
Susan. It doesn’t look like the picture.
Ms. K. If you explain how you thought about it, maybe
we can help you figure out where you’re making a
mistake. I see some other towers around the room
that don’t look like the picture. As you think aloud,
maybe together we can figure out how to build it.
Though this is a brief exchange, the messages sent
by the teacher are clear. Mistakes are an opportunity for learning, and the learning is a collaborative
process in which all students are expected to participate. Conversely, nonsupportive motivational
discourse occurs when teachers emphasize getting
the right answers without mistakes, compare or
highlight individual successes or failures, or use
sarcasm or humiliation (Turner et al. 2003).
Vol. 101, No. 4 • November 2007 | Mathematics Teacher 287
Table 1
Levels of Discourse in a Mathematics Classroom
Levels
Characteristics of Discourse
0
The teacher asks questions and affirms the accuracy of answers or introduces and explains
mathematical ideas. Students listen and give short answers to the teacher’s questions.
1
The teacher asks students direct questions about their thinking while other students listen.
The teacher explains student strategies, filling in any gaps before continuing to present
mathematical ideas. The teacher may ask one student to help another by showing how to do
a problem.
2
The teacher asks open-ended questions to elicit student thinking and asks students to comment on one another’s work. Students answer the questions posed to them and voluntarily
provide additional information about their thinking.
3
The teacher facilitates the discussion by encouraging students to ask questions of one
another to clarify ideas. Ideas from the community build on one another as students thoroughly explain their thinking and listen to the explanations of others.
Adapted from Hufferd-Ackles, Fuson, and Sherin (2004)
The following encounter provides an example of
nonsupportive motivational discourse in which the
teacher is more concerned with the right answer
than with the student’s thinking about the task.
Ms. D. Explain how you built the fourth tower.
Bill. [Holds up the tower he built with his partner.]
Ms. D. This is not the fourth tower in the pattern.
Does it look like it should be? You should be
able to build it with a picture. The first level has
1 cube; the second level has 5; the third level has
9. It’s going up by 4 each time. So, how many
cubes in the fourth level?
Bill. Thirteen.
Ms. D. Nine plus four equals thirteen. Now build
it like the picture. Who thinks they have it right?
As these examples indicate, the teacher’s role in discourse is complex. Teachers must be conscious about
the statements they make and the questions they ask
so that all students are encouraged to participate.
ASSESSING DISCOURSE
How do I know if the discourse in my classroom is
successful?
Mathematics discourse does not happen overnight,
particularly if students have experienced only
teacher-directed, procedure-oriented mathematics
classrooms. As a result, mathematics classroom
discourse is a dynamic process that is often hard to
assess. Hufferd-Ackles, Fuson, and Sherin (2004)
created a framework to describe and evaluate the
process a class goes through when discourse is
introduced. Four categories are examined—questioning, explanation of mathematical thinking,
288 Mathematics Teacher | Vol. 101, No. 4 • November 2007
source of mathematical ideas, and responsibility
for learning. A scale of 0 to 3 is used, where level
0 refers to a traditional, teacher-directed class, and
level 3 is reached when the teacher participates
as a member of the community and assists only as
needed (see table 1). Although this framework
serves as a good indicator for assessing the discourse level of the whole class, it does not assess
individual students. Teachers need to be aware of
how individual students are participating so that
they can encourage and scaffold students who are
not participating in the discourse.
CONCLUSION
Participating in a mathematical community through
discourse is as much a part of learning mathematics
as the conceptual understanding of the mathematics
itself. As students learn to make and test conjectures,
question, and agree or disagree about problems, they
are learning the essence of what it means to do mathematics. If all students are to be engaged, teachers must
foster classroom discourse by providing a welcoming
community, establishing norms, using supportive
motivational discourse, and pressing for conceptual
understanding. As Johnston (2004) puts it, “In other
words, the language that teachers (and their students)
use in classrooms is a big deal” (p. 10).
REFERENCES
Chazan, D., and D. Ball. “Beyond Exhortations Not to
Tell: The Teacher’s Role in Discussion-Intensive
Mathematics Classes.” (Craft Paper 95-2). East
Lansing, MI: National Center for Research on
Teacher Learning, 1995.
Hufferd-Ackles, K., K. Fuson, and M. Sherin.
“Describing Levels and Components of a Math-
Talk Learning Community.” Journal for Research in
Mathematics Education 35 (March 2004): 81–116.
Johnston, P. Choice Words: How Our Language Affects
Children’s Learning. Portland, ME: Stenhouse Publishers, 2004.
Kazemi, E., and D. Stipek. “Pressing Students to Be
Thoughtful: Promoting Conceptual Thinking in
Mathematics.” Paper presented at the annual meeting of the American Educational Research Association, Chicago, 1997.
Lampert, M. “When the Problem Is Not the Question
and the Solution Is Not the Answer: Mathematical Knowing and Teaching.” American Education
Research Journal 27 (Spring 1990): 29–63.
National Council of Teachers of Mathematics
(NCTM). Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000.
Phillips, E., with T. Gardella, C. Kelly, and J. Stewart.
Patterns and Functions: Addenda Series, Grades
5–8. Reston, VA: National Council of Teachers of
Mathematics, 1991.
Stoker, J. “Promoting the Professional Development of
Mathematics Teachers through Aligned Assessment
Tasks.” 2006. www.aare.edu.au/01pap/sto01354.htm.
Turner, J. C., D. K. Meyer, C. Midgley, and H. Patrick.
“Teacher Discourse and Sixth Graders’ Reported
Affect and Achievement Behaviors in Two HighMastery/High-Performance Mathematics Classrooms.” Elementary School Journal 103 (March
2003): 357–82.
Turner, J., C. Midgley, D. Meyer, M. Gheen, E. Anderman, Y. Kang, and H. Patrick. “The Classroom
Environment and Students’ Reports of Avoidance
Strategies in Mathematics: A Multimethod Study.”
Journal of Educational Psychology 94 (March 2002):
88–106.
Yackel, E., and P. Cobb. “Sociomathematical Norms,
Argumentation, and Autonomy in Mathematics.”
Journal for Research in Mathematics Education 27
(July 1996): 458–77. ∞
CATHERINE STEIN, ccstein@uncg
.edu, is a doctoral student at the
University of North Carolina at
Greensboro, Greensboro, NC 27403.
She is interested in fostering student participation in mathematics classroom communities.
Vol. 101, No. 4 • November 2007 | Mathematics Teacher 289
UNPACKING THE
NATURE OF DISCOURSE
in Mathematics
Classrooms
AND
DOMINIC PERESSINI
PHOTOGRAPH BY SCOTT DARSNEY; ALL RIGHTS RESERVED
ERIC KNUTH
ERIC KNUTH, knuth@education.wisc.edu, has professional
interests in mathematics discourse, proof, and the use of
technology in teaching mathematics. DOMINIC PERESSINI,
dominic.peressini@colorado.edu, teaches at the University
of Colorado, Boulder, CO 80309. He is especially interested in classroom discourse, assessment, and educational
reform.
320
T
HE ROLE OF DISCOURSE, ALTHOUGH
always central in education and learning, is
receiving increased attention in classrooms
today as mathematics educators strive to
better understand the factors that lead to increased
learning. Indeed, scholars have argued for—and reform initiatives underscore—the importance of
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
Copyright © 2001 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
A Framework for Examining Discourse
in a Mathematics Classroom
WE HAVE WORKED WITH CLASSROOM TEACHERS
over the last four years in a longitudinal professional development project funded by the Colorado
Commission of Higher Education. Our focus has
been to help teachers better understand and implement reform-based mathematics instruction and assessment. One of our central goals was to help
teachers foster more meaningful discourse in their
classrooms. We soon realized, however, the difficulties involved in discussing “meaningful mathematical discourse.” We drew on the work of Yuri
Lotman to develop a framework to help us make
sense of the different roles that discourse plays.
Lotman (1988) suggested that all discourse is
distinguished by two very different functions: to
convey meaning and to generate meaning. Wertsch
(1991) used the terms univocal and dialogic, respectively, to represent these two functions. Univocal discourse is characterized by communication in
which the listener receives the “exact” message
that the speaker intends for the listener to receive.
Once the speaker’s intended meaning has been
conveyed, the episode of univocal communication
is considered to be successfully finished. Dialogic
discourse, in contrast, is characterized by give-andtake communication in which the listener initially
receives the “exact” message sent by the speaker.
At this point, univocal discourse ends, but dialogic
discourse has just begun. Dialogic discourse generates meaning by using dialogue as a “thinking device” (Lotman 1988, p. 36). The visions of reformbased mathematics education embody dialogic
discourse in which both teachers and students are
responsible for contributing to discussions.
The classroom vignettes in the next section portray the same teacher using the same task to foster
both types of classroom discussions. These vignettes, which illustrate the interactions between a
teacher and her students and the interactions
among the students, are based on classroom observations of mathematics teachers participating in
our professional development project and on our
experiences as former secondary school mathematics teachers. Each vignette describes identical
solution approaches, but the teacher’s and students’ responses are different and illustrate distinctions between the two types of discourse. Our primary focus is on the nature of the discourse
demonstrated in each vignette.
The Dual Role of Discourse
in Mathematics
STUDENTS IN MS. BEE’S SEVENTH-
Different types
of discourse
emerge as
students solve
different tasks
grade mathematics class had spent
the previous two weeks working
with patterns. One of the goals of
this work was for students to develop the ability to generalize relationships found in the patterns. For
this activity, the students were
challenged to find the sum of the
first one hundred positive integers (i.e., 1 + 2 + 3 + . . .
+ 100). After introducing the problem, Ms. Bee asked
the students to work on determining a solution in
small groups. The following vignettes illustrate two
possible ways that this activity could unfold.
PHOTOGRAPH BY SCOTT DARSNEY; ALL RIGHTS RESERVED
teachers’ and students’ engaging in discourse of
various kinds (e.g., Ball 1991; NCTM 1991, 2000;
Steinbring, Bussi, and Sierpinska 1998). These calls
for more meaningful discourse are grounded in the
social nature of mathematics learning, a vision of
school mathematics practices that reflects both the
essence of practices in the discipline itself and the
need for students to be able to communicate their
mathematical knowledge in a technological society.
As we move into the twenty-first century, efforts
to enhance school mathematics teaching and learning continue. Building on research findings, previous reform recommendations, and lessons learned
from past reform efforts, the NCTM’s Principles
and Standards for School Mathematics (2000) offers
a renewed vision of school mathematics. Fostering
meaningful mathematical discourse in classroom
settings continues to be a central focus. This article
describes a framework for examining mathematical
discourse and shows how to apply this framework
to appreciate the complex relationship between discourse and understanding in mathematics. In particular, we focus our attention on the role of discourse and the different types of discourse that
emerge as students solve a particular task.
V O L . 6 , N O . 5 . JANUARY 2001
321
An example of univocal discourse
(1) “What number are you on, Helen?” asked Andy
as he stopped entering numbers into his calculator.
(2) “Thirty-two,” she replied.
(3) “Tell me what your total is when you get to 41,
so we can check our answers.”
(4) Andy waited while Helen continued to enter the
numbers on her calculator. The two other students
in the group, Barney and Gomer, were working together to look for possible patterns. They compared the sum of the first ten numbers (1 + 2 + . . . +
10 = 55) with the sum of the next ten numbers (11 +
12 + . . . + 20 = 155) and made a conjecture that the
sum of the numbers 21 through 30 would also increase by 100. At this point, Ms. Bee walked up to
the group.
(15) “You can see that your sum, Andy, is greater
than this number. You might want to double-check
your calculations. That’s one disadvantage of using
the calculator; it’s easy to make an entry mistake.
I’d like to see both you and Helen try to use some of
what we’ve been learning about patterns in the past
couple of weeks to find the sum. The two of you
might check in with Barney and Gomer to see what
they are doing.”
(16) Ms. Bee next turned her attention to Barney
and Gomer. “Now, Barney, you said that the two of
you thought you may have found a pattern. Tell me
what you’re doing.” At this point, both Andy and
Helen have gone back to entering numbers on their
calculators.
(17) “We found the sum of the first ten numbers,
then the next ten numbers. We were just getting
ready to check the sum of the next ten numbers
when you came up.”
(5) “How are the four of you coming along?”
Univocal
discourse
focuses on
sending an
exact message
(6) Andy quickly spoke up,
“We’re almost halfway there. I’ve
added the first forty-one numbers
and am waiting to check my total
with Helen’s.”
(18) “It sounds as if you are trying to find a pattern
by looking at the sums of different groups of ten
numbers. What you might try is a similar approach
of breaking down the one hundred numbers into
smaller sums, but start by looking at the sum of the
first two numbers, then the first three numbers, and
so on. See if this results in some kind of pattern.”
(7) “How about the two of you?
Are you also using the calculator
to find the sum?”
(19) Gomer looked a bit puzzled. “Do you mean do
1 + 2, then 1 + 2 + 3, and keep going like that?”
(8) “No, we’ve been looking for a
pattern, and I think we found
one,” Gomer responded.
(20) “Right. Remember how we found some of the
patterns for the problems last week? Try that same
approach, and see what you come up with.”
(9) “Great. First of all, Andy, why don’t you tell me
what sum you have found?” asked Ms. Bee.
(21) “OK. You want us to make a chart showing
each total, then look for a pattern in the chart,” Barney responded.
(10) Andy replied, “1720.”
(11) “That seems a little high. Have you found your
sum yet, Helen?”
(12) “No, I had to start over because I hit a wrong
key.”
(13) “Let’s think about what the sum must be less
than. For example, suppose we were adding fortyone 41s, then the sum would be 41 times 41. Barney
or Gomer, can you use your calculator to find that
product?”
(14) Barney entered the numbers into his calculator and responded, “The answer is 1681.”
322
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
(22) “Right,” Ms. Bee replied as she began to move
toward another group of students. Barney and
Gomer each pulled a new sheet of paper from their
notebooks and began to work on the approach that
Ms. Bee suggested.
Throughout this episode, Ms. Bee attempts to
both hear what the students are doing (lines 5, 7,
and 16) and to direct them to solve the problem in a
specific way that she has emphasized for similar
problems (lines 15 and 18). Ms. Bee first acknowledges the calculator approach used by both Andy
and Helen, but she does not give them an opportunity to make sense of Andy’s apparent miscalculation. Rather, she comments that his sum for the
first forty-one numbers “seems a little high” and
suggests that the students think about an upper
limit for the sum (line 13). In asking the students to
consider what the upper limit might be (“Let’s
think”), she encourages them to treat her suggestion as a “thinking device”; yet she suggests a procedure for determining the limit—perhaps in an effort to make sure that the students’ understanding
coincides with her own. Ms. Bee does not ask the
students to consider her suggestion or to explain
their understanding of the relationship of her calculation to Andy’s miscalculation. Finally, Ms. Bee directs the students toward the method that she
would like for them to use (last two sentences in
line 15) instead of allowing them to generate their
own approach.
Ms. Bee also elicits the approach that Barney
and Gomer have been using (line 16). Her initial response (first sentence of line 18) verifies the accuracy of the received message (line 17). Next, she attempts to reorient the students toward her
preferred approach (second sentence of line 18).
This exchange is univocal because Ms. Bee does
not try to understand the students’ method—she
does not hear what she wants to hear, thus she
redirects the students. In Lotman’s (1988) terms,
she noted a difference between the expected message (her solution approach) and the received message (the students’ solution approach) and perceives the discrepancy as a “defect in the
communications channel” (p. 36). Gomer’s question (line 19) indicates that he received her message as intended, and Ms. Bee confirms that their
understandings match. To ensure that her message
has been adequately conveyed, she then attempts
to strengthen the match by providing a shared reference point: “Remember how we found some of
the patterns for the problems last week?” Finally,
Barney recognizes this shared reference by further
explaining how he and Gomer will follow Ms. Bee’s
suggestion by using a chart (line 21). Ms. Bee then
acknowledges that the students understand what
she intended to communicate and that her message
has been successfully conveyed (line 22).
This passage is primarily univocal because the
teacher’s intention is to convey the message that
the students should use a particular approach. Ms.
Bee does her best to move the students toward her
solution method as she strives to align the students’
thinking about the problem with her own. She
makes sure that her intended message for this particular lesson, as well as for their work during the
preceding two weeks, is adequately conveyed. Her
focus is on how well everyone understands her perspective rather than on making sense of the students’ unique approaches to the problem. In con-
trast, the following vignette highlights the role of
the task, the students’ solution approaches, and the
teacher’s comments as generators of meaning—the
essence of dialogic discourse.
An example of dialogic discourse
(1) “What number are you on, Helen?” asked Andy
as he stopped entering numbers into his calculator.
(2) “Thirty-two,” she replied.
(3) “Tell me what your total is when you get to 41,
so we can check our answers.”
(4) Andy waited while Helen continued to enter the
numbers on her calculator. The two other students
in the group, Barney and Gomer, were working together to look for possible patterns. They compared the sum of the first ten numbers (1 + 2 + . . . +
10 = 55) with the sum of the next ten numbers (11 +
12 + . . . + 20 = 155) and made a conjecture that the
sum of the next ten numbers, 21
through 30, would also increase
by 100. At this point, Ms. Bee
walked up to the group.
(5) “How are the four of you
coming along?”
(6) Andy quickly spoke up,
“We’re almost halfway there.
I’ve added the first forty-one
numbers and am waiting to
check my total with Helen’s.”
Dialogic
discourse
focuses on
two-way
communication
(7) “How about the two of you? Are you also using
the calculator to find the sum?”
(8) “No, we’ve been looking for a pattern, and I
think we found one,” Gomer responded.
(9) “Great. First of all, Andy, why don’t you tell me
what sum you have found?” asked Ms. Bee.
(10) Andy replied, “1720.”
(11) “That seems a little high. Have you found your
sum yet, Helen?”
(12) “No, I had to start over because I hit a wrong
key.”
(13) Addressing the entire group, Ms. Bee asked,
“Why do you think I said that Andy’s sum seems a
little high?”
V O L . 6 , N O . 5 . JANUARY 2001
323
(14) “Because you already know the answer,”
Helen suggested, smiling.
numbers, we think that the sum of the next set of
ten numbers will also go up by 100.”
(15) “Well, let’s think about what the sum must be
less than. Any ideas on how we might determine an
upper limit for the sum? Barney and Gomer, what
do you think?”
(26) “So what will that tell you?”
(16) “It’s probably less than a million,” Gomer
suggested.
(17) “OK, I’d agree with that. Can we think of a way,
using mathematics, to find an upper limit?”
(18) “What if we only added half of the numbers
and then multiplied it by 2?” suggested Helen.
(19) “That would not be an upper limit because the
upper half of the numbers is greater than the lower
half of the numbers,” argued Barney.
(20) “So what if we took the upper half of the numbers and multiplied that total by 2?” Andy asked.
Communication
mismatch can
be a point of
departure for
inquiry
(21) At this point, Ms. Bee decided to involve the rest of the
class in this discussion. She
asked for the students’ attention and described what this
group was attempting to figure
out. After a brief discussion,
Ms. Bee turned her attention
back to Barney and Gomer and
asked about the pattern that
they claimed to have found.
(22) “Now, Barney, you said that the two of you
thought you may have found a pattern. Tell me
what you’re doing.” At this point, both Andy and
Helen have gone back to entering numbers into
their calculators.
(23) Barney described their approach: “Well, we
found the sum of the first ten numbers, then the
sum of the next ten numbers. We were just getting
ready to check the sum of the next ten numbers
when you came up.”
(24) “So it sounds as if you are trying to find a pattern by looking at the sums of different groups of
ten numbers. So what is it you are looking for with
these sets of numbers?” Ms. Bee asked.
(25) “We are checking for a possible pattern. Since
the sum of the second set of ten numbers went up
by 100 compared with the sum of the first set of ten
324
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
(27) Barney began to respond, “If it works . . . ,”
when Gomer interrupted, “We’ll know what the pattern is, then it will be easy to find the final total.”
(28) “Cool. I haven’t seen that approach before in
any of my other classes. It will be interesting to try
to figure out why that works.” At this point, Ms. Bee
moved on to another group of students.
A number of aspects of this dialogue embody
characteristics of dialogic discourse. Again, Ms.
Bee first attempts to hear what the students are
doing (lines 5, 7, 9, and 22), but then she continues
to listen and uses the students’ discourse to generate meaning for both herself (lines 24 and 26) and
her students (lines 13, 15, and 17). Ms. Bee again
acknowledges the calculator approach used by
both Andy and Helen; however, she takes advantage of a learning opportunity by using Andy’s apparent miscalculation as a point of departure to look
for an upper limit. In Ms. Bee’s mind, Andy’s error
offered an unexpected, yet appropriate, avenue to
further explore important mathematics. Rather
than tell Andy and Helen her procedure for finding
an upper limit, Ms. Bee turns to the whole class for
further inquiry and discussion. She prompts students three times to treat her suggestion as a thinking device and gives them repeated opportunities to
consider her suggestions (lines 13, 15, and 17). She
also directs students to use mathematics (line 17),
at which point we begin to see students generate
their own meanings by treating one another’s statements as thinking devices (lines 18 through 20).
As in the previous episode, Ms. Bee again elicits
the solution approach that Barney and Gomer have
been working on (line 22). Her initial response
(first sentence of line 24) verifies the accuracy of
the received message (line 23)—an aspect of univocal discourse necessary for clear communication.
Rather than guide Barney and Gomer to her solution method, however, Ms. Bee asks what they
were looking for with their approach (line 24) and
how their approach will help them solve the original problem (line 26). Ms. Bee is listening dialogically, using the students’ responses as generators
of meaning to better understand what they are
thinking. In Lotman’s terms, she notes a difference
between the expected message (her solution approach) and the received message (the students’
solution approach). She perceives this mismatch
not as a “defect in the communications channel”
(Lotman 1988, p. 36) but rather as a point of departure to generate new meanings for herself and her
students; thus, the students’ and teacher’s utterances function as thinking devices. When Ms. Bee
leaves Barney and Gomer alone to explore how
their pattern might be used to solve the problem,
she and her students also seem comfortable in
sharing the mathematical authority in the classroom (lines 27 and 28).
This passage is primarily dialogic because (a)
the teacher intends to understand her students’
thinking, (b) the teacher uses her students’ statements as thinking devices, and (c) the students use
Ms. Bee’s suggestion and their classmates’ statements as thinking devices. Ms. Bee does not attempt to convey a particular message, that is, to engage students in a specific approach. Instead, she is
open to her students’ ideas and is willing to pursue
unexpected approaches to generate new mathematical understanding—the core of dialogic discourse.
Meaningful Mathematical Discourse
in Classrooms
WE REALIZE THAT THE DISTINCTION BETWEEN
univocal and dialogic discourse is at times difficult
to discern. Indeed, in any social interaction involving spoken communication, each individual must
both decipher what is said and generate his or her
own meaning from it. Consequently, all discourse
is, to some degree, both dialogic and univocal. In
other words, discourse may be thought of as a continuum that is more or less dialogic or univocal. We
find that most discourse, however, is characterized
primarily by one of these functions. We often look
to the speaker’s intent—to transfer meaning or generate new meaning—to determine which function
is more prevalent. In a similar fashion, we also examine the listener’s intent in making sense of classroom discourse.
To recognize the dual role of discourse in classrooms, we as teachers must reflect on our instructional goals and how they relate to our intentions as
speakers and listeners and to our students’ intentions. In the first vignette, the goal in Ms. Bee’s
mind was for her students to arrive at a solution
using what they had previously learned about patterns. Ms. Bee’s students understood that she had a
particular idea of what mathematical approach she
wanted them to use. When she and her students engaged in discourse, the intentions of both teacher
and students as they spoke and listened reflected
this overall understanding. The vignette, therefore,
reflects the goals of the teacher and, as a result, is
mostly univocal. In the second vignette, Ms. Bee’s
goals were clearly different because she was more
interested in pursuing students’ ideas and seeing
where those ideas led mathematically. The teacher
and students understood that the classroom goals
were much more dynamic, and although the focus
was still on finding patterns, the speakers’ and listeners’ intentions had changed because the class
was not searching for the teacher’s way to solve the
problem. The students in the second scenario were
sharing the mathematical authority of the classroom, using the teacher’s and one another’s statements as thinking devices to generate new, and
sometimes unexpected, mathematical meaning.
We are not arguing that one of these vignettes is
necessarily better than the other. Both univocal and
dialogic can be seen as appropriate forms of discourse, depending on the daily instructional goals.
We also recognize the reality of a teacher’s classroom, which includes the competing demands of
depth versus breadth in content coverage, the presence of students of dissimilar abilities and interests,
and time constraints. These factors often influence
the classroom goals, which in turn influence the nature of discourse. Nevertheless, we recognize the
need for students and teachers to
engage in more dialogic discourse; students will acquire a
deeper understanding of mathematics when they use their own
statements, as well as those of
their peers and teacher, as thinking devices. This goal speaks to
the heart of reform-based mathematics instruction, which is the
hope that the accompanying pedagogical approaches and strategies will lead students to acquire a deeper understanding of mathematics.
Understanding
deepens when
students use
their own
statements
References
Ball, Deborah. “Implementing the Professional Standards
for Teaching Mathematics: What’s All This Talk about
Discourse?” Arithmetic Teacher 39 (November 1991):
44–48.
Lotman, Yuri. “Text within a Text.” Soviet Psychology 24
(1988): 32–51.
National Council of Teachers of Mathematics. Professional Standards for Teaching Mathematics. Reston,
Va.: NCTM, 1991.
———. Principles and Standards for School Mathematics.
Reston, Va.: NCTM, 2000.
Steinbring, Heinz, Maria Bartolini Bussi, and Anna Sierpinska, eds. Language and Communication in the
Mathematics Classroom. Reston, Va.: NCTM, 1998.
Wertsch, James. Voices of the Mind. Cambridge: Harvard
University Press, 1991. C
V O L . 6 , N O . 5 . JANUARY 2001
325
Question Prompts and Stems to Support Mathematical Discourse





Revoicing – So you are saying that . . .
Restate someone else’s reasoning – Can you repeat what she just said in your own words?
Apply their own reasoning to someone else’s – What do you think about that? Do you agree or disagree? Why?
Prompt for further participation – Would someone like to add on?
Respond neutrally to errors – What do you think about that? (to whole class)
 Help students rely more on themselves to determine whether something is mathematically correct –
o How did you reach that conclusion?
o Does that make sense?
o Can you make a model and show that?
 Help students learn to reason mathematically –
o Why does . . . work?
o Does that always work?
o Is that true for all cases?
o Can you think of a counterexample?
o How could you prove that?
 Help students to learn to conjecture, invent, and solve problems –
o What would happen if?
o Do you see a pattern?
o Can you predict the next one? What about the last one?
o When does . . . work?
o When will . . . be (larger, smaller, equal to, exactly twice, etc.) compared to . . .?
o When will . . . be as large (small) as possible?
o How are they alike? How are they different?
o Describe how to find . . .?
o What do I do if I want . . . to happen?
 Help students connect mathematics, its ideas and applications –
o How does this relate to . . .?
o What ideas that we have learned were useful in solving this problem?
o What advantages does this strategy have?
Bloom's Revised Taxonomy
Bloom created a learning taxonomy in 1956. During the 1990's, a former student of
Bloom's, Lorin Anderson, updated the taxonomy, hoping to add relevance for 21st
century students and teachers. This new expanded taxonomy can help instructional
designers and teachers to write and revise learning outcomes.
Bloom's six major categories were changed from noun to verb forms.
The new terms are defined as:
Remembering
Retrieving, recognizing, and recalling relevant knowledge
from long-term memory.
Understanding
Constructing meaning from oral, written, and graphic
messages through interpreting, exemplifying, classifying,
summarizing, inferring, comparing, and explaining.
Applying
Carrying out or using a procedure through executing, or
implementing.
Analyzing
Breaking material into constituent parts, determining how
the parts relate to one another and to an overall structure or
purpose through differentiating, organizing, and
attributing.
Evaluating
Making judgments based on criteria and standards through
checking and critiquing.
Creating
Putting elements together to form a coherent or functional
whole; reorganizing elements into a new pattern or
structure through generating, planning, or producing.
S. DeMatteo, 8/13/2014
Because the purpose of writing learning outcomes is to define what the instructor wants
the student to do with the content, using learning outcomes will help students to better
understand the purpose of each activity by clarifying the student’s activity. Verbs such as
"know", "appreciate", "internalizing", and "valuing" do not define an explicit
performance to be carried out by the learner. (Mager, 1997)
Unclear Outcomes
Revised Outcomes
Students will know described
cases of mental disorders.
Students will be able to review a
set of facts and will be able to
classify the appropriate type of
mental disorder.
Students will understand the
relevant and irrelevant numbers
in a mathematical word problem.
Students will distinguish between
relevant and irrelevant numbers
in a mathematical word problem.
Students will know the best way
to solve the word problem.
Students will judge which of the
two methods is the best way to
solve the word problem.
Figure 2: Examples of unclear and revised outcomes.
References
Anderson, L. W., & Krathwohl, D. R. (Eds.). (2001). A taxonomy for learning, teaching
and assessing: A revision of Bloom's Taxonomy of educational outcomes:
Complete edition, New York : Longman.
Cruz, E. (2003). Bloom's revised taxonomy. In B. Hoffman (Ed.), Encyclopedia of
Educational Technology. Retrieved August 22, 2007, from
http://coe.sdsu.edu/eet/articles/bloomrev/start.htm
Forehand, M. (2005). Bloom's taxonomy: Original and revised.. In M. Orey (Ed.),
Emerging perspectives on learning, teaching, and technology. Retrieved August
22, 2007, from http://projects.coe.uga.edu/epltt/
S. DeMatteo, 8/13/2014
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Implementing “Math
Talk” in Your
Classroom
Sonia Dupree, Sr. Administrator for High School Math
Anna Jackson, Coordinating Teacher for High School Math
Wake County

Defining Mathematical
Discourse

Brainstorm
What is mathematical discourse?
What teacher and student behaviors occur
in a classroom where the teacher
promotes discourse?
Defining Mathematical Discourse
 Discourse: written or spoken communication
or debate
- Oxford Dictionary
What does NCTM say?
Communication
Instructional programs from prekindergarten through grade 12 should
enable all students to—
 Organize and consolidate their mathematical thinking through
communication
 Communicate their mathematical thinking coherently and clearly
to peers, teachers, and others
 Analyze and evaluate the mathematical thinking and strategies of
others;
 Use the language of mathematics to express mathematical ideas
precisely.
See more at: http://www.nctm.org/standards/content.aspx?id=322#sthash.rEE2w8Ms.dpuf
What does the
Common Core say?
Understanding Mathematics
These Standards define what students should understand and be able to do in
their study of mathematics. Asking a student to understand something means
asking a teacher to assess whether the student has understood it. But what
does mathematical understanding look like? One hallmark of mathematical
understanding is the ability to justify, in a way appropriate to the student’s
mathematical maturity, why a particular mathematical statement is true or
where a mathematical rule comes from. There is a world of difference
between a student who can summon a mnemonic device to expand a product
such as (a + b)(x + y) and a student who can explain where the mnemonic
comes from. The student who can explain the rule understands the
mathematics, and may have a better chance to succeed at a less familiar task
such as expanding (a + b + c)(x + y). Mathematical understanding and
procedural skill are equally important, and both are assessable using
mathematical tasks of sufficient richness. – CCSSM, p. 4
What does Common Core say?
Common Core Standards for Mathematical Practice
 Skim through the Standards.
Underline or highlight
everything that is related to
discourse.
 Talk with a shoulder buddy:
What stands out to you? What
kinds of discourse are already
taking place in your classroom?
What are areas of need?
Standards for Math Practice
3. Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions,
and previously established results in constructing arguments. They make conjectures
and build a logical progression of statements to explore the truth of their conjectures.
They are able to analyze situations by breaking them into cases, and can recognize and
use counterexamples. They justify their conclusions, communicate them to others, and
respond to the arguments of others. They reason inductively about data, making
plausible arguments that take into account the context from which the data arose.
Mathematically proficient students are also able to 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. Elementary students can
construct arguments using concrete referents such as objects, drawings, diagrams, and
actions. Such arguments can make sense and be correct, even though they are not
generalized or made formal until later grades. Later, students learn to determine
domains to which an argument applies. Students at all grades can listen or read the
arguments of others, decide whether they make sense, and ask useful questions to
clarify or improve the arguments.
Definition of Mathematical
Discourse
A process by which students use discourse, both verbal and
written, to reflect on the mathematics they have engaged with
in order to discover important mathematical concepts and to
develop mathematical thinking.
Teaching Practices and
the Teacher’s Role

So now that I know what it is,
how do I do it?
How to Get Students Talking!: Generating Math Talk That Supports
Math Learning by Lisa Ann de Garcia.
 “Common Core . . . make[s] it clear that conceptual
understanding must be connected to the procedures, and
that one way to deepen conceptual understanding is through
the communication students have around concepts,
strategies, and representations.”
 “Children do not naturally engage in this level of talk.”
Practice 1: Talk Moves That
Engage Students in Discourse
 Revoicing – So you are saying that . . .
 Restate someone else’s reasoning – Can you repeat what she
just said in your own words?
 Apply their own reasoning to someone else’s – What do you
think about that? Do you agree or disagree? Why?
 Prompt for further participation – Would someone like to add
on?
 Use wait time!
Practice 2: The Art of
Questioning
 Help students work together to make sense of mathematics (Practice 1
questions)
 Help students rely more on themselves to determine whether something is
mathematically correct – How did you reach that conclusion? Does that make sense?
Can you make a model and show that?
 Help students learn to reason mathematically - Does that always work? Is that
true for all cases? Can you think of a counterexample? How could you prove that?
 Help students learn to conjecture, invent, and solve problems – What would
happen if ? Do you see a pattern? Can you predict the next one? What about the last
one?
 Help students connect mathematics, its ideas and applications – How does this
relate to . . .? What ideas that we have learned were useful in solving this problem?
Practice 3: Using Student
Thinking to Propel Discussions
 Be an active listener
 Respond neutrally to errors – What do you think about that? (to whole
class)
 Be strategic about who shares during the discussion
 Choose ideas, strategies, and representations in a purposeful way
Practice 4: Set Up a Supportive
Environment
 Have students facing each other – e.g. desks in groups for partner or
small group discussions; students sitting in a circle for whole group
 Place visual aids and vocabulary where they can be easily accessed
 Create a safe emotional environment where the value is on learning,
challenging each other, and working together to solve problems as
opposed to just getting the right answer
Practice 5: Orchestrating the
Discourse
The Five Practices Model
The teacher’s role is to:
1. anticipate student responses to challenging mathematical
tasks;
2. monitor students’ work on and engagement with the tasks;
3. select particular students to present their mathematical work;
4. sequence the student responses that will be displayed in specific
order; and
5. connect different students’ responses and connect the
responses to key mathematical ideas.
Hold Students Accountable
 Explicitly teach students how to engage in each level of
discussion: whole group, small group, partnerships
 Model the behavior – e.g. do a fishbowl of a small group or
partnership discussion, show video clips of discussions and
debrief
 Address not only content but also behavior when summarizing
– I liked how Sarah asked Tom to explain what he meant, That group
did a great job with listening to each other, etc.
 Do a plus/delta on the discussion – What went well? Where do
we need to improve?
Hold Students Accountable
 Let them know exactly what they should be saying when they
are talking in their partnerships or small groups – Today, when
you are talking to your partners and describing ______, I expect to
hear you using the words ______.
 Let students know what to focus on when someone is sharing a
strategy – When Maria is sharing her thinking, I want you to be
thinking of how her way is similar to or different from your way.
Hold Students Accountable
 Heighten students awareness of themselves as learners
through self-evaluation and goal setting
 have students set and track personal goals related to
participation in mathematical discussions – e.g. exit ticket of a
plus/delta on their participation
 support students in being open with each other regarding their
strengths and weaknesses so they can improve their
communication skills and behaviors – e.g. hold a class meeting
that focuses on this
Experience
Mathematical Discourse
from a Student’s
Perspective

What does it feel like?
The Tower Problem
Use the blocks to build the
fourth tower in the sequence.
How many cubes did you use?
How many cubes would you
need to build the fifth tower?
The 12th tower? The 20th tower?
The 100th tower? Write a rule
to help you find the number of
cubes for the nth tower.
Take a break as needed while your group works on this problem.
Example Discourse:
The Good, the Bad, and
the Ugly

Example Discourse – the Good,
the Bad, and the Ugly
Read the “Facilitating Discourse” section p. 286-288 of Let’s
Talk: Promoting Mathematical Discussions in the Classroom by
Catherine C. Stein. Discuss with your shoulder buddy:
 What is the difference between cognitive and motivational
discourse? Why are both important?
 What is the difference between low-press and high-press
classrooms? How does the level of “press” affect student
learning?
Example Discourse – the Good,
the Bad, and the Ugly
Read “An example of univocal discourse” on p. 322 of
Unpacking the Nature of Discourse in Mathematics Classrooms by
Eric Knuth and Dominic Peressini.
In your group:
 Identify any missed opportunities (give specific line number
and explain).
 How could the discourse be improved?
Keys to Mathematical
Discourse

The Keys to Mathematical
Discourse
 Authentic, Rich Tasks
 Level of Questioning
The only reasons to ask
questions are:
To PROBE or uncover students’ thinking.
• understand how students are thinking about the
problem.
• discover misconceptions.
• use students’ understanding to guide instruction.
To PUSH or advance students’ thinking.
• make connections
• notice something significant.
• justify or prove their thinking.
(Black et al., 2004)
Question Prompts and Stems
Question Analysis
 Revised Bloom’s Taxonomy
 Question Analysis Activity
Authentic, Rich Tasks
Current research evidence indicates that students who are
given opportunities to work on their problem solving skills
enjoy the subject more, are more confident and are more likely
to continue studying mathematics, or mathematically related
subjects, beyond the age of 16. Most importantly to some,
there is also evidence that they do at least as well in standard
tests such as GCSEs and A-levels.
http://nrich.maths.org/6299
Authentic, Rich Tasks
Rich tasks (or good problems):
 are accessible to a wide range of learners,
 might be set in contexts which draw the learner into the mathematics
either because the starting point is intriguing or the mathematics that
emerges is intriguing,
 are accessible and offer opportunities for initial success, challenging the
learners to think for themselves,
 offer different levels of challenge, but at whatever the learner's level there is
a real challenge involved and thus there is also the potential to extend
those who need and demand more (low threshold - high ceiling tasks),
 allow for learners to pose their own problems,
 allow for different methods and different responses (different starting
points, different middles and different ends),
http://nrich.maths.org/5662
Authentic, Rich Tasks
 offer opportunities to identify elegant or efficient solutions,
 have the potential to broaden students' skills and/or deepen and broaden
mathematical content knowledge,
 encourage creativity and imaginative application of knowledge.
 have the potential for revealing patterns or lead to generalizations or
unexpected results,
 have the potential to reveal underlying principles or make connections
between areas of mathematics,
 encourage collaboration and discussion,
 encourage learners to develop confidence and independence as well as to
become critical thinkers.
http://nrich.maths.org/6299
How do I incorporate this?
Start Simple (KISS!)
 Take current problems and make them better
 Set a goal: I will incorporate the use of a rich
task once a week, once every two weeks, etc.
 Stick with it – it won’t be easy for you or your
students; lean on each other in your PLT
 Don’t reinvent the wheel – there are plenty of
resources out there
Typical Problem
The children in the Wright family are aged
3, 8, 9, 10, and 5. What is their average age?
Better Problem
There are five people in a family and their
average age is 7. What might their ages be?
Typical Problem
Round 11.8 to the nearest whole number.
Better Problem
My coach timed me running 100 meters in
about 12 seconds. What numbers might have
been on the stopwatch?
How About This?
There are 6 birds and 2 cats. If the
answer is . . .
a. 20
b. 8
c. 4
What could the question be?
Better Questions, Better Results
Rich Tasks – Where do I find
them?
 Core Plus is full of them
 NCTM: Illuminations
 Mathematics Assessment Project
 Ohio Resource Center
 University of Cambridge: NRICH Project
Assessing Discourse

Basic Rubric for Assessing Levels of
Discourse in a Math Classroom
http://www.nctm.org/publications/mt.aspx?id=8594
Observation Tools
 Scripting of Questions/Question Analysis Tool
 Classroom Discourse Data Tool
 Student Discourse Observation Tool
 Video Modeling
 OMLI Classroom Observation
Shifting Our Perspective
When students don’t seem to understand
something, my instinct is to consider how I can
explain more clearly. A better way is to think
“They can figure this out. I just need the right
question.” - D. Kennedy (2002)
Never say anything a kid can say. - Reinhart (2000)
Thank you!
 Sonia Dupree: sdupree@wcpss.net
 Anna Jackson: ajackson1@wcpss.net