THE MATHEMATICS EDUCATOR
Volume 21 Number 2
Winter 2011/2012
MATHEMATICS EDUCATION STUDENT ASSOCIATION
Editorial Staff
Editors
Allyson Thrasher
Catherine Ulrich
Associate Editors
Amber G. Candela
Tonya DeGeorge
Erik D. Jacobson
Kevin LaForest
Laura Lowe
David R. Liss, III
Patty Anne Wagner
Advisor
Dorothy Y. White
MESA Officers
2011-2012
President
Tonya DeGeorge
Vice-President
Shawn Broderick
Secretary
Jenny Johnson
Treasurer
Patty Anne Wagner
NCTM
Representative
Clayton N.
Kitchings
Colloquium Chair
Ronnachai Panapoi
A Note from the Editors
THE MATHEMATICS EDUCATOR
Dear TME readers,
On behalf of the editorial staff and the Mathematics
Education Student Association at The University of
Georgia, I am happy to present the concluding issue of
the 21st volume of The Mathematics Educator. This issue
also marks the conclusion of my tenure as Co-Editor. I
have learned a great deal from my time with TME.
Authors, fellow editors, and reviewers have helped me
become a more critical researcher and I thank them all for
their contributions to TME and my continuing
professional growth. This issue showcases several
familiar ideas in novel contexts intended to push the
thinking of our readers, just as working on TME has
pushed my thinking in new directions.
In our editorial, Kyle T. Schultz and LouAnn Lovin
explore an emerging framework for unpacking
specialized disciplinary knowledge. They provide
compelling examples of using a Decoding Disciplines
Model in their work with preservice teachers. Michelle
Cirillo and Patricio G. Herbst offer new ways to
incorporate proving in mathematics classes that goes
beyond the two-column proof, and their examples easily
translate to high school geometry. Lu Pien Cheng and
Lee Peng Yee take a new look at Lesson Study in the
context of a primary school in Singapore. They describe
the lesson study process and its influence on teacher
participants as they revised a second grade fraction
lesson. Anderson Norton and Michael Baldwin discuss
student struggles with accepting the equality of 0.999…
and 1 and the consequences of rejecting this equality.
Our loyal readers will notice a change to the style
and format of this issue of TME. This year, TME staff
will continue these upgrades, including overhauling our
website. We ask for your patience and feedback as we
implement these changes. Katy and I hope that you enjoy
this issue and share it with your colleagues.
An Official Publication of
The Mathematics Education Student Association
The University of Georgia
Winter 2011/2012
Volume 21 Number 2
Table of Contents
2
Guest Editorial… Examining Mathematics
Teachers’ Disciplinary Thinking
KYLE T. SCHULTZ & LOUANN LOVIN
11
Moving Toward More Authentic Proof
Practices in Geometry
MICHELLE CIRILLO & PATRICIO G.
HERBST
34
A Singapore Case of Lesson Study
LU PIEN CHENG & LEE PENG YEE
58
Does 0.999… Really Equal 1?
ANDERSON NORTON & MICHAEL
BALDWIN
68
A Note to Reviewers
70
Submission Guidelines
Allyson Hallman Thrasher
Cover Art:“Metacognition Mandala” by Kylie Wagner inspired by Schultz and Lovin’s
editorial of expert mathematics educators researching their own thinking.
© 2012 Mathematics Education Student Association
This publication is supported by the
College of Education at The University of Georgia.
All Rights Reserved
The Mathematics Educator
2011/2012 Vol. 21, No. 2, 2–10
Guest Editorial…
Examining Mathematics Teachers’
Disciplinary Thinking
Kyle T. Schultz and LouAnn Lovin
Shulman’s (1986) seminal paper on subject-matter
knowledge in teaching brought attention to different domains
of teacher knowledge and how that knowledge might be
cultivated. In particular, he described a “reflective awareness”
(p. 13), developed from analysis of discipline-focused teaching
and learning. This reflective awareness enables professionals to
perform tasks in their particular disciplines but also enables
them to communicate their thinking, rationales, and judgments
as they do so. For mathematics teacher educators, being able to
articulate our thinking, rationales, and judgments with respect
to doing and teaching mathematics is extremely important as
we attempt to help prospective teachers develop their own
reflective awareness. In order to do so, we must have a welldefined sense of what the disciplinary thinking about teaching
mathematics entails.
Although we have different focuses within mathematics
education, with LouAnn teaching PreK–8 mathematics content
courses and Kyle teaching middle grades and high school
mathematics methods and practicum courses, we have found
commonalities in the ways that our prospective teachers
Kyle T. Schultz, a former high school mathematics teacher, is an Assistant
Professor of mathematics education at James Madison University in
Harrisonburg, Virginia. His work focuses on teachers' decision making
with respect to mathematics curriculum, instruction, and technology.
LouAnn Lovin, a former classroom teacher, is an Associate Professor in
mathematics education at James Madison University. She teaches
mathematics content and methods courses for practicing and prospective
PreK-8 teachers. She is interested in learner-centered mathematics
instruction and conducts research investigating the mathematical
knowledge needed to teach for understanding.
Disciplinary Thinking
struggle to develop the disciplinary thinking processes that are
integral to understanding mathematics and teaching it
effectively. For example, prospective teachers in mathematics
content courses often cannot make sense of their classmates’
solutions when the method of solution differs greatly from their
own. Similarly, prospective teachers in methods courses
struggle when identifying and sequencing appropriate
mathematical tasks for instruction. These skills are examples of
specialized content knowledge (SCK), mathematical
knowledge of particular importance to PreK–12 teachers (Ball,
Thames, & Phelps, 2008). We have made our prospective
teachers’ development of SCK an important focus of our
programs due to its positive correlations with student
achievement (Hill, Rowan, & Ball, 2005). For example, we
have attempted to situate activities, assignments, and
assessment items in mathematical tasks of teaching (Ball,
Thames, & Phelps, 2008)—everyday tasks of teaching that
require the use of SCK. Such tasks include “choosing and
developing usable definitions,” “responding to students’ ‘why’
questions,” and “asking productive mathematical questions” (p.
400).
As mathematics teacher educators, we have found it
difficult to pin down and articulate in detail the disciplinary
thinking used by mathematics teachers when enacting their
SCK. The general nature of characterizations of critical
thinking, such as focusing on the obscure notion of “concept”
and practices such as brainstorming, making comparisons, and
questioning, prompted us to seek a more discipline-specific
solution. A program sponsored by our institution’s Center for
Faculty Innovation introduced us to a model aimed at decoding
disciplinary thinking, that is, the thinking specifically used by
experts in their discipline. Middendorf and Pace (2004)
characterized this kind of thinking as something that is rarely
presented to students explicitly.
Decoding the Disciplines Model
Middendorf and Pace (2004) presented a model based on
seven questions (see Figure 1) that guides university faculty
through a process to better understand the implicit ways of
thinking exhibited within their disciplines and how to make
3
Kyle T. Schultz & LouAnn Lovin
those ways of thinking explicit to students. Rather than
focusing on the general goal of critical thinking, the Decoding
the Disciplines Model (DDM) targets specific bottlenecks to
student learning, instances during the learning process where a
significant number of students falter. Once a bottleneck is
identified, the faculty member attempts to unpack how he or
she might navigate through it. This results in a list of ideas and
tasks used by the faculty member to work through the
bottleneck. This list of ideas and tasks can serve as a heuristic
guide for novices. The first six questions of this model form a
cycle of inquiry, with the seventh question serving as an
offshoot from the sixth. Through using the DDM, students are
provided opportunities to practice and receive feedback on
discipline-specific ways of reasoning or skills.
1.
2.
3.
4.
What is a bottleneck to learning in this class?
How does an expert do these things?
How can these tasks be specifically modelled?
How will students practice these skills and get
feedback?
5. What will motivate the students?
6. How well are the students mastering these learning
tasks?
7. How can the resulting knowledge about learning be
shared?
Figure 1. The seven questions of the Decoding the
Disciplines Model (Middendorf & Pace, 2004).
Our efforts to address the initial questions of the DDM
were supported by a self-study methodology in which we acted
as “critical friends” (Loughran, 2004, p. 157) by challenging
each other’s claims and pushing for more explicit clarification
of ideas. In addition, we shared the products of our work with a
colleague outside of mathematics education but familiar with
the DDM as a way to ensure we were “constantly asserting
ideas and interrogating them, inviting alternative interpretations
and seeking multiple perspectives” (Pinnegar & Hamilton,
4
Disciplinary Thinking
2009, p. 165). To illustrate our use of the DDM, we will focus
on a bottleneck for prospective teachers in the middle grades
mathematics methods course, developing a sequence of tasks
used to teach a new concept.
Identifying Bottlenecks
To identify bottlenecks, we examined prospective teachers’
work on assessments from their previous courses to determine
specific instances where a majority demonstrated difficulty
with key ideas of the course. For elementary and middle grades
teacher candidates, we also considered data from a programwide multiple-choice assessment of prospective teachers’ SCK
of K–8 mathematics, which was modeled after the Learning
Mathematics for Teaching assessment developed at The
University of Michigan (Hill, Schilling, & Ball, 2004) as well
as focus group interview data about the tasks on this
assessment. Although it was easy to identify instances where
our students struggled, it was often difficult to articulate
precisely what that struggle entailed. To hone this precision, we
strove to push each other for further clarification of our ideas
by asking questions such as “How would you reason through
that task?” and “What do you mean by that terminology?” For
this process, we attempted to set aside our knowledge of
familiar concepts and jargon-laden terms to clarify our own
understanding of them. Repeating this process with our out-ofdiscipline colleague reinforced this push for a layman’s view,
improving our ability to better articulate how one might
navigate through a given bottleneck.
One bottleneck was identified using a methods course
assessment on lesson planning. In this assessment, many
prospective teachers struggled to use and sequence tasks within
the targeted students’ zones of proximal development. For
example, in an introductory lesson about fraction division, one
prospective teacher began his lesson by asking students to
solve the task 53 ÷ 12 using manipulatives and, from this solution,
independently develop an algorithm to divide any two
fractions. Although this task has the desired goal of students
understanding the underlying mechanics of the division
algorithm, it uses a relatively difficult quotient, provides only
one concrete example, and does not provide a context for the
5
Kyle T. Schultz & LouAnn Lovin
quotient, focus on the meaning of fraction, or connect to
previously learned computation strategies (recommendations
offered by Van de Walle, Karp, & Bay-Williams, 2010). Other
prospective teachers provided multiple contextual tasks to
develop the concept, but struggled to sequence them in an order
that would build understanding. In each of these cases, the
prospective teachers lacked the SCK needed to identify the
subtle mathematical differences between similar tasks and
distinguish between the relative complexities caused by these
differences. For example, some began their progressions using
non-unit-fraction divisors before those with unit fractions.
Therefore, we identified the development of a sequence of
tasks used to teach a new concept as a bottleneck for the
prospective teachers.
An Expert’s View
For each identified bottleneck for prospective teachers, we
sought to write a detailed description of what we, as expert
mathematics teachers, would do to navigate through it. Because
some of these processes were automatic or almost instinctual
for us, we found it difficult to articulate our thinking without
glossing over subtle nuances that might be crucial for a novice
teacher. Using the discourse strategy previously described, we
challenged each other to define and clarify our own
disciplinary thinking.
To identify the thinking one might use to create a sequence
of tasks used to introduce a new mathematical concept, Kyle
looked to recreate the experience of a novice by working with a
mathematical concept with which he was familiar as a learner,
but not as a teacher (mirroring the situation faced by
prospective teachers). Because he had never taught calculus, he
focused on the steps he would undertake to design a sequence
of tasks to teach the concept of related rates. This process
involved unpacking the mathematics found in textbook
examples, identifying the relationships between them, and
using these relationships as a foundation for developing student
understanding. From this work, the disciplinary thinking was
generalized into a set of small incremental steps (see Figure 2)
that could guide prospective teachers during their initial
attempts to navigate the bottleneck.
6
Disciplinary Thinking
Bottleneck: Developing a sequence of tasks used to teach a
new concept.
1. Examine the curriculum framework goal(s) to be
addressed.
2. Determine the big idea(s) (Charles, 2005) associated
with these goals.
3. Write learning objectives for the lesson that relate back
to the big ideas.
4. Work each example task in the book. In this process,
note:
a. Different representations that might be productively
used in a solution
b. Connections or common themes between the tasks,
objectives, and big mathematical idea(s)
c. Prerequisite knowledge needed to engage in each
task
d. Non-contextual differences between the tasks
(changes in mathematical complexity or required
level or type of thinking)
5. Identify stages of development needed to understand
the concept and perform related skills.
6. Identify existing tasks corresponding to these stages.
For example, could the provided textbook examples
serve this purpose? Would additional tasks be needed?
7. Brainstorm possible student strategies or solutions for
these tasks.
8. Evaluate and modify the identified tasks to optimize
student strategies and misconceptions.
Figure 2. A list of the small incremental steps for navigating
the bottleneck of developing a sequence of tasks used to
teach a new concept.
7
Kyle T. Schultz & LouAnn Lovin
Modeling and Practice
Once we had achieved a sense of the disciplinary thinking
needed to navigate a particular bottleneck, our attention shifted
to designing course activities that would enable prospective
teachers to learn and practice that thinking themselves.
Examining the prospective teachers’ work during these
activities has helped us to identify additional bottlenecks and
provided further insight into our view of disciplinary thinking.
For example, Kyle’s prospective teachers struggled with
identifying big mathematical ideas, the second step in the
process shown in Figure 2. Given the struggles of his
prospective teachers, Kyle returned to the literature and found
evidence that might support his observations in class:
Some mathematical understandings for Big Ideas can
be identified through a careful content analysis, but
many must be identified by “listening to students,
recognizing common areas of confusion, and analyzing
issues that underlie that confusion” (Schifter, Russell,
and Bastable 1999, p. 25).
Research and classroom experience are important
vehicles for the continuing search for mathematical
understandings. (Charles, 2005, p. 10)
The possibility that his prospective teachers’ difficulties
with big ideas may stem from a lack of teaching experience has
prompted Kyle to plan experiences for his class using
classroom data (video, written cases, vignettes, etc.) to provide
his prospective teachers with opportunities to listen to students,
to recognize common misconceptions, and to analyze issues
that help to create these misconceptions.
Looking Ahead
This work is an iterative process. As we continue working
with our prospective teachers, we further refine our bottleneck
articulations, descriptions of our unpacked disciplinary
thinking, and the associated classroom activities whose purpose
is to help our learners navigate through the identified
bottlenecks. As we implement our work in our classrooms,
assessment plays a key role in shaping future iterations in two
8
Disciplinary Thinking
ways. First, using pre- and post-assessments will quantify
prospective teachers’ gains in mastering disciplinary thinking.
Second, qualitatively examining their responses may enable us
to identify other bottlenecks (Kurz & Banta, 2004).
As discussed, we have found that some of the steps we
have identified to illuminate our disciplinary thinking for
prospective teachers are in fact bottlenecks themselves,
requiring further unpacking and clarification. For example,
determining big mathematical ideas and brainstorming possible
student strategies or solutions for a task, two processes
identified as key steps for developing a sequence of tasks to
teach a new concept, are not trivial. As a result, we have
labeled these skills as bottlenecks as well and have undertaken
defining the disciplinary thinking needed for each. In this way,
focusing on bottlenecks as a fundamental idea has enabled us
to better define our course objectives and hone our instruction
and assessment, with the goal of ultimately improving our
prospective teachers’ performance in their future classrooms.
References
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for
teaching: What makes it special? Journal of Teacher Education, 59,
389–407.
Charles, R. I. (2005). Big ideas and understandings as the foundation for
elementary and middle school mathematics. Journal of Mathematics
Education Leadership, 7(3), 9–24.
Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’
mathematical knowledge for teaching on student achievement. American
Educational Research Journal, 42, 371–406.
Hill, H.C., Schilling, S.G., & Ball, D.L. (2004). Developing measures of
teachers’ mathematics knowledge for teaching. Elementary School
Journal, 105, 11-30.
Kurz, L., & Banta, T. W. (2004). Decoding the assessment of student
learning. In D. Pace & J. Middendorf (Eds.), Decoding the disciplines:
Helping students learn disciplinary ways of thinking (pp. 85–94). San
Francisco, CA: Jossey-Bass.
9
Kyle T. Schultz & LouAnn Lovin
Loughran, J. (2004). Learning through self-study: The influence of purpose,
participants, and context. In J. Loughran, M. L. Hamilton, V. LaBoskey,
& T. Russell (Eds.), International handbook of self study of teaching
and teacher education practices (pp. 151–192). London, England:
Kluwer.
Middendorf, J., & Pace, D. (2004). Decoding the disciplines: A model for
helping students learn disciplinary ways of thinking. In D. Pace & J.
Middendorf (Eds.), Decoding the disciplines: Helping students learn
disciplinary ways of thinking (pp. 1–12). San Francisco, CA: JosseyBass.
Pinnegar, S., & Hamilton, L. (2009). Self-study of practice as a genre of
qualitative research. London, England: Springer.
Shulman, L. S. (1986). Those who understand: Knowledge growth in
teaching. Educational Researcher, 15(2), 4–14.
The Mathematics Educator
2011/2012 Vol. 21, No. 2, 11–33
Moving Toward More Authentic Proof
Practices in Geometry
Michelle Cirillo and Patricio G. Herbst
Various stakeholders in mathematics education have called for
increasing the role of reasoning and proving in the school
mathematics curriculum. There is some evidence that these
recommendations have been taken seriously by mathematics
educators and textbook developers. However, if we are truly to
realize this goal, we must pose problems to students that allow them
to play a greater role in proving. We offer nine such problems and
discuss how using multiple proof representations moves us toward
more authentic proof practices in geometry.
Over the past few decades, proof has been given increased
attention in many countries around the world (see, e.g.,
Knipping, 2004). This is primarily because proof is considered
the basis of mathematical understanding and is essential for
developing, establishing, and communicating mathematical
knowledge (Hanna & Jahnke, 1996; Kitcher, 1984; Polya,
1981; Stylianides, 2007). More specifically, in describing proof
as the “guts of mathematics,” Wu (1996, p. 222) argued that
anyone who wants to know what mathematics is about must
learn how to write, or at least understand, a proof. This
comment complements the call to bring students’ experiences
in school mathematics closer to the discipline of mathematics,
that is, the practices of mathematicians (Ball, 1993; Lampert,
1992; National Council of Teachers of Mathematics [NCTM]
Michelle Cirillo is a former classroom teacher who is now an Assistant
Professor in the Department of Mathematical Sciences at the University of
Delaware. Her research interests include proof in geometry, classroom
discourse, and teachers’ use of curriculum materials.
Patricio Herbst is a former classroom teacher who is an Associate
Professor in the School of Education at the University of Michigan. His
research interests include teacher decision making in geometry classrooms
and the use of reasoning and proof to solve problems and develop new
ideas.
10
Michelle Cirillo & Patricio G. Herbst
2000). This idea is not new: A number of curriculum theorists
from Dewey (1902) to Schwab (1978) have argued that the
disciplines should play a critical role in the school curricula.
Thus, by engaging students in authentic mathematics, where
they are given opportunities to refute and prove conjectures
(Lakatos, 1976; Lampert, 1992; NCTM, 2000), teachers can
create small, genuine mathematical communities in their
classrooms (Brousseau, 1997).
Through the introduction of the Standards documents
(1989, 2000), NCTM put forth some significant
recommendations related to the Reasoning & Proof and
Geometry standards that have had the potential to impact the
high school geometry curriculum. First, it has been
recommended that reasoning and proof should not be taught
solely in the geometry course, as it typically has been done in
the United States. Rather, instructional programs in all grade
bands
•
•
•
•
should enable students to recognize reasoning and proof
as fundamental aspects of mathematics;
make and investigate mathematical conjectures;
develop and evaluate mathematical arguments and
proofs; and
select and use various types of reasoning and methods of
proof. (NCTM, 2000, p. 56)
Other calls to increase attention to reasoning and proof
come from descriptions of mathematical proficiency. For
example, the National Research Council (2001) recommended
that students develop the capacity to think logically, to justify,
and, ultimately, to prove the correctness of mathematical
procedures or assertions (i.e., adaptive reasoning). More
recently, the U.S. Common Core State Standards document
(National Governors Association Center for Best Practices &
Council of Chief State School Officers, 2010) included, as one
of their Standards for Mathematical Practice, the ability to
construct viable arguments and critique the reasoning of others.
Despite these recommendations, in the United States the
high school geometry course continues to be the dominant
place where formal reasoning and the deductive method are
12
More Authentic Proof
learned (Brumfiel, 1973; Driscoll, 2011; Yackel & Hanna,
2003). One reason for this is practical: After students
conjecture about the characteristics and relationships of
geometric shapes and structures found in the real world,
geometry offers a natural space for the development of
reasoning and justification skills (NCTM, 2000). However,
even in the high school geometry course, students are typically
not provided the kinds of experiences recommended in the
standards documents. For example, in her study on teachers’
thinking about students’ thinking in geometry, Lampert (1993)
outlined what doing a proof in high school geometry typically
entails. According to Lampert, students are first asked to
memorize definitions and learn the labeling conventions before
they can progress to the reasoning process. They are also taught
how to generate a geometrical argument in the two-column
form where the theorem to be proved is written as an if-then
statement. After students write down the “givens” and
determine what it is that they are to prove, they write the lists
of statements and reasons to make up the body of the proof. In
this context, there is never any doubt that what needs to be
proved can be proved, and because teachers rarely ask students
to write a proof on a test that they have not seen before,
students are not expected to do much in the way of independent
reasoning. Similarly, through their analyses, Herbst and Brach
(2006) argued that the norms of the situation of doing proofs do
not necessarily support students through the creative reasoning
process needed to come up with arguments on their own.
Another recommendation that has had the potential to
impact the high school geometry curriculum is related to the
modes of representation that are used to communicate
mathematical proof. In the 1989 NCTM Geometry Standard,
two-column proofs (which have typically been the proof form
presented in U.S. textbooks) were put on the list of geometry
topics that should receive “decreased attention” (p. 127). In the
2000 Standards, NCTM clarified its position, stating, “The
focus should be on producing logical arguments and presenting
them effectively with careful explanation of the reasoning
rather than on the form of proof used (e.g., paragraph proof or
two-column proof)” (p. 310). In other words, it is the argument,
not the form of the argument, that is important.
13
Michelle Cirillo & Patricio G. Herbst
Since these recommendations have been published, we
have begun to see some changes to the written curriculum (i.e.,
textbooks). For example, many authors have addressed the
proof form recommendation by promoting paragraph and flow
proofs in their textbooks (see, e.g., Larson, Boswell, & Stiff,
2001). Discovering Geometry (Serra, 2008) is another example
of a curricular shift in which the author expanded the role of
the students by asking them to discover and conjecture through
investigations but delays the introduction of formal proofs until
the final chapter of the textbook. Most recently, the CME
(Center for Mathematics Education) Project’s Geometry
(Education Development Center [EDC], 2009) asks students to
conjecture and analyze arguments, proposes a variety of ways
to write and present proofs, and asks students to identify the
hypotheses and conclusions of given statements.
While we do not necessarily endorse all of these changes,
we see these curricular adjustments as evidence that
mathematics educators and textbook developers are, in fact,
rethinking the geometry course. Through our research,
however, we have noticed that even when teachers share this
goal, many find it difficult to move away from the two-column
proof form where students are provided with “givens” and a
statement to prove (Cirillo, 2008; Herbst, 2002). In fact, the
two-column form is so prominent that some research has
shown that when proofs are written in other forms (e.g.,
paragraphs), high school students are unsure of their validity
(McCrone & Martin, 2009).
One reason that the two-column form continues to
dominate geometry proof is likely related to the
“apprenticeship of observation” (Lortie, 1975) where teachers
tend to teach in ways that are similar to how they were taught
as students. We argue that this version of “doing proofs” does
not do enough to involve students in the manifold aspects of
proving that are found in the discipline of mathematics. This is
important because, unless we expand our vision of proving in
school mathematics, we cannot fully realize the
aforementioned goals for mathematical proficiency and of
NCTM’s Reasoning & Proof and Geometry Standards. The
focus of this article is on NCTM’s recommendations for
students to make and investigate conjectures, develop and
14
More Authentic Proof
evaluate mathematical proofs, and select and use various types
of reasoning and methods of proof. Through our examples, we
focus on the recommendation to expand the role of the student
in the work of developing proofs and support this work through
the selection of various proof representations.
In this paper, we first provide some historical context that
sheds light on the prominent position that the two-column
proof form holds in the geometry course. We do this in order to
show how the student’s role in proving has been narrowed over
time. We then present a set of problems that are intended to
expand the role of students by providing them with
opportunities to make and investigate conjectures and develop
and evaluate mathematical proofs. Finally, we discuss various
proof forms as representations used to communicate
mathematics. We conclude with a brief discussion of how these
activities allow students to participate in more authentic proof
practices in geometry.
Historical Context
A second reason that the two-column proof holds such a
prominent position in the geometry course is historical. A
perusal of American geometry textbooks covering the last 150
years reveals that problems where students are expected to
produce a proof have changed considerably. As Herbst (2002)
noted, the custom of using a two-column proof developed
gradually. Before the 20th century, students were expected to
prove statements in which geometric objects are referred to by
their general names (e.g., triangle, angle) rather than by the
labels for specific objects (e.g., ABC, ∠ABC). Students also
had the chance to use deductive reasoning to determine the
claim of their proof. For example, in response to a question
about a generally described figure, they might be expected to
develop a conjecture and prove it. Although less common,
some problems (those problems left for independent
exploration) included finding the conditions or hypotheses (i.e.,
the “givens”) on which basis one could claim a certain
conclusion.
During the 20th century, the student’s role in proving
substantially narrowed. It is interesting that this narrowing
occurred simultaneously with the standardization of the two15
Michelle Cirillo & Patricio G. Herbst
column form for writing proofs. If a goal for our students is
simply to use the “givens” to construct the statements and the
reasons that prove a conclusion, then the two-column form
offers a useful scaffold to assist students in this work. Were we
to increase the share of labor that students do when proving,
however, we might have to think of other types of problems
and forms of representation to support and scaffold their work.
In thinking about expanding the student’s role in the proof
process, two questions are important to consider: What kinds of
problems might be posed to increase students’ share of the
labor? What kinds of support, other than the traditional twocolumn scaffold, could be provided to students to do this work?
We address these two questions in the sections that follow.
Expanding the Role of the Student Through Alternative
Problems
One reason that the two-column form has come under so
much scrutiny in recent times is related to the belief that it is
not an authentic form of mathematics. For example, in A
Mathematician’s Lament, after presenting a two-column proof
(that demonstrates that an angle inscribed in a semicircle where
the vertex is on the circle is a right angle), Lockhart (2009)
stated, “No mathematician works this way. No mathematician
has ever worked this way. This is a complete and utter
misunderstanding of the mathematical enterprise” (pp. 76–77).
A critical piece that has been lost in our modern version of
what doing proofs is like in school mathematics today is related
to conjecturing and setting up the proof. This is important if
you believe, as Lampert (1992) argued, that “conjecturing
about…relationships is at the heart of mathematical practice”
(p. 308). Related to this is the importance of determining the
premises (“givens”) and statements to be proved:
Many people think of geometry in terms of proofs, without
stopping to consider the source of the statements that are to
be proved….Insight can be developed most effectively by
making such conjectures very freely and then testing them
in reference to the postulates and previously proved
theorems. (Meserve & Sobel, 1962, p. 230)
16
More Authentic Proof
Because we believe that students should play a larger role
in the important work of setting up and carefully analyzing
proofs, we present problems that are reminiscent of the
historical problems described above in that they do not simply
provide students with the given hypotheses and ask them to
prove particular statements. Rather, we propose nine different
problems (presented in no particular order) that illustrate how
students may be provided with opportunities to expand their
role in the process of proving.
In the first three problems, students are asked to participate
in setting up the proof by either providing the premises, the
conclusion, and/or the diagram for the proof. In Problem 1, the
student is provided with a conjecture (i.e., the diagonals of a
rectangle are congruent) and a corresponding diagram and
asked to write the “Given” and the “Prove” statements. In
contrast, in Problem 2, the student is provided with the “Given”
and the “Prove” statements but is asked to draw the diagram.
PROBLEM 1: Writing the “Given” and “Prove” from a
conjecture
Suppose you conjectured that the diagonals of a rectangle
are congruent and drew the diagram below.
A
B
D
C
Write the “Given” and the
“Prove” statements that you
would need to use to prove your
conjecture.
PROBLEM 2: Drawing a diagram when provided with
the “Given” and the “Prove”
Draw a diagram that could be used to prove the following:
Given: Parallelogram PQRS where T is the midpoint of PQ
and V is the midpoint of SR .
Prove: ST ≅ QV
17
Michelle Cirillo & Patricio G. Herbst
Finally, in Problem 3, when provided with a particular
theorem, the student is asked to do all three of these tasks (i.e.,
write the “Given,” the “Prove,” and draw the diagram).
PROBLEM 3: Setting up the “Given,” the “Prove,” and
the diagram when provided with the theorem
Determine what you have been given and what you are
being asked to prove in the theorem below. Mark a diagram
that represents the theorem.
More Authentic Proof
Given: ___________
PROBLEM 4: Determining the “Given” from a Flow
Proof
1. Provide the two missing “Given” statements for the
proof shown on the next page.
2. Write a single statement that could replace these two
given statements.
B
___________
1
Prove: CL ≅ MB
J
2
L
Theorem: If the diagonals of a quadrilateral bisect each
other, then the quadrilateral is a parallelogram.
Problem 4 is similar to the first three in that students are
invited to determine the “Given,” but this time they are also
provided with the statement to be proved as well as the proof of
that statement. Students are asked to determine what would
have been “Given” in order to develop the proof that is
provided. They are then asked to condense those two “Givens”
into a single, more concise statement. This exercise asks
students to reflect on two different ways that the line segment
bisector premise might be handled. Problem 4 is similar to the
“fill in” type proofs that we have seen in some textbooks (e.g.,
Larson et al., 2001; Serra, 2008), except that rather than having
students fill in the statements or reasons, they are filling in the
premises.
C
M
?
?
(Given)
(Given)
CJ ≅ MJ
JL ≅ JB
∠1 ≅ ∠2
(Definition of
Midpoint)
(Definition of
Midpoint)
(Intersecting lines
form congruent
vertical angles)
∆CJL ≅ ∆MJB
(SAS ≅ SAS)
CL ≅ MB
(CPCTC)
(Adapted from Serra, 2008, p. 239)
18
19
Michelle Cirillo & Patricio G. Herbst
More Authentic Proof
Next, in Problem 5, students are asked to draw a
conclusion or determine what could be proved when provided
with particular “Given” conditions and a corresponding
diagram. This type of problem can be a useful scaffold in that it
isolates particular geometric ideas such as definitions or
postulates of equality, for example.
PROBLEM 5: Drawing Conclusions from the “Given”
What conclusions can be drawn from the given information?
be drawn (Herbst & Brach, 2006). We view these first six
problems as scaffolds that could eventually allow students to
conjecture and set up a proof on their own.
PROBLEM 6: Drawing an auxiliary line.
What auxiliary line might we draw in to construct this
proof?
Is it possible to construct the proof without considering an
auxiliary line?
A
Given: Kite ABCD with
Given: ABC , DEF
AD ≅ AB and
AB ≅ DE
BC ≅ EF
B
D
DC ≅ BC
A
B
C
D
E
F
Prove: ∠B ≅ ∠D
C
Given: Quad ABCD where FG is bisected by diagonal AC
B
G
A
E
D
F
C
(Adapted from Lewis, 1978, pp. 135 & 68)
In Problem 6, students are asked to determine what
auxiliary line might be drawn in order to construct the proof
that two angles are congruent. This is not a common problem
posed to students because, typically, teachers either construct
the auxiliary lines for their students or a hint is provided in the
textbook that helps students determine where this line should
20
Problem 7 is unique in the sense that the student is asked
what could be proved, but the givens are ambiguous. Leaving
the problem more open-ended affords students opportunities to
write conjectures. It is expected that the student will consider
two different cases corresponding to whether the quadrilateral
is concave or convex. In both cases the student could argue that
the remaining pair of sides are congruent to each other.
PROBLEM 7: Solving a problem that involves writing a
conjecture (i.e., deciding what to prove)
Consider a quadrilateral that has two congruent consecutive
segments and two opposite angles congruent. The angle
determined by the two congruent sides is not one of the
congruent angles. What else could be true about that
quadrilateral? What could you prove in this scenario? What
are the “Given” statements?
21
Michelle Cirillo & Patricio G. Herbst
More Authentic Proof
Finally, in Problems 8 and 9, students have the opportunity
to take part in analyzing proofs. In Problem 8, a paragraph
proof is provided, and students are asked to find the error. In
this proof, the corresponding parts that are proved to be
congruent are two pairs of angles and one pair of sides. The
student author determined that the triangles were congruent by
Angle-Side-Angle (ASA) based on the order that these
corresponding parts were proved congruent, rather than
attending to how these parts are oriented in the triangles. In
Problem 9, students are provided with a proof and asked to
determine what theorem was proved.
PROBLEM 9: Determine the theorem that was proved by
the given proof.
C
Write the theorem that was
proved by the proof below.
1 2
A
Statements
D
B
Reasons
1. ∆ACB with CA ≅ CB
1. Given.
2. Let CD be the bisector of
vertex ∠ACB , D being the
point at which the bisector
intersects AB .
2. Every angle has one and
only one bisector.
3. ∠1 ≅ ∠2
3. A bisector of an angle
divides the angle into two
congruent angles.
4. CD ≅ CD
4. Reflexive property of
congruence.
Explain why his paragraph proof is incorrect and give a
reason why he may have made this error.
5. ∆ACD ≅ ∆BCD
5. Side-Angle-Side ≅ SideAngle-Side
Proof:
6. ∠A ≅ ∠B
6. Corresponding parts of
congruent triangles are
congruent.
PROBLEM 8: Finding the error in a proof.
In the figure to the right,
D
E
AB || ED and
AB ≅ ED
.
F
A
B
Luis uses this information to prove that ∆ABF ≅ ∆DEF .
It is given that AB || ED so ∠DEB ≅ ∠ABE because parallel
lines cut by a transversal form congruent alternate interior
angles. It is also given that AB ≅ ED . And ∠AFB ≅ DFE ∠
because they are vertical angles, and vertical angles are
congruent. So ∆ABF ≅ ∆DEF by ASA.
(Adapted from EDC, 2009, p. 122)
22
(Adapted from Keenan & Dressler, 1990, p. 172)
In this section, we proposed nine problems that illustrate
how teachers could increase their students’ involvement in
proving by having them make reasoned mathematical
conjectures, use conjectures to set up a proof, and evaluate
mathematical proofs by looking for errors and determining
what was proved. In the next section, we address the issue of
23
Michelle Cirillo & Patricio G. Herbst
supporting students in proving by commenting on multiple
proof representations.
Proof Representations that Support Developing and
Writing Proofs
Representation is one of the five Process Standards which
highlight the ways in which students acquire and make use of
content knowledge (NCTM, 2000). In particular, various proof
forms can be considered as representations of geometric
knowledge. Providing students with access to various proof
representations is useful because “different representations
support different ways of thinking about and manipulating
mathematical objects” (NCTM, 2000, p. 360). Although it is
important to encourage students to represent their ideas in ways
that make sense to them, it is also important that they learn
conventional forms of representation to facilitate both their
learning of mathematics and their communication of
mathematical ideas (NCTM, 2000). The purpose of this section
is to highlight four different ways that proofs can be
represented in geometry and discuss how these various
representations have the potential to facilitate proving.
As pointed out by Anderson (1983), successful attempts at
proof generation can be divided into two major episodes—“an
episode in which a student attempts to find a plan for the proof
and an episode in which the student translates that plan into an
actual proof” (p. 193). We refer to these two activities as
developing and writing a proof, respectively. The proof forms
that we highlight include proof tree, two-column proof, flow
proof, and paragraph proof. Descriptions and examples of each
representation can be found in the appendix. In this section we
briefly discuss the ways in which these proof representations
can support students in developing and writing a proof.
Two-Column Proof
A two-column proof lists the numbered statements in the
left column and a reason for each statement in the right column
(Larson et al., 2001). The two-column form requires that
students record the claims that make up their argument (in the
statements column) as well as their justifications for these
claims (in the reasons column). In this sense, the two-column
24
More Authentic Proof
form appears to be a rigid representation. This could be
intimidating to students. However, students can be flexible
when using this representation. For example, they might leave
out a reason that they do not know but still move ahead with
the rest of the proof; the incomplete form reminds them that
they still have something to complete (Weiss, Herbst, & Chen,
2009) However, the consecutively numbered steps of the proof
may lead students to believe that the deductive process is more
linear than it actually is. The deductive process, in general,
hides the struggle and the adventure of doing proofs (Lakatos,
1976).
Paragraph Proof
A paragraph proof describes the logical argument using
sentences. This form is more conversational than the other
proof forms (Larson et al., 2001). Paragraph proofs are more
like ordinary writing and can be less intimidating (EDC, 2009).
For this reason, they look more like an explanation than a
structured mathematical device (EDC, 2009). However the lack
of structure could also be a detriment. In particular, some
teachers have concluded that the paragraph form was not
appropriate for high school students because students tended to
leave out the reasons that justified their statements. As a result,
students would often come to invalid conclusions (Cirillo,
2008). Yet, if a goal is to help students develop mathematical
literacy, this proof form most closely resembles the
representation that a mathematician would use to write up a
proof. Another advantage of this form is that when writing a
proof by contradiction, the paragraph form seems a more
sensible choice than some of the other options (Lewis, 1978).
Proof Trees
The proof tree is an outline for action, where the action is
writing the proof. Anderson (1983) described the proof tree as
follows:
The student must either try to search forward from the
givens trying to find some set of paths that converge
satisfactorily on the statement to be proven, or [s/he]
must try to search backward from the statement to be
25
Michelle Cirillo & Patricio G. Herbst
proven, trying to find some set of dependencies that
lead back to the givens. (p. 194)
In other words, students might begin by asking
themselves, “What would I need to do in order to prove this
statement?” Using a proof tree to think through a proof could
be a useful scaffold to support students in developing a proof.
The proof tree could also be a useful tool to scaffold the work
of determining what the given premises are or what conclusion
can be proved.
Flow Proof
A flow proof uses the same statements and reasons as a
two-column proof, but the logical flow connecting the
statements is indicated by arrows (Larson et al., 2001) and
separated into different “branches.” The flow proof helps
students to brainstorm, working through the most difficult parts
of solving a proof: (1) understanding the working
information—analyzing the given and the diagram—and (2)
knowing what additional information is needed to solve the
proof—analyzing what is being proved (Brandell, 1994). The
flow proof form also allows students to see how different
subarguments can come together to make the overarching
argument (i.e., the “prove” statement). A disadvantage to this
proof form might be that students are not required to supply
reasons that justify their statements in the way that the
“Reasons” column of the two-column proof forces them to do.
For that reason, however, it allows students to focus on
organizing the argument and thus could be particularly useful
toward developing a proof.
The Teacher’s Role in Managing Proof Activity
Through his work, Stylianides (2007) concluded that
teachers must play an active role in managing their students'
proving activity by making judgments on whether certain
arguments qualify as proofs and selecting from a repertoire of
courses of action in designing instructional interventions to
advance students' mathematical resources related to proof. One
way that we can see teachers playing this active role is through
their use and allowance of various representations of proof.
More specifically, acceptance of these various representations
26
More Authentic Proof
of proof allows teachers and their students to focus more on the
argument rather than its form. This can be done through the
side-by-side presentation of a flow proof and a two-column
proof that presented the same argument, as we observed in one
secondary classroom. In this case, the teacher emphasized to
his students that he was not as concerned with the form of the
proof as he was with the presentation of valid reasoning
(Cirillo, 2008).
Lampert (1992) noted:
Classroom discourse in ‘authentic mathematics’ has to
bounce back and forth between being authentic (that is,
meaningful and important) to the immediate participants
and being authentic in its reflection of a wider
mathematical culture. The teacher needs to live in both
worlds in a sense belonging to neither but being an
ambassador from one to the other. (p. 310)
If teachers can be flexible in their thinking about the form
that proofs might take, while at the same time concerning
themselves with the content of the argument, then students may
have more success in learning to prove. Furthermore, the
examples we provide suggest that teachers could also enrich
students’ proving experiences by creating opportunities for
students to do more than producing an argument that links the
givens and the prove. The experiences of students can be more
authentic if they have opportunities to hypothesize the premises
needed to prove a conclusion, to make deductions from a set of
premises so as to find an unanticipated conclusion, and so
forth. This affords students opportunities to learn about proof
as a mathematical process and participate in mathematics in
ways that are truer to the discipline.
Conclusion
Various stakeholders in mathematics education have called
for reasoning and proof to play a more significant role in the
mathematics classroom. There is some evidence that these
recommendations have been taken seriously by mathematics
educators and textbook developers. In this paper, however, we
argue that if we are truly to realize the goals of these standards,
we must pose problems to our students that allow them a
27
Michelle Cirillo & Patricio G. Herbst
greater role in proving. We presented problems that asked
students to write the premises, write the statements to be
proved, as well as construct the diagrams. We suggest that
students should be provided with opportunities to make
reasoned conjectures and evaluate mathematical arguments and
proofs. Last, we suggest that teachers promote and allow
various types of reasoning and methods of proof. We believe
that this is important because adherence to a specific proof
format may elevate focus on form over function. A focus on
form potentially obstructs the creative mix of reasoning habits
and ultimately hinders students' chances of successfully
understanding the mathematical consequences of the
arguments.
As Lakatos (1976) described using the dialectic of proofs
and refutations, mathematicians do not just prove statements
given to them, they also use proof to come up with those
statements. Teaching practices that involve students in solving
problems, conjecturing, writing conditional statements to
prove, and then explaining and verifying their conjectures can
provide students with more authentic opportunities to engage in
mathematics.
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More Authentic Proof
APPENDIX
Proof Representations
THEOREM: If a parallelogram is a rectangle, then
the diagonals are congruent.
Given: Rectangle ABCD
with diagonals
AC and BD .
Prove: AC ≅ BD
A
B
D
C
A proof tree is an outline or plan of action that specifies a set of
geometric rules that allows students to get from the givens of the
problem, through intermediate levels of statements, to the to-be-proven
statement.
(Adapted from Anderson, 1983)
A two-column proof lists the numbered statements in the left column
and a reason for each statement in the right column.
Statements
1. Rectangle ABCD with
Reasons
1. Given
diagonals AC and BD
2. AD ≅ BC
2. Opposite sides of a rectangle
are congruent.
3. DC ≅ DC
4. ∠ADC and ∠BCD are right
angles.
3. Reflexive Postulate
4. All angles of a rectangle are
right angles.
5. ∠ADC ≅ ∠BCD
5. All right angles are congruent.
6. ∆ADC ≅ ∆BCD
6. Side-Angle-Side ≅ Side-Angle-Side
7. Corresponding Parts of Congruent
Triangles are Congruent (CPCTC)
7. AC ≅ BD
A flow proof uses the same statements and reasons as a two-column
proof, but the logical flow connecting the statements is indicated by
arrows. Depending on whether it is the plan or the proof itself, students
may or may not choose to write the reasons beneath the statements.
A paragraph proof describes the logical argument with sentences. It
is more conversational than a two-column proof.
Since ABCD is a rectangle with diagonals AC and, BD then
AD ≅ BC because opposite sides of a rectangle are congruent. By the
reflexive postulate DC ≅ DC . Since all angles in a rectangle are right
angles, then ∠ADC and ∠BCD are right angles. Thus,
∠ADC ≅ ∠BCD . By Side-Angle-Side, ∆ADC ≅ ∆BCD . Thus,
AC ≅ BD .
32
33
The Mathematics Educator
2011/2012 Vol. 21, No. 2, 34–57
A Singapore Case of Lesson Study
Lu Pien Cheng & Lee Peng Yee
In this article, we present a case study of six Singaporean
elementary school teachers working in a Lesson Study team that
prepared them for problem solving instruction. The Lesson Study
process included preparing, observing, and critiquing mathematics
lessons in the context of solving fractions tasks. By conducting Lesson
Study, we anticipated that these teachers would develop greater
insight into students’ mathematics, which would influence their
classroom practices. Through the process of planning, observing and
critiquing and by purposefully listening to students’ explanations, the
teachers began to better understand their students’ learning, which in
turn could help them develop their students’ mathematical
knowledge.
In Singapore, a range of professional development courses
for mathematics teachers are available, from one-session
workshops and whole-day conferences to certification
programs. Though many commercial providers offer short
courses, the main providers of mathematics professional
development courses are the National Institute of Education,
the Ministry of Education’s Curriculum Planning and
Development Division, school- or cluster-organized
customized sessions, and professional bodies (Lim, 2009).
In addition to the wide range of professional development
courses offered to mathematics teachers in Singapore, the
concept of learning communities has been encouraged since
1998. Schools in Singapore are grouped into clusters or
learning communities according to their geographical locations
Lu Pien Cheng is an assistant professor with the Mathematics and
Mathematics Education Academic Group at the National Institute of
Education (NIE), Nanyang Technological University, Singapore. Her
research interests are in teacher education.
Lee Peng Yee is an Associate Professor with the Mathematics and
Mathematics Education Academic Group at NIE, Nanyang Technological
University, Singapore. His research area is real analysis, and he teaches
mathematics courses for mathematics education students.
Singapore Lesson Study
to enhance teachers’ effectiveness as professionals and this
grouping is to encourage teachers to learn and inquire together
in order to become more effective in their teaching practices
(Chua, 2009). When teachers are engaged in learning
communities, they are more likely to innovate their teaching
practice as they continually rethink their practice based on how
students learn (McLaughlin & Talbert, 2006; Vescio, Ross &
Adams, 2008). Lesson Study has traditionally been one of the
professional development processes used to encourage teachers
to work together in teams to become more effective teachers. In
Singapore, mathematics Lesson Study has been adopted by
some schools as a school-based professional development
program or as a cluster-initiated program. At least 60 schools
out of 328 primary and secondary schools in Singapore were
attempting Lesson Study in 2009 (Fang & Lee, 2010). Schools
reported the use of Lesson Study across various subjects in
both the primary and secondary schools. Lesson Study efforts
in Singapore have been reported in research briefs, newsletters,
school reports, action research projects and book chapters. In
this article, we examine teachers’ learning and teaching as a
result of their experience in one Lesson Study cycle.
Lesson Study
Lesson Study (LS) is a form of teacher professional
development that originated in Japan and has been cited as a
key factor in the improvement of their mathematics and science
education (Stigler & Hiebert, 1999). LS is the primary form of
professional development in Japanese elementary schools and
its use has been increasing across North American since 1999
(Lewis, Perry, & Hurd, 2009). Through LS, teachers in Japan
work together to improve their teaching in the context of a
classroom lesson. Perry and Lewis (2009) describe the LS
process as follows:
Lesson Study is a cycle of instruction improvement in
which teachers work together to: formulate goals for
student learning and long-term development;
collaboratively plan a “research lesson” designed to
bring to life these goals; conduct the lesson in a
classroom, with one team member teaching and others
gathering evidence on student learning and
35
Lu Pien Cheng & Lee Peng Yee
development; reflect on and discuss the evidence
gathered during the lesson, using it to improve the
lesson, the unit, and instruction more generally. (Perry
& Lewis, 2009, p. 366)
Japanese teachers followed eight steps to achieve unified
effort in collaborative Lesson Study; (1) define a problem to
guide their work, (2) plan the lesson, (3) teach and observe the
lesson, (4) evaluate and reflect on the lesson, (5) revise the
lesson, (6) teach and observe the revised lesson, (7) evaluate
and reflect a second time, and (8) share the results (Stigler &
Hiebert, 1999). Rock and Wilson (2005) claimed that
completing these steps “requires a group of teachers to
collaborate and share their ideas, opinions, and conclusions
regarding the research lesson. This process requires substantial
time and commitment” (p. 79). They also asserted that the LS
process serves as a catalyst to encourage teachers to become
more reflective practitioners and to use what they learned to
collegially revise and implement future lessons.
Japanese educators have conducted LS at the school,
regional, and national level (Stigler & Hiebert, 1999). At the
national level, LS may be used to explore new ideas about
teaching and curriculum (Murata & Takahashi, 2002). Teachers
in the same subject matter or who have common professional
interests may form regional or cross-district LS groups (Murata
& Takahashi, 2002; Shimizu, 2002). Individual schools may
also form their own LS group to serve their school-based
professional development needs.
Because LS is a locally designed process, the forms may
vary. Across the different variations in LS, four key features
can be identified: investigation, planning, research lesson, and
reflection (Lewis, Perry, & Hurd, 2009). Another distinctive
feature of LS is its constant focus on student learning (Stigler
& Hiebert, 1999). In any LS effort, the shared purpose is
improved instruction (Fernandez & Yoshida, 2004; Lewis,
2002a, 2002b; Lewis & Tsuchida, 1997, 1998; Yoshida, 1999).
Research on Lesson Study
LS has been implemented widely across Asia, but under
several different monikers: in Hong Kong as Learning Study,
in China as Action Education, and in many Asia-Pacific
36
Singapore Lesson Study
Economic Cooperation (APEC) member countries as LS (Fang
& Lee, 2009). Researchers have reported that, in the United
States, LS improved teachers’ instruction and offered them
opportunities to learn (Rock & Wilson, 2005; Lewis, Perry,
Hurd, & O’Connell, 2006). Perry, Lewis and Hurd (2009)
reported a successful adaptation of mathematics LS in a US
school district. They provided an “existence proof” of the
potential effectiveness of LS in North America, noting in their
case that “teachers used Lesson Study to build their knowledge
of mathematics and its teaching, their capacity for joint work,
and the quality of the teaching materials” (Lewis, Perry, &
Hurd, 2009, p. 302).
Research studies have shown that one way LS improves
instruction is through building learning communities.
Lieberman (2009) reported a case study of a middle school
mathematics department, comprised of seven teachers that had
been participating in LS for seven years and found that one
“pathway by which LS results in instructional improvement is
in increasing teachers’ community...Teachers learn that being a
teacher means opening their practice to scrutiny, and thinking
critically about their lesson plans” (p. 97). Research on
mathematics teachers from nine independent school districts in
Texas, who participated in three consecutive lesson studies,
showed that LS activities “promoted interactions among
members within this community of mathematics educators that
offered occasions for teachers to explicitly think about their
views, influences on instructional choices, and possible
changes in practice” (Yarema, 2010, p. 15). In Hong Kong
where LS involved secondary English language teachers, Lee
(2008) reported that LS “creates a culture of peer learning and
learning from actual classroom practice.…[and] provides
opportunities for a free discussion of ideas, with participants
able to challenge others’ and their own way of thinking, and
seeing learning from students’ perspectives” (p. 1124). In a
two-year intervention study for six teachers in one primary
Singaporean school, Fang and Lee (2009,) reported that
“Lesson Study is a powerful tool to bring together knowledge
from diverse communities” (p. 106).
In mathematics LS, the participation of a person more
knowledgeable in mathematics teaching and learning has been
37
Lu Pien Cheng & Lee Peng Yee
reported to enhance the pedagogical content knowledge of the
learning community. Findings from a case study of two
primary school mathematics LS teams highlighted that “the
knowledge contribution from the experienced teachers and
subject specialists from NIE was significant in developing the
pedagogical content knowledge in the community of practice”
(Fang & Lee, 2010, p. 3). Lewis, Perry, and Hurd (2009)
reported similar findings: “Lesson Study groups may need
someone sufficiently experienced in mathematics learning to
ensure such [learning] opportunities arise and are used
productively” (p. 301).
Research findings also showed that LS affects teachers’
instruction in mathematics in particular areas; instructional
vocabulary, differentiated instruction, instruction using
manipulatives, knowledge of mathematical learning stages, and
the establishment of high student expectations (Rock &
Wilson, 2005). Similarly, teams in Singapore schools reported
that LS “holds tremendous potential in uncovering both
students’ and teachers’ conceptions of and approaches to
learning” (Yoong, 2011, p. 4). According to Fang and Lee
(2009), participants in their study “developed a well-blended
form of pedagogical content knowledge that is directly
applicable to improve pupil’s understanding of these topics” (p.
126).
The main challenge of implementing LS in Singapore was
the time needed for its many iterative cycles (Fang & Lee,
2009). Lee (2008) also reported that the “time constraints and
pressure faced by many school teachers” (p. 1124) would
diminish interest in LS. He further added that “although Lesson
Study is time-consuming, it can be highly rewarding. What is
needed is teachers’ commitment to the practice, and the support
of school administrators and the government” (p. 1123).
Research Questions
The main intent of this study was to gain an in-depth
understanding, from the teachers’ perspectives, of the LS
process used in Singapore. This article examines aspects of
teacher professional development through LS and seeks to
build upon the previous investigations of LS in Singapore.
Several questions regarding the use of LS in Singapore are
38
Singapore Lesson Study
important to consider: What are the concerns of teachers in
Singapore when implementing LS? What type of support is
needed for LS to be effective in Singapore? To what extent
might we expect other LS groups in Singapore to conduct LS
similar the one discussed here? In particular, we are interested
in finding what teachers can learn from the LS experience and
if, from the teachers’ perspective, LS can be used effectively in
Singapore for mathematics lessons. This article presents a
school-based professional development initiative using the
Japanese lesson-study model described by Stigler and Hiebert
(1999) based on a university-school partnership. We report the
results of conducting a LS with a group of six elementary
school teachers in Singapore. The following research question
guides our study: What did the teachers learn as a result of their
experience in one LS cycle? In the next section, we outline a
theoretical framework of teachers’ learning to teach along with
our methods of data collection and analysis. Finally we present
the teachers’ perspectives of their experiences in the LS cycle.
Theoretical Framework of Teachers Learning to Teach
The framework used in this study was described by Lin
(2002). According to this framework, teachers learn through
reflection, cognitive conflict, and social interaction. Vygotsky’s
zone of proximal development is used to explain the difference
between what teachers can do alone and what they can do with
assistance from others. Cognitive conflicts caused by observing
students, discussing, critiquing, and negotiating during
interactions among the teachers, their peers, and professional
developers served as catalysts to progress to a higher
developmental level. The teachers in the study were involved
in a school-based professional development where knowledge
is generated from social interaction within a learning
community. Similar to Lin’s (2002), this study was designed to
create opportunities for teachers to develop more specific and
deeper mathematical and pedagogical content understanding
through observation and discussion.
39
Lu Pien Cheng & Lee Peng Yee
Research Design and Data Collection
Spring Hill Elementary1, a neighborhood public school,
serves as the setting of this research study. The mathematics
department head, who had an interest in using LS as a
professional development tool, invited one of the researchers, a
university faculty member, to be an LS consultant in 2008. The
resulting professional development emphasized deepening the
teachers’ pedagogical knowledge on mathematics by focusing
on students’ mathematical thinking. The project started in 2008
and was ongoing during the preparation of this paper. As the
LS coordinators and facilitators, the researchers provided
strategies to team members to consider before the actual
planning of classroom instruction. They listened to the team’s
input and, if needed, shared insights and posed additional
questions to push the team members to think more deeply
about what they observed. The team consisted of four teachers
(Mabel, Zoe, Jade, and Sarah), the department head of
mathematics (Rose) and level head of mathematics (Mary).
Rose and Mary were the team leaders for this mathematics LS
and they were also considered to be the more knowledgeable in
terms of teaching mathematics. They taught upper elementary
grade mathematics (which, in Singapore, includes sixth grade)
and the rest of the teachers taught first and second grade
mathematics during the study. Mabel, Zoe, Jade and Sarah
worked very closely together because they shared common
interests in enhancing their pedagogies in teaching
mathematics. The teachers’ role in the LS was to gain better
understanding of effective pedagogies through the process of
planning, research lesson, and reflection. They volunteered to
participate in the LS project as a team when approached by
their department head. In this manuscript, we report the results
of the first LS cycle conducted by the six teachers.
Lesson Study Procedure
This manuscript focuses on a professional development
using the following eight steps for collaborative LS by Stigler
1
Singapore Lesson Study
and Hiebert (1999). Table 1 describes the schedule for the eight
steps.
1. Define a problem during the first meeting. The team
decided to work on recognizing and naming unit fractions
up to 121 in various contexts involving squares, rectangles
and triangles because they found that fractions are generally
a difficult topic for second grade students.
2. Plan the mathematics lessons. Two full days were used to
plan a lesson for second grade students on reading fractions.
Six elementary school internal faculty members participated
in the discussions. By the end of the day, the teachers
completed the initial lesson plan (Figure 1) and listed some
of the expected student responses.
3. Teach and observe the lesson in the classroom. Mabel
executed the lesson while the rest of the teachers observed.
4. Critique and reflectively discuss the lesson after classroom
observations. Following Mabel’s lesson, the LS group spent
approximately one hour critiquing and reflecting on the
lesson. The participants shared and discussed issues of
pedagogy and students’ learning. Mabel was asked to
reflect on her own teaching of the lesson and the rest of the
participants were asked to articulate their observations after
reflecting generally on their own teaching practices.
5. Revise the lesson. Immediately following the critique, the
participating teachers spent another hour revising the
fraction lesson. The teachers incorporated what was learned
from the critique into revised lesson plan.
6. Teach and observe the revised lesson. Zoe taught the
revised lesson the next day while the rest of the team
members observed.
7. Critique, reflect, and revise. The team met to critique and
reflect on the revised lesson taught by Zoe and the lesson
plan was revised again.
8. Share the results. The head of department arranged to share
results of the LS cycle with the rest of the teachers in the
school.
Pseudonyms were assigned to the school and the participants to ensure
confidentiality.
40
41
Lu Pien Cheng & Lee Peng Yee
Singapore Lesson Study
Table 1
Lesson Study Cycle Schedule at Spring Hill Elementary School
Meeting
Purpose
Data
Duration
1-2
• Discuss the
mathematical
concept
• Discuss how
concept is linked to
other mathematical
topic
• Anticipate
students’
misconceptions on
that topic
• Identify key factors
leading to students’
misconceptions or
learning difficulties
• Plan a mathematics
lesson to address
the problem
Lesson plan
12 hours
(2 full
days)
3
4
5
6
Follow
-up
C
D
G
B
E
F
Diagram 1
1 hours
Critique & revise
lesson plan
Audio
recording
2 hours
Audio
recording and
student work
1 hours
Reflect on LS
experience
in
Prerequisite Knowledge:
Pupils need to be able to use shapes to represent one whole and
fractions with denominators of up to 12 and identify parts and whole
of a given situation.
Development
A
Audio
recording and
student work
Critique & revised
lesson plan
1
12
Introduction to the Problem
Using fraction strips (rectangular, triangular, circular), the teacher
recapitulates reading unit fractions.
Observe lesson
(taught by Mabel)
Observe lesson
(taught by Zoe)
Fractions for Primary Two
Specific Instructional Objectives:
Pupils will be able to recognize and name unit fractions up to
various contexts involving squares, rectangles, and triangles.
Audio
recording
Recording
and
questionnaire
Diagram 2a
Diagram 2b
Key Teacher Questions
T: Look at Diagram 1 and take out the yellow cut outs. Where is the
whole? This is the whole of the figure. Let us take a look at A. What
shape is A? It is a rectangle. What shape is B? It is a square. What
shape is C? It is a square. What shapes make up the figure? 1
rectangle and 2 squares. The teacher points and goes around the
respective parts as the teacher introduces shapes A, B, and C to
students. T: Now with your partner, discuss what fraction of the
whole square is square C.
Expected Student Responses:
1
3
, 12 , No answer
The teacher asks the students to explain how they arrive at
why there is no answer to the problem.
2 hours
Teacher addresses
1
3
1
3
, 12 , and
as an incorrect answer: What is the simple rule
that you must remember for fractions? They must have equal parts.
1
Does this figure have equal parts? Do you think your answer 3 is
correct?
Using the above structure, the teacher continues with the following
problems of similar nature as shown in Diagram 2a and 2b.
Figure 1. The problem solving lesson plan.
42
43
Lu Pien Cheng & Lee Peng Yee
Data Collection and Analyses
A qualitative design was selected to be the most
appropriate research approach for this study because the main
intent was to gain an in-depth understanding, from the
teachers’ perspectives, of the LS process when used in
Singapore. Table 1 illustrates the data collection schedule. The
data collected in this study consisted of audio recordings of the
observed lessons and critiques, questionnaires, a focused group
interview, lesson plans, and student work from the observed
research lessons. The audiotaped meetings captured the
teachers’ conversations about their understanding of students’
thinking, important suggestions teachers provide to revise the
mathematics lesson, and what they learn from the LS cycle.
These discussions provided the platform for teachers to
constantly reflect on their teaching practices. The researchers
administered a questionnaire (Figure 2) at the end of the LS
cycle in order to document the teachers’ experiences. The
focused group interview (Patton, 2002) was conducted with the
teachers at the end of the study to consolidate the teachers’
reactions from the LS cycle (Figure 3). Interviews were
audiotaped and transcribed. The LS team analysed the students’
work during the cycle to provide evidence of student learning.
1. What did you learn when you planned the lesson with
your colleagues?
2. Did your students respond to the lesson the way you
anticipated? (Give specific examples to justify your
observations)
3. What did you learn when you observed the
mathematics lesson?
4. What did you learn when you critiqued the
mathematics lesson with your colleagues?
5. How is the Lesson Study cycle helpful to you as a
teacher?
6. How can Lesson Study be best implemented?
Figure 2. Sample of questionnaire conducted at the end of
Lesson Study cycle.
44
Singapore Lesson Study
1. What did you learn from the Lesson Study cycle?
2. How has participating in the Lesson Study cycle
impacted your instructional practice?
Figure 3. Focused group interview questions.
A qualitative approach was used for the data analysis. An
explanatory effects matrix (Miles & Huberman, 1994) was
used to analyze the data. Data were collected and analyzed
mainly to determine what the teachers learned and what they
considered to be the effects of the LS. First, we entered quotes
from the questionnaire and analyzed the data from Question 1
(see Appendix for a sample of the results). In the last column of
the explanatory effects matrix, we added a general explanation
of our observations of the data entered (Miles & Huberman, p.
148). During the data entry, we picked out chunks of material
and developed codes, such as language, understanding
students, teaching style and, manipulatives, by moving across
each row of the matrix. We repeated the process for the rest of
the questions and once each row was filled in for all the
participants, we had an initial sense of emerging themes and
patterns. Next, we sought confirming evidence by entering
quotes and paraphrases from the interview and analyzing this
data for each question. The students’ work helped us follow
and understand the taped discussions and interviews. Next, we
organized and collapsed some of the codes into a theme. For
example, understanding students and learning styles were
regrouped and renamed learning from students. In the next
section, we summarize our findings for each major theme. Our
numerous data sources (discussions, focus group interviews,
questionnaires, and student work) allowed us to triangulate our
findings and provided greater confidence in our interpretations.
Results and Discussion
In the following paragraphs, we present the teachers’
reports of what they learned during one LS cycle. In all the
meetings, the teachers shared their opinions and observations
openly. Our generalizations are not applicable to all the
elementary schools in Singapore, but our work can be
45
Lu Pien Cheng & Lee Peng Yee
compared to existing theories of how LS cycles work in
neighbourhood public schools in Singapore. We include
representative student responses from Mabel’s lesson to
support this discussion.
Instructional Improvements: Instructional Vocabulary
Instructional vocabulary was one of the key issues brought
up for discussion during the critique. Mathematical language
was mentioned 11 times in the questionnaire by four of the
participants. Jade wrote in her questionnaire that “mathematical
language is important and the teacher must be consistent in
using the language.” Mabel wrote, “I think I have learnt a lot in
being more careful in the terms used and more aware of the
need in reiterating the terms or concepts that I want the pupils
to retain.” During the fourth meeting, Rose and Mary pointed
out that fractions were read in multiple ways by students and
the classroom teacher in Mabel’s lesson. The rest of the
teachers revealed that they used the fraction language based on
their familiarity of it and were unaware of the implications of
the differing language for student learning. The team decided
to list all the different ways that they posed a fraction question.
Table 2 shows the multiple ways that the teachers posed
fraction questions, read fractions, and used fraction
terminology.
The teachers were all amazed with the repertoire of
terminologies they each had for just reading fractions. At this
point, Mary commented that if students are unfamiliar with the
terms their teachers use in teaching mathematics, they are
likely to struggle with their teachers’ language. If this occurs,
the students become more preoccupied with this struggle than
with the thinking processes embedded in the mathematics
lessons. Zoe added that, in addition to this problem, students
may also encounter challenges when they enter the next grade,
in which a new mathematics teacher might use a different term
to describe the same idea. At this point, Jade said with
excitement:
I didn’t think that saying 3 quarters or 3 out of 4 equal
parts matter to the students because I thought they are all
common language that second graders should know.
Shouldn’t we have a vocabulary list clearly listed out for
46
Singapore Lesson Study
each topic so that students know what the mathematical
terms they should know?
Everyone agreed that such a list would be very helpful. Mary
suggested that the vocabulary list should be undertaken as a
project across all grade levels so that students could focus on
learning concepts without being confused by new terminology.
can move beyond learning the basic mathematical knowledge.
Table 2
A Summary of Different Ways That the Teachers Posed a
Fraction Question
Posing fraction
questions
What fraction is shaded?
What fraction of the figure is shaded?
What is the shaded fraction?
Which part is shaded?
Reading
fraction
3 out of 4 equal parts
3 fourths
3 over 4
3 quarters
Numerator
The number above
The number on top
The number above the line
The number that represents what the question asks
The number that is not downstairs
Denominator
The number below
The number below the line
The number downstairs
The number that represents the total number of
parts in one whole
Zoe also brought up the necessity of being precise about
the referents of our mathematical terminology. During her
lesson, Mabel, referring to the figure in Figure 3, asked the
students, “What fraction of the square is shaded?” She asked
this without realizing that the square can be the whole figure
composed of parts A, B, and C; just part B; or just part C.
47
Lu Pien Cheng & Lee Peng Yee
Singapore Lesson Study
Student A treated the smaller square as one whole and was
consistent in using the smaller square as one whole throughout
the entire worksheet (Figure 4). On the basis of observation and
analysis of Student A’s work and responses in class, the
teachers learned that it is important to be specific when
referring to elements of figures, such as the big or the small
square. If this is not done, student errors might occur.
What fraction of the square is shaded?
1 whole of the
square is shaded
2 wholes of the
square is shaded.
Figure 4. Sample of Student A’s written seat work.
The above discussion led the team to realize that having a
repertoire of mathematical terms for the same mathematical
concept may be counterproductive if students are unfamiliar
with some terms. This problem becomes even more significant
when the teacher does not help the students relate the terms
used in different grades. The LS discussion also challenged
teachers to translate their observations into tangible classroom
aides—in this case a mathematical language reference sheet
integrated with appropriate terminology—which otherwise
might not have occurred. The discussion also led the team to be
more aware of the role of accurate and precise language as a
tool to minimize students’ learning difficulties.
Professional Development Through Lesson Study:
Learning From Students
The teachers already knew the importance of listening to
students, but, from LS, they gained a deeper and richer
perspective of what their students perceive about the classroom
instruction. For example, in observing Mabel’s lesson, Jade,
Mabel, Mary, and Zoe said that they were amazed by some of
the interesting, but incorrect, interpretations that students
48
developed for the concept of one whole. In this discussion,
Mary referred to Student B’s response and Jade referred to
Student C’s response (Table 3). The observation and
discussions led the team to realize that a focus on student
thinking can compel them to listen more closely to their
students and that teachers should expect multiple
interpretations of mathematical concepts. In the focused group
interview, the teachers all claimed that the main benefit of
participating in the LS was the opportunity to closely examine
and analyze students’ learning. They believed that by listening
more carefully to their students’ responses, they were able to
identify factors that might give rise to student learning
difficulties. This new understanding led the teachers in the
team to recognize the importance of carefully planning every
mathematics lesson using the knowledge they built through LS
as the basis for making instructional decisions. This result
affirms that LS leads to a focus on student learning. The
mistakes the students made were directly used to improve
classroom instruction (Stigler & Hiebert, 1999) in that the
teachers took note of the understanding students demonstrated
and the solutions they offered to the fraction problem.
Table 3
Description of Students’ Verbal Responses
Student
B
A
C
Student
C
B
Squares B and C have equal parts.
Rectangle A does not have equal parts so it
cannot be a part of one whole ... one whole
makes up of squares B and C. Square B is
½, and Square C is ½.
The parts are not equal so no fraction of the square is
shaded.
We have already shown that the teachers learned that
mathematical language may be a potential barrier to students’
learning of mathematics. In addition, the team became
increasingly aware of other possible causes of student learning
difficulties. During the focused group interview, all of the
teachers agreed that their teaching pedagogies grew
49
Lu Pien Cheng & Lee Peng Yee
exponentially at the end of the first LS cycle as a result of their
collaborative effort to understand student learning.
Professional Development Through Lesson Study:
Learning From Colleagues
During the focused group interview, the teachers expressed
appreciation that LS offered a structured system for
professional development within the school context. The
teachers also shared that their colleagues’ observations of the
lesson contributed directly to the richness of their critiques
because of the variety of student thinking captured. They added
that colleagues may also offer new points of view when
observing the students. For example, during the sixth meeting,
Sarah said “I was hoping Zoe would notice Student D’s
misconception and ask Student D to explain how they got two
sixths during whole class discussion” (Table 4). In another
incident, Rose said, “For figure 2(b) Student E and Student F
actually wrote one half as an answer, but after Zoe said the
correct answer is two fourths, the two students hurriedly
changed their answers to two fourths” (Figure 5). Rose felt that
Student E and Student Fs’ responses provided a great
opportunity to connect reading fractions (Grade 2 topic) to
equivalent fractions (Grade 3 topic). Such peer observations
and critiques offered more feedback to detect and follow up
important teachable moments, which would otherwise go
unnoticed.
What fraction of the figure is shaded?
½ of the square is shaded.
2
4
Figure 5. Sample of Student E’s and Student’s F’ written
seat work
50
Singapore Lesson Study
Table 4
Description of Student D’s Verbal Responses and
Corresponding Written Seat Work
Verbal Responses
Written Seat Work
B
C
There are 6 parts. A horizontal
dotted line should be drawn ...
that is how part C was cut ...
likewise for part B. A vertical
line should be drawn because
that was how part B was cut.
What fraction of the figure is
shaded?
Dotted line drawn by Student D
Teaching can be extremely private because teachers
typically work only with their own students and have little
collegial interaction (Lortie, 1975). Through their participation
in LS, the participants were able to work in teams to challenge
their own and their peers’ use of instructional vocabulary. We
have already discussed how this affirms Rock and Wilson’s
(2005) findings that LS affects instructional vocabulary. This
result also supports Lee’s (2008) findings in that the LS created
the opportunities for the teachers to freely discuss, as part of a
learning community, ideas rooted to classroom practices. In
this case study, the teachers organized and built their repertoire
of instructional vocabulary in order to attend to student
misconceptions. This result affirms that LS offers the teachers
a community in which to open the teachers’ practice to
scrutiny, and together with their community assist one another
51
Lu Pien Cheng & Lee Peng Yee
to think critically about their lessons, resulting in the teachers’
instructional improvement (Lieberman, 2009).
During the focused group interview, the teachers said that
they were planning and critiquing their daily lessons
individually. According to the teachers, observing a live lesson
and critiquing the lesson together with their team members
gave them opportunities to challenge their hypotheses of
students’ thinking during lesson planning and test and verify
those hypotheses during lesson observation and critique.
Furthermore, observing live lessons allowed the teachers to
capture more efficiently how students of different ability
groups react to different segments of the lesson.
Rich mathematical tasks. Fang and Lee (2009) found that
“pedagogical practices in Singapore are dominated by
traditional forms of teacher-centred and teacher as authority
approaches with little attention to the development of more
complex cognitive understanding” (p. 106). Our participants
wanted to focus on their pedagogical practices that developed
complex cognitive understanding. Hence, the teachers in the
team did not want to use the textbook or activities suggested in
the teachers’ guide. Instead, students explored a task which is
usually not found in the Singapore textbook. They did so with
the help of teachers who lead the entire class through the
exploration by using focused questions. The teachers
responded positively to the task on the questionnaire including
Mabel who wrote that the task “enables pupils to apply
mathematical concepts to solve new problems.” Zoe
commented that the task “brings about a refreshing way of
acquiring the necessary knowledge and concept for the
children” and that the unique task required the children to think
rather than just be fed information. In addition, all the teachers
agreed that the tasks enabled them to study how children learn.
For example, Rose said that by analyzing the children’s
common errors, by utilizing strategies to help those children,
and by being able to realize the effectiveness of such strategies,
the teachers gained a better understanding of how children
learn.
Nonetheless, the teachers had several concerns about
implementing the fraction tasks in their own classrooms. The
greatest concern the teachers had was the extensive time
52
Singapore Lesson Study
required for students to fully explore and investigate the
problems. In addition, teachers were not convinced that their
students were ready to explore and investigate the problems on
their own. Due to the aforementioned situations, the teachers in
the team felt that they were likely to have insufficient time to
accomplish the designed target stipulated in the syllabus. Given
the constraint and tight curriculum, the teachers believed in
providing more structure when implementing the fraction task.
Concerns about Lesson Study. Teachers felt similarly
constrained by time when implementing the cycles required of
LS. Zoe wrote, “Time is the greatest constraint. Even if there is
a culture of sharing ... we lack the time to do so.” This supports
Lee’s (2008) finding on time constraints faced by teachers
involved in LS. This also affirms Rock and Wilsons’ (2005)
report that LS process requires substantial time and
commitment. Rose suggested that schools could support the LS
effort by arranging timetables to include more common time
for teachers of the same grade level to meet. Sarah suggested
that LS needed to be one of the school’s top training plans in
order to embed LS as a permanent professional development
tool. Although the LS cycle was time-consuming, the results of
our case study showed that the teachers found the whole
process highly rewarding in terms of enhancing their
instructional effectiveness.
Concluding Remarks
In this study, we examined teachers’ experiences in one LS
cycle. Our findings indicate many positive outcomes: The
teachers are more aware of their instructional vocabulary. In
particular, they are sensitive to the fact that inconsistencies and
inaccurate use of mathematical terms may pose an extra
challenge for the students. The LS cycle impacted the teachers’
ability to think about the effects on children’s learning when
mathematical terms are read in multiple ways. Such
observations were translated directly into useful resources for
the teachers (e.g., a mathematical terminology reference sheet
for students across all grades). The LS cycle also motivated the
teachers to reconstruct students’ thinking and to plan lessons
that address students’ misconceptions based on their models of
student thinking.
53
Lu Pien Cheng & Lee Peng Yee
During the focused group interview, the teachers in the
study said they generally felt that the LS inspired the team to
experiment with new tasks and provided them opportunities to
evaluate and improvise those tasks. We suggest that LS
facilitates the teachers’ research on the efficacy of different
types of tasks and the teaching approach required by those
tasks, and we hypothesize that this enhances the teachers’
pedagogical practices. The teachers were able to explicitly
think about their views of new tasks, new pedagogies, their
influences on instructional choices, and possible changes in
practice, similar to the findings reported by Yarema (2010). By
providing teachers with such a support system, allowing them
to lay the groundwork for rich mathematical learning through
reflective and critical thinking, we suggest that LS can serve as
a platform to helps teachers cultivate good pedagogical habits.
Because LS requires a significant commitment of teachers’
time and energy, the greatest challenge in adopting LS as a
school-based professional development approach is time. In
order to facilitate teachers’ engagement with LS “school
administrators can show their support in terms of timetabling…
and providing staff development time” (Lee, 2008, p.1123), as
suggested by Rose and Sarah.
In this study, when LS was used as a professional
development tool, it improved the teachers’ reflective thinking
about teaching, especially when the teachers worked in a
learning community. They were not only there to teach but also
to plan, observe, and critique common lessons. Such a platform
also provided an avenue of support for teachers to experiment
with different teaching approaches. When professional
development was embedded in these teachers’ practice that
included planning, observing, critiquing, and, collaborating, it
led to their professional growth. The participants in this study
believed that such growth will have lasting impact on their
instructional practices.
Singapore Lesson Study
References
Chua, P. H. (2009). Learning communities: Roles of teachers network and
zone activities. In K. Y. Wong, P. Y. Lee, B. Kaur, P. Y. Foong & S. F.
Ng (Eds.), Series on Mathematics Education: Vol. 2. Mathematics
education: The Singapore journey (pp. 85–103). Singapore: World
Scientific.
Fang Y., & Lee, C. (2009). Lesson study in Mathematics: Three cases from
Singapore. In K. Y. Wong, P. Y. Lee, B. Kaur, P. Y. Foong & S. F. Ng
(Eds.), Series on Mathematics Education: Vol. 2. Mathematics
education: The Singapore journey (pp. 104–129). Singapore: World
Scientific.
Fang, Y., & Lee, C. (2010). Lesson study and instructional improvement in
Singapore (Research Brief No. 10-001). Singapore: National Institution
of Singapore.
Fernandez, C., & Yoshida, M. (2004). Lesson study: A Japanese approach to
improving mathematics teaching and learning. Mahwah, NJ: Erlbaum.
Lee, J. F. K. (2008). A Hong Kong case of lesson study: Benefits and
concerns. Teaching and Teacher Education, 24, 1115–1124.
Lewis, C. (2002a). Does lesson study have a future in the United States?
Nagoya Journal of Education and Human Development 1(1), 1–23.
Retrieved from http://www.lessonresearch.net//nagoyalsrev.pdf
Lewis, C. (2002b). Lesson study: A handbook of teacher-led instructional
change. Philadelphia, PA: Research for Better Schools.
Lewis, C., & Tsuchida, I. (1997). Planned educational change in Japan: The
shift to student-centered elementary science. Journal of Educational
Policy, 12, 313–331.
Lewis, C., & Tsuchida, I. (1998). A lesson is like a swiftly flowing river:
Research lessons and the improvement of Japanese education. American
Educator, 22(4),12–52.
Lewis, C., Perry, R., Hurd, J., & O’Connell, M. P. (2006). Lesson study
comes of age in North America. Phi Delta Kappan, 88, 273–281.
Lewis, C., Perry, R., & Hurd, J. (2009). Improving mathematics instruction
through lesson study: A theoretical model and North American case.
Journal of Mathematics Teacher Education, 12, 285–304.
Lieberman, J. (2009). Reinventing teacher professional norms and identities:
The role of lesson study and learning communities. Professional
Development in Education, 35, 83–99.
Lim, S. K. (2009). Mathematics teacher education: Pre-service and in-service
programmes. In K. Y. Wong, P. Y. Lee, B. Kaur, P. Y. Foong & S. F.
Ng (Eds.), Series on Mathematics Education: Vol. 2. Mathematics
54
55
Lu Pien Cheng & Lee Peng Yee
Singapore Lesson Study
education: The Singapore journey (pp. 104–129). Singapore: World
Scientific.
APPENDIX
Explanatory Effects Matrix: Lessons learned
Lin, P. J. (2002). On enhancing teachers’ knowledge by constructing cases in
classrooms. Journal of Mathematics Teacher Education, 4, 317–349.
Yoong, J. I. (2011, September/October). Let the students tell us how they
learn. SingTeach, 32. Retrieved from
http://singteach.nie.edu.sg/files/SingTeach_Issue32.pdf
Yoshida, M. (1999). Lesson study: A case of a Japanese approach to
improving instruction through school-based teacher development
(Doctoral dissertation.) University of Chicago. Available from ProQuest
Dissertations and Theses database. (UMI No. 9951855)
56
More careful in the Compile a
Teachers have a
use of mathematics mathematics
reservoir of
vocabulary for the terminologies
vocabulary
school
Researcher
explanation
Differentiated
instruction is a
topic of great
interest to this
group of teachers
Need to explore
differentiated
instruction to cater
to the different
learning styles
More aware of
different learning
styles
Zoe
Mathematical language is
Mathematical
important and the teacher must be Language
consistent in using the language.
(questionnaire)
Yarema, C. H. (2010). Mathematics teachers’ views of accountability testing
revealed through lesson study. Mathematics Teacher Education and
Development, 12(1), 3–18.
There is a need to bring in various Learning styles
strategies in a single lesson to
accommodate the various learning
styles of the pupils in order to
better achieve a higher percentage
of pupils grasping the concepts
taught. (questionnaire)
Vescio, V., Ross, D., & Adams, A. (2008). A review of the impact of
professional learning communities on teaching practice and student
learning. Teaching and Teacher Education, 24, 80–91.
Mathematical
Language
Stigler, J., & Hiebert, J. (1999). The teaching gap. New York, NY: The Free
Press.
Mathematical
communication
Shimizu, Y. (2002). Lesson study: What, why, and how? In H. Bass, Z. P.
Usiskin, & G. Burrill (Eds.), Studying classroom teaching as a medium
for professional development: Proceedings of a U.S.—Japan workshop
(pp. 53–57). Washington, DC: National Academy Press.
Establish the need Finding ways to
for mathematical support
communication
mathematical
communication
Rock, T., & Wilson, C. (2005). Improving teaching through lesson study.
Teacher Education Quarterly, 32(1),77–92.
Understanding
students
Perry, R., & Lewis, C. (2009). What is successful adaptation of lesson study
in the US? Journal of Educational Change, 10, 365–391.
Pupils had difficulties in
expressing themselves using the
appropriate language when asked
to explain or justify their answers.
Although they know the reason,
they need to be taught the proper
language so as to be able to
support their answers.
(questionnaire)
Patton, M. Q. (2002). Qualitative research and evaluation methods (3rd ed.).
Thousand Oaks, CA: Sage.
Longer-run
consequences
Murata, A., & Takahashi, A. (2002, October). Vehicle to connect theory,
research, and practice: How teacher thinking changes in district-level
lesson study in Japan. Paper presented at the annual meeting of the
North American Chapter of the International Group for the Psychology
of Mathematics Education, Athens, GA.
Short-run effects
Miles, M. B., & Huberman, A. M. (1994) Qualitative data analysis: An
expanded sourcebook (2nd ed.). Thousand Oaks, CA: Sage.
Code
McLaughlin, M., & Talbert, J. (2006). Building school-based teacher
learning communities: Professional strategies to improve student
achievement. New York, NY: Teachers College Press.
The mathematical
task chosen
allowed teachers
to see the need to
foster
mathematical
communication
Question 1: What did you learn when you observe the
mathematics lesson?
Lortie, D. (1975). School teacher: A sociology study. Chicago, IL: The
University of Chicago Press.
Jade
57
The Mathematics Educator
2011/2012 Vol. 21, No. 2, 58–67
Does 0.999… Really Equal 1?
Anderson Norton and Michael Baldwin
This article confronts the issue of why secondary and postsecondary students resist accepting the equality of 0.999… and 1,
even after they have seen and understood logical arguments for the
equality. In some sense, we might say that the equality holds by
definition of 0.999…, but this definition depends upon accepting
properties of the real number system, especially the Archimedean
property and formal definitions of limits. Students may be justified in
rejecting the equality if they decide to work in another system—
namely the non-standard analysis of hyperreal numbers—but then
they need to understand the consequences of that decision. This
review of arguments and consequences holds implications for how we
introduce real numbers in secondary school mathematics.
Whenever the equality of 0.999… and 1 arises, teachers
can expect a high degree of disbelief from students, and proofs
may do little to abate their skepticism (Sierpinska, 1994). This
equality challenges students’ conceptions of the real line,
limits, and decimal representation, but students have a strong
historical and intuitive basis for their resistance. The purpose of
this paper is to investigate the reasons students reject the
equality and to consider the consequences of this rejection.
With this purpose in mind, we have organized the paper in the
following way: We begin by outlining various arguments
supporting the equality and then review some of the
pedagogical struggles noted in research that explain students’
resistance. Next, we justify students’ intuitive resistance by
presenting a system of hyperreal numbers in which the equality
Equality of 1 and 0.999…
does not necessarily hold. Finally, we consider the implications
of adopting such a system, which forces students to choose
between conflicting properties; we offer as an example the
conflict between the Archimedean property for real numbers
and the existence of infinitesimals.
Arguments for the Equality
There are many arguments that support the equality of
0.999… and 1. Here we present four of these arguments.
Relying on the Decimal Expansion for 1/3
A common argument for the equality goes as follows: If
0.333…= 1/3 then digit-wise multiplication by 3 would imply
that 0.999…= 1. Of course, this argument relies on students’
acceptance of the equality of 0.333… and 1/3. Research has
shown that students generally accept this equality, even while
rejecting the equality of 0.999… and 1 (Fischbein, 2001).
Students might resolve this tension by asserting, “Well, then,
maybe 0.333… doesn’t equal 1/3.”
Subtracting Off the Infinite Sequence
Figure 1 outlines a more formal argument that does not
depend on similar equalities. Yet students might still object.
If x = 0.999 then
10 x = 9.999
9x + x = 9 + x
−x =
9x =
−x
9
Therefore x = 1
Anderson Norton is an Associate Professor in the Department of
Mathematics at Virginia Tech. He teaches math courses for future
secondary school teachers and conducts research on students'
mathematical development.
Michael Baldwin is a PhD candidate in Mathematics Education at
Virginia Tech. His research interests include students' conceptions of
the real number line.
Figure 1. A proof of the equality.
The issue with this argument is whether x can be canceled.
Richman (1999) asserted that skeptics might reject the equality
by claiming that not all numbers can be subtracted from one
another! Moreover, if we consider 0.999… as the limit of the
59
Anderson Norton & Michael Baldwin
sequence 0.9, 0.99, 0.999, … then we see that the
corresponding products, using the standard algorithm for
multiplication of 9 by 0.999… produces a limit of 8.999…,
which leads back to the same central issue that x might not be 1
after all.
Generating a Contradiction
A third argument for the equality works by contradiction:
If 1 and 0.9 are not equal, then we should be able to find a
distinct number in between them (their average), but what
could that number be other than 0.9 itself? Still, students
might argue that some pairs of distinct numbers simply do not
have averages; some students have even argued that there are
numbers between 1 and 0.9—namely, ones represented by a
decimal expansion that begins with an infinite string of 9’s and
then ends in some other number (Ely, 2010). Even when
students cannot find fault with the argument, they still might
not believe the result. After reproducing the proof illustrated in
Figure 1, one frustrated student sought help from Ask Doctor
Math (www.mathforum.com): “The problem I have is that I
can't logically believe this is true, and I don't see an error with
the math, so what am I missing or forgetting to resolve this?”
Defining the Decimal Expansion with Limits
Since Balzano formalized the definition of limits in the
early 19th century, Calculus has been grounded in the formal
definitions of limits that we teach in Precalculus and many
college-level mathematics courses. Figure 2 lays out Balzano’s
formal ε − N definition for limits of sequences.
Formally, a sequence S n converges to a the limit S
Sn = S
lim
n→∞
if for any ε > 0 there exists an N such that
S n − S < ε for n > N
Figure 2. Definition of the limit of a sequence (Weisstein,
2011)
60
Equality of 1 and 0.999…
This definition amounts to a kind of choosing game: Assuming
S is the limit of a sequence, {Sn}, for any positive distance, ε,
you choose, I can find a natural number, N, so that whenever
the sequence goes beyond the Nth term, the distance between
any of those terms and S is less than ε. The definition says that
if the tail of a sequence gets arbitrarily close to a number, then
that number is the limit of the sequence.
We can think about the decimal representation, 0.999…, as
the limit of an infinite series:
9/10 + 99/100 + 999/1000 + …
Thus, we arrive at the following conclusion:
∞
n
9
9
=
lim
∑
k
k = 1.
n→∞
k =1 10
k =1 10
0.9 = ∑
The equality holds because for any real value of ε that you
choose, I can find a natural number N such that 1 is within ε of
n
9
∑ 10
k
whenever n > N.
k =1
This means that we have devised a way to answer the question,
“How close is close enough?” The answer is that we are close
enough to the number 1 if, when given an ε neighborhood
extending some distance about the number 1, we can find a
number N such that the terms at the tail end of the series are
inside that neighborhood. When this happens, we no longer
distinguish between the terms of the series and the number 1.
Why Students Remain Skeptical
There is a historical basis for students’ skepticism in
accepting any of the arguments above, and researchers have
found several underlying reasons for why students reject the
equality—some more logical than others (Ely, 2010; Fischbein,
2001; Oehrtman, 2009; Tall & Schwarzenberger, 1978). For
example, many students conceive of 0.999… dynamically
rather than as a static point; they interpret the decimal
expansion as representing a point that is moving closer and
closer to 1 without ever reaching 1 (Tall & Schwarzenberger,
1978). Starting from 0, the point gets nine-tenths of the way to
1, then another nine-tenths of the remaining distance, and so
61
Anderson Norton & Michael Baldwin
on, but there is always some distance remaining (cf. Zeno’s
paradox). This conception aligns with Aristotle’s idea of
potential infinity and his rejection of an actual infinity: 0.999…
is a process that never ends, producing a decimal expansion
that is only potentially infinite and not actually an infinite
string of 9’s (see Dubinsky, Weller, McDonald, & Brown,
2005, for an excellent discussion of historical struggles with
infinity and related paradoxes). This issue points to a confusion
between numbers and their decimal representations: Would
students be inclined to say that one-third is a process that never
ends simply because its decimal expansion is 0.333…?
Tall and Schwarzenberger (1978) analyzed student reasons
for accepting or rejecting the equality and found that they
generally fit into the following categories:
•
•
Sameness by proximity: The values are the same
because a student might think, “The difference
between them is infinitely small,” or “At infinity it
comes so close to 1 it can be considered the same” (p.
44).
Infinitesimal Difference: The values are different
because a student might think “0.999… is the nearest
you can get to 1 without actually saying it is 1,” or
“The difference between them is infinitely small” (p.
44).
It is interesting that students in the two categories draw
different conclusions using the same argument. Each uses a
non-standard, non-Archimedean distance from the number one
as an argument in their favor. In other words, each believes that
there is some unmeasurable space between the two numbers, as
in the number “next to” one.
In his research involving 120 university students,
Oehrtman (2009) found that mathematical metaphors had
significant impact on claims and justifications. With regard to
the mathematical equality, 0.999… = 1, Oehrtman found that
students were likely to use what he called an “approximation
metaphor.” Student comments referred to “approximations that
could be made as accurate as you wanted” and the
“irrelevance” of “negligible differences” or “infinitely small
errors that don’t matter” (p. 415). Although the students were
62
Equality of 1 and 0.999…
asked to explain why 0.999… = 1, most students disagreed
with the equality. Many students referred to 0.999… as the
number next to 1, or as a number touching 1.
Oehrtman (2009) went on to suggest that there is potential
power in the approximation metaphor because this type of
thinking closely resembles arguments for the formal definition
of a limit. In fact, early definitions of limit by mathematicians
such as D’Alembert included the language of approximation:
“One magnitude is said to be the limit of another magnitude
when the second may approach the first within any magnitude
however small, though the first magnitude may never exceed
the magnitude it approaches” (Burton, 2007, p. 603). Although
the modern definition reflects an attempt to remove temporal
aspects (see Figure 2), such ideas still underlie our conceptions
of limit. And although students might make incorrect
metaphorical statements, these metaphors often provide a
gateway for deeper understanding of corresponding concepts.
The Hyperreals
The argument that 0.999… only approximates 1 has
grounding in formal mathematics. In the 1960’s, a
mathematician, Abraham Robinson, developed nonstandard
analysis (Keisler, 1976). In contrast to standard analysis, which
is what we normally teach in K–16 classrooms, nonstandard
analysis posits the existence of infinitely small numbers
(infinitesimals) and has no need for limits. In fact, until
Balzano formalized the concept of limits, computing
derivatives relied on the use of infinitesimals and related
objects that Newton called “fluxions” (Burton, 2007). These
initially shaky foundations for Calculus prompted the following
whimsical remark from fellow Englishman, Bishop George
Berkeley: “And what are these fluxions? … May we not call
them ghosts of departed quantities?” (p. 525). Robinson’s work
provided a solid foundation for infinitesimals that Newton
lacked, by extending the field of real numbers to include an
uncountably infinite collection of infinitesimals (Keisler,
1976). This foundation (nonstandard analysis) requires that we
treat infinite numbers like real numbers that can be added and
multiplied. Nonstandard analysis provides a sound basis for
treating infinitesimals like real numbers and for rejecting the
63
Anderson Norton & Michael Baldwin
Equality of 1 and 0.999…
equality of 0.999… and 1 (Katz & Katz, 2010). However, we
will see that it also contradicts accepted concepts, such as the
Archimedean property.
Consequences of Accepting Infinitesimals and Rejecting the
Equality
Consider the argument for equality that uses limits outlined
in the previous section. What if you were allowed to choose ε
to be infinitely small? Then the game is up; one cannot
possibly hope to bring the sequence within such an intolerant
tolerance! However, you should beware that, in order to win
(i.e. choosing a value for ε that makes the limit argument fail,
thus proving 0.999… does not equal 1), you have violated the
Archimedean property.
The Archimedean property states that, for any positive real
number, r, we can choose a natural number, N, large enough so
that their product is greater than 1. That means any real number
is farther from 0 than 1/N for some N. To visualize what this
means, consider the illustration in Figure 3. No matter how
close r is to 0, if we zoom in on 0 enough, the two numbers
will be visibly separate. In other words, there is no number
“next to 0,” or infinitely close to 0. If r were allowed to be an
infinitesimal, this would not be the case; r would be less than
1/N for all N, or stay perpetually next to 0, which violates the
Archimedean property. Thus, the only way to maintain this
intuitive property of the real line is to reject infinitesimals, as
we have done in the historical development of the real line
(standard analysis).
Ely (2010) described a case study of a college student who
argued that there is no number next to zero but that there are
numbers infinitely close to 0. This argument aligns with
nonstandard analysis and presents the greatest challenge to the
Archimedean property and other concepts from standard
analysis. In particular, the student argued that one could zoom
in infinitely to separate 0 from an infinitesimal number. Note,
however, that the Archimedean property insists that positive
real numbers be separable from 0 when zooming by a finite
value, specified by the natural number N.
64
Figure 3. The Archimedean property.
Conclusions and Implications
The Archimedean property captures one of the most
intuitive ideas about the real line (Brouwer, 1998). Starting
from that property, we can use the definition of limits to show
that the equality of 0.999… and 1 must hold. Thus, we can see
that the Archimedean property and the formal definition of
limits imply the equality. The only way to reject the equality is
to reject the property or to reject our definition of limits.
As our investigation affirms, “attempts to inculcate the
equality in a teaching environment prior to the introduction of
limits appear to be premature” (Katz & Katz, 2010, p. 3). Yet a
meaningful introduction of limits at the K–12 level is
problematic. Bezuidenhout (2001) discusses difficulties in
introducing limits even at the college level. Similar issues arise
with the introduction of irrational numbers in the K–12
curriculum. It may be useful for students to recognize that
some numbers (such as the length of the diagonal on the unit
square) cannot be written as the ratio of two integers, but state
standards demandmore. Consider the following example from
the Common Core State Standards (National Governors’
Association and Council of Chief State School Officers, 2010):
“In eighth grade, students extend this system once more,
augmenting the rational numbers with the irrational numbers to
65
Anderson Norton & Michael Baldwin
form the real numbers.” Are middle school teachers prepared to
meaningfully address the formation of the real number system,
and is this an important requirement for eighth graders?
In the history of mathematics, the development of calculus
prompted speculation about the existence of infinitesimals,
while motivating the construction of limits (Burton, 2007).
Even the Archimedean property arose from a pre-calculus
concept—namely Archimedes’ method of exhaustion. If
history is any guide for motivating and developing ideas in the
classroom, then Katz and Katz (2010) draw a natural
conclusion in suggesting that we delay the discussion of
irrational numbers and infinite decimal expansions until after
limits are formally addressed. An equally natural conclusion is
that, when we do introduce students to limits, we should take
advantage of intriguing problems, such as the (in)equality
discussed here, so that students will understand why we might
want to reject infinitesimals and, as a consequence, why we
need limits.
Whereas Common Core State Standards ask students to
consider infinite decimal expansions as early as eighth grade,
many students are never asked to seriously consider whether
0.999… really does equal 1. Consideration of this equality
might generate meaningful discussion about students’ intuitive
concepts. Imagine a Precalculus classroom full of students who
have studied decimal expansions but have never studied
irrational numbers except to prove that some numbers (such as
the square root of 2) cannot be expressed as a ratio of two
integers. Some students might have wondered, but none had
formally studied whether this property is related to repeating or
terminating decimal expansions. On the first day of a unit on
limits, the teacher could ask whether 0.999… equals 1. This
paper outlines potential connections students might make
through arguments about this equality—connections between
decimal expansions, the real number system, and limits. It
seems that this kind of discussion does not typically happen
because we ask some questions too early and others not at all.
66
Equality of 1 and 0.999…
References
Bezuidenhout, J. (2001). Limits and continuity: Some conceptions of firstyear students. International Journal of Mathematics Education in
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Brouwer, L. E. J. (1998). The structure of the continuum. In P. Mancosu
(Ed.), From Brouwer to Hilbert (pp. 54-63). Oxford, England: Oxford
University Press.
Burton, D. M. (2007). The history of mathematics: An introduction (6th ed.).
New York, NY: McGraw Hill.
National Governors’ Association and Council of Chief State School Officers.
(2010). Common core state standards for mathematics. Retrieved from
http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
Dubinsky, E., Weller, K., McDonald, M., & Brown, A. (2005). Some
historical issues and paradoxes regarding the concept of infinity: An
APOS-based analysis: Part I, Educational Studies in Mathematics, 58,
335–359.
Ely, R. (2010). Nonstandard student conceptions about infinitesimals.
Journal for Research in Mathematics Education, 41, 117–146.
Fischbein, E. (2001). Infinity: The never-ending struggle. Educational
Studies in Mathematics, 48, 309–329.
Katz, K. U., & Katz, M. G. (2010). When is .999… less than 1? The Montana
Mathematics Enthusiast, 7, 3–30.
Keisler, H. J. (1976). Foundations of infinitesimal calculus. Boston, MA:
Prindle, Weber & Schmidt.
Oehrtman, M. (2009). Collapsing dimensions, physical limitation, and other
student metaphors for limit concepts. Journal for Research in
Mathematics Education, 40, 396–426.
Richman, F. (1999). Is 0.999… = 1? Mathematics Magazine, 72, 396–400.
Sierpinska, A. (1994). Understanding in mathematics (Studies in
Mathematics Education Series: 2). Bristol, PA: Falmer Press.
Tall, D. O., & Schwarzenberger, R. L. E. (1978). Conflicts in the learning of
real numbers and limits. Mathematics Teaching, 82, 44–49.
Weisstein, E. W. (2011). Convergent Sequence. In Wolfram MathWorld.
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http://mathworld.wolfram.com/ConvergentSequence.html
67
REVIEWERS FOR
THE MATHEMATICS EDUCATOR, VOLUME 21, ISSUE 2
The editorial board of The Mathematics Educator would like to take
this opportunity to recognize the time and expertise our many
volunteer reviewers contribute. We have listed below the reviewers
who have helped make the current issue possible through their
invaluable advice for both the editorial board and the contributing
authors. Our work would not be possible without them.
Kimberly Bennekin
Behnaz Rouhani
Georgia Perimeter College
Stephen Bismarck
Keene State College
Laurel Bleich
The Westminster Schools
Margaret Breed
RMIT University
Rachael Brown
Knowles Science Teaching
Foundation
Günhan Çağlayan
Columbus State University
Samuel Cartwright
Fort Valley State University
Alison Castro-Superfine
Danny Martin
Mara Martinez
University of Illinois, Chicago
Lu Pien Cheng
National Inst. of Singapore
Nell Cobb
DePaul University
68
Shawn Broderick
Tonya Brooks
Victor Brunaud-Vega
Amber Candela
Nicholas Cluster
Anna Marie Conner
Zandra DeAraujo
Tonya DeGeorge
Jackie Gammaro
Eric Gold
Erik Jacobson
Jeremy Kilpatrick
Ana Kuzle
Kevin LaForest
David Liss
Kevin Moore
Ronnachai Panapoi
Laura Singletary
Ryan Smith
Denise A. Spangler
Leslie P. Steffe
Dana TeCroney
Kate Thompson
Patty Wagner
The University of Georgia
Kelly Edenfield
Filyet Asli Ersoz
Kennesaw State University
Ryan Fox
Penn. State, Abington
Brian Gleason
University of New Hampshire
Hulya Kilic
Yeditepe University
Hee Jung Kim
Louisiana State University
Yusuf Koc
Indiana University, Northwest
Carmen Latterell
U. of Minnesota, Duluth
Anderson Hassell Norton, III
Virginia Tech
Molade Osibodu
African Leadership Academy
Drew Polly
UNC Charlotte
Ginger Rhodes
UNC Wilmington
Kyle Schultz
James Madison University
Ann Sitomer
Portland Community College
Susan Sexton Stanton
East Carolina University
Brian Lawler
Cal. State U., San Marcos
Erik Tillema
Indiana U.-Purdue U.
Indianapolis
Soo Jin Lee
Montclair State University
Andrew Tyminski
Clemson University
Norene Lowery
Houston Baptist University
Catherine Vistro-Yu
Ateneo de Manila University
Michael McCallum
Georgia Gwinnett College
Bill D. Whitmire
Francis Marion University
If you are interested in becoming a reviewer for The
Mathematics Educator, contact the Editor at tme@uga.edu.
Jill Cochran
Texas State University
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Manuscript Submission Guidelines
The Mathematics Educator (ISSN 1062-9017) is a
biannual publication of the Mathematics Education Student
Association (MESA) at The University of Georgia and is
abstracted in Zentralblatt für Didaktik der Mathematik
(International Reviews on Mathematical Education). The
purpose of the journal is to promote the interchange of ideas
among students, faculty, and alumni of The University of
Georgia, as well as the broader mathematics education
community.
The Mathematics Educator presents a variety of viewpoints
within a broad spectrum of issues related to mathematics
education. Our editorial board strives to provide a forum for a
developing collaboration of mathematics educators at varying
levels of professional experience throughout the field. The
work presented should be well conceptualized; should be
theoretically grounded; and should promote the interchange of
stimulating, exploratory, and innovative ideas among learners,
teachers, and researchers. The Mathematics Educator
encourages the submission of a variety of types of manuscripts
from students and other professionals in mathematics education
including:
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reports of research (including experiments, case studies,
surveys, and historical studies),
descriptions of curriculum projects, or classroom
experiences;
literature reviews;
theoretical analyses;
critiques of general articles, research reports, books, or
software;
commentaries on research methods in mathematics
education;
commentaries on public policies in mathematics
education.
The work must not be previously published except in the case
of:
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translations of articles previously published in other
languages;
abstracts of or entire articles that have been published in
journals or proceedings that may not be easily available.
Guidelines for Manuscript Specifications
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Manuscripts should be typed and double-spaced, 12point Times New Roman font, and a maximum of 25
pages (including references and endnotes). An abstract
(not exceeding 250 words) should be included and
references should be listed at the end of the manuscript.
The manuscript, abstract, references and any pictures,
tables, or figures should conform to the style specified in
the Publication Manual of the American Psychological
Association, 6th Edition.
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An electronic copy is required. The electronic copy
must be in Word format and should be submitted via
an email attachment to tme@uga.edu. Pictures, tables,
and figures should be embedded in the document and
must be compatible with Word 2007 or later.
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The editors of TME use a blind review process.
Therefore, to ensure anonymity during the
reviewing process, no author identification should
appear on the manuscript.
A cover age should be submitted as a separate file
and should include the author’s name, affiliation,
work address, telephone number, fax number, and
email address.
If the manuscript is based on dissertation research, a
funded project, or a paper presented at a
professional meeting, a footnote on the title page
should provide the relevant facts.
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In This Issue,
Guest Editorial… Examining Mathematics Teachers’
Disciplinary Thinking
KYLE T. SCHULTZ & LOUANN LOVIN
Moving Toward More Authentic Proof Practices in
Geometry
MICHELLE CIRILLO & PATRICIO G. HERBST
A Singapore Case of Lesson Study
LU PIEN CHENG & LEE PENG YEE
Does 0.999… Really Equal 1?
ANDERSON NORTON & MICHAEL BALDWIN
The Mathematics Education Student Association is an official
affiliate of the National Council of Teachers of Mathematics.
MESA is an integral part of The University of Georgia’s
mathematics education community and is dedicated to serving
all students. Membership is open to all UGA students, as well
as other members of the mathematics education community.
Visit MESA online at http://www.ugamesa.org