____ THE
______ MATHEMATICS____
_________ EDUCATOR _____
Volume 18 Number 2
-
Fall 2008
MATHEMATICS EDUCATION STUDENT ASSOCIATION
THE UNIVERSITY OF GEORGIA
Editorial Staff
A Note from the Editor
Editors
Ryan Fox
Diana May
Dear TME Reader,
Along with my co-editor Diana May, I welcome you to the second, and final, issue of the
18th volume of The Mathematics Educator (TME). It is my hope that the articles found in this
issue engage and sustain the discussion among members of our audience and the larger
mathematics education community.
Associate Editors
Tonya Brooks
Allyson Hallman
Catherine Ulrich
Advisor
Dorothy Y. White
As has been our mission at TME, we aim to provide a variety of perspectives on issues
within the mathematics education community. For this issue’s invited editorial, Azita
Manouchehri presents a study on using tasks to understand how individuals preparing to become
mathematics teachers learn mathematics content. In this issue there are three articles that provide
a variety of perspectives on issues within mathematics education. In our first piece, Alison Castro
Superfine shows a model for teachers’ planning of instruction as teachers are using a reformbased curriculum. Denise Forrest’s piece discusses the connection between communication theory
and mathematics instruction, using the theory to help explain teachers’ beliefs. Finally, Elliott
Ostler, Neal Grandgenett, and Carol Mitchell show us a new approach for analyzing assessment
instruments, one that takes a critical use at the use of rubrics.
MESA Officers
2008-2009
President
Brian Gleason
Vice-President
Sharon O’Kelley
Secretary
Allyson Hallman
There are many people involved here at MESA that have helped in making this issue
possible, and I would like to thank them at this time. I want to thank my colleagues who worked
as Co-Editor and Associate Editors for this issue; their names appear at the top of the column to
the left. Additionally, there are many individuals who have provided additional assistance this
semester in a unique capacity. These individuals have been part of a seminar that introduces them
to the work of our journal. They have done great work in whatever task we have given them:
Zandra de Araujo, Eric Gold, Erik Jacobson, Hee Jung Kim, Hulya Kilic, Ana Kuzle, Laura
Lowe, Anne Marie Marshall, and Laura Singletary. I cannot thank you enough for all that you
did!
Treasurer
Richard Francisco
Colloquium Chair
Dana TeCroney
NCTM
Representative
Ryan Fox
Undergraduate
Representative
Emily Ferris
Cynthia Thomas
Ryan Fox
105 Aderhold Hall
The University of Georgia
Athens, GA 30602-7124
tme@uga.edu
www.coe.uga.edu/tme
About the Cover
On the front cover of this issue, we include sketches from Elliott Ostler, Neal Grandgenett, and Carol Mitchell’s piece
on new forms of assessment. In their article, readers are asked to reflect upon ways to assess that students can verify
the area of a circle: one such way is presented here on the front cover.
This publication is supported by the College of Education at The University of Georgia
____________ THE ________________
___________ MATHEMATICS ________
______________ EDUCATOR ____________
An Official Publication of
The Mathematics Education Student Association
The University of Georgia
Fall 2008
Volume 18 Number 2
Table of Contents
3 Guest Editorial… Motivating Growth of Mathematics Knowledge for Teaching: A
Case for Secondary Mathematics Teacher Education
AZITA MANOUCHEHRI
11 Planning for Mathematics Instruction: A Model of Experienced Teachers’
Planning Processes in the Context of a Reform Mathematics Curriculum
ALISON M. CASTRO SUPERFINE
23 Communication Theory: Another Perspective to Think About for Mathematics
Teachers’ Talk
DENISE B. FORREST
33 Rethinking mathematics and science assessment: Some reflections on Solution
Dynamics as a way to enhance quality indicators
ELLIOTT OSTLER, NEAL GRANDGENETT, CAROL MITCHELL
40 Submissions information
41 Subscription form
© 2008 Mathematics Education Student Association
All Rights Reserved
2
The Mathematics Educator
2008, Vol. 18, No. 2, 3–10
Guest Editorial…
Motivating Growth of Mathematics Knowledge for Teaching: A
Case for Secondary Mathematics Teacher Education
Azita Manouchehri
There is consensus within the teacher education
community that effective teaching hinges upon several
factors. Those factors include the teacher’s knowledge
of the subject matter, ways the subject matter could be
manipulated to be made meaningful and accessible to
learners, a deep understanding of learners and their
developmental trajectories, and a perspective on short
and long term trajectory of curriculum. Teachers need
to learn how to select appropriate strategies by
reflecting on what factors influence the adaptation of
particular approaches when teaching specific concepts.
They also need to develop a disposition of inquiry and
a professional attitude that allows them to continue to
learn from practice (Hiebert et al., 2003). A major
challenge in mathematics teacher education is fostering
prospective teachers’ knowledge base in all these
domains. As a means to meet this challenge, scholars
have proposed that case-based tasks can serve as a
powerful vehicle for advancing teacher learning and
nurturing the desired dispositions (Richardson, 1996).
It is suggested that as teachers examine dilemma
driven tasks and analyze teaching actions they not only
learn about teaching but also develop conditional
knowledge that is crucial to effective practice (Kishner
& Whitson, 1997). In light of these perceived benefits,
the use of written, video, or animated case studies in
methods courses designed for teachers has gained
considerable momentum in the past decade (Merseth,
2003). Certainly, sound analysis of teaching actions
calls for deep reflection on the subject matter, the
structure of the discipline, and its associated
ontological and epistemological obstacles and issues.
This specialized body of teaching knowledge can be
better nurtured when the contexts for learning are
presented to students of teaching at the appropriate
time and juncture. As such, content courses for
teachers present an ideal environment for raising
teachers’ awareness of the complexities of teaching the
subject matter to children. When used in a mathematics
Azita Manouchehri is a professor of mathematics education at
Ohio State University. Her research has focused on the study of
classroom interactions and evolution of mathematical discourse
in presence of problem based instruction.
Azita Manouchehri
content course however, the tasks need to be crafted
carefully so as to ensure that mathematics is treated
soundly while allowing for the development of insight
in both areas. Our research was an attempt at first
developing and then examining the potential of the
type of case-based tasks that could be used in a content
course designed for teachers. One research question
guided our research efforts: What impact do case-based
tasks have on prospective teachers’ mathematics
learning when used as instructional tools in a content
course required for prospective secondary mathematics
teachers?
In this article, we will first describe the task we
designed and used as the research instrument in our
study. Drawing from data collected from two teaching
experiments, we will outline ways in which the task
seemingly enhanced mathematical and pedagogical
development of the participants.
Task Design Issues: Goals and Considerations
Our task design was guided by two prominent
scholarly voices as they pertain to the scope, goals, and
the audience of this project: recommendations of
literature for development of contexts for learning in
mathematics teacher education and a situated cognition
perspective on task design. In listening to the voice of
mathematics teacher education scholars, both
mathematical and pedagogical goals were considered.
Mathematically, we wanted the task to motivate
reflection on connections among various (seemingly
disjointed) mathematical ideas, engaging them in
mathematical
problem
solving
and
critical
mathematical analysis. Pedagogically, we wanted to
increase teachers’ awareness of ways in which
children’s work could impact instruction and
curriculum decision making. The content of the task
was chosen, bearing in mind these recommendations.
The structure of the task was chosen so as to align with
the constitutive elements of situated learning. We
wanted it to be authentic, dilemma driven, in order to
be conducive for the development of discourse,
collaboration, reflection, and critical thinking.
3
In search of the center: Analyzing construction algorithms
7. Examine the three attached chapters from three
The following problem was assigned to a group of
different textbooks on triangles and circles. How
geometry students who had used GSP in exploring
does the content of these chapters differ from the
geometry concepts in class. The students had spent five
mathematics content that the group is addressing?
instructional periods learning about and working on
How do you account for these differences? What
explorations concerning centers of triangle.
assumptions can you make about the teacher of this
Additionally, the group had learned in the previous
group and her choice of curriculum? What is your
session that in a circle, if a chord is the perpendicular
assessment of these assumptions?
bisector of another chord, then it is a diameter of the
8. Study the article: “A journey with circumscribable
circle. The teacher of this group posed the following
quadrilaterals” by Charles Worrall in the October
problem in order to see whether students could use
2004 issue of Mathematics Teacher Journal
their knowledge of diameter in a novel context.
[Volume 98(3)]. How would you compare the
Problem:
investigations that the author described in his
article with those of students in this class? If you
The center of a circle was accidentally erased.
had to use an activity from that article to use with
Define a procedure for locating the center.
this group of students, which would it be and why?
9. The NCTM Professional Standards (1991) suggest
A list of responses offered by several students
that the teachers of mathematics must provide
during a whole group sharing time of strategies is
opportunities for students to engage in building
presented below.
conjectures, verifying their conjectures and
debating the accuracy of those conjectures. In such
1. Test each of the methods presented below, and
a setting, the role of the teacher is to facilitate the
decide whether the suggested procedure leads to
students’ development by orchestrating tasks and
locating the center. Make a list of “mathematical
assignments that extend the students’
assumptions” that each child has seemingly made
understanding of the concepts, while helping them
and issues that might need to be resolved in each
realize the efficiency and elegance of ideas.
case.
Imagine that you are the teacher of this group.
2. How are the students’ responses similar? How are
Write an outline of a whole group classroom
they different? Which of these responses draw on
discussion that you might lead, along with any
the same mathematical concepts? Justify your
tasks that you will use to structure their work.
responses.
Justify why the exploration would be helpful in
3. Rank the student responses along a continuum
addressing the mathematical needs of the group.
ranging from Irrelevant to Sophisticated (you may
choose your own ranking categories if you find this
Student 1:Inscribe a triangle ABC in the circle.
range inappropriate). Explain the basis for your
Then construct the perpendicular bisectors of the sides.
ranking (as well as your categories, if you have
The point of intersection of these perpendicular
chosen to use a different ranking scheme).
bisectors is the center of the circle (the circumcenter of
4. Hypothesize about the mathematical issues that the
the triangle).
teacher needs to address with the group. That is,
Student 2:First inscribe an equilateral triangle in
what are the central mathematical topics that she
the circle. Since in an equilateral triangle incenter,
needs to bring up and synthesize? What is the basis
orthocenter, circumcenter, and centroid coincide, once
for your choice?
we find one of them then that point is the center of the
5. Decide the type of feedback the teacher might
circle. So, construct the medians and mark the point of
provide to each child based on his/her suggested
intersection of the medians. That point is the center of
method. How could the teacher expand the
the circle.
thinking of each student and help him/her justify
Student 3:Locate three points on the circumference
her/his approach? For instance, what questions
of the circle. From those points construct tangent lines
could the teacher ask? What extensions could the
to the circle. In this way the circle becomes the incircle
teacher offer? What example could the teacher use
of the triangle we drew. Now, if we construct angle
to counter the false assumptions?
bisectors of triangle ABC, the point of intersection is
6. What other methods could be used to find the
the center of the circle we started with.
center of the circle?
4
Motivating Growth of Mathematics Knowledge for Teaching
Student 4:Consider circle c. Draw chord AB of
circle c. From either B or A, construct a line
perpendicular to AB to form an inscribed 90-degree
angle. Mark the point of intersection of the
perpendicular line and the circle, label it as C. Now, we
know the side opposite an inscribed right angle is the
diameter of the circle that circumscribes the angle. All
we need now is another diameter which we can find
using the same procedure as before. The point of
intersection of these two diameters is the center of the
circle.
Student 5:First fold the circle in half. Then fold it
again. Now we have two diameters that intersect at the
center.
Student 6:Consider circle j and chord AB of j. Find
the midpoint of segment AB, label it as M. Construct a
circle centered at M with radius MB. What we have
now is two circles that intersect at two points. The
segment connecting the centers of the two circles is the
perpendicular bisector of the line segment that
connects the points of intersection of the two circles.
So, if we construct a perpendicular to AB from M, we
know that line contains the center of j. If we connect A
and B, to the point of intersection of perpendicular line
and circle j, we get two right triangles (or one isosceles
triangle). If we construct the perpendicular bisectors of
the sides, we get the center (their point of intersection).
Student 7:Draw a regular polygon in the circle.
Actually draw a rectangle. Construct the diagonals, and
the point of intersection of diagonals is the center.
Figure 1. Description of case-based task.
In light of these considerations, we designed
“Locating the center,” a case-based task to be used as
the research instrument in our study (see Figure 1). The
case contained a range of student responses to a
geometry problem. The reader was then asked to
analyze the children’s work and to hypothesize about
how the teacher of the group could proceed with her
lesson in the presence of the children’s diverse ideas.
Mathematical Goals of the Task
The mathematical focus of the task is on two
topics central to the study of Euclidean and NonEuclidean geometries: triangles and circles. These
topics are usually addressed as separate chapters in
standards textbooks. Frequently, students leave a
geometry course without realizing the connections
between them. The goal of the task is to help students
develop a sense for how concepts that make up the
field are closely related to each other and are
sufficiently self-contained. Usually if not always, this
kind of conceptual unity is not nurtured in
undergraduate mathematics programs. The following
list summarizes specific goals addressed by each
guiding question.
Question 1
Question 2
Question 4
Question 6
Question 8
Mathematical problem solving
Mathematical connections
Content coherence and unity
Mathematical analysis
Extending mathematical inquiry
and
content
specific
pedagogical reasoning
The task contains a deliberate range of learner
responses. Each of these responses could lead to a
Azita Manouchehri
series of important mathematical explorations as listed
below.
• Exploring properties of centers of a triangle
(Response #1)
• Relationship between the lengths of
inradii/circumradii and the area of triangle
(Response #1, #2)
• Trisecting an arc (Response #3)
• Construction of tangents to circles and tangent
circles (Response #3)
• Relationship between the size of the equilateral
triangles and their respective inradius and
circumradius (Response #1,2, 3)
• Inversions on circles (Response #4)
• Angular measures of chords of circle
(Response #4, 5)
• Properties of tangent circles (Response #6)
• Circumscribed
quadrilaterals and
their
properties (Response #7)
Pedagogical Goals of the Task
Pedagogical goals of the tasks include assisting
future teachers to develop an understanding of the
connections between student learning and instructional
decision making. The task was structured to allow for
pedagogical problem solving. The reader is asked to
analyze learners’ responses, hypothesize teaching
actions, design assessment tasks, and develop
activities. Furthermore, by asking them to explain,
justify, and defend their choice of representations,
assessment, and intervention, we envisioned that the
task would provide opportunities for the readers to
engage in pedagogical reasoning.
5
Lastly, the task introduces teachers to professional
journals and national professional standards (in the US
this includes documents published by the National
Council of Teachers of Mathematics). The following
list summarizes the pedagogical goals addressed by
each guiding question.
Question 3
Question 5
Question 7
Question 9
Content specific pedagogical
decision making
Content specific pedagogical
analysis
Curricular
analysis
and
connections to student learning
Content specific pedagogical
reasoning and decision making
Context and Research Design
The primary goal of our exploratory study was to
investigate teacher learning in the presence of casebased tasks when used in a mathematics content
course. Using a teaching experiment methodology
(Steffe & Thompson, 2000), data was collected in two
different sections of a course titled Modern Geometry,
an advanced mathematics course required of all
undergraduate and graduate mathematics majors
pursuing a degree in secondary mathematics teaching.
Each teaching experiment lasted two and half weeks
(five, 75-minute long sessions) and involved 40
prospective secondary teachers. The students enrolled
in this course were either of junior or senior academic
standing. All students had completed a minimum of 12
hours of coursework in general pedagogy, and a
minimum of 21 hours of mathematics coursework prior
to taking this class. The teaching experiment
commenced during the second month of instruction,
after the participants had examined both topics
involved in the task.
The teaching sequence in each class consisted of a
particular routine. The participants were assigned the
task as a homework activity. The follow-up discussion
in each teaching experiment included a large group
discussion at the beginning of the session. During this
time the participants were encouraged to offer their
initial reactions to the case, share their responses to the
guiding questions, ask any questions that they might
have regarding the content and/or expectations of the
task, their impressions of the children’s work and ways
in which the teacher of the group could proceed with
her instruction. A small group activity followed the
large group discussion. A final whole group discussion
allowed for observable evidence of learning including,
synthesizing and formalizing processes that could
determine a shared level of mathematical and
6
pedagogical analysis by the group. We used the initial
sharing time as an opportunity to collect base line data
on the participants’ initial approach to task analysis
and used that data as a means to trace changes in their
work as their interactions with the task intensified.
The decision to rely on two different teaching
experiments was to verify our interpretations of the
participants’ work. By collecting data from two
different groups we would be in a better position to
account for multiple variables that could impact the
participants’ interactions with the task including: their
mathematical background and experiences, interest
levels, classroom routines and instructional practices to
which they might have been accustomed.
In each class, two video cameras were set to
capture both large group interactions as well as two
targeted small groups. The small groups were selected
randomly and the same groups were videotaped
throughout the teaching experiment. All videotapes
were transcribed and used in the data analysis.
Data Analysis
In analyzing the impact of the task on the
participants’ activities and learning, we considered two
intertwined aspects of their work, including: (1)
Interactions with the task—Issues that the participants
raised about and/or extracted from the activity; (2)
Mathematical activities of the participants—Ideas and
problems the participants explored. Hence, data
analysis was organized around these two key
categories.
Participants’ Interactions with and Reactions to the
Task
In determining the participants’ particular
approach to task analysis, we focused on their verbal
exchanges during the small and large group
discussions. We considered whether the participants
showed an interest in learning about teaching, the
learners, curriculum and mathematics by the type of
questions they asked the facilitator or each other. Their
comments regarding children’s work were also coded
in order to trace sensitivity, or lack thereof, to relevant
mathematical and pedagogical issues.
Participants Mathematical Analysis and Learning
In seeking evidence of learning, we considered the
participants’ modes of production (Balacheff, 2000)
during the discussions as an indicator of learning.
Accordingly, we sought instances of mathematical
action, conjecture formulation, and validation
processes during each session. We considered
situations of action to include instances of problem
Motivating Growth of Mathematics Knowledge for Teaching
solving, problem posing, attempts at theoretical
constructions, testing a method, or judging merit of
ideas by reference to mathematical knowledge.
Instances of conjecture formulation included
articulation of relationships among mathematical ideas,
children’s solutions, and suggestions that peers offered.
Additionally, in seeking evidence of validation
processes, we considered whether the participants
referenced mathematical theory when analyzing the
problem and its extensions, examining children’s work
or judging the quality of mathematical arguments
offered in groups.
Results
In both sites, the classroom activities followed four
phases: Primitive pedagogical theorizing, facilitator
modeling, mathematical problem solving and
curriculum analysis, and pedagogical inquiry.
Phase I (Approximately 35 Minutes)
During the first phase of the activity, the
participants were reluctant to engage in discussions
about mathematics, showing a tendency to focus on
pedagogical theorizing. Their suggestions, however,
were not grounded in evidence or supported by a
rationale for choice. They seemed confident in their
assessment of children’s strategies and rarely elicited
explanations and/or guidance from each other or the
facilitator. They made brief and trivial references to
children’s solutions and characterized them as right or
wrong without offering a rationale for their choice.
None of the participants tried to formalize or justify
their assessment beyond stating their personal
preferences regarding the classroom environment they
felt were conducive to building children’s confidence.
When asked to comment on how the teacher might
decide which of the students’ responses to pursue in
class, only one participant in both cohorts was willing
to commit to a particular method. The facilitators’
comments during this phase were aimed at confronting
the participants’ assessment of children’s work and the
suggestions they offered for how the teacher might
organize subsequent classroom instruction.
Phase II (Approximately 40 Minutes)
In structuring individual and group analysis, the
second phase consisted of facilitator modeling.
Choosing one of the children’s solutions the
participants had labeled as incorrect (student 5) as an
example, the facilitator spent approximately 40
minutes of the first session in each class describing her
interpretation of one child’s method, and ways in
which it connected to other solutions as well as
Azita Manouchehri
different mathematical concepts. She listed additional
questions the teacher could ask the child either to gain
additional insight into his thinking or to advance his
work.
Phase III (Approximately 220 Minutes)
Following the modeling episode, the participants
were instructed to examine children’s solutions again
in small collaborative groups. A structure for group
deliberation was also set to reach consensus on their
assessment of children’s work as well as their
hypothesis concerning pedagogy. They stated and
explored several extension problems that could be
shared with children. They also examined different
textbook chapters to find places where specific topics
could be shared with children in instruction. Three
class sessions (210 minutes) were devoted to working
on specific problems that different individuals had
proposed as extensions to be used with children. The
facilitator guided the discussions, offered explanations
when asked, and continued to challenge over
generalizations that the participants made.
Phase IV (Approximately 75 Minutes)
The last phase of the participants’ activities
focused on synthesizing and formalizing pedagogical
and mathematical analysis of the case. The participants
began to ask each other and the facilitator questions
about curricular guides that could inform their practice.
During the last cycle of the activity, a major
component of the participants’ discourse included
articulation of concerns about their own knowledge of
mathematics.
Tables 1 and 2 summarize the different types of
comments that the participants made during each phase
of their case analysis experience. The major categories
of
comment
types
included:
Pedagogical,
mathematical, eliciting support and feedback, and
declarative.
Pedagogical themes included instances of: (1)
hypotheses about instructional moves (i.e. the teacher
should ask that each child go to the board and explain
his method to the group), (2) references to the impact
of instruction on children’s work (i.e. “Maybe the
teacher should have told them to use only one method
for solving the problem”), (3) references to the type of
evidence they used to support their pedagogical
decision making (i.e. The drawing is not accurate so
the teacher should be sure to help them draw it right)
and, (4) references to the impact of children’s work on
instruction (i.e. children are not ready to move on to a
7
Table 1
Typology of Comments: Pedagogical and Mathematical
Comment Type
Episodes of mathematical question posing
(Is there a way to locate the incircle of a quadrilateral?)
Group 1
61
Group 2
54
Episodes of conjecturing about new mathematical relationships
(I think there is a relationship between the area of inscribed regular polygon and its
circumradius)
82
96
Episodes of eliciting explanations concerning connections among mathematical concepts
(Can we connect the study of triangles to other polygons?)
75
82
Episodes of eliciting explanations concerning relevant mathematical theorems they could
use
(Is there a way to find the area of pentagon?)
234
186
Episodes of confronting peer’s analysis when discussing children’s work or extension
problems
(But this works for only one case! You need to generalize it)
187
198
Referencing evidence from children’s work when hypothesizing about teaching actions
(It would be useful to start with the informal approach first, the folding paper part and
then connect it to S7’s suggestion before moving on to 1st and 2nd methods)
56
61
Offering mathematical explanations on children’s solutions
(When he says fold it twice he is finding two diameters of the circle, so he is right)
78
72
Statements indicating having gained new mathematical insights—Aha!
(I know better why triangles are so important to geometry)
83
72
Number of mathematical problems on which participants worked*
28
25
Table 2
Declarative Statements
Eliciting Guidance
Typology of Comments: Eliciting Support and Declarative Statements
8
Statement of need for additional guidance on pedagogical decision
making
Statement of need for additional information on curriculum
Statement of need for additional guidance on mathematics
Statement of need for additional information on learners and how they
learn
Statements of concern about the ability to teach
Phase I & II
M=2
SD = 4.6
M = 12
SD = 4.2
M=3
SD = 9.08
M=0
M=0
Statements of concern about the ability to make sense of children’s work
M=0
Statements of concern for knowledge about appropriate decision making
M=0
Statements of concern about finding appropriate resources
M=0
Statement of concern about the quality of their teacher training
M=0
Phase III
M = 32
SD = 4.2
M = 21
SD = 0.04
M = 89
SD = 0.09
M = 13
SD = 0.19
M = 12
SD = 0.16
M=0
M = 24
SD = 2.8
M = 18
SD = 1.88
M = 19
SD = 0.92
Phase IV
M = 48
SD = 0.92
M = 52
SD = 4.8
M = 22
SD = 0.87
M = 76
SD = 2.11
M = 43
SD = 5.03
M = 76
SD = 6.4
M = 32
SD = 4.11
M = 29
SD = 2.01
M = 81
SD = 3.3
Motivating Growth of Mathematics Knowledge for Teaching
different topic, the teacher should go back and review
her lesson).
Eliciting support included instances of statements
of need for additional information on: (1) classroom
conditions (i.e. How did the teacher organize
classroom activities? What is the teacher’s
curriculum?), (2) mathematics (i.e. Is this
mathematically sound?), (3) learners’ thinking (i.e. Is
this method common to all children this age?) and (4)
pedagogical decision making (i.e. Is this approach
developmentally appropriate? Is this question adequate
to be posed to this group?).
Mathematical themes included instances of
mathematical analysis consisting of: (1) references to
mathematical theory in analyzing children’s work (i.e.
What this student is suggesting is related to the
theorem that says the diameter is the perpendicular
bisector of a chord), (2) references to the impact of
children’s work on instruction with a focus on
identifying mathematical significance of the ideas
presented by a child (i.e. I think the teacher should ask
the students to study this solution first cause they can
see there are several things that need to be resolved),
(3) references to mathematical connections among
ideas (i.e. Circumscribing a triangle is the same as
finding the perpendicular bisector of three chords, so
the sides of the triangle are actually three chords of the
circle that intersect at vertices).
Declarative statements included instances of
volunteered remarks regarding self knowledge (i.e. I
am not sure I know the subject well enough now; how
do we decide which textbook to use?), learning or lack
thereof (i.e. I used the last student’s method to solve
this problem), particular needs (i.e. Why can’t we do
this type of activity more often?), and projected plans
for professional development either in mathematical or
pedagogical domains (i.e. Maybe we should start
building a resource book; I really should work on
learning the software more.)
Following the modeling episode by the facilitator
(Phase II), mathematical references that the
participants made when they analyzed children’s work
increased significantly. The participants asked more
questions about theories they could use or ways in
which children’s methods related to other
mathematical ideas. Indeed, the participants’
declarative statements were indicative of the impact of
the task on raising the participants’ sensitivity to the
quality of their own knowledge and ways in which they
could improve their capacity to teach.
Azita Manouchehri
The Participants’ Mathematical Work
The number of mathematical problems on which
the participants worked averaged 21 per group. The
sheer number of problem posing, conjecturing, and
explaining episodes is remarkable considering that the
participants had rarely practiced such processes in their
regular classroom. The children’s solutions were used
as a springboard for extending the study of triangles to
circumscribed and inscribed polygons. The following
is a partial list of common problems on which the
participants worked during the extended problem
solving episode.
1. Construction of tangent line to a circle, and tangent
circles
2. Constructing three kissing circles
3. Relationship among the radii of kissing circles
4. Properties of
quadrilaterals
circumscribed
and
inscribed
5. Finding areas of regular polygons using the
measure of inradius and circumradius
6. Star Trek Lemma (formulated by the group)
7. Bow-tie theorem (formulated by the group)
8. Determining the interior angle sum of polygons
using the Star Trek lemma
9. Properties of Pedal and Orthic triangles
10. Derivation of the extended law of sines using
circumcircles
11. Describing the area of triangle and quadrilateral
using the law of cosines
Discussion
Our data indicate that the case-based task
successfully engaged teacher candidates in doing
mathematical inquiry and pedagogical analysis.
Evidence of mathematical learning from the task was
manifested not only in the number of problems that the
participants explored in the course of their case
analysis sessions, but also in the amount of
mathematical information shared as they made and
verified conjectures. Further evidence was evident in
how the participants justified their assessment of
children’s work. They elicited and articulated
connections among solutions and detailed ways in
which these connections could be made public in
instruction. They elicited information about, and also
identified mathematical structures that could be used
when discussing specific problems with children.
9
The participants’ comments revealed that their
experience with the task helped them to realize
connections between a teacher’s own mathematical
knowledge and his or her pedagogical choices. The
significant number of participants’ requests for
theoretical guidance on both mathematics and
pedagogy is a strong indication of their interest in
learning. Additionally, the large number of statements
of concerns they raised about their own knowledge, the
quality of their preparation, and their ability to teach
further support our perspective that the goal of the task
in raising teachers’ sensitivity to complexities
associated with pedagogical decision making.
Lastly, a careful analysis of case-based tasks in
order to maintain the integrity of both mathematics and
pedagogy is a time intensive process. Discussion of the
center activity took five class sessions. Indeed, if we
had the opportunity to pursue each of the mathematical
and pedagogical questions that the participants raised,
we could have easily doubled the length of time spent
on each case. In making decisions about what to pursue
with the teacher candidates, we focused mainly on
mathematical objectives of the course. Considering the
amount of learning developed as the result of exposure
to only one case-based task, we are prepared to
conjecture that if used systemically facilitators will be
10
in a better position to strike a balance in instruction
when simultaneously addressing mathematical and
pedagogical issues in a course.
References
Hiebert, J. , Morris, A. K., & Glass, B. (2003). Learning to learn to
teach: An “experiment” model for teaching and teacher
preparation in mathematics. Journal of Mathematics Teacher
Education, 6(3), 201–222.
Kirshner, D. & Whitson, J. A. (1997). Situated cognition: Social,
semiotic, and psychological perspectives. Mahwah, NJ:
Erlbaum.
Merseth, K.K. (2003). Windows on teaching math: Cases of middle
and secondary classrooms. New York: Teachers College
Press.
National Council of Teachers of Mathematics (2000). Principles
and standards for school mathematics. Reston, VA: Author.
Richardson, V. (1996). The case for education: Contemporary
approaches for using case methods. Boston: Allyn & Bacon.
Steffe, L. P. and Thompson, P.W. (2000). Teaching experiment
methodology: Underlying principles and essential elements in
Research design in mathematics and science education. In R.
Lesh & A. E. Kelly (Eds.), Research design in mathematics
and science education (pp. 267-307), Hillsdale, NJ: Erlbaum.
Motivating Growth of Mathematics Knowledge for Teaching
The Mathematics Educator
2008, Vol. 18, No. 2, 11–22
Planning for Mathematics Instruction: A Model of Experienced
Teachers’ Planning Processes in the Context of a Reform
Mathematics Curriculum
Alison Castro Superfine
Planning is an important phase of teaching, during which teachers make decisions about various aspects of
instruction that ultimately shape students’ opportunities to learn. Prior research on teacher planning, however,
fails to adequately describe experienced teachers’ planning decisions, and is unclear about the extent to which
teachers use curriculum materials to inform their decisions. Using data from 6th grade mathematics teachers’
use of curriculum materials, this study presents a discipline-specific model of experienced mathematics
teachers’ planning. The proposed model provides a lens for understanding the nature of teachers’ planning
decisions, and the conditions under which such decisions change over time.
Planning
is
an
important
and
often
underappreciated aspect of teaching practice, when
teachers make decisions that ultimately impact
students’ opportunities to learn (Clark & Peterson,
1986; Floden, Porter, Schmidt, Freeman, & Schwille,
1980; Stigler & Hiebert, 1999). Planning commonly
refers to the time teachers spend preparing and
designing activities for students. From tasks and
activities to instructional practices employed during
lessons, teachers need to consider a variety of aspects
of their instruction before students even enter the
classroom. Teachers need to pay careful attention to
designing their lessons; “effective teachers understand
that teaching requires a considerable effort at design.
Such design is often termed planning, which many
teachers think of as a core routine of teaching.”
(Kilpatrick, Swafford, & Findell, 2001, p. 337).
Reviews of teacher planning and decision-making
further emphasize the centrality of planning processes
in teachers’ practice (Clark & Peterson, 1986; Clark &
Yinger, 1977; Shavelson & Stern, 1981). Despite this
general agreement about the importance of planning,
few researchers have explicitly examined the precise
ways in which teachers plan for mathematics
instruction.
Prior research related to teacher planning presented
a “linear” or “rational” model of teacher planning by
delineating the various lesson elements teachers
Alison Castro Superfine is Assistant Professor of Mathematics
Education and Learning Sciences at the University of Illinois at
Chicago. Her current work focuses on teacher-curriculum
interactions and elementary preservice teacher education..
The research reported in this article is based upon the author’s
doctoral dissertation under the direction of Dr. Edward A. Silver
at the University of Michigan.
Alison Castro Superfine
generally considered when planning their lessons
(Popham & Baker, 1970; Taylor, 1970; Tyler, 1950).
Under this model, teachers first consider the learning
activities that take into account students’ interests and
abilities, then the learning goals and objectives of the
lesson, and finally the evaluation procedures to be used
during the lesson. Some researchers later argued that
linear models of teacher planning do not adequately
describe experienced teachers’ planning processes and
do not account for the complexities inherent in
mathematics teaching. Rather, a variety of additional
factors, such as teachers’ experiences and conceptions
of mathematics teaching and learning, also influence
the ways in which teachers plan their lessons
(Kilpatrick et al., 2001; Leinhardt & Greeno, 1986;
Yinger, 1980).
More recent research on teachers’ planning does
not clearly indicate the extent to which teachers draw
from curricular resources when making planning
decisions. Moreover, there is even less research that
focuses explicitly on teachers’ planning in the context
of the reform mathematics curricula that provide much
of the instructional design for teachers (Kilpatrick et
al., 2001; Trafton et al., 2001). Such reform curricula
are increasingly prevalent in classrooms in the United
States, embodying new modes of instruction (Reys,
2002). The challenges of planning lessons using such
curricula may be somewhat different from the
challenges of planning lessons with more conventional
mathematics curricula. Thus, exploring how teachers
plan in the particular context of reform curricula is
critical if mathematics educators want to understand
this important phase of teaching.
In order to explore the theoretical considerations
presented in this article, the author has selected
11
examples of teachers’ planning routines taken from a
larger study examining experienced 6th grade teachers’
use of the Connected Mathematics Project (CMP)
materials. CMP is a middle school reform curriculum
developed in response to the National Council of
Teachers of Mathematics (NCTM, 1989) Curriculum
and Evaluation Standards. By emphasizing the
discovery of mathematical ideas through tasks, CMP
encourages students to make connections between
topics and important mathematical ideas in order to
help them apply their learning to real-world contexts.
The larger study focused on how four teachers used the
CMP teacher’s guide in both the planning and
enactment of their lessons. The teacher’s guide
includes summaries of the mathematical content,
specific questions to ask students throughout a lesson,
and examples of student errors.
Teacher Planning
Past research on teacher planning focused on the
broad features and order of teachers’ planning
decisions and considerations, with minimal attention
given to the particular ways that teachers considered
engaging students with the content. Adhering to a
linear model of teacher planning, Tyler (1950) and
Popham and Baker (1970) found that teachers specified
ordered objectives, selected learning activities,
organized learning activities, and specified evaluation
procedures. Similarly, Taylor (1970) found that
teachers sequentially considered four aspects of a given
lesson when planning: materials and resources,
students’ interests, the aims and purposes of teaching,
and evaluation. Implicit in these studies is the notion
that teachers create their own objectives and activities
for students, which may reflect the design of the types
of curriculum materials available to teachers at the
time in which these studies were conducted.
In a later study on teacher planning, Brown (1988)
examined the extent to which 12 teachers adhered to a
linear model of planning. Focusing on teachers’
planning in different subject areas, Brown found that
teachers tend to use curriculum materials and the
objectives expressly stated in these resources as a
starting point for their planning. She noted, “teachers
operate as curriculum implementers and not curriculum
planners as they consider objectives already written in
curriculum guides” (p. 79). Yackel and Cobb (1996)
noted that planning decisions about ways of facilitating
students’ activity in a history or English classroom are
considerably different from those in a mathematics
classroom. Nevertheless, Brown’s (1988) study points
to the integral role of curriculum materials in the
12
process of teachers’ planning, which was not clearly
addressed by proponents of the linear planning model.
Some researchers have focused on the role of
curriculum materials as a resource for teachers to draw
upon when making planning decisions. For example,
McCutcheon (1981) found that when planning for daily
lessons, teachers tend to rely heavily on suggestions in
the teacher’s guide. In a study of one teacher’s
planning throughout the school year, Clark and Elmore
(1981) found that curriculum materials are primary
resources in the teacher’s planning. Similarly, Smith
and Sendelbach (1979) studied this issue at the level of
teachers’ unit planning. They found that teachers tend
to construct a mental image or plan of the unit and then
supplement their plan with notes based on the
suggestions in the teacher’s guide. Additional research
has highlighted various factors that influence how
teachers use these curricular resources in the planning
and instructional processes (Ben-Peretz, 1990; Cohen
et al., 1990).
Research points to experience as a potential factor
that influences teachers’ planning. Such research
suggests that experienced teachers have more extensive
and well-organized knowledge of both pedagogy and
student learning, making them more flexible and
attentive to the nature of the students’ learning
opportunities that they create (Borko & Shavelson,
1990; Leinhardt & Greeno, 1986; Livingston & Borko,
1989). For example, researchers found that when
planning, experienced teachers make more extensive
mental plans than written plans and rely less on
curriculum materials than their less experienced
counterparts (Bush, 1986; Leinhardt, 1983; Livingston
& Borko, 1990).
As teachers’ experience with a particular
curriculum program increases, they become more
familiar with the details, nuances, and presentation of
the specific mathematics content in the curriculum.
Thus, they may have developed daily routines in
planning and engaging with the curriculum in
particular ways. For the purposes of this study,
experienced teachers are identified as having at least
five years of teaching experience and at least three
years experience using a certain curriculum program.
Other possible factors influencing teachers’
planning decisions are the various conceptions teachers
bring to bear on their practice. These conceptions
contribute to what Remillard and Bryans (2004) refer
to as teachers’ orientation toward the curriculum.
Experienced teachers have refined their conceptions of
mathematics teaching, learning, and curricula because
they have spent considerable time formulating and
Teacher Planning
applying these conceptions in the classroom. Teachers’
conceptions of mathematics content are also important
for understanding how teachers engage with the
curriculum (Lloyd, 1999; Remillard, 1999; Remillard
& Bryans, 2004). Moreover, it is important to
understand the extent to which teachers’ conceptions of
mathematics teaching and learning align with the ideas
about teaching and learning underlying the curriculum
(Manouchehri & Goodman, 1998; Remillard &
Bryans, 2004).
While linear models are useful for capturing
certain basic elements of teacher planning, these
models fail to account for an array of factors that have
been identified as influencing teachers’ planning
processes, such as curriculum materials, teaching
experience, and the various conceptions teachers have
about teaching and learning. Although teachers have a
variety of conceptions, this article will focus on
teachers’ conceptions of mathematics teaching,
learning, and curricula for the purposes of this study.
The Work of Reform-Oriented Mathematics
Teaching
Although the various factors highlighted by
researchers as influencing teachers’ planning are
essential to consider when developing a new planning
model, it is also important to consider the demands and
characteristics of the particular discipline in which
such planning occurs. Most of the research discussed
previously does not explicitly focus on planning or
instruction in the context of a specific discipline.
Mathematics teaching, specifically in the context of
reform mathematics curricula, involves particular
demands and challenges that may shape teachers’
planning processes.
The model developed in this study is grounded in a
specific conception of mathematics teaching, drawing
from the works of Stein, Smith, Henningsen, and Silver
(2000), Lampert (1992, 2001), Clark and Elmore
(1981), and Lampert and Ball (1998). These
researchers describe in detail the nuances and
complexities of mathematics teaching in a way
typically embodied in reform curricula. Teachers need
to consider the mathematical content and ways to
engage students in discussion about the content, while
simultaneously guiding students towards a particular
goal. For example, during planning and instruction
teachers modify tasks and ask high-level questions in
order to promote students’ understanding of the
underlying ideas and concepts.
To support students’ understanding, teachers need
a variety of pedagogical skills. For example, teachers
Alison Castro Superfine
need to be able to resist the urge to tell students how to
work on the content so that they provide students with
adequate time to think through what they are asked to
do (Donovan, Bransford, & Pellegrino, 2000).
Anticipating student responses and having an
awareness of common errors can also help teachers
effectively respond to and redirect students’
discussions (Chazan & Ball, 1999; Fennema, Franke,
Carpenter, & Carey, 1993). In addition, modifying
tasks based on students’ current knowledge and
abilities may help teachers to be mindful of the
cognitive activity in which students should be engaged
(Stein et al., 2000). Employing such instructional
practices to facilitate student learning in accordance
with the principles of reform mathematics, however,
requires extensive and demanding work on the part of
teachers. Therefore, teachers face several challenges
supporting students at such a high level of
mathematical activity.
These challenges to teachers’ work are considered
the “problems” in mathematics teaching (Lampert,
1992, 2001). “Problems” in mathematics teaching refer
to the work teachers do to further students’
understanding of mathematics. This includes
facilitating students’ discussion around the content,
continuously pressing students to explain their ideas
and to communicate with each other, posing questions,
and selecting solution strategies to present to the class.
Teachers need to make important and often
simultaneous decisions in ways that do not undermine
students’ thinking or the mathematical opportunities
afforded by the content in reform curricula. Hereafter,
“problems” will be used to refer to the challenges and
decisions teachers face during mathematics teaching,
as described in Lampert (1992, 2001).
Teaching includes not only the physical act of
teaching, during which teachers interact with students,
but also includes the time teachers spend preparing for
these interactions (Jackson, 1966, 1968). Planning for
the demands and challenges of mathematics instruction
requires teachers to engage in a planning process that
involves the development of skeletal frameworks
rather than detailed scripts for teaching lessons
(Rosebery, 2005). In particular, teachers must identify
a particular mathematical topic to discuss and the
means necessary to cover that topic, without
necessarily delineating the precise steps needed to
teach that topic. Therefore, planning for reformoriented instruction requires teachers to select specific
topics or concepts and to identify particular activities,
instructional strategies, and suitable materials for
discussing and engaging students with the topics or
13
concepts. In addition, during planning, teachers must
anticipate potential problems that may arise during
instruction and then make decisions regarding how to
manage these problems. Planning problems refer to the
considerations and decisions teachers face when both
planning for and anticipating what will happen during
a specific lesson.
A Model of Teacher Planning
There are several elements that a model of teacher
planning in the specific context of reform-oriented
mathematics instruction must capture. These elements
include the approaches to mathematics teaching and
learning embodied in reform curricula and teachers’
various experiences and conceptions they bring to their
planning decisions. Such a model must particularly
capture how these elements interact with each other
and ultimately influence teachers’ planning decisions.
The concept of planning problems is well suited for
developing a conceptualization of planning because it
incorporates the influence of these different elements.
The model developed in this study draws heavily on
the notion of planning problems and highlights the
various elements that drive the emergence of such
planning problems in teachers’ practice.
As discussed previously, the work of reformoriented mathematics teaching includes facilitating and
supporting students’ understanding in ways that will
neither constrain students’ opportunities to learn nor
undermine students’ thinking. Consequently, teachers
need to plan for engaging in this sort of work during
instruction. For example, teachers must plan questions
they will ask students that will guide students’ thinking
about the content without giving them too much
information, while also encouraging students to explain
their ideas (Hiebert & Wearne, 1993; Maher &
Martino, 1992; Moyer & Milewicz, 2002). Teachers
need to anticipate different solutions students may
offer, as well as alternative ways of thinking about a
task, in order to facilitate students’ learning and
discussion of these strategies in ways that foster a
shared understanding of the ideas (Kilpatrick, 2003;
NCTM, 1991). Teachers also need to anticipate
potential errors in order to respond appropriately and
help students learn from incorrect solutions. Finally,
teachers should be prepared to change or modify a
task, in the case that students are struggling with a
14
concept, in ways that both preserve the task’s
complexity and help students learn from working on
the task (Stein et al., 2000). Though their intended
plans often differ from their enacted plans, teachers
need to carefully plan their lessons and anticipate how
students will interact with the content during
implementation in order to further students’
understanding of different mathematical ideas. In this
way, planning problems can be considered to be the
anticipation of instructional problems.
Planning problems are inherently different for each
teacher depending on the teacher’s experiences, ideas,
and conceptions, as well as the curriculum being used.
For instance, asking higher-order questions that press
students to justify and explain their thinking is also
only a planning problem for teachers who view the use
of such questions as contributing to student
understanding. Planning problems also may be quite
different for a teacher who adheres to a more
conventional conception of mathematics teaching.
Such a teacher may need to determine how to
incorporate opportunities for students to practice the
application of certain skills and procedures within a
curriculum the teacher perceives as deficient.
Furthermore, a teacher with experience implementing
multiple curriculum programs must consider how to
apply what they know of other mathematics curricula
to their planning with a specific Standards-based
curriculum. Therefore, planning problems appear to be
a useful lens for understanding the relationship
between teachers’ experiences, conceptions of
mathematics teaching and learning, and the curriculum
used in the planning process.
As the Planning for Mathematics Instruction (PMI)
Model in Figure 1 illustrates, teachers’ various
conceptions influence their engagement with
curriculum materials during planning. Additionally, the
conceptions influence the type of planning problems
teachers encounter in the course of their work. These
planning decisions then influence teachers’ lesson
enactment and the types of learning opportunities they
create with students during instruction. This lesson
enactment then informs experiences and shapes the
information teachers have to draw upon when they
plan for and enact the lesson in subsequent classes or
school years.
Teacher Planning
Figure 1. Planning for Mathematics Instruction Model.
As teachers’ conceptions help to frame the
planning problems they encounter, their various
conceptions also serve as a resource for managing
planning problems that arise in the course of their
planning. When confronted with planning problems,
teachers draw upon their previous experiences with the
task and their ideas about what it means to learn and
teach mathematics in order to make decisions about
ways of managing these different planning problems.
In some cases, teachers may also draw from the
information and support provided within the actual
mathematics curriculum materials. Notably, the CMP
teacher guide provides the means for teachers to
manage certain planning problems, such as anticipating
solution methods students may generate, questions to
ask students, and errors and misconceptions students
may have in relation to a task. Though, the extent to
which teachers use curriculum materials to inform their
planning decisions is largely dependent on the nature
of their conception of the curriculum.
Teachers’ Planning Practices
The following teacher examples illustrate the
various ways in which experienced teachers, with
distinct conceptions towards the CMP curriculum, can
engage with reform curriculum materials in the course
of their planning and demonstrate how the PMI Model
applies to actual teaching practices. Alicia, Richard,
and Susan were selected for the present study because
their planning decisions and considerations captured
the range of variation in planning routines and
problems encountered.1 All three teachers were
teaching sixth grade at the same middle school at the
time the study took place. As this study uses teaching
and curricular experience to define experience, Alicia
has been teaching 16 years, using CMP for 3 years;
Richard has been teaching 17 years, using CMP for 10
Alison Castro Superfine
years; Susan has been teaching 6 years, using CMP for
3 years. These teachers were observed planning and
enacting the same unit, Bits and Pieces III, which
focuses on operations with rational numbers (Lappan et
al., 2006). The data collected includes interviews with
teachers prior to and immediately following classroom
observations to understand teachers’ lesson plans and
their reflections on their lesson enactments. Although
post-hoc examinations of teachers’ planning were
conducted, teachers were interviewed the same day of
the observation in order to increase the accuracy of
teachers’ responses. Field notes from classroom
observations and artifacts from teachers’ lesson
planning are additional data sources.
Prior to using CMP, Alicia used a more
conventional mathematics curriculum for 13 years and
claims to strongly believe that CMP does not provide
students with sufficient opportunities to practice skills
and procedures. She views her role as a teacher as that
of an intervener, providing direct guidance and explicit
instructions for students, which is evident in her
planning decisions. Alicia first reads through the entire
lesson in the student book and solves the task: “And I
do the whole [lesson] myself, you know without
looking at the teacher’s guide or anything…And then I
have an idea about what might be tricky and what
might not be.” Alicia says she then looks through the
accompanying teacher’s guide and decides the most
appropriate course of action, taking into consideration
both who her students are as learners and the
constraints of class time. “And I kind of pick and
choose what I think will work best with my students…
And most times I won’t use all of it.” In general, Alicia
claims to regularly modify the suggested content of the
lesson and the suggestions for how students should
engage with the content during the lesson. Her view of
CMP, and ultimately her conception of what students
15
should learn, is readily apparent in the nature of her
lesson additions and deletions: “I’m usually thinking
about what I need to add…Like word problems I
usually skip…and substitute other practice problems
that I feel like need to be emphasized more.”
Richard, on the other hand, claims to be a strong
proponent of the instructional approach embodied by
the program. Having taught CMP for over 10 years, he
believes that CMP is a desirable alternative to a
conventional mathematics curriculum because it allows
him as a teacher to facilitate student discussion and to
play less of a central role in the classroom discussion:
“I would characterize it less as teacher-driven…and
more kid-driven…I think the focus becomes--to me
I’m giving up being the center of attention.” When
planning, to make sure that he understands the content
for himself, Richard says he first reads through the
lesson and solves the task that he will use with
students. He then tries to ascertain, from reading the
student book, exactly what the lesson is about, the “big
idea” students are to come away with, and also places
where students may struggle or misconceptions
students may have while working on the lesson:
“…[J]ust so I get a sense…of what problems they’re
going to struggle with. Just looking at the answer
doesn’t help me out. If I actually work through the
problem, that gives me a sense of where they’re going
to struggle.” Richard says he uses the teacher’s guide
during planning only when an idea or concept or even
the wording of a particular question in the student book
is unclear. The student book provides him with the
necessary information he needs for enacting the task
during instruction.
I think if I would read through the [teacher] manual
and not read through the kid edition, I’d just feel
like I’d be at a disadvantage to know what to
expect from the kids…It tells me as a teacher what
to know conceptually…but it doesn’t help me
understand quite how I think the kids are going to
react to what I’m asking them to do.
In general, Richard claims to plan for using most, if not
all, of the lesson elements as described in the
curriculum.
Although both Alicia and Richard use the
curriculum materials for the content features of the
lessons (albeit to a limited degree in Alicia’s case),
both teachers appear to have different views of the
curriculum, and thus use the materials in very different
ways when planning. While Alicia “picks and chooses”
from the lesson suggestions what she considers
relevant and important for her students, Richard tends
to plan for enacting the lesson as described in the
16
materials. Notably, both of these teachers do not follow
the lesson suggestions in a prescriptive fashion,
including Richard who seems to only rely on the
materials for content features of the lessons and not for
suggestions as to how to engage students with the
content. By engaging with the materials in such an
adaptive, or even modified fashion, Richard and Alicia
leave the lesson open to interpretation, making room
for their conceptions of mathematics teaching and
learning to inform their lesson planning.
In contrast to the other two teachers, Susan, having
taught for 6 years and used CMP for 3 years, appears
to adhere closely to the lesson and corresponding
suggestions in the curriculum. She believes the
instructional approach espoused in CMP is an ideal
way to support students’ mathematical development
and their ability to communicate their understanding.
She views her role as that of a referee, mediating
students’ discussions of their proposed solution
strategies during instruction:
And so sort of mediating that discussion is sort of
the biggest part because…learning of the strategies
is supposed to take part amongst themselves. So
it’s like I have very little, this is how it works, you
know, it’s more, ok what are your ideas? Let’s put
them together.
When planning for a lesson, Susan follows the
suggestions in the teacher’s guide almost as a script for
the lesson. She will first read through and solve the
task in order to think about how students will approach
and solve the task, as well as to consider potential
solution strategies that may arise. After working
through the content, Susan says she reads through the
longer lesson summary in the teacher’s guide to
understand the overall direction and purpose of the
lesson. In addition to reading the detailed teaching
notes, she reads both the suggested questions and
answers to the task, and then includes these elements in
a slideshow presentation she uses during instruction.
Though, she only includes suggested questions in her
lesson if she decides they are worthwhile and
appropriate for her students: “So…whenever there are
suggested questions, I see ok, is this a meaningful
question for my students?” Susan says she reads
through the lesson suggestions to ensure she does not
extend the discussion of a concept further than what is
expected or stated in the materials, and also to gain an
overall sense of the lesson and the mathematical ideas
embedded in the lesson: “So, it’s good to read that
sometimes just to see ok, this is only where they need
to get to at the end.”
Teacher Planning
Table 1
Summary of planning problems teachers encountered during unit
Planning Problem
Alicia
Richard
Susan
Anticipating students’ work on
task
Treatment
of
content
in
curriculum
Recalled previous experience
with lesson
Read teacher guide to clarify
content, but focused only on
“important” aspects
Read student book, solved task
himself
Read teacher guide to clarify
content, but focused only on
“important” aspects
Planned for more teacherdirection of task
Read teacher guide to clarify
content, and planned to follow
lesson suggestions
Susan also says the teacher’s guide provides her
with an image of how students will engage with the
task: “…it gets me ready for what they might say.
What’s the book going to be after? You know, what’s
sort of the big idea that they want to come away with.”
She states that her purpose for using the student book
is to understand the task for herself. She uses the
teacher’s guide, on the other hand, to understand how
students may approach and solve the task, including
potential misconceptions students may have in relation
to the task.
In contrast to Alicia, Susan appears to agree with
the principles underlying CMP, and accordingly plans
for enacting lessons in the unit largely as described in
the materials. Moreover, unlike Richard, Susan uses
the curriculum not only for the content features of the
lesson, but also she uses the suggestions for how to
engage students with the content during the lesson.
Thus, Susan, the teacher with less teaching experience
as compared to Alicia and Richard, draws heavily from
the suggestions in the curriculum when planning, and
plans to enact the lesson largely as described in the
materials. By adhering to the lesson suggestions in an
almost prescriptive fashion, Susan leaves little room
for her own interpretation of the lesson.
As these examples illustrate, the nature of teachers’
engagement with curriculum materials during planning
is determined by a variety of factors. Although Richard
agrees with CMP’s overall approach to teaching and
learning, he largely relies on the curriculum materials
solely for its content. It appears that he does not require
much pedagogical support when planning; instead, he
typically limits himself to reading and working through
the student book. Susan, on the other hand, seems to
rely on the materials for both content and pedagogical
purposes during her planning for the unit, closely
following the lesson suggestions. In contrast to Richard
and Susan, Alicia does not seem to believe that CMP
provides students with sufficient opportunities to
practice basic skills and procedures, and therefore
Alison Castro Superfine
modifies the lesson suggestions as needed when
planning. In fact, all three teachers held varied
conceptions of the curriculum – curriculum as a guide
to varying extents (Alicia and Richard) and curriculum
as a script (Susan). As these examples illustrate,
teachers’ various conceptions influence the types of
planning problems these teachers encounter and the
ways in which teachers manage these problems as they
arise during planning for the unit.
Planning Problems
As discussed previously, planning problems
constitute a fundamental structural component of the
PMI Model because they highlight the relationships
between teachers’ experience, conceptions of
mathematics teaching and learning, and the actual
curriculum program used by teachers. Applying this
model to teachers’ practices requires close analysis of
the planning problems experienced by these teachers
and the factors underlying the emergence of these
planning problems. The teachers in this study primarily
encountered two different planning problems – (1)
anticipating potential errors and misconceptions
students may have in relation to a task and (2)
treatment of content in the curriculum. Although
teachers encountered several planning problems
throughout the unit, these two problems were selected
for analysis because they illustrate how the PMI Model
depicts teachers’ planning practices. Though the two
types of planning problems teachers primarily
encountered during their planning were quite similar,
teachers varied considerably in the ways in which they
managed these two planning problems. Table 1
summarizes the planning problems these three teachers
encountered during their planning for the unit, and
briefly describes the ways in which the teachers
managed these different problems. Table 1 does not
reflect the frequency in which participating teachers
encountered planning problems during the unit.
17
Anticipating Students’ Work on Tasks
A specific planning problem all three teachers
encountered during their planning for the unit was
anticipating how students would work on the content
of the lesson. Alicia, for example, relied on her
experience from previous classes to anticipate how
students would engage with the content of several
lessons in the unit. Based on experiences with classes
in previous years, Alicia anticipated that students
would struggle with lining up the decimal points when
adding and subtracting decimals in a certain lesson.
She therefore planned to enact the entire lesson as a
whole-class activity to help students through the
lesson, as opposed to providing opportunities for
students to work collaboratively in groups during the
lesson, as was the suggested organization. Similarly,
she anticipated students would struggle with another
task involving computing discounts by drawing upon
her previous experience with that lesson, and again
planned the lesson as a whole class activity. As she
described in her planning, Alicia believed that by
implementing lessons as either whole-class or
individual activities, she was better able to address
student difficulties and “guide them in the right
direction.” Richard regularly encountered this same
planning problem but managed it quite differently than
Alicia. Rather than relying on his previous experiences,
Richard anticipated how students would work on the
different tasks by working through the lessons himself
in the unit – he read the student book and solved the
task while thinking about how students would
approach the task and what potential aspects might
confuse students. In doing so, Richard attempted to
forecast the various ways in which students could
engage with the content, which reflected his more nonconventional conception of mathematics learning.
Susan also anticipated how students would work
on the task, primarily drawing upon what she knew of
her students’ previous work throughout the unit, but
also the information included in the teacher’s guide.
She became aware of this planning problem by not
only reading the teacher’s guide, but also from her
previous experience with a particular lesson involving
decimal division in which students seemed to struggle
with long division. Although the materials alerted her
to this potential source of confusion for students, she
did not seem to use the suggestions in the teacher’s
guide to support students’ understanding of long
division. Instead, she used her view of how students
should learn in order to address students’ difficulty
with the content and planned to enact the lesson in
particular ways to lessen the likelihood that students
18
would struggle. This was also the case in Susan’s
planning for a lesson involving computing discounts.
In both situations, Susan’s previous experience, her
view of how struggling students should learn, and her
proclivity to follow the curriculum suggestions closely,
influenced how she managed the problem of
anticipating how students would work on the lesson.
She often reduced the complexity of the tasks by
telling students how to solve them, taking away
students’ opportunities to wrestle with the central
ideas, but still enacted the lesson largely as written.
Treatment of Content in Curriculum
The treatment of the content within two lessons
dealing with long division also emerged as a planning
problem for these teachers. All three teachers
considered long division as a particularly important
concept for students to know and to be able to do.
However, the long division algorithm was not
explicitly presented in the unit; it was presented as a
set of two interrelated lessons in order for students to
understand the underlying rationale of the algorithm
and the role of place value when dividing decimals. All
three teachers had to consider how to enact these two
lessons in light of their conceptions towards the
content and the curriculum. Alicia and Richard
modified the lesson to focus on the procedural aspects
of decimal division in these two lessons. This
modification reflected their conceptions about what
they considered to be the most important aspects of the
content. Moreover, this modification comported with
their conceptions of the curriculum as a guide rather
than a script for their lesson planning. This particular
conception of the curriculum left room for the
teachers’ conceptions toward the content to dictate how
teachers planned to enact the lessons.
While Susan also encountered this planning
problem, she planned to enact the lesson largely as
written in the curriculum despite her reluctance to do
so. Her conceptions toward both the curriculum and the
content influenced how she framed and managed this
problem. Susan felt inclined to change the treatment of
the content because her conception of the content
clashed with the treatment of the content in the
curriculum. However, her desire to plan her lessons
largely in accordance with the lesson suggestions
provided a push in the opposite direction to teach the
content as written: “I don’t know about this lesson
because students have always struggled with
division….Though [the lesson] helps students
understand, so I just have to be patient.” Susan had to
consider what content to enact with students in light of
Teacher Planning
these conflicting conceptions. Her conception to plan
for lessons in accordance with the lesson suggestions
ultimately outweighed her conception of the content.
The shape of Susan’s planning problem contrasts with
that of Alicia’s and Richard’s in that they did not
negotiate conflicting conceptions. In summary, the
three teachers in this study encountered different
problems in the course of their planning for the unit.
Despite the fact that the CMP materials provided the
means to manage some potential planning problems,
teachers seemed to rely largely on their previous
experiences and particular conceptions to manage their
planning problems. Therefore, in the case of all three
teachers, the PMI Model highlights how teachers with
diverse conceptions and experiences frame and manage
particular planning problems.
Discussion and Conclusion
The
previous
discussion
highlights
the
interrelationship among curriculum materials, teachers’
various conceptions, and the types of and ways in
which teachers frame and manage planning problems
that arise in the course of their work. During planning,
teachers often use curriculum materials as a starting
point for their lesson planning. The nature and extent
of teachers’ engagement with the curriculum materials,
however, is determined primarily by their various
conceptions. Teachers’ various conceptions then
influence the type of planning problems they
encounter, and also how teachers manage these
planning problems.
Teachers’ lesson enactment also contributes to the
pool of knowledge and information they have to draw
from in subsequent years, thereby influencing their
conceptions. As Figure 1 illustrates, the PMI Model
represents an iterative process that is continuously
shaped by teachers’ experiences over the course of
their careers. With every lesson, teachers potentially
encounter unanticipated questions or new strategies
that contribute to the knowledge they can draw from
when planning the same or related lessons in
subsequent years. The ways in which teachers’ enact
lessons with students over time can also inform how
teachers conceive of what it means to teach and learn
school mathematics. The proposed model provides a
way to understand how teachers’ planning practices
change, or fail to change, over the course of their
careers.
The PMI model suggests a possible cause for the
“experience problem,” perhaps one of the most
significant problems teachers face as they advance
through their careers. Unlike their less experienced
Alison Castro Superfine
colleagues, experienced teachers have to consider how
to make use of their prior knowledge and experience.
With regard to teachers’ various conceptions, the
“experience problem” consists of how to utilize
experienced teachers’ assumptions about and prior
knowledge of mathematics curricula, and their ideas
about what it means to learn and teach mathematics.
The planning routines of Alicia, Richard, and
Susan reflect how the experience problem plays out in
actual teaching practice. Experienced teachers do not
face significantly fewer or different planning problems
as compared to less experienced teachers. On the
contrary, all three teachers anticipated that a group of
students would struggle with a particular aspect of a
lesson, or even struggled themselves with certain
aspects of the mathematics content. Yet, these teachers
encountered these planning problems differently. The
differences among these teachers seem to be in their
conceptions of the curriculum and content, the
prevalence of their conceptions in their planning
decisions, and ultimately their instructional decisions.
Regardless of their conceptions, teachers’
conceptions of curriculum and mathematics teaching
and learning can become calcified over time
(Leinhardt, 1983; Leinhardt & Greeno, 1986). As a
result, teachers may become inattentive to how their
planning decisions influence students’ opportunities to
learn and they may become resistant to external
influences such as new curriculum programs or
professional development experiences. Consider
Alicia, who seemed to adhere to a more conventional
conception of mathematics teaching and learning
during her planning. Because Alicia has quite
extensive teaching and curricular experience in using
more conventional mathematics curricula, she planned
to focus students’ work on practicing computations and
procedures and planned to modify lessons as wholeclass discussions rather than collaborative work
groups. It appears that Alicia’s conceptions of teaching
and learning have become somewhat cemented
throughout her teaching career and seemed to have
hindered her from planning for enacting CMP lessons
in this unit in accordance with the curriculum’s
underlying principles. The PMI Model also captures
this common aspect of teachers’ practice – as this
model represents an iterative process, allowing for
teachers’ conceptions to become reinforced as teachers
amass an increasing amount of knowledge and
experience.
The PMI Model has the potential for even broader
utility because the planning problems and teacher
conceptions discussed here constitute only a handful of
19
the planning problems and conceptions that might
influence teachers’ lesson planning. For example,
teachers’ perceptions of limited time during a lesson
may prove problematic for some teachers when
deciding how much of the lesson to cover in the time
allotted. Another planning problem may arise when
teachers have to anticipate how to orchestrate the use
of multiple solution strategies during a given lesson,
thinking carefully about the order in which to present
certain strategies and the mathematical affordances of
discussing different strategies. In addition to
accounting for a broad range of planning problems, the
proposed model also can account for a broad range of
teachers’ conceptions. For example, teachers’
conceptions of their role as teachers may influence
how they engage with the curriculum materials during
planning. This engagement, in turn, will give rise to
new planning problems and ways of managing these
problems. Regardless of the precise planning problems
and conceptions that may influence teachers’ work,
these key elements of the PMI Model help to explain
teachers’ considerations and decisions made during the
planning process.
Given the PMI Model and the notion of planning
problems that provides its underlying structure, the
question for future research becomes how reformers
can work to improve teachers’ practice, and ultimately
student learning. Viewed by many as a driving force of
reform, mathematics curriculum materials have the
potential to boost educational achievement while
embodying new modes of instruction. However,
teachers can hold diverse conceptions that stand in
contrast to the conceptions of teaching and learning
underlying the curriculum, which can hinder teachers
from planning in accordance with the curriculum.
Teachers who have more experience with conventional
curricula and exhibit a more conventional conception
of teaching are desensitized to the modes of instruction
entailed in implementing reform curricula. Researchers
have found that for teachers with extensive experience
teaching with more conventional methods and
curricula questioned the value and relevance of reform
curricula (Manouchehri & Goodman, 1998; Preston &
Lambdin, 1995). Still, reformers can target teachers’
conceptions directly by designing professional
development experiences that are aimed at helping
teachers shift their views of what it means to know,
learn, and teach mathematics. This is not to say that
teachers should participate in the equivalent of a
philosophy course, but rather reformers can situate
teachers’ learning in the actual practice of teaching,
wherein teachers can experience what learning and
20
teaching mathematics in reform oriented ways entails.
At the very least, the PMI Model underscores teachers’
conceptions as a target for reform efforts because they
structure and provide a major resource for managing
planning problems that arise in the course of teachers’
work.
In summary, the PMI Model highlights how
teachers’ various conceptions frame and influence how
teachers’ manage planning problems that arise when
preparing for mathematics instruction. The model is
structured around Lampert’s (2001) notion of teaching
problems, which can illuminate processes teachers
engage in during their daily planning, thus providing a
useful lens to understand the nature of teachers’
planning routines and reasons underlying their
decisions during this phase of teaching. The concept of
teaching problems is useful for understanding teachers’
practice because it captures the interactions among
teachers’ various conceptions, their engagement with
actual curriculum materials, and their previous
experiences. Although the discussion of the PMI
Model is specific to teacher planning in a reform
mathematics context, such a model of teacher planning
is applicable to the planning that occurs in the context
of conventional curricula as well, though the nature of
teachers’ planning problems may be different. As the
examples presented in this article illustrate, despite the
principles of teaching and learning underlying a
curriculum, teachers’ various conceptions heavily
influence teachers’ engagement with the materials
during planning, thereby influencing the ways in which
teachers manage problems, and the types of planning
problems teachers encounter. Applying the PMI Model
to understand planning problems, how these problems
change over time, and under what conditions they
change highlights important elements in mathematics
teachers’ planning processes.
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22
Teacher Planning
The Mathematics Educator
2008, Vol. 18, No. 2, 23–32
Communication Theory Offers Insight into Mathematics
Teachers’ Talk
Denise B. Forrest
This article discusses how communication theory is used to understand the thoughts mathematics
teachers employ when creating messages intended for students. According to communication theory,
individuals have different premises about the act of communicating, and these thoughts, called
message design logics, guide the process of reasoning from goals or intentions to actual messages
(O’Keefe, 1988, 1990). Three distinct message design logics have been identified by communication
theorists: expressive, conventional, and rhetorical. Depending upon which logic an individual
employs, a very different message is said and heard. This theory was used to investigate the message
design logics of 15 secondary mathematics teachers. It was found that teachers have varying logics in
their message production and, depending upon the logic used, distinct characteristics correspond to
different teacher premises for classroom communication.. The logic employed also results in different
ways teachers encourage mathematical learning and evaluate classroom interactions.
In the last twenty years, a considerable literature
base has been created by mathematics educators that
describes effective verbal exchanges for classroom
instruction and the critical role the teacher has in that
process (Cobb, Wood, & Yackel, 1990, 1993; Cobb,
Wood, Yackel, & McNeal, 1992; Cohen & Ball, 1990,
2000; Hiebert et al., 1997). The teacher uses verbal
communication to articulate expectations, show care
for students, and encourage discussion of specific
content knowledge. During instruction, the teacher
uses verbal communication to initiate questions and
describe tasks in order to elicit, engage, and challenge
student thinking. The teacher decides what topics to
pursue in depth based on student feedback and content
objectives, how to encourage every student to
participate, and how to integrate further mathematical
connections and representations of the topic.
Research on teaching and learning supports
classroom discussions where the teacher focuses on
students’ mathematical thinking and guides the
discussion so the group can reach a consensus on an
understanding of the particular mathematical content.
However, these interactions have not been typically
found in mathematics classrooms (Goos, 1998; Jacobs,
Hiebert, Givvin, Hollingsworth, Garnier, & Wearne,
2006; Weiss, Pasley, Smith, Banolower & Heck,
2003). Specifically in the United States, the Third
Denise Forrest is an assistant professor of secondary/middle
mathematics education at Coastal Carolina University. Her
current research focuses on classroom verbal interactions for
learning and specifically how teachers develop the skills and
strategies for these interactions. For further contact, her e-mail
is dforrest@coastal.edu..
Denise B. Forrest
International Mathematics and Science Study (TIMSS,
Jacobs et al., 2006) reported that 78% of the topics
covered during the eighth grade lessons were
procedural, without ideas being explained or
developed. Also in that report, 96% of eighth grade
teachers stated that they had some awareness of current
recommendations for mathematics education and 76%
said that they kept up with these recommendations.
This inconsistency between research and practice
needs more research; the National Research Council
(Kilpatrick, Swafford, & Findell, 2001) called this area
of research incomplete. Researchers should continue to
make visible teachers’ decisions, and their
consequences for students’ learning, as they manage
classroom discourse.
Communication theory offers a different approach
for mathematics educators to understand classroom
interactions.
Communication
researchers
have
developed a body of research describing how
individuals create and understand verbal messages.
They view verbal communication as a strategic type of
social interaction where “conversationalists create and
modify their individual interpretations of their social
world” (Stamp, Vangelists, & Knapp, 1994, p. 194).
According to communication theory, message design
logics (MDLs) are systematic thoughts about a
communication situation that an individual relies on
when creating a verbal message (O’Keefe, 1988,
1990). Depending upon the logic used by an
individual, a very different message is said and heard.
This paper examines how message design logic theory
provides insight into secondary mathematics teachers’
verbal messages.
23
Message Design Logic Theory
Researchers have found evidence for three
different message design logics used by leaders in
specific communication situations (Hullman, 2004;
Lambert & Gillespie, 1994; Lambert, Street, Cegala,
Smith, Kurtz & Schofield, 1997; O’Keefe &
McCornack, 1987; Peterson & Albrecht, 1996; Street,
1992). These logics are identified as expressive,
conventional, and rhetorical and are developmentally
ordered. Each has a constellation of related beliefs that
describes the individual’s purpose for the message,
choice of message context, management of the
interaction, and evaluation of the interaction.
Expressive Message Design Logic
Individuals employing expressive design logic
operate
under
an
assumption
that
verbal
communication is a medium for expressing thoughts
and feelings. When these individuals hear or see an
event, they respond verbally with their immediate
thoughts, conveying a clear and honest reaction.
Individuals using this logic believe listeners will
understand the message provided that they speak
openly, directly, and clearly. The conversation, being
organized around immediate reactions, is quite literal,
with little distinction between what is objectively and
subjectively relevant in the situation (O’Keefe, 1988).
If another person in the exchange challenges the
communication, the individual will again respond
verbally, including some editing of previously stated
messages. On average, 22% of participants in message
design logic studies employ this type of logic in their
verbal communication (Hullman, 2004; Lambert &
Gillespie, 1994; Lambert et al., 1997; O’Keefe &
McCornack, 1987; Peterson & Albrecht, 1996; Street,
1992). In the mathematics classroom when students
ask clarifying questions, a teacher employing
expressive design logic reacts by stating his or her
immediate thoughts. These thoughts will likely focus
on the teacher’s thinking, not the student’s thinking. As
a result, this teacher will tend to simply repeat what
was said earlier, attempting to be more clear and
organized.
Conventional Message Design Logic
An individual employing conventional message
design logic believes communication is a cooperative
“game” to be played using conventional rules and
procedures. The individual organizes messages for the
purpose of achieving a particular response, and expects
everyone to play the game by listening to the
communication context and inferring the individual’s
24
intentions. Communicators who employ conventional
message design logic try to say things they believe are
appropriate, coherent, and meaningful for the situation.
These messages are coherent and meaningful only
when all parties involved agree on the same rules and
norms. The individual hears and sees the response of
others, assesses the response in the context of the
situation, and continues the conversation using
conventionally defined actions that they feel are
appropriate. The individual judges the communication
successful when he or she achieves the desired
response, provided that everyone agrees on the
communication rules and norms. This is the most
common message design logic individuals employ in
conversations, with studies reporting that 42% to 58%
of individuals use the conventional message design
logic (Lambert & Gillespie, 1994; O’Keefe &
McCornack, 1987; Peterson & Albrecht, 1996).
The mathematics teacher employing conventional
design logic will focus on using conventional norms
and practices for communication in the mathematics
classroom. Upon hearing and evaluating students’
responses, the teacher says what is needed to move
them in the direction he or she thinks is appropriate. As
teachers develop their professional expertise they learn
responses they should employ in various situations;
this newly developed expertise guides their
communication. Unlike the expressive design logic,
where responses are immediate, this communication is
more purposeful and guided by conventional rules for
communicating, though it may not necessarily address
the students’ needs or questions.
Rhetorical Message Design Logic
Rhetorical message design logic is based on the
belief that “communication is the creation and
negotiation of social selves and situations” (O’Keefe,
1988, p. 87). The individual employing this message
design logic realizes that the intended meanings of his
or her messages are not fixed, but are part of the social
reality being created with others. Rather than merely
being immediate reactions or conventional responses to
situations, messages are explicitly designed toward the
achievement of goals. Words shared in the exchange
are not treated as givens, but as resources that can be
called on in transforming the situation towards
attaining the desired goal. These communicators use
language to transform the situation to be more
motivational and to give explicit re-descriptions of the
context so that goals are achieved.
Communicators using rhetorical message design
logic will also modify their language style to define a
Communication Theory
symbolic reality so that listeners can make an
acceptable interpretation and be motivated to give an
acceptable response. Successful communication is
viewed as a smooth and coherent negotiation among all
participants towards a desired goal. Although this logic
is used by 22% to 32% of adults, researchers have
found that individuals typically preferred messages
consistent with a rhetorical message design logic
(Lambert & Gillespie, 1994; O’Keefe & McCornack,
1987; Peterson & Albrecht, 1996).
Of the three message design logics, rhetorical
message design logic seems to best resemble the
current literature describing preferred classroom
communication (Franke, Kazemi, & Battery, 2007).
The rhetorical message design logic emphasizes a
dynamic negotiation in communication. Mathematics
education
literature
describes
classroom
communication where the teacher, as facilitator,
focuses on student thinking and encourages dialogue so
that students negotiate mathematical understanding.
Teachers who use this logic realize that communication
is a dynamic negotiation process and that the students’
thoughts, the current situation, and the teacher’s goals
must all be taken into account. They do not respond
with prescribed statements, but are more reflective in
their interactions with students.
In summary, expressive message design logic is a
system of talk that simply reacts to circumstances,
whereas conventional message design logic is a system
that responds to exigencies with some appropriate
preconceived remedy. In conventional message design
logic, responses are limited by historically evolved
structures. Rhetorical message design logic, on the
other hand, draws on a wider range of structures, while
containing within it the knowledge of conventional
social forms and relations. Further, rather than seeing
people and situations as givens in a conventional
system of rules or seeing meaning as fixed in messages
by their form and context, “meaning is instead treated
as a matter of dramaturgical enactment and social
negotiation” (O’Keefe, 1988, p. 87). The relation of
message and context is reversed in the conventional
and the rhetorical view. In the conventional view,
context is given and the relevant features of the context
anchor meaning, but in the rhetorical view, context is
created by the message or the process of
communication. Table 1 summarizes the three message
design logics.
Message Design Logics of Secondary Mathematics
Teachers
Message design logic theory provides a framework
for studying classroom interactions. Consider the
following description of classroom communication
summarized from the National Council of Teachers of
Mathematics’ Principles and Standards for School
Mathematics (NCTM, 2000):
Students should engage in conversations in which
mathematical ideas are explored from multiple
perspectives. They should participate in
discussions where they are expected to justify
solutions—especially in the face of disagreement.
This will allow them to gain better mathematical
understanding and develop the ability to acquire
and recognize conventional mathematical styles of
dialogue and argument. Through the grades, their
arguments should become more complete and
should draw directly on the shared knowledge in
the classroom. The role of the teacher is to support
classroom discourse by building a community
where students feel free to express their ideas. (pp.
60 – 61)
.
Table 1
Characteristics of Message Design Logics
Fundamental Premise
Key Message Function
Message/Context Relationship
Method of Managing the
Interactions with Other(s)
Evaluation of Communication
Denise B. Forrest
Expressive
Verbal communication is a
medium for expressing thoughts
and feelings.
Self-expression
Little attention to context
Editing
Conventional
Verbal communication is a game
played cooperatively by social
rules.
Secure desired response
Action and meaning determined
by context
Politeness forms
Rhetorical
Verbal communication is for the
creation and negotiation of social
selves and situations.
Negotiate social consensus
Communication process creates
context
Context redefinition
Expressive clarity, openness and
honesty, unimpeded signaling
Appropriateness, control of
resources, cooperativeness
Flexibility, symbolic
sophistication, depth of
interpretation.
25
The teacher’s role in this communication is to
create the opportunity for students to talk and share
their ideas. From the message design logic perspective,
the emphasis is not on the teacher being clear and
organized in presenting the mathematics, nor on
securing a desired response from students. Instead the
focus is to allow a dynamic conversation to take place
where negotiations and consensus by all parties is the
desired outcome; this idea is consistent with rhetorical
message design logic.
The present study was designed to investigate
message design logics of secondary mathematics
teachers. Because message design logic theory informs
us that individuals hear and say different messages
depending upon which message design logic they use,
this could be an informative perspective for
mathematics educators who are trying to better
understand the verbal communication practices in
mathematics classrooms. In particular, this study used
the message design logic framework to identify a) the
fundamental purpose for secondary mathematics
teachers’ verbal messages to students, b) the key
reasons teachers gave for their verbal messages, c) the
primary ways secondary mathematics teachers account
for students and content in their messages, and d) the
perceived success of teachers’ verbal messages.
Methods
Fifteen
secondary
mathematics
teachers
participated in this study. They were purposefully
chosen (Patton, 1990) to reflect a range in experience,
school setting (urban and suburban), and education. In
interviews, teachers were asked to 1) provide a sample
verbal message to address two hypothetical classroom
vignettes, 2) recollect two classroom situations where
they felt their verbal communication with students was
successful and two situations where they felt it was not
and 3) provide general information about their
experiences with classroom verbal communication and
4) discuss their development of verbal communication
skills.
The first part of the interview, responding to the
written hypothetical vignettes, was consistent with
other message design logic studies. The vignettes in
this study were designed with the guidance of
mathematics educators, previous message design logic
studies (Lambert & Gillespie, 1994; O’Keefe &
McCornack, 1987; Peterson & Albrecht, 1996), and a
communication professor who has formally studied
message design logics (Kline, 1984, 1988, 1991; Kline,
Hennan-Floyd & Farnell, 1990). The vignettes needed
26
to contain three key features in order to elicit a variety
of responses and determine the message design logic
being used by the teacher. First, there is a lack of
conformity in the expected response to the situation,
allowing salient beliefs from the past that may not
particularly be relevant for dealing with the present
situation to be accented. Second, actions or processes
are included that could be subject to renegotiation but
that are relevant to the current situation. Third, the
subject is assigned an authoritative role in the group.
This last criterion was easily met in this study, because
teachers are assumed to be the leaders of classroom
instruction. Factors relative to the other two criteria
were incorporated into the vignettes, by embedding
two to four problem situations that require teachers
make decisions about curriculum and instruction.
This paper focuses on the first vignette, where an
algebra class is working on the following open-ended
problem on the board: ‘If the value of -7abc2 is
negative, what do you know about the signs (positive
or negative) of a, b, and c?’ While the students begin
working, the teacher walks around monitoring their
work and checking homework. The teacher notices that
students are struggling with the problem as a number
of them had not completed the homework assignment
and some were socializing. (See Appendix A for
vignette.)
Teachers were asked to state whether they thought
the given vignette was realistic, and to provide a
sample response message for the vignette. Though in
previous message design logic studies the participants
were asked to give their response in writing, in this
study the response was audio-taped in order to use a
cued-recall procedure (Waldron & Applegate, 1994;
Waldron & Cegala, 1992). This procedure entails
playing back the response and stopping intermittently
to get the participant to share thoughts that are relevant
to the specific statements. (See Appendix A for
interview protocol.) A member check was conducted
following each interview. Teachers were supplied with
their sample messages and reasons for each message,
their positive and negative classroom communication
experience, along with general information provided in
the interview. After all member-check documents were
validated, each message was coded separately by two
researchers as reflecting either an expressive,
conventional, or rhetorical message design logic; there
was 100% agreement between the two coders.
Findings
All three message design logics were found to exist
amongst the secondary mathematics teachers. Twenty
Communication Theory
percent of the messages were coded as employing an
expressive design logic, 53% conventional, and 26%
rhetorical. Sample messages given in response to the
first vignette representing each message design logic
follow.
Mathematics Teachers Using an Expressive Design
Logic
Twenty percent of the teacher responses provided
for the above vignette were coded as expressive. These
messages were characteristically a set of statements in
reaction to the situation that often included
observations by the teacher that were irrelevant points
to solve the immediate mathematical tasks. Here is an
example of one such response.
Folks, we need to get on task here, I need everyone
working on this problem. That’s important because
math is not a spectator sport, you just can’t listen to
me talk and expect to understand it. Now, get to
work so we can get going with this lesson. (If
students continue to be off task then I’ll tell them
I’m going to grade this problem.)
This message was a reaction to the students’ offtask behavior. The teacher said the first idea that comes
to mind, with little attempt to reorganize or address the
students’ understanding of the mathematics. There was
some irrelevant information in the message about
mathematics not being a spectator sport and there were
consequences for students who continued not to work
on the problem.
The teachers who used this message design logic
seemed to have a genuine desire to get students to
learn. They designed their messages to guarantee a
certain responses, but these messages were
predominantly past-oriented, incoherent, and might
have failed to engage the immediate mathematical
problem at hand. In summary, these teachers used their
messages to express their immediate reaction to the
current situation.
Mathematics Teachers Using a Conventional Design
Logic
Conventional message design logic was employed
in 53% of the messages. These messages focused on an
appropriate action in the current situation in order to
get students engaged with the mathematics. The
teacher’s main purpose was to secure a desired
response from the students, manage the situation, and
encourage student cooperation. The context of the
message was centered on the action, meaning, and
justification of the students’ response. A sample of a
conventional message is:
Denise B. Forrest
OK class, there seems to be some confusion with
the problem. Let’s work it out together and we’ll
talk about the thinking I am asking you to do and
why that might be valuable. (After working on the
problem together) The problem involved using
some critical thinking which is an important part of
mathematics. Looks to me like we need to think
about and work out more problems like this. (Make
up several other problems that are similar.)
When these teachers were asked why they chose to
say this message, responses were consistent with
O’Keefe’s (1988) interpretation: “Either the speaker
said this because he or she wants X and saying this is a
normal way to obtain X in this situation; or the speaker
is responding to prior message M, and the relevant
response to M” (p. 87). In summary, the speaker said
what was believed to be appropriate to accomplish the
intended purposes. When one of the teachers was asked
to clarify why he gave a message that was coded as
conventional design logic, he stated,
I want them to think about the logic, the problem
solving; it’s going through a situation where
something is given to you, here’s a problem, now
what do you do, and they have to realistically think
through it, think about what are the things I need in
order to solve this problem, do I need to converse
with someone else about it, do I need to get input,
do I not, do I have the material in front of me, what
are my resources, you know, there are three
different variables here and so I go through that
with them, I say ‘OK those are the kind of things
you need to be thinking about in this’. And then
after that, then they start to understand.
The teachers who used this message design logic
also expressed a genuine desire to encourage student
learning. They talked about saying what needed to be
said in order to accomplish specific learning goals. The
teachers, not the students, defined the direction of
classroom discussion and activity. These teachers
assumed that they knew what the students needed to
hear to move students closer to the desired outcome.
Mathematics Teachers Using a Rhetorical Message
Design Logic
Rhetorical message design logic was employed in
26% of the messages. These messages allowed for
student input, setting the stage for negotiation. The aim
of the verbal communication was to build a social
consensus. The teacher tried to manage the situation
and move the communication strategically towards a
desired context. An example of a message employing
rhetorical message design logic is:
27
(Moves to front of class, and asks for everyone’s
attention.) “I’m noticing that there is something
about this problem that is causing confusion for
some of you. Take a minute and write down at least
one thing that confuses you, or the rest of you write
down at least one key thought that helped you get
started on it. (Listen to responses and depending
upon what was said would determine what I do
next.)
The teachers who employed this design logic in
their message were cognizant of the social negotiation.
For example, one teacher described the reasoning
behind her message as follows:
I have some ideas about what’s going on in this
situation, but it’s always good to get the students’
input first, you know, it could be something I
haven’t thought of at the time. I don’t just want to
assume I have all the facts.
Rhetorical message producers placed importance
on harmony and consensus. They tended to ignore
power and resource control as a means in conflict
resolution. They persistently underestimated the force
of social convention and routine, and overestimated
individuality and creativity. (O’Keefe, 1988). This was
also evident when these teachers clarified the reasoning
for their messages. One teacher said,
Students should be given a voice in the classroom,
it’s so easy to answer and speak for them and move
on, when in fact they have a lot to say and
contribute, and if we just listen, we learn a lot from
them.
These messages were neither a reaction to some prior
condition nor a taken-for-granted feature of the
classroom. Rather than being a conventional response
to some prior state of affairs, they were forwardlooking and goal-connected.
Discussion
The main finding of this study is that mathematics
teachers have varying knowledge and beliefs about
verbal communication and these seem to influence
what teachers hear and say when they talk to students.
This is a notion to consider as mathematics educators
try to understand classroom discourse better. When
mathematics teachers have the common goal of
engaging students in learning mathematical content,
message design logics provide an explanation for the
different paths a teacher’s verbal message can take
towards achieving this goal. In particular, these logics
help explain the possible thoughts teachers use as they
communicate with students. Because this study found
that all three message design logics could be identified
in secondary mathematics teachers’ verbal messages, it
is natural to consider how these message design logics
might influence classroom interactions.
Message Design Logics and Classroom Interactions
A teacher employing expressive design logic
generally creates messages in response to what is heard
and seen in the current situation. The teacher responds
with the thoughts that come to mind based on what is
happening at that moment. A figure representing this
situation is shown below. (See Figure 1.) Even though
the classroom interaction includes student talk, the
diagram is focused on the teacher’s verbal message and
the space the teacher provides for students to interact in
the discussion. Students’ mathematical learning may be
the teacher’s desired outcome, but the verbal path
towards that learning is viewed as more random. The
random arrows represent the messages that are
expressions of the teacher’s initial thoughts. The path
from teacher message to desired learning outcome is
implicit, as indicated by the dashed line. The space
available for students to interact in the discussion is
also indicated by the dashed rectangle.
Figure 1. An illustration of expressive design logic.
28
Communication Theory
Figure 2. An illustration of conventional design logic.
The teacher who employs conventional design
logic, where the verbal exchange is more controlled
and fixed, uses thoughts that move students one step
closer to the goal of mathematical learning. The figure
below represents this classroom interaction. (See
Figure 2.) The teacher focuses his or her message on a
piece of information, determining the appropriateness
of responses to the next piece of information. Each
message is an effort to move the students closer to the
desired learning goal. The space for student interaction
can be narrow or broad depending upon the teacher’s
intention for that piece of information. The teacher’s
message encompasses elements that relate to the
desired outcome for the current piece of information.
For a teacher who employs rhetorical message
design logic during classroom instruction, the goal is to
create verbal messages that allow students to have
space to discuss their thinking, and allow the teacher to
redirect the conversation when needed in order to
achieve mathematical learning goals. (See Figure 3.)
With these goals in mind, the teacher creates his or her
verbal messages. The teacher begins the interaction
with a message to open the space for negotiation.
Focusing on the learning goal, the teacher creates
messages to collect everyone’s thoughts and directs the
student interactions as needed.
In summary, the theory of message design logics
provides mathematics teacher educators a way to
explain teachers’ knowledge and beliefs about
communication: what they believe is important to say
and why it is important. These different beliefs
influence the message design logic used, thereby
impacting how the message is stated and heard by the
teacher. This perspective can inform mathematics
teacher educators’ thinking about teachers’ classroom
communication.
Further Research Using Message Design Logic
Theory
Because this study established that different
message design logics do exist in secondary
mathematics teachers’ communication, this theory is
being used to study interactions in the classroom
setting. For a study in progress, teachers have agreed to
have their classroom interactions audio recorded and
follow up with an interview similar to the protocol,
shown in the Appendix. This investigation aims to
identify how the teacher verbally addresses the
challenges that arise in the classroom. Other issues
being considered are the consistency of a teacher’s
message design logic across conversations, and the
identification of the influence of contextual factors.
Preliminary findings indicate that there is a consistent
message design logic that a teacher uses during
classroom conversations.
A second study in progress is investigating how
preservice teachers develop their logical reasoning for
classroom verbal interactions while participating in
their teacher preparation program. Data has been
collected throughout the preservice teachers’ university
experiences during related coursework, field
experiences, and student teaching. This data include an
initial survey, written responses to classroom episodes,
a self-evaluation of classroom discussion, student
teaching evaluations, and interviews at the conclusion
of their program. In these two studies described above,
message design logic theory continues to provide an
effective lens for studying teacher communication.
Figure 3. An illustration of rhetorical message design logic.
Denise B. Forrest
29
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Communication Theory
Appendix: Interview Documentation
A classroom vignette and a set of semi-structured interview questions were among the documentation taken to
each interview. An abridged interview outline focusing on the cued recall questions and general questions is shown
below. It should be noted the interview outline here is the one used during the study discussed in the article, but
changes have been made in subsequent studies to reduce the amount of time needed to study each individual and allow
for a greater number of participants. For example, the evaluation of another teacher’s classroom transcript has been
omitted because this data focused less on the verbal interaction and more on the environmental and situational factors.
I. Introduction, Expectations, Info Sheet
II. Vignette 1/Vignette 2:
• Directions
Here is a hypothetical classroom situation. Please take a minute to read this and think about how you would
respond. When you are ready, tell me exactly what you would say in this situation to the student and/or students.
•
How realistic is this situation likely to happen in a mathematics classroom?
• Cued-Recall Task
Now we are going to play back pieces of your response OR Now I am going to review some of the things you said.
Try to remember what you were thinking at that time. You are going to be asked to answer three questions the best you
can about your thoughts during this period of the response. If you cannot remember or are not sure, just indicate so, do
not try to guess.
1) What were your reasons for saying that?
2) Were you thinking about other things that you might do or say in the near future or later in the conversation?
3) Was there something you thought about saying but didn’t? Why?
•
1)
2)
3)
4)
After Cued-Recall
What would you like your students to think about/do/say after hearing your message?
What do you believe students thought were your reasons for saying that?
In summary, what do you believe are the most important ideas needing responded to in this situation?
What do you think will happen next?
• Critique
Participant was given a transcript of an actual lesson where the teacher employs an expressive design logic. That
is, the teacher in the lesson just reacts to the questions being asked, focusing on one student at a time. Participants
were asked to evaluate the transcript, providing examples from their classroom experiences.
• General communication questions
We have been talking about particular messages you would create in the classroom, based on specific situations.
Now I’d like to step back and ask some general questions about this.
1) On a scale of 1 – 5, one being lowest and five highest, how would you rank your classroom communication
and tell me why.
2) Can you remember a time when it was higher than this number and describe that situation to me?
3) What made that situation “better”?
4) Can you remember a time when it was lower than this number and describe that situation to me?
5) What made that situation “worse”?
6) What is your role in classroom communication?
7) What is the students’ role?
8) What factors do you think effect your verbal communication in the classroom?
If extended message stated, use cued-recall
Denise B. Forrest
31
Now we are going to play back pieces of your response OR Now I am going to review some of the things you said.
Try to remember what you were thinking at that time. You are going to be asked to answer three questions the best you
can about your thoughts during this period of the response. If you cannot remember or are not sure, just indicate so, do
not try to guess.
1) What were your reasons for saying that?
2) What would you like your students to think about/do/say after hearing your message?
3) What do you believe students thought were your reasons for saying that?
Vignette #1: The bell has rung, you asked students to get out last night’s homework and while you go around and
check to see if they have it done, students are to work on a problem you’ve written on the board to start the day’s
lesson.
If the value of –7abc2 is negative, what do you know about the signs (positive or negative) of a, b, and c?
As you walk around, you notice there are many of them who had not completed the homework assignment, and
even more are taking this time to socialize instead of work on the problem. You remind them to work on the board
problem. Some students begin working on the problem, others just sit there, and Max, a student on the other side of the
room says “Why do we have to do this?” Another student, sitting right next to you adds, “This problem is stupid.”
Describe exactly what you would say to the student(s).
32
Communication Theory
The Mathematics Educator
2008, Vol. 18, No. 2, 33–39
Rethinking Mathematics Assessment: Some Reflections on
Solution Dynamics as a Way to Enhance Quality Indicators
Elliott Ostler
Neal Grandgenett
Carol Mitchell
This paper is intended to offer some reflections on the difficulties associated with the appropriate use
of rubric assessment in mathematics at the secondary level, and to provide an overview of an
assessment technique, hereafter referred to as solution dynamics, as a way to enhance popular rubric
assessment techniques. Two primary aspects of solution dynamics are presented in this manuscript.
The first aspect considers how the tasks assigned in mathematics classrooms might be better
organized and developed to demonstrate an evolving student understanding of the subject. The
second aspect illustrates how revised scoring parameters reduce the potential for scoring
inconsistencies stemming from the non-descript language commonly used in rubrics.
Introduction
Professional teacher organizations have established
the importance of assessment as the vanguard of
instructional decision making. Specifically, in
mathematics, the Principles and Standards for School
Mathematics (National Council of Teachers of
Mathematics [NCTM], 2000), emphasize assessment
as a cornerstone to effective instruction and illustrate
the need for teachers to have a solid grasp of what it
means to effectively assess their students’ abilities. Of
course, how specific assessments are carried out in
different environments will always vary according to
individual needs; nevertheless, the authors still see a
great need for innovation in assessment, both in
interpretation and in technique.
The U.S. educational industry makes a staggering
number of decisions, fiscal and otherwise, based on the
“snapshot” results of standardized tests. These tests
Dr. Elliott Ostler is a professor of mathematics education at the
University of Nebraska at Omaha. He has interests in assessment
and technology as they relate to mathematics education and has
consulted with institutions such as the College Board to develop
teams for implementing curriculum for vertical articulation.
Dr. Neal Grandgenett is a professor of mathematics education at
the University of Nebraska at Omaha. He has extensive expertise
in technology based learning in mathematics and has authored
more than 100 articles in mathematics education. He is currently
assisting in building a curriculum for robotics in mathematics.
Dr. Carol Mitchell is a professor of science education at the
University of Nebraska at Omaha. She has been involved public
education for the past 37 years and is an alumna of the 2002
Oxford Roundtable, Oxford, England. She has numerous
publications and provided leadership for several National
Science Foundation Grants.
Denise B. Forrest
cause part of the assessment dilemma, forcing a teacher
to decide whether to use the results of standardized
measures or focus on assessment methods that are
more contemporary and meaningful. The standardized
assessments most appropriate for large-scale policy
decisions are not necessarily those most suitable for
instructional decision making. Ostensibly, the most
appropriate small-scale assessments would be those
allowing teachers to make decisions about their
instruction (NCTM, 2000). Yet standardized test
results continue to capture the lion’s share of attention
even for teachers gauging their own success. In fact,
despite the research-supported utility of rubric-based
assessments that allow teachers to examine quality
indicators (Arter & McTighe, 2001; Goodrich, 2000;
Stiggins, 2001; Wiggins, 1998), there still appears to
be great resistance to transferring the scope of
pedagogical decisions made from standardized tests to
those more appropriate for evaluating the quality of
students’ mathematics work.
The purpose of this manuscript is to offer some
reflections on item selection and scoring difficulties
associated with appropriate use of rubric assessment in
secondary mathematics and to introduce an interpretive
assessment strategy, hereafter referred to as solution
dynamics, as a way to enhance popular rubric
assessment techniques. Two primary aspects of
solution dynamics are presented in this article: first,
how mathematical tasks might be better organized and
developed to allow students to demonstrate evolving
understanding as they progress through the subjects,
and, second, how revised scoring parameters reduce
the potential for scoring inconsistencies stemming
from the non-descript language commonly used in
rubrics.
33
Figure 1.
Solution Dynamics Defined
Solution dynamics can be thought of as a way to
analyze, organize, and rank student solutions based on
the inherent level of sophistication represented in the
tasks. This is akin to how a performance rubric might
be used, but instead of measuring student performance
with vague descriptors, we will make statements
concerning the complexity of the tasks. Specifically,
solution dynamics considers what that complexity
implies for student understandings needed for
completing the tasks. In some sense, the analysis of
mathematical tasks for solution dynamics assessment
will also determine which tasks are most effective for
instructional purposes.
The solution dynamics process uses the same
general techniques for ranking the complexity of
problems that are used to rank the difficulty of
problems in standardized tests, but the nature of the
tasks require that student solutions be more openended. For example, if a student correctly completes a
math problem of moderate difficulty on a standardized
test, we may come to the conclusion that the student
understands the nature of mathematics related to
solving such problems. However, given the
opportunity to investigate further, we may find that the
student took a long time to solve the problem by using
a low level trial-and-error technique, or that he or she
may even have simply guessed. A rubric assessment of
the same type of problem could possibly determine that
the correct solution illustrates some understanding of
how to complete the task, but this type of scoring
would not necessarily be able to provide specific
references to quality because of the nature of the way
the task was presented. For example, the student’s
solution may receive a score of “progressing,” or a 1
34
on a 0-3 scale, which is actually no more effective for
instructional decision making than a multiple choice
answer to such a question.
On the other hand, in the solution dynamics model
a group of teachers would first look specifically at the
task and provide an organizational structure of possible
solution techniques, each of which would be ranked by
the complexity of the mathematics needed. Student
solutions would then be mapped to the ranked structure
template (See Figure 1) for a score. At first glance, this
may appear to simply be a subtle new twist on an
existing rubric technique, and to some extent, it is;
however, by creating a ranked structure of possible
solutions for a given task, teachers have not only been
forced to analyze the importance and validity of the
task, but also to review a template which provides the
vertically articulated concepts immediately above and
below what the student’s solution illustrates.
Example Solution Dynamics Task: Optimizing the
Volume of a Box
Problem:
Suppose
a
rectangular
(threedimensional) box is to be created by using a 20-inch by
20-inch square sheet of plastic (See Figure 2). Square
corners will be cut from the original sheet of plastic
and the rectangular tabs on each side will be folded up
to create the sides of the box as illustrated below. What
size corner pieces need to be removed so that the box
will have the greatest possible volume? This is
considered a good solution dynamics task because
there is great potential for a number of possible unique
solutions, starting at an arithmetic level and ending at a
calculus level. The same problem can be used in each
of a number of successive courses but the solutions
will change (become dynamic) as the material in the
courses becomes more sophisticated.
Rethinking Mathematics Assessment
Figure 2.
Level 1 (arithmetic-based) solution. The student
creates a chart (See Table 1) that records the volumes
of all possible boxes with whole number increments
being removed from the corners. Such a chart might
look something like Table 1.
Table 1
Example of arithmetic-based solution
Corner
Removed
1 x 1 inch
2 x 2 inch
3 x 3 inch
4 x 4 inch
5 x 5 inch
Resulting
Bases
18 x 18 inch
16 x 16 inch
14 x 14 inch
12 x 12 inch
10 x 10 inch
Height
Volume
1 inch
2 inch
3 inch
4 inch
5 inch
342 inch3
512 inch3
588 inch3
576 inch3
500 inch3
By the time the student has reached the fifth entry in
the chart, they will probably be able to recognize that
the volume is decreasing and that the optimal corner
piece to remove is a 3-inch by 3-inch section. This kind
of solution indicates the student recognizes that the
corner piece removed has the same dimension as the
height of the box and that the base of the box decreases
steadily as larger and larger corner pieces are removed.
They are likely to make a number of other observations
as well; however, at this level they may not yet have
the ability to efficiently test fractional increments,
making their solution incomplete.
Level 2 (algebraic) solution. Students will use the
same basic diagram to provide context, but this
representation of the solution indicates that they
recognize the volume of the box is a function of the
corner piece removed. When examining the pattern
Elliott Ostler, Neal Grandgenett, Carol Mitchell
that emerges from the chart in the first level, students
may derive the following formula: V = (x)(20 – 2x)2.
Using this formula, students can test both whole
number and fractional increments of corner piece
dimensions much more efficiently than was possible
with a chart. Yet this solution is still limited in that it
does not allow for an efficient determination of an
exact solution.
Level 3 (advanced algebra/calculus-based)
solutions. Once again, students will use the same
diagram to provide context for the problem. An
advanced understanding of this problem will illustrate
that students not only recognize the functional
relationship between the volume of the box and the
dimension of the corners removed, but that they
understand that the volume can be graphed as a
function of the dimension of that corner. They may
also recognize that a maximum volume can be
determined by closely examining the resulting graph or
that by calculating the derivative of the function, they
can determine an exact maximum point, which would
represent a maximum volume of the box.
By analyzing the solution to a problem in terms of
levels of sophistication, not only can we place a
student on a scale, we can surmise with some accuracy
what they know, and what they need to know in order
to achieve the next level of complexity. The general
tree diagram in Figure 1 adapted from Craig (2002) can
help determine the complexity of mathematical tasks
based on a continuum, which progresses from simple
to complex. Galbraith and Haines (2000) conducted
research that clearly indicated that mechanical
35
processes, here referred to as algorithmic processes,
were easier than interpretive problems, which, in turn,
were easier than constructive problems. Algorithmic
processes consisted of mechanical solutions where
students needed only to follow a sequenced set of steps
to solve a problem. Interpretive problems were those
problems presented in more abstract forms (i.e. word
problems) from which the correct processes had to be
interpreted. Constructive problems were those that
required a combination of the two lower categories.
Certainly the use of the model in Figure 1 does not
allow for the ranking of mathematical tasks to be an
exact science, but it does guide teachers to focus on the
hierarchy of difficulty innate to a task.
The following example illustrates how solutions on
another simple mathematical task might be ranked on a
solution dynamics rubric as the mathematics used to
solve the problem becomes more sophisticated. Note
that the same problem is used year after year so that
growth in the understanding of the processes related to
this specific problem can be tracked. The differences in
the complexity of the mathematics at each scoring
level have been greatly exaggerated in this example in
order to help differentiate between the elements in
Figure 1. With an actual solution dynamics task, the
differences would be more subtle and require the
attention of a team of mathematics teachers to study
the nuances of expected students’ solutions.
The levels of the task shown in Figure 3 are
certainly subject to interpretation, but illustrate how
solutions become dynamic by focusing on the
sophistication of the mathematics and the process of
derivation rather than on the actual formula for the area
of the circle as an answer. This particular task is one of
the most basic examples of solution dynamics and one
that has been used successfully by the authors in
calculus courses. Allowing students to observe the
evolution of complexity in a mathematical task
provides context to the processes of integration.
The mathematical tasks assigned would be used to
help reinforce concepts being taught at each course
level. Certainly a teacher would not expect a student to
use a complex mathematical technique to solve a very
simple problem, but often a simple problem can
provide a very powerful context for illustrating how
complex mathematical ideas can be applied to various
situations. In the example above we saw that a simple
task can be used to demonstrate how both simple and
complex mathematics can be applied to a situation. The
derivation of the formula for the area of a circle is
simply a convenient task that can be repeated through
multiple levels of instruction to allow students to
36
demonstrate an understanding of increasingly
sophisticated thinking within a familiar context.
Because the task remains the same, teachers can get a
sense of what students know about evolving levels of
mathematics based on how they might approach the
solution.
Why Not Rubrics Alone
Rubrics are popular tools for assessment and can
no doubt provide insight to student understanding in a
variety of subjects and contexts if they are carefully
constructed. Rubrics by themselves, however, have
some inherent flaws that inhibit consistent scoring and
decision making (Popham, 1997). The three most
problematic flaws are as follows: rubrics are actually
secondary scoring instruments but are often
misunderstood to be the primary instrument; the
language used in the quality descriptors, although
consistent, is too vague to make meaningful decisions;
and quantity indicators are often mistaken for quality
descriptors. We elaborate on each of these three flaws
below.
Rubrics are secondary scoring instruments
Students do not perform on a rubric. Students
perform a task that is then scored by a rubric. This
simple misunderstanding creates confusion about the
nature of rubrics and how they should be used. It is not
unusual to hear people talk about how students
performed on the rubric, when in fact they mean how
students scored on a preset task as interpreted by the
rubric. This being the case, it should be at least as
important to consider the innate value of the
mathematical task as it is to consider the performance
level descriptors used to rank the students’
understanding of the task. Unfortunately, task
considerations tend to be passed over in lieu of more
careful consideration of the rubric scale.
The language used in the quality descriptors of rubrics
is too vague
Tierney & Simon (2004) argue for the need to state
the performance criteria and the attributes clearly.
They also argue for the need to describe the qualitative
degrees of performance more consistently between the
performance levels of the rubric. They indicate that
these modifications make the task, criteria, and
attributes clearer to students and allow a broader use of
the rubric. These are noble concepts, and the claim
they make about clarity may be true, but the
terminology they suggest is part and parcel of the
problem with broad-use assessments: non-descript
language. In one example, the terms they suggest using
Rethinking Mathematics Assessment
Figure 3. Levels of solutions for the task of deriving the area of a circle.
to provide consistency and clarity are few, some, most,
and all. These are not bad terms, but they are only
indications of clarity or quality when antecedent to
some very specific requirements provided in the initial
task. For example suppose a timed, 100-item, singledigit multiplication test were being used as an
assessment. A student answering 45 items correctly
would probably fall into the “few” or “some” category
of the rubric. We might surmise from that score that
the student has difficulty with multiplication. However,
suppose the student only answered 45 questions and
was correct on all completed items. It is possible that
the student simply writes slowly but knows the
information very well. The terms few, some, most, and
Elliott Ostler, Neal Grandgenett, Carol Mitchell
all generally do nothing more than a checklist would,
particularly when they are applied in the manner
indicated above. If however, the assessment instrument
included items that gradually became more difficult,
the terms few, some, most, and all would be more
appropriate because they would be antecedent to levels
of difficulty within the test rather than just looking at
quantity of similar items completed. This idea leads
into the next point.
Quantity indicators are often mistaken for quality
descriptors
As far as student performance on a given task is
concerned, the demonstration of basic knowledge does
not necessarily require a rubric. Once again, the nature
37
of the task needs to be a primary consideration. For
instance, if a teacher wants students to know basic
facts like multiplication tables, a rubric is probably not
necessary. If a teacher were to create a rubric where the
scale indicators showed increased student performance
by the number of problems they correctly answered
(i.e. “Beginning” = 20 problems, “Progressing” = 30
problems, “Advanced” = 40 problems, etc.) the
categories would not be indications of conceptual
quality nor are the descriptors assigned to the scales
necessarily set by any externally valid criteria. It is
therefore unnecessary to provide a rubric scale that
counts or quantifies the number of correct answers. For
a task such as this, a checklist would be more
appropriate. Quality indicators are more appropriate to
tasks that require some higher-level thinking and rubric
levels that clearly indicate the quality of thinking, or
the lack thereof.
Conclusion
Ultimately, there are two primary factors that make
a solution dynamics approach a potentially effective
way to clarify and increase the accuracy of rubricbased assessment. First, a solution dynamics model
considers the evolution of a mathematical solution over
time. Second, this approach specifically considers the
quality of the student performance and the difficulty of
the task within the same instrument. Both of these
factors, though somewhat obvious, emphasize ideas
that are generally absent in the explanation of rubric
assessment. Problems such as the derivation of the area
formula for a circle, as illustrated earlier, have been
used with great success in a solution dynamics format
by the authors to show not only the evolution of
students’ simple mathematical models to complex
ones, but also to illustrate natural connections and
applications between scientific and mathematical
content. This has been particularly true in calculus
courses where students tend to lack the conceptual
understanding behind processes like integration.
Though it is probably not realistic to expect large
gains in mathematical understanding to come in a
single academic year for every student, the selection of
the right kinds of dynamic mathematical problems can
better illustrate the dynamic nature of the mathematics
the students are learning and therefore help facilitate
the conceptual evolution of mathematical knowledge
that represents a transition from algorithmic to abstract
thinking. It is important and appropriate to engage in
assessment techniques that measure students’ progress
over a successive period of years. Attention to evolving
representations of student solutions allow for this to
38
happen. A focused effort on vertical articulation, and in
particular, efforts to build dynamic solution exercises
(specific mathematical tasks that lend themselves well
to solution dynamics assessment) will provide a more
comprehensive view about students’ understanding of
mathematics and its various components, concepts, and
skills.
Romberg (2000) argues that, with appropriate
guidance from teachers, students can build a coherent
understanding of mathematics and that their
understanding about the symbolic processes of
mathematics can evolve into increasingly abstract and
scientific reasoning. This, of course, happens through
opportunities to participate in appropriate kinds of
mathematical tasks. As mentioned previously, a
coherent understanding of anything does not happen
with most students over the course of a single
academic year. The evolution in a student’s thinking
that allows them to demonstrate a transition from
algorithmic to abstract semiotics presumably happens
over a period of years. It follows then that developing
the kinds of appropriate mathematical assessments, the
dynamic kinds that allow for this transition to be
measured over time, can most appropriately be done by
a team of mathematics educators. Each considers the
nuances of what the others do, and then documents
their part in the process through thoughtful solution
dynamics assessments.
References
Arter, J., & McTighe, J. (2001). Scoring rubrics in the classroom:
Using performance criteria for assessing a improving student
performance. Thousand Oaks, CA: Corwin Press/Sage
Publications.
Craig, T. (2002). Factors affecting students’ perceptions of
difficulty in calculus word problems. 2 nd International
Conference on the Teaching of Mathematics. Crete, Greece,
July 1-6, 2002. Retrieved November 28, 2008 from
http://www.math.uoc.gr/~ictm2/Proceedings/pap411.pdf.
Galbraith, P., & Haines, C. (mis(understandings) of beginning
undergraduates. International Journal of Mathematical
Education in Science and Technology. 31(5). 651–678.
Goodrich, A. (2000). Using rubrics to promote thinking and
learning. Educational Leadership, 57, 13–18.
National Council of Teachers of Mathematics. (2000). Principles
and standards for school Mathematics. Reston, VA: Author.
Popham, W. J. (1997). What’s wrong—and what’s right—with
rubrics. Educational Leadership, 55, 72–75.
Romberg, T. (2000). Changing the teaching and learning of
mathematics. Australian Mathematics Teacher, 56(4), 6–9.
Stiggins, R.J. (2001). Student-involved classroom assessment (3rd
ed.). Upper Saddle River, NJ: Prentice-Hall.
Rethinking Mathematics Assessment
Tierney, R. & Simon, M. (2004). What's still wrong with rubrics:
focusing on the consistency of performance criteria across
scale levels. Practical Assessment Research & Evaluation,
9(2). Retrieved November 28, 2008 from
http://PAREonline.net/getvn.asp?v=9&n=2 .
Elliott Ostler, Neal Grandgenett, Carol Mitchell
39
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In this Issue,
Guest Editorial… Motivating Growth of Mathematics Knowledge for Teaching: A
Case for Secondary Mathematics Teacher Education
AZITA MANOUCHEHRI
Planning for Mathematics Instruction: A Model of Experienced Teachers’ Planning
Processes in the Context of a Reform Mathematics Curriculum
ALISON CASTRO SUPERFINE
Communication Theory: Another Perspective to Think About for Mathematics
Teachers’ Talk
DENISE B. FORREST
Rethinking mathematics and science assessment: Some reflections on Solution
Dynamics as a way to enhance quality indicators
ELLIOTT OSTLER, NEAL GRANDGENETT, CAROL MITCHELL