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THE MATHEMATICS EDUCATOR

Volume 21 Number 2

Winter 2011/2012

MATHEMATICS EDUCATION STUDENT ASSOCIATION

This publication is supported by the

College of Education at The University of Georgia.

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

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.

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 All Rights Reserved

THE MATHEMATICS EDUCATOR

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

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 well-

defined 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.


3

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

Kyle T. Schultz & LouAnn Lovin

4

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. What is a bottleneck to learning in this class?

2. How does an expert do these things?

3. How can these tasks be specifically modelled?

4. 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,


5

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 program-

wide 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-of-

discipline 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 2

1

3

5÷ 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


6

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.


7

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.


8

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


9

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.


10

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: Jossey-

Bass.

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.


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.

Michelle Cirillo & Patricio G. Herbst

12

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

More Authentic Proof

13

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.


14

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


15

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 two-


16

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 two-

column 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)


17

B

D C

A

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.

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: QVST ≅


18

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.

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.

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.


19

21

ML

J

BC

Given: ___________

___________

Prove: MBCL ≅

?

MBCL ≅

(CPCTC)

(Adapted from Serra, 2008, p. 239)

(Given) (Given)

MJCJ ≅ JBJL ≅ 21 ∠≅∠

(Definition of

Midpoint)

(Definition of

Midpoint)

(Intersecting lines

form congruent

vertical angles)

MJBCJL ∆≅∆

(SAS ≅ SAS)

?


20

A C

FD

B

E

G

F

E

B

D C

A

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?

Given: ABC , DEF

DEAB ≅

EFBC ≅

Given: Quad ABCD where FG is bisected by diagonal AC

(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


21

BD

A

C

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?

Given: Kite ABCD with

ABAD ≅ and

BCDC ≅

Prove: DB ∠≅∠

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?


22

F

D

A B

E

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 8: Finding the error in a proof.

In the figure to the right,

EDAB || and

EDAB ≅ .

Luis uses this information to prove that DEFABF ∆≅∆ .

Explain why his paragraph proof is incorrect and give a

reason why he may have made this error.

Proof:

It is given that EDAB || so ABEDEB ∠≅∠ because parallel

lines cut by a transversal form congruent alternate interior

angles. It is also given that EDAB ≅ . And DFEAFB ≅∠ ∠

because they are vertical angles, and vertical angles are

congruent. So DEFABF ∆≅∆ by ASA.

(Adapted from EDC, 2009, p. 122)


23

21

DA

C

B

PROBLEM 9: Determine the theorem that was proved by

the given proof.

Write the theorem that was

proved by the proof below.

Statements Reasons

1. ACB∆ with CBCA ≅

2. Let CD be the bisector of

vertex ACB∠ , D being the

point at which the bisector

intersects AB .

3. 21 ∠≅∠

4. CDCD ≅

5. BCDACD ∆≅∆

6. BA ∠≅∠

1. Given.

2. Every angle has one and

only one bisector.

3. A bisector of an angle

divides the angle into two

congruent angles.

4. Reflexive property of

congruence.

5. Side-Angle-Side ≅ Side-

Angle-Side

6. Corresponding parts of

congruent triangles are

congruent.

(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


24

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


25

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


26

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


27

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


28

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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32

B

D C

A

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 two-column proof lists the numbered statements in the left column

and a reason for each statement in the right column.

Statements Reasons

1. Rectangle ABCD with

diagonals AC and BD

2. BCAD ≅

3. DCDC ≅

4. ADC∠ and BCD∠ are right

angles.

5. BCDADC ∠≅∠

6. BCDADC ∆≅∆

7. BDAC ≅

1. Given

2. Opposite sides of a rectangle

are congruent.

3. Reflexive Postulate

4. All angles of a rectangle are

right angles.

5. All right angles are congruent.

6. Side-Angle-Side ≅ Side-Angle-Side

7. Corresponding Parts of Congruent

Triangles are Congruent (CPCTC)

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

BCAD ≅ because opposite sides of a rectangle are congruent. By the

reflexive postulate DCDC ≅ . Since all angles in a rectangle are right

angles, then ADC∠ and BCD∠ are right angles. Thus,

BCDADC ∠≅∠ . By Side-Angle-Side, BCDADC ∆≅∆ . Thus,

BDAC ≅.


33

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 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.


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

35

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


36

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


37

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


38

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


39

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.


40

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 Pseudonyms were assigned to the school and the participants to ensure

confidentiality.


41

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 12

1 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.


42

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 Observe lesson

(taught by Mabel)

Audio

recording and

student work

1 hours

4 Critique & revise

lesson plan

Audio

recording 2 hours

5 Observe lesson

(taught by Zoe)

Audio

recording and

student work

1 hours

6 Critique & revised

lesson plan

Audio

recording 2 hours

Follow

-up

Reflect on LS

experience

Recording

and

questionnaire


43

Fractions for Primary Two

Specific Instructional Objectives:

Pupils will be able to recognize and name unit fractions up to 12

1 in

various contexts involving squares, rectangles, and triangles.

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.

Introduction to the Problem

Using fraction strips (rectangular, triangular, circular), the teacher

recapitulates reading unit fractions.

Development

Diagram 1 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: 2

1

3

1 , , No answer

The teacher asks the students to explain how they arrive at 2

1

3

1 , , and

why there is no answer to the problem.

Teacher addresses 3

1 as an incorrect answer: What is the simple rule

that you must remember for fractions? They must have equal parts.

Does this figure have equal parts? Do you think your answer 3

1 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.

C B

A

G E

D

F


44

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.


45

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


46

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


47

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.


48

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 2 wholes of the

square is shaded 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


49

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

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.

Student

C

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

C B

A


50

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.

4

2

Figure 5. Sample of Student E’s and Student’s F’ written

seat work


51

C

B

Table 4

Description of Student D’s Verbal Responses and

Corresponding Written Seat Work

Verbal Responses Written Seat Work

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


52

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


53

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.


54

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.


55

References

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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

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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.

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concerns. Teaching and Teacher Education, 24, 1115–1124.

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Lewis, C., Perry, R., & Hurd, J. (2009). Improving mathematics instruction

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57

APPENDIX

Explanatory Effects Matrix: Lessons learned

Question 1: What did you learn when you observe the

mathematics lesson?

Researcher

explanation

The mathem

atical

task

chosen

allowed

teach

ers

to see the need to

foster

mathem

atical

communication

Differentiated

instruction is a

topic of great

interest to this

group of teachers

Teach

ers hav

e a

reservoir of

term

inologies

Longer-run

consequen

ces

Finding w

ays to

support

mathem

atical

communication

Need to explore

differentiated

instruction to cater

to the different

learning styles

Compile a

mathem

atics

vocabulary for the

school

Short-run effects

Estab

lish

the need

for m

athem

atical

communication

More aware of

different learning

styles

More careful in the

use of mathem

atics

vocabulary

Code

Understan

ding

studen

ts

Mathem

atical

communication

Mathem

atical

Lan

guag

e

Learning styles

Mathem

atical

Lan

guag

e

Pupils had

difficu

lties in

expressing them

selves using the

appropriate languag

e when

ask

ed

to explain or justify their an

swers.

Although they

know the reason,

they

need to be taught the proper

languag

e so

as to be ab

le to

support their an

swers.

(questionnaire)

There is a need to bring in various

strategies in a single lesso

n to

acco

mmodate the various learning

styles of the pupils in order to

better achieve a higher percentage

of pupils grasp

ing the co

ncepts

taught. (questionnaire)

Mathem

atical lan

guag

e is

importan

t an

d the teacher m

ust be

consisten

t in using the languag

e.

(questionnaire)

Zoe Jade


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 post-

secondary 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

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.

Equality of 1 and 0.999…

59

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 999.0=x then

99

99

999.910

=

−=−

+=+

=

x

xx

xxx

x

Therefore 1=x

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

Anderson Norton & Michael Baldwin

60

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 and0.9 —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 sequencen

S converges to a the limit S

SS nn

=→∞

lim

if for any 0>ε there exists an N such that

ε<− SS n for Nn >

Figure 2. Definition of the limit of a sequence (Weisstein,

2011)


61

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:

0.9 =9

10k= lim

n→∞

9

10k= 1

k=1

n

∑k=1

∞

∑ .

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

9

10kk=1

n

∑ whenever n > N.

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


62

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


63

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


64

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.


65

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


66

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.


67

References

Bezuidenhout, J. (2001). Limits and continuity: Some conceptions of first-

year students. International Journal of Mathematics Education in

Science and Technology, 32, 487–500.

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.

Retrieved from

http://mathworld.wolfram.com/ConvergentSequence.html

68

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

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

Jill Cochran

Texas State University

69

Kelly Edenfield

Filyet Asli Ersoz

Kennesaw State University

Ryan Fox

Penn. State, Abington

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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

Brian Lawler

Cal. State U., San Marcos

Soo Jin Lee

Montclair State University

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Houston Baptist University

Michael McCallum

Georgia Gwinnett College

Anderson Hassell Norton, III

Virginia Tech

Molade Osibodu

African Leadership Academy

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UNC Charlotte

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UNC Wilmington

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Indianapolis

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If you are interested in becoming a reviewer for The

Mathematics Educator, contact the Editor at [email protected].

70

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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:

• 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.

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The work must not be previously published except in the case

of:

• 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

• Manuscripts should be typed and double-spaced, 12-

point 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.

• An electronic copy is required. The electronic copy

must be in Word format and should be submitted via

an email attachment to [email protected]. Pictures, tables, and figures should be embedded in the document and

must be compatible with Word 2007 or later.

• 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.

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

the mathematics educatormath.coe.uga.edu/tme/issues/v21n2/v21n2_online.pdfthe mathematics educator...

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