____ THE

_____ MATHEMATICS ___

________ EDUCATOR _____ Volume 20 Number 2

Winter 2011 MATHEMATICS EDUCATION STUDENT ASSOCIATION

THE UNIVERSITY OF GEORGIA

Editorial Staff

Editor

Catherine Ulrich

Allyson Hallman

Associate Editors

Zandra deAraujo

Erik D. Jacobson

Laura Lowe

Laura Singletary

Patty Wagner

Advisors

Dorothy Y. White

MESA Officers

2003-2004

President

Zandra deAraujo

Vice-President

Tonya DeGeorge

Secretary

Laura Lowe

Treasurer

Anne Marie Marshall

NCTM

Representative

Allyson Hallman

Undergraduate

Representative

Derek Reeves

Hannah Channel

Derek Reeves

A Note from the Editor

Dear TME readers,

This issue closes out the twentieth volume of The Mathematics Educator. In the

first issue of TME, on the inside front cover, the editorial panel laid out their

motivation for starting a new mathematics education journal:

The purpose of our journal is to fill a perceived need for providing students, faculty,

alumni, and the broader mathematics education community a medium for more

localized communication. We can foresee other journals improving as a result of our

publication which provides current and future contributing authors and editors with

additional experience in communicating ideas.

This issue’s table of contents reveals important qualities of the role The

Mathematics Educator currently plays in the mathematics education community 20

years after its inception. Namely, in this issue, a broad array of the mathematics

education community is represented. In fact, the vast majority of our contributions are

now from outside of the UGA community. In this issue only one contributor has any

direct tie to UGA; Sybilla Beckmann is a member of UGA’s Department of

Mathematics, but her editorial is a call for action to all mathematics educators as the

Common Core Standards are rolled out across the nation. In addition, the first authors

on all of the other four articles are all emerging researchers. That is, they are graduate

students or relatively recent graduates of mathematics education programs. Also, this

issue communicates across a range of issues; elementary education (Inoue &

Buczynski; Mueller, Yankelewitz, & Maher) to secondary education (Evans) to post-

secondary mathematics education (Smith & Powell). In addition, some articles report

on research studies focusing on student learning (Mueller, Yankelewitz, & Maher),

others on teacher education (Evans), while one article is about a classroom experience,

not a research project (Smith & Powell). In the end, there is no clear pattern to which

issues of mathematics education that TME articles address. Instead, TME sets itself

apart as a platform for emerging researchers to communicate about current issues in

mathematics education, while also providing them experience in all facets of

publication: submitting articles, reviewing articles, editing for the journal, and

publishing the journal.

As TME moves forward into its next decade of publication, we will strive to

continue this service to the mathematics education community. Many thanks to all the

people who have made The Mathematics Educator possible over the years. In

particular, thank you to all of the contributors, reviewers, and editors who have helped

shape the current issue. We hope you enjoy the results of our efforts!

Sincerely,

Catherine Ulrich

Allyson Hallman

105 Aderhold Hall tme@uga.edu

The University of Georgia www.math.coe.uga.edu/TME/TMEonline.html

Athens, GA 30602-7124

About the cover The cover art shows a student representation exploring combinatorial patterns. To learn more about this, please see the article by

Mueller, Yankelewitz, and Maher.

This publication is supported by the College of Education at The University of Georgia

____________THE ________________

___________MATHEMATICS________

______________EDUCATOR ____________

An Official Publication of

The Mathematics Education Student Association

The University of Georgia

2011 Volume 20 Number 2

Table of Contents

3 Guest Editorial… From the Common Core to a Community of All

Mathematics Teachers SYBILLA BECKMANN

10 You Asked Open-ended Questions, Now What? Understanding the

Nature of Stumbling Blocks in Teaching Inquiry Lessons NORIYUKI INOUE & SANDY BUCZYNSKI

24 Secondary Mathematics Teacher Differences: Teacher Quality and

Preparation in a New York City Alternative Certification Program BRIAN R. EVANS

33 Sense Making as Motivation in Doing Mathematics: Results From Two Studies

MARY MUELLER, DINA YANKELEWITZ, & CAROLYN MAHER

44 An Alternative Method to Gauss-Jordan Elimination: Minimizing

Fraction Arithmetic LUKE SMITH & JOAN POWELL

52 Subscription form 53 Submissions information

© 2011 Mathematics Education Student Association

All Rights Reserved

The Mathematics Educator

2011, Vol. 20, No. 2, 3–9

3

Guest Editorial…

From the Common Core to a Community of All Mathematics Teachers

Sybilla Beckmann

As I write now, early in 2011, over 40 states have

adopted the Common Core State Standards in

Mathematics (National Governors Association Center

for Best Practices and the Council of Chief State

School Officers, 2010). This is a strong, coherent set of

standards that asks students to understand and explain

mathematical ideas and lines of reasoning. These

standards should act as a framework to support vibrant

teaching and learning of mathematics, in which

students actively make sense of mathematics, discuss

their reasoning, explore and develop ideas, solve

problems, and develop fluency with important skills.

Calls for vibrant mathematics teaching and

learning and improved student proficiency in

mathematics have been steady for a number of years

(e.g., National Commission on Excellence in

Education, 1983; National Council of Teachers of

Mathematics [NCTM], 2000; National Commission on

Mathematics and Science Teaching for the 21st

Century [NCMST], 2000; National Mathematics

Advisory Panel [NMAP], 2008). This new set of

standards is one of many initiatives and projects that

answer this call. But as strong as the Common Core

standards are, they cannot improve students’

understanding of mathematics on their own—the

standards will not teach themselves. Teachers are

certainly key to enacting the standards as they are

intended. They need to know the mathematics well,

and they need to how to teach it in engaging and

effective ways.

Thinking about how to improve mathematics

teaching and learning has led me to consider the larger

environment in which this teaching and learning takes

place. This, in turn, has led me to think about several

interconnected groups and communities that are related

to PreK-12 mathematics: the group of all mathematics

teachers from pre-kindergarten through the college

level; the community of mathematics researchers; and

the community of mathematics educators, which

includes teacher educators and mathematics education

researchers. I am a member of all three groups and as I

write I am drawing on my own experience as a

mathematics researcher and member of a mathematics

department; my experience teaching a variety of

college-level mathematics courses, in particular,

courses for prospective teachers; and my one year of

teaching sixth grade mathematics.

In this editorial, I want to make the case for the

group of all mathematics teachers—from early

childhood, to the elementary, middle, and high school

grades, through the college and graduate levels, and

including mathematics educators who teach teachers—

to form a cohesive community that works together with

the common goal of improving mathematics teaching

at all levels. Although all parts of this community work

individually towards improvement, I believe this

community should take collective responsibility for

improving the quality of all mathematics teaching. In

making the case for the community of all mathematics

teachers, I will draw on my knowledge of the

mathematics research community and how it is set up

to work towards excellence in mathematics research. I

will also contrast research in mathematics and teaching

of college-level mathematics, much of which is done

by the same group of people.

What Can Mathematics Research Tell Us About

Mathematics Teaching?

Why is it that at no level of mathematics

teaching—from elementary school, to middle and high

school, to the college level—do we have widespread

excellence in mathematics teaching in this country? Of

course, there are many examples of outstanding

mathematics teaching and mathematics teachers, but,

on the whole, there is cause for concern. At the K-12

level, mathematics teaching in the US is widely

regarded as needing improvement (NCTM, 2000;

NCMST, 2000; NMAP, 2008). Nor does it compare

favorably with teaching in other countries, such as in

Japan, where students perform well on international

Sybilla Beckmann is a Professor of Mathematics at The University

of Georgia. She is the author of a textbook for preservice

elementary school teachers of mathematics, and her research

interests are in the mathematical education of teachers and

arithmetic geometry/algebraic number theory.

Community of Math Educators

4

comparisons of mathematics achievement (Hiebert et

al., 2003). At the college level, strong students who

decide to leave the fields of mathematics, science,

technology, and engineering often cite the quality of

instruction as a key factor in their decision

(Undergraduate Science, 2006; Seymore & Hewitt,

1997).

The state of mathematics teaching in the US is

especially perplexing in light of the strong state of

mathematics research. The mathematics research

community in this country is vibrant and active; it

attracts students and researchers from all over the

world. Unlike in the case of mathematics teaching,

there are no calls for improving the quality of

mathematics research. Yet the vibrant mathematics

research community is also heavily involved in

teaching: For many mathematics researchers, 50% of

their job consists of teaching. It is perhaps surprising

that mathematicians’ excellence in mathematics

research has generally not translated into excellence in

teaching.

Could the differences in the way mathematics

researchers conduct their research and their teaching

shed light on why mathematics research is so full of

vitality yet mathematics teaching seems to be suffering

from malaise? If the conditions that lead to vibrancy in

mathematics research could be adapted for and applied

to mathematics teaching, could this lead to a similar

vibrancy in mathematics teaching? This may seem like

a preposterous question to ask, but there are some good

reasons to believe the answer may be yes.

What Conditions Make Mathematics Research

Strong?

Mathematics research is done within a cohesive

community in which members share their work and

build on each other’s ideas. Five factors strike me as

key in making mathematics research so strong. First,

mathematics researchers share their work, they discuss

it in depth, and they built upon each other’s work.

Second, the quality of a community member’s work is

judged from within the community based on peer

recognition and admiration, not from outside the

community. Third, the mathematics research

community is a meritocracy. Leaders in the community

are active, enthusiastic community members whose

work is admired within the community. Fourth,

mathematics researchers have sufficient time to think

about their research. And fifth, entry into the

mathematics research community requires a high level

of education and accomplishment. These five factors

combine to create a highly motivating professional

environment. Peer admiration within a cohesive,

meritocratic community of accomplished professionals

provides a strong incentive for developing creative new

approaches, sharing good ideas, and building upon

each other’s work. In such a community, mathematics

researchers are motivated to work in an especially

deliberate and focused way.

The mathematics research environment helps

mathematics researchers to do more than just put in

long hours of work; the very nature of the environment

fosters an intense kind of work, a deliberate practice of

honing and refining, of building on what others have

done, and of looking for gaps and weaknesses.

According to research on the development of expertise,

it is precisely such a deliberate practice, done over a

period of ten or more years, which is required for

expertise (Ericsson, Krampe, & Tesch-Roemer, 1993;

popularized by Colvin, 2008).

Motivation research done over several decades and

validated repeatedly in a variety of settings has shown

that systems that fulfill people’s basic psychological

needs for competence, autonomy, and relatedness lead

to more internalized forms of motivation, which lead to

more successful outcomes. In contrast, systems that

people experience as externally controlling by such

means as external evaluations, rewards, or

punishments, lead to less internalized motivation and

less successful outcomes (Deci & Ryan, 2008a, 2008b;

Greene & Lepper, 1974; popularized by Pink, 2009)1.

The mathematics research community fosters

relatedness, namely the feeling of being involved with

and related to others, because mathematicians share

and discuss their work and build on each other’s ideas.

In the process, the mathematics research community

forms opinions about the quality of work, and

community members attain a certain standing based on

the community’s views about the quality of the work.

The mathematics research community fosters

competence because the quality of work matters in the

community. The community fosters autonomy because

finding innovative ideas and lines of reasoning leads to

peer admiration. For mathematicians, the possibility of

raising one’s standing within one’s community through

the judgment of one’s peers—as opposed to through

evaluation from outside of the community—may

contribute to internalized motivation and a strong drive

and desire to excel.

Comparing Mathematics Teaching With

Mathematics Research

Now consider mathematics teaching with respect

to the five factors— collaboration, internal evaluation,

internal leadership, time, and high standards for

entry—which make mathematics research so strong.

Sybilla Beckmann

5

First, mathematics teaching is often an isolated

activity: Most teachers in the US do not share or

discuss their practice in depth and do not have

systematic ways of learning from each other. In the

US, at the K-12 level, there is a “low intensity of

teacher collaboration in most schools” and “the kind of

job-embedded collaborative learning that has been

found to be important in promoting instructional

improvement and student achievement is not a

common feature of professional development across

many schools” (Darling-Hammond, Wei, Andree,

Richardson, & Orphanos, 2009, pp. 23, 25). In

contrast, the tradition of Lesson Study in Japan, in

which groups of teachers collaborate to create, teach,

revise, and publish research lessons, is an important

factor in the high quality of teaching in Japan (Stigler

& Hiebert, 1999; Lewis, 2002). Lesson Study has been

specifically recommended for the new Common Core

State Standards (Lewis, 2010). In the current system, at

both the K-12 and college levels there is not a culture

of looking for and using mathematical and pedagogical

knowledge that has been developed by others to help

improve mathematical understanding and teaching.

Such knowledge does exist (although, of course, we

still need more), but the lack of intellectual vigor

concerning teaching sometimes makes mathematics

teachers at all levels uninterested in considering new

ideas. I have heard prospective elementary teachers

claim that they do not need to know some

mathematical concepts that directly relate to the school

mathematics they will teach because the mathematical

ideas are unfamiliar to them. Similarly, I have heard

mathematicians express disdain for all mathematics

education research.

Second, because mathematics teachers do not

routinely have opportunities to share or discuss

findings about their teaching with any depth, they

cannot develop good judgments about each other’s

teaching. Also, teaching is usually evaluated from

outside of the mathematics teaching community. At the

college level, student evaluations are commonly used

to evaluate teaching; K-12 teachers are evaluated by

administrators, who are typically not active

mathematics teachers and may have limited knowledge

about mathematics teaching. Soon K-12 teachers may

be evaluated and rewarded or rated based on their

students’ performance on standardized tests (Duncan,

2009; Hearing on FY 2011, 2010).

Third, it is not clear who the leaders are in

mathematics teaching. Textbook authors and

professional developers are sources of leadership;

individuals may also think of a favorite teacher to

emulate. But, in the US, we do not seem to have a

detailed and widely shared view of what constitutes

effective teaching (Jacobs & Morita, 2002). In contrast,

there is evidence that Japanese teachers do have a

refined, shared conception of high-quality mathematics

instruction (Corey, Peterson, Lewis, & Bukarau, 2010).

Highly accomplished teachers in Japan become known

through the public research lessons they teach during

Lesson Study, thereby becoming leaders in teaching

(Lewis, 2002).

Fourth, teachers at all levels have many demands

on their time. Most K-12 teachers do not have much

time built into their demanding schedules for

collaborative planning and thinking, for learning from

and with outside experts, and for sharing, testing, and

refining lessons or teaching ideas. According to

Darling-Hammond et al. (2010, p. 20), “few of the

nation’s teachers have access to regular opportunities

for intensive learning” and “mathematics teachers

averaged 8 hours of professional development on how

to teach mathematics and 5 hours on the ‘in-depth

study’ of topics in the subject area during 2003-04.” At

the college level, the requirement to publish, the

prestige of publishing research findings, and the dearth

of opportunities to write in a scholarly way about

teaching leave little or no time for serious, deliberate

work that is devoted to teaching improvement. Fifth,

as I will discuss below, the mathematical preparation

of teachers is often weak.

In sum, at both the college and K-12 levels,

mathematics teachers are often not part of a strong

professional community that promotes sharing and

refining their practices or thinking deeply about

mathematics teaching. Mathematics teaching is simply

not set up to foster the development of internal

motivation and deliberate practice towards expertise in

the same way that mathematics research is.

Entry Into the Mathematics Teaching Community

A strength of the mathematics research community

is the high standard for entry, namely, a PhD in

mathematics, which involves intensive mathematics

coursework, rigorous qualifying exams, and original

research. In contrast, entry into the mathematics

teaching profession is currently varied and often

inadequate. Although some teachers receive excellent

preparation for teaching mathematics, others are

allowed to teach with very little mathematical

preparation. The problem is especially severe for

elementary teachers. The importance of

mathematically knowledgeable teachers has been

emphasized (NMAP, 2008), and there are

recommendations that teachers take sufficient

Community of Math Educators

6

coursework to examine the mathematics they will teach

in detail, with depth, and from the perspective of a

teacher (Conference Board of the Mathematical

Sciences, 2001; Greenberg & Walsh, 2008). But, in

practice, the number and nature of the courses that are

required often deviate considerably from these

recommendations, as documented by Lutzer, Rodi,

Kirkman, and Maxwell (2007, tables SP.5 and SP.6).

Their research does not even take into account

alternative routes to certification, which could require

fewer courses still.

The Common Core Standards in Mathematics are

rigorous and will put a high demand on teachers. Many

of us who teach teachers believe that most will need a

much stronger preparation than they are currently

getting to be ready to teach these new standards. What

constitutes sufficient preparation? Based on my many

years as a teacher of mathematics content courses for

elementary teachers, I know that it takes far more work

than most people realize to be ready to teach

mathematics to children. My students (prospective

teachers) are bright, hard working, and dedicated; I am

not dealing with unmotivated or dull students. Yet it

takes a full three semesters of courses (nine semester

hours total) for us to discuss with adequate depth the

ideas of PreK through grade five mathematics. In

addition, I think that further content-heavy

mathematics methods courses are necessary for such

activities as examining curriculum materials used in

elementary school, for studying how children solve

mathematics problems, which may include examining

videos and written work and interviewing children, and

for learning how to question and lead discussions.

It may seem surprising that so much coursework is

needed in preparation for teaching elementary school.

Yet even the mathematics that the very youngest

children learn is surprisingly deep and intricate, and

much is known about how children learn this

mathematics (see Cross, Woods, & Schweingruber,

2009, for a summary about early childhood

mathematics). Even mathematically well-educated

people who have not specifically studied early

childhood and elementary school mathematics from the

perspective of teaching are unlikely to know it well

enough to teach it. For example, if a child can count to

five, and is shown five blocks in a row, will she

necessarily be able to determine how many blocks

there are, and, if not, what else does she need to know

to do so? Why do we multiply numerators and

denominators to multiply fractions, but we do not add

numerators and denominators when we add fractions?

Where do the formulas for areas and volumes come

from? Where does the formula for the mean come

from? To teach the Common Core State Standards for

Mathematics adequately, teachers will need to have

studied all these details and many more. Children

deserve to be taught by teachers who have studied such

intricacies, inner workings, and subtle points that are

involved in teaching and learning mathematics.

If we think of other important professions, such as

those in medicine, it is hard to imagine that doctors or

nurses would be allowed to enter their professions

without taking required coursework that focuses

specifically on the knowledge these professionals rely

on in their work. Yet in mathematics teaching, there

are not such requirements. Would we be comfortable

with doctors who had not had courses in chemistry and

human anatomy, which underlie their work? Similarly,

we should not be comfortable with teachers who have

not studied the essential ideas they will need in their

work. These essential ideas involve much more than

being able to carry out procedures and solve problems

in elementary mathematics or even in advanced

mathematics.

Governing boards and agencies set a bare

minimum of coursework that is required for

certification, but currently, the requirements do not

ensure adequate coursework in mathematics before

teaching. In my experience, without requirements from

governing boards or agencies, it is difficult to ensure

that individual certification programs will require

prospective teachers to complete a sufficient amount of

suitable mathematics coursework. Without changes, I

believe that many teachers will not be ready to teach

the Common Core State Standards when they begin

teaching.

A Community of All Mathematics Teachers

Working Together Towards Excellence

I have argued that mathematics research is strong

along five factors—collaboration, internal evaluation,

internal leadership, time, and high standards for

entry—and that research in psychology indicates that

these factors may play an important role in the success

of mathematics research. I have also argued that

mathematics teaching has considerable weaknesses in

the five factors. Therefore it seems that mathematics

teaching could benefit from an environment more like

the environment of mathematics research. How could

we create such an environment?

First, suppose that all of us who teach mathematics

could work within collaborative communities in which

we share ideas and learn from each other about

mathematics and about teaching. A number of small

professional learning communities (including Lesson

Sybilla Beckmann

7

Study groups and Teacher Circles) exist. But, such

smaller professional communities should also band

together into a larger community—the community of

all elementary, middle grades, high school, and college

mathematics teachers and teachers of mathematics

teachers. Why should the group of all mathematics

teachers view itself as a cohesive community? One

reason is the interconnectedness of mathematics

teaching. At each grade level, mathematics teaching is

intertwined with the teaching at all other grade levels.

The mathematics teaching that students experience in

elementary school influences what those students learn,

which influences what the students will be ready to

learn in later grades, which in turn influences the

teaching that is possible and appropriate at those higher

grade levels. In addition, the mathematics teaching that

teachers experience in college surely influences their

own understanding of mathematics and their

subsequent mathematics teaching.

Suppose that we—the community of all

mathematics teachers—were to take collective

responsibility for the quality of all mathematics

teaching. The judgments we form about each other

through the process of sharing our insights, ideas, and

successes in improving our students’ performance

could create a viable system of internal evaluation, so

that, as with mathematics research, we might not need

to be evaluated from outside the community. Sharing

our knowledge within a strong professional community

may motivate us to work deliberately, intensively, and

continuously over the long term towards excellence in

mathematics teaching. Given the electronic means of

communication that are now available, we may have

opportunities for sharing our work in teaching that

were not available in the past. There may be new ways

of organizing ourselves and working together that

would help us learn useful information from each other

and join together as we think about specific areas we

are trying to improve in our teaching.

Suppose that leadership within the community of

all mathematics teachers were to evolve internally by

peer recognition and admiration. Some intriguing

research indicates that successful teaching

communities that lead to improvements in student

outcomes depend on certain kinds of leadership (Bryk,

Sebring, Allensworth, Luppescu, & Easton 2010;

Penuel, Riel, Krause, & Frank, 2009). So developing

appropriate leadership could be important to

developing effective communities of teachers.

Suppose that all mathematics teachers had time

built into their schedules to work together and to learn

from each other and from outside experts, as

envisioned by Collins (2010, pp. 27, 36), in which

teaching improvement is driven by “the kind of deep

focus on content knowledge and innovations in

delivery to all students that can only come when

teachers are given opportunities to learn from experts

and one another, and to pursue teaching as a scientific

process in which new approaches are shared, tested,

and continually refined across a far-flung professional

community.”

Suppose that the community of all mathematics

teachers were to set professional standards for entry

into the community. Although the relationships among

teachers’ mathematical knowledge and skill,

instructional quality, and student learning are not yet

well understood and are a matter for research (NMAP,

2008), the mathematics teaching profession has the

responsibility of setting reasonable standards for entry

that fit with the duties of the profession. We should

separate the need for research that can inform and

guide us in making improvements in the preparation of

teachers from making reasonable demands for entry

into the profession, as is common in other professions.

Doctors are required to study chemistry and biology

because a certain level of knowledge of these subjects

is a foundation for practicing medicine. Such a

requirement is reasonable even though there may not

be research evidence linking the study of biology and

chemistry to good practice in medicine. To become a

cosmetologist in Georgia requires at least 1500 credit

hours of coursework in addition to passing written and

practical exams (O.C.G.A., 2011). If we care about

mathematics and about students, and if we want

mathematics teaching to be treated as the serious

profession it is, then we need to insist on higher

minimum required coursework for entry into the

profession even as we continue to study how to

improve teacher preparation. We must also insist that

agencies and boards in positions of responsibility for

teachers honor our standards.

One final thought about the evaluation of work

from within a community by one’s peers: Albert

Einstein supposedly had a sign outside his office

saying, “Not everything that counts can be counted,

and not everything that can be counted counts.”

Although mathematicians do care about numbers of

papers published and numbers of presentations,

standing within the community is not determined

purely by the numbers. An important component is the

judgment of quality by one’s peers. Similarly, although

it makes sense to find out how a teacher’s students do

on common tests compared to other teachers’ students,

Community of Math Educators

8

evaluating teachers purely in this way, without peer

judgment in the mix, is counterproductive.

The judgment of one’s peers is, of course,

subjective and far from perfect, but it might be just

what makes us try harder and look more closely at

what other people have done. The process of looking

closely at what others have done, trying to make

improvements upon prior work, and bringing new

ideas and insights to this work is precisely the process

by which a field advances.

Concluding Remarks

The Common Core State Standards provide all of

us with an opportunity for renewal, revision, and

transition, and an opportunity to address the call for

improving mathematics education that has been loud

and clear over many years. But, in this process, two

things seem certain: the first is that it will be tempting

to make only superficial changes that merely repackage

what we are already doing; the second is that we

cannot create a top-notch system of mathematics

education immediately and in one fell swoop. To create

substantive improvements we must be in a system that

helps us develop an authentic desire to improve and

that promotes our internal motivation to do the hard

work it will take to move towards excellence over the

long term.

I have argued that a key component in the success

of the Common Core State Standards in Mathematics

will be teaching and that in order to improve

mathematics teaching, we must band together to form a

cohesive community of mathematics teachers. Such a

community should set standards for entry into the

community, as do other important professions. I have

argued that the possibility of raising one’s standing

within the community through the judgment of one’s

peers is likely to be a key driver of excellence. A

stronger sense of community among all mathematics

teachers, in which we challenge and support each other

as we work together towards excellence in teaching,

seems like a wonderful and exciting possibility. It is a

vision for enlivening mathematics teaching from within

through peer interactions rather than from without

through external evaluations that will pit us against

each other and sap our motivation. With apologies to

John Lennon, you may say I’m a dreamer, but I hope

I’m not the only one.

Acknowledgements

I would like to thank Kelly Edenfield, Francis

(Skip) Fennell, Christine Franklin, and Tad Watanabe,

for helpful comments on a draft of this paper.

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The Mathematics Educator 2011, Vol. 20, No. 2, 10–23

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You Asked Open-Ended Questions, Now What? Understanding the Nature of Stumbling Blocks in Teaching Inquiry Lessons

Noriyuki Inoue & Sandy Buczynski

Undergraduate preservice teachers face many challenges implementing inquiry pedagogy in mathematics lessons. This study provides a step-by-step case analysis of an undergraduate preservice teacher’s actions and responses while teaching an inquiry lesson during a summer math camp for grade 3-6 students conducted at a university. Stumbling blocks that hindered achievement of the overall goals of the inquiry lesson emerged when the preservice teacher asked open-ended questions and learners gave diverse, unexpected responses. Because no prior thought was given to possible student answers, the preservice teacher was not equipped to give pedagogically meaningful responses to her students. Often, the preservice teacher simply ignored the unanticipated responses, impeding the students’ meaning-making attempts. Based on emergent stumbling blocks observed, this study recommends that teacher educators focus novice teacher preparation in the areas of a) anticipating possibilities in students’ diverse responses, b) giving pedagogically meaningful explanations that bridge mathematical content to students’ thinking, and c) in-depth, structured reflection of teacher performance and teacher response to students’ thinking.

The things we have to learn before we do them, we

learn by doing them.

-Aristotle

Many school reform efforts confirm the importance of inquiry-based learning activities in which students serve as active agents of learning, capable of constructing meaning from information, rather than as passive recipients of content matter (Gephard, 2006; Green & Gredler, 2002; National Council of Teachers of Mathematics, 1989, 2000; National Research Council [NRC], 2000). In inquiry-based mathematics lessons, students are guided to engage in socially and personally meaningful constructions of knowledge as they solve mathematically rich, open-ended problems.

Van de Walle (2004) emphasizes that conjecturing, inventing, and problem solving are at the heart of inquiry-based mathematics instruction. In inquiry-based lessons, students develop, carry out, and reflect

on their own multiple solution strategies to arrive at a correct answer that makes sense to them, rather than following the teacher’s prescribed series of steps to arrive at the correct answer (Davis, Maher, & Noddings, 1990; Foss & Kleinsasser, 1996; Klein, 1997). Inquiry-based lessons can be structured on a continuum from guided inquiry, with more direction from the teacher and a small amount of learner self-direction, to open inquiry, where sole responsibility for problem solving lies with learner.

In order to deliver an effective inquiry lesson, a set of general principles typically suggested in pedagogy textbooks are (a) to start the lesson from a meaningful formulation of a problem or question that is relevant to students’ interests and everyday experiences; (b) to ask open-ended questions, thus providing students with an opportunity to blend new knowledge with their prior knowledge; (c) to guide students to decide what answers are best by giving priority to evidence in responding to their questions; (d) to promote exchanges of different perspectives while encouraging students to formulate explanations from evidence; and (e) to provide opportunities for learners to connect explanations to conceptual understanding (e.g., NRC, 2000; Ormrod, 2003; Parsons, Hinson, & Sardo-Brown, 2000; Woolfolk, 2006). In effective mathematics inquiry lessons, students are supported in reflecting on what they encounter in the environment and relating this thinking to their personal understanding of the world (Clements, 1997).

Noriyuki Inoue is an Associate Professor of Educational

Psychology and Mathematics Education at the University of San

Diego. His recent work focuses on inquiry pedagogy, Japanese

lesson study, action research methodology, and cultural

epistemology and learning.

Sandy Buczynski is an Associate Professor in the Math, Science

and Technology Education Program at the University of San

Diego. She is the co-author of recently published: Story starters

and science notebooking: Developing children’s thinking through

literacy and inquiry. Her research interests include professional

development, inquiry pedagogy, and international education.

Noriyuki Inoue & Sandy Buczynski

11

Preservice Teachers’ Difficulties with Inquiry-

Based Lessons

Though research indicates the importance of students’ construction of knowledge, multiple research reports show that preservice teachers are poor facilitators of knowledge construction in inquiry-based lessons, and that this persists even when they have gone through teacher-training programs focused on inquiry-centered pedagogy (Foss & Kleinsasser, 1996; Tillema & Knol, 1997). These research reports suggest that preservice teachers have a tendency to duplicate traditional methods, rather than implement the inquiry-based pedagogy they experienced in their teacher education programs. Traditional pedagogy is typically associated with a style of direct instruction that is teacher-centered and front-loaded with subject matter. It is characterized by the teacher reviewing previously learned material, stating objectives for the lesson, presenting new content with minimal input from students, and modeling procedures for students to imitate. Throughout the lesson, the teacher periodically checks for learners’ understanding by assessing answers to closed-ended tasks and providing corrective feedback. In contrast, inquiry pedagogy is student-centered and allows time for metacognitive development. In an inquiry classroom, the teacher presents an open-ended problem, and the learners explore solutions by defining a process, gathering data, analyzing the data and the process, and developing an evidence-supported claim or conclusion.

Preservice teachers’ tendency to duplicate traditional methods has been attributed to a lack of a sound understanding of the mathematics content that they teach (Kinach, 2002a; Knuth, 2002), an inability to consider various ways students construct mathematical knowledge during instruction (Inoue, 2009), and a failure to consider how the content, curriculum map, and classroom situations contribute to students’ understanding (Davis & Simmt, 2006). Other researchers report that preservice teachers’ reluctance to stray from traditional methods is originates in the difficulty that they feel in conceptualizing their teaching in terms of the classroom culture and its social dynamics (Cobb, Stephan, McCain, & Gravemeijer, 2001; Cobb & Bausersfeld, 1995). These researchers suggest that preparing a non-traditional lesson requires the teacher to predict the possibilities of classroom interactions and carefully consider ways to shape the social norms of the classroom to facilitate student-centered thinking. However, many preservice teachers go into teaching believing that knowledge transmission and teacher authority take precedence over students

constructing ideas (Klein, 2004). Even if preservice teachers learn about inquiry lessons in their teacher-training programs and believe students’ construction of ideas should take priority, they struggle to consider the multiple issues that are key for a successful inquiry lesson, limiting their ability to implement effective inquiry lessons.

Current literature on inquiry learning focuses on identifying and theorizing various psycho-social factors that contribute to teachers’ ability to deliver an effective mathematics inquiry lesson in the classroom. Some researchers stress the importance of transforming teachers’ perceptions and understanding of inquiry teaching (Bramwell-Rejskind, Halliday, & McBride, 2008; Manconi, Aulls, & Shore, 2008; Stonewater, 2005) and transforming teachers’ beliefs (Robinson & Hall, 2008; Wallace & Kang, 2004). Others examine teachers’ personally constructed pedagogical content knowledge (PCK) that stems from their experiences as learners and their perceptions of students’ needs (Chen & Ennis, 1995). Wang and Lin (2008) add that students’ conception and understanding of inquiry lessons needs attention as well. Though some of these research findings are based on studies of inservice teachers’ struggles with implementing inquiry lessons, we believe that a majority of these research findings are applicable to preservice teachers as well.

Rationale for Study

Though the literature provides many insights on preservice teachers’ struggles in implementing inquiry-based lessons, it is also essential to obtain a practice-linked understanding of why and how preservice teachers, particularly those who are motivated to teach mathematical inquiry lessons, encounter difficulty in authentic teaching contexts. This approach, taken together with the theoretical knowledge the literature provides, strengthens our understanding of how preservice teacher training should be improved. In this paper we address this identified need by presenting the results of one representative case study in which we analyzed a preservice teacher’s inquiry-based lesson taught in a mathematics classroom. Obtaining a practice-linked understanding of the nature of the difficulties that a preservice teacher might encounter in an inquiry lesson provides detailed insight into how specific contexts affect inquiry pedagogy.

Research Questions

In the process of implementing inquiry lessons, many interactions can serve as stumbling blocks to the inquiry process. Here, a stumbling block refers to instances where a teacher poses an open-ended

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question, the students respond (or fail to respond), and the teacher does not know how to reply to students’ comments or questions and, therefore, fails to guide the learning activity towards the rich inquiry investigation initially envisioned. With this in mind, the questions guiding this investigation are: 1) What instances serve as stumbling blocks for preservice teachers motivated to teach inquiry lessons? 2) How do preservice teachers respond to stumbling blocks and how do those responses influence the direction of the lesson?

Any preservice teacher who crafts an inquiry lesson could encounter these types of stumbling blocks. Therefore, the knowledge gained from this study can inform preservice teacher education in two ways: It can increase teacher educators’ awareness of preservice teachers’ issues in implementing inquiry-based lessons, and it can guide teacher educators in helping preservice teachers deliver effective mathematics lessons that are characterized by meaningful construction of knowledge through mathematics inquiry activities.

Methodology

Context

University faculty from the Mathematics Department in the School of Arts and Science were joined by faculty from the Learning and Teaching Department in the School of Leadership and Education Sciences to conduct a summer mathematics camp for third- through sixth-grade students. This cross-campus collaboration provided an opportunity for the faculty to mentor undergraduate preservice teachers to help them bridge mathematical content with pedagogical practice and knowledge of context. Preservice teachers were offered the opportunity to serve as camp instructors in order to gain experience teaching inquiry lessons. We then observed their inquiry-based lessons in order to answer our research questions.

The summer mathematics camp served as an ideal environment for this investigation since the camp’s novice teachers could practice implementing inquiry lessons free from the pressure of supervisor evaluation and externally imposed state standards or tests. The camp also created an environment where learners were given time to be curious and to develop positive attitudes toward learning mathematics. The mission of math camp was two-fold: to provide mathematical enrichment for a diverse group of children and to support the mathematical and pedagogical development of preservice elementary school teachers.

The summer math camp had unique contextual constraints that distinguished it from a traditional

classroom. The mathematics instruction was embedded in a thematic context of Greek mathematicians. Each class included combined grade levels; one for rising second through fourth graders and one for rising fifth through sixth graders. Students from across the city attended the camp. While this context diverged from a typical classroom, some features of the camp provided a context similar to a typical mathematics class: both classes had a heterogeneous mix of diverse students and class periods lasting 90 minutes. We believe that the educational context also highlighted opportunities for a preservice teacher to implement a quality inquiry-based lesson because the students attended voluntarily and were not pressured to perform on tests or homework. Similarly, there was little pressure on the instructors to cover certain material or deliver inquiry lessons with the goal of students’ performing well on tests.

Camp instructors (preservice teachers)

University mathematics professors recruited camp instructors from an undergraduate elementary mathematics methods course. The professors informed preservice teachers enrolled in the course about the opportunity to practice inquiry-based lessons in this summer camp, and a number of them applied to be camp instructors. As part of the recruitment process, the candidates were informally interviewed about their interests and goals in mathematics teaching. Eight preservice teachers were selected to serve as camp instructors based on their enthusiasm and willingness to work in the team. All the eight camp instructors were female undergraduates working towards a bachelor’s degree in liberal studies combined with an elementary teacher credential. During the interview, all of the camp instructors professed an interest in developing their teaching skills and math content knowledge in an activity-rich environment and were willing to commit to one week of camp preparation mentoring and one week of classroom teaching during camp. Each camp instructor’s experience working with children varied, as did their time in the teacher education program. Two were sophomores, three were juniors, and three were seniors. Though they were at different points in the program, half of the camp instructors had completed foundation courses in education, and all had completed the mathematics teaching methods course.

Camp students

Because the camp was advertised in the local newspaper, children from across the city, as well as faculty and university-neighborhood children, applied

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and were accepted on a first-come, first-served basis. The price of the camp for each child was approximately $300. The university helped cover the operational cost of the camp with a $7,490 academic strategic priority fund award which applied to the camp instructors’ salaries, classroom resources, and tuition reduction for eligible children. Each of the two classes enrolled 30 students with approximately ten each of rising second, third, and fourth graders in the lower grade class and approximately 15 each of rising fifth and sixth graders in the upper grade class. Caucasian, Latino, and Asian students made up approximately 60%, 30%, and 10% of the student campers respectively. Because of the age range in each class, a wide range of skill levels was observed.

Undergraduate preservice teacher preparation

For entering the undergraduate elementary teacher education program preservice teachers must be in the university’s Bachelor’s degree program in a content area of their choice. To become a licensed elementary teacher they must then complete the 33-credit hour multiple-subject education program and pass a standardized state content exam. Most of the students who enroll in the undergraduate credential program are liberal studies majors with a concentration in one of the content areas. The credential program includes coursework in educational psychology, content pedagogy (including elementary mathematics teaching methods taught by mathematics faculty with expertise in pedagogy), educational theory, and courses on children’s learning. Through this coursework, the students gain field experience through a series of practicum placements in K-6 schools. In these placements they observe classroom instruction and teach inquiry lessons under the guidance of a school-based and a university-based supervisor.

Camp instructor preparation

Before the math camp program began, the camp instructors attended a required week-long preparation program focused on deepening their mathematics content knowledge, as well as mathematics pedagogy. Camp instructors learned about key developmental and learning theories and were exposed to current research on K-12 learners’ social and personal construction of meaning. They also learned how to develop lesson plans using a wide variety of instructional approaches that focused on helping students construct knowledge. Because exposure to inquiry-based lesson development differed across camp instructors, faculty mentors provided both group and one-on-one instruction and mentorship in this pedagogy.

Four faculty mentors led seminars on the general principles of inquiry lessons. These faculty members also taught in the university’s regular preservice credential program, therefore, the seminars were highly comparable to the university’s regular preservice program. Constructivist philosophy influenced the design of the seminars. Preservice teachers were taught to encourage children to actively make sense of mathematics instead of teachers presenting and modeling procedures for solving problems. In other words, giving authoritarian feedback to students was not a pedagogical strategy valued by the math camp faculty mentors.

The camp instructors were also taught lesson planning based on detailed task analyses of instructional goals called “backward design” (Wiggins & McTighe, 2005). In backward design, the teacher begins with the end in mind, deciding how learners will provide evidence of their understanding, and then designs instructional activities to help students learn what is needed to meet the goals of the lesson. Based on this model, the camp instructors started designing a camp lesson with an initial mathematical idea and then discussed with their peers how students’ understanding of this idea could be gauged. During the process, camp instructors were introduced to strategies including cooperative learning, active learning, mathematical modeling, and the use of graphic organizers. The instruction in these strategies emphasized inquiry pedagogy with the goal of learners developing understanding beyond rote knowledge.

Faculty members also guided camp instructors in how to navigate the disequilibrium between what children want to do versus what they can do. Though the camp preparation lasted only one week, students instructors reviewed the basic principles of learning and designed a camp lesson based on pragmatic instructional fundamentals. They learned what to include in a lesson plan, how to pace activities within the 90-minute class period, how to pose appropriate questions, how to make use of wait time, how to manage the classroom, and what to consider in a thoughtful reflection on teaching experience. Camp instructors’ lessons were required to (a) provide a mathematically rich problem allowing for open-ended inquires of mathematical ideas, (b) ask open-ended questions, (c) encourage students to determine answers with rationales in their responses for problem solving activities, (d) and elicit exchanges of different ideas.

Faculty mentorship

Though the camp instructors had a theoretical understanding of how students make sense of

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mathematical ideas and lesson planning, they did not have any practical experience in planning appropriate inquiry-based mathematics lessons for students. To guide and support them through this process, faculty mentors were available to provide generous assistance and offer advice. Two mathematics professors and two education professors, one specializing in educational psychology and the other in curriculum design and STEM education, served as mentors. During the pre-camp training session, the eight student instructors were paired into four teams of two instructors each. All four mentor professors worked with each team. Mentors met individually with each team to discuss their proposed lesson activities in terms of developmental appropriateness, mathematics content, and pedagogy. At the end of the preparation week, a survey developed by the education faculty members (see Appendix A) was administered to get a sense of teachers’ beliefs and attitudes toward inquiry learning after the camp instructor training program. According to this survey, all eight camp instructors had positive views about inquiry-based lessons and were motivated to deliver effective inquiry-based, activity-rich lessons in the camp.

Each camp instructor team member designed one inquiry lesson for the lower grade class and then one for the upper grade class, or vice versa. These two lessons focused on the same content, but were modified to be appropriate for each age range. For instance, one camp instructor of each team-taught her lesson for the lower grade class during the morning session and the other taught her lesson for this class in the afternoon session. The teams then presented the upper grade lessons in the same manner later in the week. The camp instructors were completely responsible for classroom instruction, however, mentor professors were present in the classroom for additional support as needed. When camp instructors were not teaching, they were observing their peer camp instructors’ lessons. At the end of each day, all camp instructors met as whole group with all of the faculty mentors. These whole group meetings included discussions of how the day went and what aspects of the lesson were effective or ineffective, what revisions could be made, and what concepts should be revisited. Following this schedule, the camp instructors taught each lesson variation during the camp week and had a chance for individual feedback and advice from a faculty member after each presentation of their lesson. A large part of the camp instructors’ experiential learning arose from their reflection on their daily

teaching experience and the mentors’ input about their classroom performance.

Data collection and analysis

During the camp session, the authors observed a total of 12 of the camp instructors’ inquiry lessons: three randomly chosen pairs of lower and upper grade lessons and six other randomly chosen lessons. These observations allowed the researchers to gain a conceptual understanding of the inquiry process that these novice teachers enacted from their lesson plans. Researchers made field notes and video-taped lessons as video cameras and audio-visual staff were available. Camp instructors also completed a post-lesson questionnaire (Appendix B) that probed their perceptions of their effectiveness as math teachers and their success with inquiry pedagogy.

The 12 observed lessons offered a wide range of information about the camp instructors’ approach to inquiry learning in elementary mathematics. The cross-case analyses of observed lessons led us to believe that the camp instructors followed the design principles of an inquiry lesson. However, camp instructors had moments of difficulty that we have termed stumbling blocks. As described earlier, in these moments, the camp teacher responded to an instructional situation in such a way that derailed the inquiry-based goals of the lesson and created moments that significantly undermined the quality of the inquiry lesson.

There were many different kinds of stumbling blocks. When we looked into the cases more closely, we found that the nature of the stumbling blocks was highly contextual and content specific. In each case, stumbling blocks emerged in math camp lessons, one after another, in ways that were nested. By nested we mean that once one stumbling block appeared in the lesson, it had the potential to contribute to the emergence of a subsequent stumbling block. For example, when a preservice teacher was faced with no student response to a question she posed, she resorted to guiding students with leading questions without giving ample opportunity for students to make sense of the concept. In this case, the initial problem that was created from the first stumbling block (i.e. not knowing how to respond when students have no input) served as a foundation for another stumbling block to emerge (i.e., guiding students with leading questions). These in-depth case study analyses revealed that each inquiry component of the lesson depended on other components of that lesson that developed from previous actions and interactions in the lesson. The only way to evaluate the inquiry process and conduct meaningful analyses of the stumbling blocks in inquiry

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pedagogy appeared to be step-by-step deconstructions of the camp instructors’ actions and utterances within each lesson.

We reasoned that presenting a representative individual camp instructor as a case study was the most effective way to capture the nature of stumbling blocks that the camp instructors encountered during the presentation of their lessons. An analysis of one camp instructor’s performance provided the best insight into strengths and weaknesses of the inquiry teaching process. The following section describes the findings of this study based on this methodological framework.

Findings

The case analyses of the observed lessons indicate that all the teachers were not successful in giving mathematically and pedagogically meaningful explanations, ignored creative responses from the students, or switched the nature of instruction to the

direct transmission model where the teacher simply gave answers to students as an authority with little attention to students’ thinking about mathematics. A variety of kinds of stumbling blocks were identified, and each type of stumbling block was found in multiple cases. The type of stumbling blocks depended on the mathematical content covered in the lessons, the students, and the particular dynamics of the interactions in the classroom. We analyzed and identified different stumbling blocks that the camp instructors encountered when teaching a mathematics inquiry-based lesson. Based on the cross-case analyses of the observed lessons, we identified a total of thirteen stumbling blocks, summarized in Table 1.

To exemplify these stumbling blocks, the following section describes an in-depth case study that illustrates the ways a preservice teacher actually encountered the stumbling blocks during the

Table 1 Stumbling Blocks

Location of Stumbling Block

Type of Stumbling Block Teacher Response

1. Problematic problem design

The teacher uses a poor or developmentally inappropriate set up of an inquiry problem or question for the lesson.

Planning the Inquiry Lesson

2. Insufficient time allocation

In the interest of time, the teacher moves on to the next planned activity scheduled in the lesson plan in spite of students’ confusion or teaching opportunities created by students’ responses.

3. Unanticipated student response

The teacher fails to anticipate students’ input and cannot give a pedagogically and mathematically meaningful response to the students.

4. No student response The teacher fails to give a meaningful response to students’ silence or lack of input in reply to the teacher’s question.

5. Disconnection from prior knowledge

The teacher’s response severs connections between the lesson and students’ prior knowledge or their attempt to make sense of the concept using their experiential knowledge.

6. Lack of attention to student input

The teacher ignores the students’ input in reply to the teacher’s open-ended questions.

7. Devaluing of student input

The teacher diminishes student input by rejecting their suggestions and shuts down their attempts at making sense of a problem.

Teacher Response to Student Input

8. Mishandling of diverse responses

The teacher does not know how to effectively manage or give meaningful traffic controls to diverse responses that the students gave for open-ended questions.

9. Leading questions The teacher’s questions directly guide learners to the answer without creating enough opportunities for learners to make sense of the concept.

10. Premature introduction of material

The teacher introduces a new concept or symbol without giving enough opportunity for students to make sense of previous content.

11. Failure to build bridges

The teacher misses important opportunities to effectively connect his or her question to the problem solving activity or the ideas that the students formulated during problem solving.

12. Use of teacher authority

The teacher uses his or her authority to impose the answer or strategy or judge the students’ answer or strategy as right or wrong.

Teacher Delivery of Inquiry Lesson

13. Pre-empting of student discovery

The teacher provides the main conclusion that students were supposed to discover.

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presentation of an upper grade lesson. This descriptive case study (Yin, 2003) illustrates a thick description of some of the issues faced in mathematics inquiry pedagogy. We chose this particular case among all the observed cases since it most vividly informs us of the nature of stumbling blocks that the camp instructors typically encountered in the inquiry lessons observed in the study. We labeled each stumbling block that the preservice teacher encountered at various points of the lesson in reference to the above table.

Case study

Jessica (pseudonym) was a university senior majoring in liberal studies and enrolled in the university’s elementary school teaching credential program. She had successfully completed an educational psychology class and other credential courses, but did not have any formal mathematics teaching experience. In the pre-survey Jessica described effective teaching as, “The teacher needs to prepare the students for what they will learn by getting them interested and providing a foundation to build on (pre-teach if necessary). Also the lesson/activity must be engaging (hands-on, collaborative).” This comment is representative of all the camp instructors’ responses to this survey item; many indicated their belief in the importance of using activities meaningful to children, eliciting children’s interest, and scaffolding students’ personal construction of knowledge that is grounded in their prior experiences. Even though camp instructors’ comments did not encompass the entirety of inquiry-based learning principles, they did show understanding of the key ideas. Jessica, in particular, showed an understanding of her intention and plan to deliver an inquiry lesson in the summer camp.

Jessica’s instruction contained a wide variety of stumbling blocks and can inform us of the nature of the difficulties that preservice teachers can encounter in teaching inquiry lessons. As discussed before, Jessica prepared her lesson plan in the pre-camp session with guidance from the faculty mentors. The objectives of Jessica’s lesson were to help children (a) understand the concept of ratio and (b) understand π as a constant ratio for any circle. As was true with the other camp instructors, Jessica was friendly and made personal contact with children very well. In the upper grade classroom, the children were divided into six groups sitting at different tables.

First, with a picture of trail mix containing M&Ms projected, Jessica asked her students if they liked M&Ms. After hearing a positive response from most of the children, she indicated that she had three brands of trail mix, each containing M&Ms, nuts, and raisins.

She said, “We need to find out which brand we should buy if we would like to get the most M&Ms.” With this problem statement, she has started with an interesting story and formulated an open-ended question relevant to students’ everyday experiences, a key component of an inquiry-based lesson.

Jessica then explained that each brand of trail mix advertised that it contained two scoops of M&Ms. She showed ladles of varying sizes and said that she was not sure which ladle each brand used to measure their two scoops. She asked the children how they might determine which brand of trail mix to purchase to maximize the amount of M&Ms. The children were listening to her attentively and appeared to be thinking about this question. Then one child answered, “What about finding how much sugar that they have on the box?” This child knew that the package should indicate its amount of sugar on the nutrition label and that this would vary directly with the amount of M&Ms. She had not anticipated the direction of this response that overall sugar content would indicate quantity of M&Ms nor had she anticipated this particular question from one of the children. Jessica did not know how to respond. If she simply said no, her inquiry lesson would have lost its real life meaningfulness and stumble just as it was starting. After a pause, Jessica responded, “But the raisins also have sugar, so we cannot compare trail mixes based on sugar [to determine amount of M&Ms in each brand].” With this clever response, the child who asked the question seemed convinced and began to consider other approaches. In responding to the child’s unexpected answer, Jessica managed to avoid using her authority as a teacher to silence the child. This child came up with a creative solution which she responded to by acknowledging his creativity while re-directing his thinking.

While the children were still considering solutions, Jessica suggested using actual trail mixes as stimuli and distributed three plastic bags that contained different brands of trail mix along with a worksheet to each group of students. She asked the children to collaborate at each table to record 1) the number of M&Ms, 2) the number of nuts, 3) and number of raisins. First through her failure to elicit additional solution strategies from students to connect their thinking to the problem and second through her imposing a particular strategy to count M&Ms for problem solving, two stumbling blocks (SB11: Failure to build bridges & SB12: Use of teacher authority) emerged. In other words, this strategy of counting pieces of trail mix did not come from the students, and

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Jessica did not help the children make sense of what they were asked to do. One thing that needs to be pointed out here is that these stumbling blocks emerged even though a) she was trying to follow some aspects of the inquiry teaching principles by having students gather evidence and by giving priority to this evidence in responding to questions (NRC, 2000) and, b) the students were given the opportunity to connect the process of problem solving with the concrete experience of counting M&Ms and comparing their results for the different brands.

After receiving the bags, the children immediately started collaborating and using various strategies to count the pieces in the trail mixes. When they finished, Jessica recorded and displayed their results to discuss with the class (Figure 1).

Brand of Trail Mix

M&Ms Nuts Raisins Total pieces in trail mix

Crunch Beans

66, 67 110, 117, 126, 111

32, 35, 36, 34

220

Sweet & Salty

30 69, 70 11 110

Snick Snack

71 167 91 329

Figure 1. Results of each group’s counting

Note: Each cell displays the counting results from the groups. If the groups’ counting results are the same, the same number was not added to the table to avoid repetition.

It was not until this point in the lesson that we realized that each group’s bag of a particular brand of trail mix had the same number of M&Ms, nuts, and raisins; Jessica had set up the brands to have no counting variations among groups. Of course, the children made minor counting mistakes and this resulted in the variations shown in Figure 1. After the completing the chart, she suggested the correct number of pieces for each brand and totaled them in the table for the children. In other words, she told them the right answers as an authority (SB12: Use of teacher authority).

After the counting activity, she asked the class, “Which one [brand] has more M&Ms compared to the whole package?” When no child responded to the question (SB4: No student response), Jessica pointed out the numbers in the table (Crunch Beans brand: 67 M&Ms in 220 pieces and Snick Snack brand: 71 M&Ms in 329 pieces). Again, she asked the question, “Which brand had more M&Ms compared to the total

number of trail mix pieces in the package?” Jessica attempted to assist children in finding the answers to her close-ended question by directing them to relevant evidence. However, the children remained confused because her explanation did not clarify that she was asking about the proportion of M&Ms compared to the total amount of trail mix. Still, with no child answering, she then asked “67 over 220 or 71 over 329?” (SB9: Leading questions). A child asked, “You mean, if the price of the packages is the same?” Again, Jessica clearly did not anticipate this question (SB3: Unanticipated student response), and responded by saying, “It's a good question,” but went on to say that price was not important here since the price of three packages of one brand could be the same as one package of another brand; she pointed out that price comparison can be very complicated, and is not what they should consider in the problem solving. Jessica’s reply indicated she did not understand the issue the student raised. The student was questioning a tacit assumption that Jessica did not address: if the prices were different then the comparison was invalid (SB5: Disconnect from prior knowledge). Jessica’s response confused this student and many students began interjecting comments about the price and taste of various trail mixes they liked. Finding out which trail mix to buy by holding the price constant is a meaningful assumption for the children since it is what shoppers (and parents) do in choosing a brand of trail mix in everyday life. However, this line of thinking was different from how Jessica’s problem set up: Her assumption was to hold the number of pieces constant, not a very meaningful set-up in everyday life. This discrepancy in interpretation of the problem served as another stumbling block for the inquiry process (SB1: Problematic problem design). She responded, “Let’s not think about the price; let’s explore this problem” (SB7: Devaluing student input). No one resisted this suggestion or asked why they needed to make such an assumption. Jessica began to subordinate children’s meaning construction with her response loaded with authority (SB12: Use of teacher authority).

Then she asked the children if they knew what a ratio was, and wrote on the board, “Ratio = The relationship between quantities” (SB10: Premature introduction of material). At this point, the children began to be increasingly quiet. Without explaining why she was introducing the concept of ratio here, Jessica indicated that the children could use calculators to divide numbers and compare the ratios. She asked, “Does anyone know why divide?” No one answered the question, but some of the children were silently

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taking a note of the formula on their notebooks. Here, Jessica did not follow up on her question or support learners’ meaning-making in the lesson (SB4: No student response and SB5: Disconnect from prior knowledge). This created another stumbling block that seems to have led the children to gradually shut down their personal construction of meaning in the following lesson segments. She pointed out to the children, “67/220 is like dividing a pizza. If you divide, you can compare, right?” (SB5: Disconnect from prior knowledge and SB9: Leading questions)

Then she told the children that she could use her calculator to execute the division. She input two numbers to the calculator and wrote on the board

(67/220) 54304.≈ .1 Here, she did not take the time to

explain what she was doing or why she was doing this procedure prematurely assuming that students knew the meaning of the mathematical symbols (SB5: Disconnect from prior knowledge and SB10: Premature introduction of material). The children became increasingly confused because she failed to give an effective explanation of the meaningfulness of the assumption (i.e., holding the number of pieces constant for ratio comparison), why they needed to divide the numbers, or why this relates to the action of dividing a pizza. It was apparent that her failure to provide a meaningful rationale for the new mathematical idea created another stumbling block in the lesson. It was no surprise that, at this point, most of the children became quiet and watched her actions rather than participating in a discussion about the mathematics, which created the atmosphere of a traditional mathematics classroom.

Once she introduced the concept of ratio she began moving forward in her lesson plan despite student confusion (SB 2: Insufficient time allocation). Jessica hesitated for a while, but, in the interest of completing her planned lesson, she proceeded to introduce a new concept, a constant. She wrote Constant on the board and said, “Let's think about a constant. What is the quantity that does not change?” (SB10: Premature introduction of material) Jessica did not relate this question about constants to the M&M problem (SB11: Failure to build bridges). However, many of the children suddenly became engaged and raised their hands. They actively responded, “speed of light”, “fingers”, “gravity.” The sudden increase in participation was possibly because they knew that they could answer the question and project personal meaning in the activity. Jessica smiled and nodded in response to each of the children’s responses, but did

not give any other reply (SB8: Mishandling diverse responses).

Next, Jessica suddenly introduced a story where Romans killed Archimedes while he was thinking about a circle he drew on beach sands. Without providing a rationale for the story (SB11: Failure to build bridges), she asked the children, “So… what's so interesting about circles? Again, given this opportunity to participate in the open-ended question-and-answer activity, children presented many different responses: “The circle is round,” and “It looks like a hole.” She responded with nodding and smiling (SB8: Mishandling diverse responses). Then one child answered, “Unlimited angle, no end, no beginning.” Jessica looked a little puzzled by this child’s answer. Clearly, she did not expect this response, and did not know how to react (SB3: Unanticipated student response). She missed this educational opportunity to discuss central angles of a circle (SB5: Disconnect from prior knowledge). The child’s creative, yet unexpected, response served as another stumbling block in the lesson. She told the child that it was an interesting idea, and asked other children for more ideas.

Without clarifying the link with her original activity (SB11: Failure to build bridges), she then distributed objects that contained circles (cans, lids, duct tape, etc.). She explained what circumference and diameter of a circle are, and asked each group to measure them on their object and explore the relationship. However, she did not give any instruction about how to measure these accurately (SB1: Problematic problem design). After some exploration, most of the groups could reason that the ratio is a little more than three (though some children already knew the ratio to be 3.14 from school mathematics classes). Then Jessica wrote on the board the symbol π and mentioned that this ratio is a constant for any circle, pointing out that the relationship values calculated were almost the same across the various groups (SB13: Pre-empting student discovery).

At this point, a child raised her hand to say that their ratio was “a little less than three” in her group. Jessica approached this group and, while other groups were waiting, realized they were saying that three times diameter is a little less than circumference and therefore the ratio of circumference to diameter is a little less than three. She needed to spend a significant amount of the time for this particular group since she could not understand the logic underlying their claim, which served as another stumbling block in the lessons (SB3: Unanticipated student response). Essentially the

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group claimed that Cd <3 implied that (C/d) < 3.

Jessica missed another opportunity to compare these different ideas, address the misconception, and help the children construct their own meaning of ratio. Without sharing this group’s information with the entire class, Jessica began to explain the next activity (SB2: Insufficient time allocation). Then she was stopped when one child suddenly asked how to find π of an oval. Again, Jessica did not expect this question and did not know how to respond. This moment served as another stumbling block in the lesson (SB3: Unanticipated student response). She simply told the child that she would think about it. She continued her lesson by drawing examples of inscribed and circumscribed triangles on the board and asked if the circumference of the circle or perimeter of the triangles are bigger. At this point, she did not have enough time to finish the lesson (SB 2: Insufficient time allocation). She thanked all the children for their interesting ideas, and the lesson ended within the class’s allotted time.

As shown in the above case study, the ways the stumbling blocks appeared and influenced the lessons and student instructor’s responses were complex. No simple descriptions seem to be able to capture the complexity and dynamics of these factors that were intertwined with each other. Please note that the camp instructors chose to teach in the inquiry-centered math camp and were highly motivated to teach inquiry lessons. This makes this group an unlikely representative of preservice teachers across the nation. However, we do not believe that this weakens our argument, but strengthens it. It highlights one of the key points of the study: Even if pre-teachers are motivated to teach inquiry lessons, they encounter stumbling blocks and often do not know how to overcome them. The following examples illustrate stumbling blocks that the camp instructors encountered in other lessons observed in the study.

1. When students explored how to expand a 2’ x 3’ picture of a face into a larger dimension without distorting the image, one of the students responded 5’ x 6’ since 2 + 3 = 5 and 3 + 3 = 6. The teacher simply responded, “That’s not quite right,” in front of all the students without explaining or examining this. (SB6: Lack of Attention to Student Input and SB7: Devaluing Student Input)

2. In a lesson to understand the effects of volume and mass on water displacement, the teacher started the lesson by asking very broad questions: “Have you ever heard the term volume?”, “How does it relate to math?”, and “What are some ways to find

volume?” When the students gave diverse responses to these broad questions, the camp teacher merely listened to them without giving any sort of meaningful response and proceeded on to the planned water displacement activity. This lack of validation or even acknowledgement of students’ responses quieted their eagerness to answer, as after that the students spoke up much less in the lesson. (SB8: Mishandling Diverse Responses and SB6: Lack of Attention to Student Input)

3. In a lesson to find the height of a pyramid, the camp teacher asked students to measure their shadows and compare the measures with their actual heights. Though the children made the connection between this activity and finding the height of a pyramid, the teacher did not make the relationship between the two activities explicit. (SB5: Disconnect from prior knowledge)

Most of the camp instructors expressed a sense of failure in their first round of teaching, but did not clearly know the reasons why their inquiry lessons did not work very well. After the first lesson, each teacher had an individual meeting with the faculty mentor(s) who observed the lesson to go over their reflection and receive suggestions for improvement. This opportunity to discuss the lesson presentation with faculty helped the camp instructors determine reasons why these stumbling blocks were encountered and provided them expert advice on what to do next.

Discussion

Teachers are known to possess personalized understanding of how to support children’s construction of knowledge based on their own learning experiences (Chen & Ennis, 1995; Segall, 2004). The weaknesses observed in the student instructors’ inquiry lessons could be seen to stem from a novice teacher’s immature understanding of how elementary school students think and understand mathematics. Ample literature supports this point that teachers’ beliefs and understanding about how children learn significantly impact the effectiveness of teaching (e.g., Kinach, 2002b; Warfied, Wood, & Lehman, 2005).

The camp instructors knew that helping children make connections between abstract concepts and the material representations of those concepts is critical to a meaningful inquiry-based lesson. However, there were many instances in our observations where such connections were not made, in spite of the camp instructors knowledge about inquiry lesson principles

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and their willingness to deliver an inquiry lesson. Through the study, we found that asking preservice teachers to take command of a full classroom of students with only a crash preparation course was insufficient to circumvent problems. In a way, it is not a surprise that the camp instructors had so much room for improvement in lesson planning, time management, and transitioning from one activity to the next. However, we believe that Jessica’s lesson was informative for any teacher, novice or veteran, who attempts to deliver an inquiry lesson because it addresses several important stumbling blocks that anyone could encounter in delivering an inquiry lesson, yet not recognize in the moment of teaching.

There are several lessons that could be learned from this case study. First, Jessica often asked open-ended questions to help the children personally construct meaning. But it was often the case that she had not considered possible learner responses, and caught off guard, did not know how to respond, as she admitted in the post-teaching interview. As a result, in the face of these rich educational opportunities provided by the diverse learner responses, Jessica was unprepared, inflexible, and unable to make use her knowledge of content and children’s thinking to improvise within the parameters of the lesson. Consequently, the children who contributed these thought-provoking answers did not receive any meaningful response or validation of their ideas from the teacher or their peers. We observed many such instances throughout the math camp and suspect this is true in many classrooms where teachers attempt to deliver an inquiry lesson.

To be fair, in preparing these camp instructors to teach a math concept, anticipating student response was not a part of their lesson plan template. Inoue (2011) points out that this should be a key component of lesson design; in his cross-cultural lesson study research, “failure to anticipate students’ diverse responses” was one of the reasons that an inquiry lesson was ineffective and deviated from the initially planned instructional goal. One solution for this could be adding a section to the lesson plan template that includes thinking through possible student answers to questions, as Japanese educators are known to include in their lesson plans (Fernandez & Yoshida, 2004). This would help them prepare for conceptual conversations in the classroom and help them evaluate the lesson by envisioning students’ diverse perspectives.

Furthermore, we discovered that preservice teachers were more focused on their own performance

than on their students’ performance in these classroom experiences. Berliner (1994) reports similar finding from his research that inexperienced teachers had a tendency to focus on teachers’ actions, rather than students’ actions, and lacked the ability to identify meaningful sub-activities integrated within a larger lesson. The camp instructors’ tendency to focus on their own performance could work in favor of their learning from their pedagogical mistakes, strengthening their content delivery, and gaining insight into the inquiry process. However, it could do little to help them learn to consider each action in the lesson in reference to the goals of student learning, a necessity for successful inquiry instruction.

Passing over or ignoring a response that has merit in the conceptual framework of the lesson could not only lower the learner’s inclination to participate in the lesson but also invalidate or devalue the learner’s prior knowledge (Cooper, 1994, 1998). If a learner’s response falls outside the realm of anticipated responses, yet presents an opportunity to expose the class to a different facet of understanding of a mathematical concept, the teacher needs to first validate a student’s legitimate response and then use that response to navigate to the instructional goals. In addition, the teacher needs to have the flexibility and confidence in content matter to build a consensus among the students and achieve the instructional objective within the allocated time. More importantly, expecting diverse and high-quality responses and knowing how to incorporate a learner’s prior knowledge in the lesson is an important skill for teachers to have when delivering an inquiry lesson. What holds the key seems to be a deep understanding of how children think and might react to concepts. For example, Lubienski (2007) points out that lower socioeconomic students are more likely to use “solid common sense” (p.54) than they are to use a sophisticated mathematical concept.

Researchers point out that mathematical word problems are often written without accurately reflecting the experiences described in the problems (Greer, 1997; Inoue, 2005; Verschaffel, Greer, & De Corte, 2000). In Jessica’s episode, this was evident when students tried to compare prices of brands of trail mix rather than use ratios of ingredients. Being aware that students often become engaged with the real-world aspects of math problems rather than focusing on the mathematical concept intended by the problem would help teachers anticipate students’ responses and prepare a means to incorporate that line of thinking into the math concept being studied.

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Second, in spite of the camp instructors’ attempt to explain abstract concepts in ways that were grounded in students’ prior experiences and concrete models, they often failed to explain mathematical concepts in pedagogically meaningful ways. Jessica simply did not know how to explain the concept of ratios effectively, and gave irrelevant, misleading, and disconnected instructions to the students. This issue could be due to her lack of deep knowledge on how to deliberately unpack a mathematical concept, as seen with her treatment of ratio, constant, and π. As Jessica herself pointed out in the post-teaching interview, it is important for teachers to possess multi-layered content knowledge in order to utilize it in a pedagogical meaningful way to make connections among concepts and to a learner’s prior knowledge. In this sense, teaching inquiry lessons effectively requires going beyond merely following the principles of inquiry lessons to developing a deep pedagogical understanding of how one could construct each mathematical concept in a meaningful way (Ball, Hill, & Bass, 2005; Ma, 1999; Mapolelo, 1998).

This point is emphasized by Shulman (1987) who claimed “the key to distinguishing the knowledge base of teaching lies at the intersection of content and pedagogy” (p. 15). Shulman (1986, 1987) described the construct of pedagogical content knowledge as an integrated synthesis of subject matter content knowledge and pedagogical knowledge that is specific to education and separates teachers from mere content experts. For instance, when the student says a circle is interesting because it has “unlimited angles, no end, no beginning,” the teacher needs to be able to confidently respond to this mathematical statement in a pedagogical meaningful way without losing the scope of the planned lesson. For instance, the teacher could have responded by first pointing out that central angles are an important concept to explore in understanding a circle. She could have instructed the students on drawing central angles and challenged them to draw a

180° central angle, the diameter, before starting the

activity to discover π, the ratio of diameter to the circumference. Preservice teachers must have meaningful criteria for suitable open-ended questions that are supported by deep pedagogical content knowledge. This will enable them to anticipate probable responses and have sufficient confidence in their content knowledge to determine which avenues are worth exploring and how best to follow up on diverse student input.

Finally, we learned that the evaluation of an inquiry lesson for teacher training requires step-by-step

analyses of the preservice teacher’s actions and utterances linked with prior actions, appropriateness of content, and students’ understanding. Instructional dialogues that teachers engage in to support students’ understanding are highly complex and do not allow linear, simplistic formulation (Inoue, 2009; Leinhardt 1989, 2001). We found this to be the case with the inquiry lessons that we observed. It is also true that we cannot expect epistemological enlightenment to arise spontaneously through two weeks of mentoring, no matter how strong the mentoring or the mentees. However, we infer that experience, combined with consistent, constructive step-by-step analysis of teacher performance, curricular materials, and learner interaction with both, are needed to support the teachers in order to build an effective teaching practice. Likewise, what is helpful to any teacher is to plan a lesson, deliver the lesson, and reflect on their step-by-step actions in the classroom. From this careful scrutiny of the meaningfulness of their every action and reaction, the teacher can become aware of possible stumbling blocks in their instructional path and use this awareness to strengthen future performances.

We do not deny the importance of learning the guidelines for delivering effective inquiry lessons. However, we also learned that actually teaching an inquiry lesson based on the guidelines had many possible pitfalls for teachers. We learned the importance of improving teachers’ ability to explain content, anticipate children’s responses, respond appropriately to children's answers, and link new content to appropriate models and experiences. For meaningful knowledge construction to occur, implementing an inquiry lesson is not enough. It is more important to effectively negotiate the topic’s meaning as different perspectives and interpretations emerge at each moment in the classroom’s instructional dialogue (Cobb & Yackel, 1998; Voigt, 1996). Without such micro-level support for students’ thinking, any attempt to deliver inquiry lessons will encounter many serious stumbling blocks.

Implications for Teacher Training

These stumbling blocks of inquiry-based lessons are not bumps to be ignored. In designing professional development for teachers or coursework for preservice teachers, highlighting the role of teacher awareness on teacher actions and re-actions to learners is critical to developing practice (Buczynski & Hansen, 2010). For example, a teacher may not spend enough time acknowledging or validating students’ responses. If, from careful examination of a teaching event, the teacher is made aware of this behavior, then this

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awareness creates a heightened sensitivity to the issue and a potential for change in the teacher’s future behavior. Teacher practice then, goes beyond principled pedagogy to conscious responsiveness.

We found that teachers asking open-ended questions instead of giving answers provided learners with an opportunity to blend new knowledge with prior knowledge. However, this approach also presented a stumbling block. The teacher opens herself up to the unexpected nuances of the mathematical concept. By being aware that posing open-ended questions can lead to uncharted territory and take extra instructional time, teachers can design a lesson plan that includes consideration of strategies for anticipating responses and allowing contingency time for the subsequent discussion that might arise. A planned approach to the student comment would allow validation of the student’s ideas and integration of student’s prior knowledge with the topic at hand, two essential components of an inquiry learning activity.

Conclusion

This close examination of a preservice teacher’s performance in math camp resulted in valuable information about potential stumbling blocks that stand in the way of effectively executing a well designed inquiry lesson. This study points teacher educators to focus teacher preparation in the areas of (a) anticipating possibilities in children’s diverse responses, (b) developing deep pedagogical content knowledge that allows them to give pedagogically meaningful responses and explanations of the content, and (c) step-by-step analysis of a teacher’s actions and responses in the classroom. Although the case study described in this article provides only a snapshot of one novice teacher’s practice, we believe that uncovering these stumbling blocks across all camp instructors overcomes this limitation. To truly transform traditional teaching toward the inquiry model, we need to make every effort to help teachers become aware of potential missteps so that they may avoid these stumbling blocks in future inquiry lessons.

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1 Though “≈” was incorrectly used, and likely unfamiliar

to the students, we do not classify this as a significant stumbling block given the context of the on-going issues in the lesson.

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2011, Vol. 20, No. 2, 24–32

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Secondary Mathematics Teacher Differences: Teacher Quality and Preparation in a New York City Alternative Certification

Program

Brian R. Evans

Providing students in urban settings with quality teachers is important for student achievement. This study

examined the differences in content knowledge, attitudes toward mathematics, and teacher efficacy among

several different types of alternatively certified teachers in a sample from the New York City Teaching Fellows

program in order to determine teacher quality. Findings revealed that high school teachers had significantly

higher content knowledge than middle school teachers; teachers with strong mathematics backgrounds had

significantly higher content knowledge than teachers who did not have strong mathematics backgrounds; and

mathematics and science majors had significantly higher content knowledge than other majors. Further, it was

found that mathematics content knowledge was not related to attitudes toward mathematics and teacher

efficacy; thus, teachers had the same high positive attitudes toward mathematics and same high teacher efficacy,

regardless of content ability.

In fall 2000, New York City faced a predicted

shortage of 7,000 teachers and the possibility of a

shortage of up to 25,000 teachers over the following

several years (Stein, 2002). In response to these

shortages the New Teacher Project and the New York

City Department of Education formed the New York

City Teaching Fellows (NYCTF) program (Boyd,

Lankford, Loeb, Rockoff, & Wyckoff, 2007; NYCTF,

2008). The program, commonly referred to as

Teaching Fellows, was developed to recruit

professionals from other fields to fill the large teacher

shortages in New York City’s public schools with

quality teachers.

The Teaching Fellows program allows career-

changers, who have not studied education as

undergraduate students, to quickly receive provisional

teacher certification while taking graduate courses in

education and teaching in their own classrooms.

Teaching Fellows begin graduate coursework at one of

several New York universities and begin student

teaching in the summer before they start independently

teaching in September. Those who lack the 30 required

mathematics course credits are labeled Mathematics

Immersion, and must complete the credits within three

years, while those with the minimum 30 required

credits are labeled Mathematics Teaching Fellows.

Prior to teaching in September, Teaching Fellows must

pass the Liberal Arts and Sciences Test (LAST) and

the mathematics Content Specialty Test (CST) required

by the New York State Education Department

(NYSED) for teaching certification. Teaching Fellows

receive subsidized tuition, earn a one-year summer

stipend in their first summer, and are eligible to receive

full teacher salaries when they begin teaching. Over the

next several years Teaching Fellows continue taking

graduate coursework while teaching in their

classrooms with a Transitional B license from the

NYSED that allows them to teach for a maximum of

three years before earning Initial Certification.

The Teaching Fellows program has grown very

quickly since its inception in 2000. According to Boyd

et al. (2007), Teaching Fellows “grew from about 1%

of newly hired teachers in 2000 to 33% of all new

teachers in 2005” (p. 10). Currently, Teaching Fellows

account for 26% of all New York City mathematics

teachers and a total of about 8,800 teachers in the state

of New York (NYCTF, 2010). Of all alternative

certification programs in New York, the Teaching

Fellows program is the largest (Kane, Rockoff, &

Staiger, 2006).

There has been concern that teachers prepared in

alternative certification programs are lower in quality

than those prepared in traditional teacher preparation

programs (Darling-Hammond, 1994, 1997; Darling-

Hammond, Holtzman, Gatlin, & Heilig, 2005; Laczko-

Kerr & Berliner, 2002); thus measures of teacher

quality are of particular concern to the Teaching

Brian R. Evans is an Assistant Professor of mathematics education

in the School of Education at Pace University in New York. His

primary research interests are in teacher knowledge and beliefs,

social justice, and urban mathematics education.

Brian R. Evans

25

Fellows Program, New York State policymakers, and

other states implementing and evaluating alternative

certification programs.

Teacher Quality

Teacher quality is one of the most important

variables for student success (Angle & Moseley, 2009;

Eide, Goldhaber, & Brewer, 2004). In this study three

variables that indicate teacher quality were analyzed:

content knowledge, attitudes toward mathematics, and

teacher efficacy.

The National Council of Teachers of Mathematics

(NCTM, 2000) defined highly qualified mathematics

teachers as teachers who, in addition to possessing at

least a bachelor’s degree and full state certification,

“have an extensive knowledge of mathematics,

including the specialized content knowledge specific to

the work of teaching, as well as a knowledge of the

mathematics curriculum and how students learn” (p. 1).

NCTM recommends that high school mathematics

teachers have the equivalent of a major in mathematics,

commonly understood in New York to be at least 30

credits of calculus and higher. For middle school

teachers NCTM recommends that mathematics

teachers have at least the equivalent of a minor in

mathematics. The NYSED requires both high school

and middle school mathematics teachers to have at

least 30 credits in mathematics.

Researchers have supported the notion that strong

mathematical content knowledge is essential for

quality teaching (Ball, Hill, & Bass, 2005; Ma, 1999;

NCTM, 2000). Teachers prepared in alternative

certification programs, such as the Teaching Fellows

program, have on average higher content test scores

than other teachers (Boyd, Grossman, Lankford, Loeb,

& Wyckoff, 2006; Boyd et al., 2007). While these

findings are encouraging, there has been a lack of

concentrated focus on the content knowledge of

secondary mathematics teachers specifically. Building

on this position, this study examined the content

knowledge of the Teaching Fellows with teacher

content knowledge defined for this study to be the

combination of knowledge, skills, and understanding

of mathematical concepts held by teachers.

Despite strong academic credentials (Kane et al.,

2006), few differences are found between the

mathematics achievement levels of students of

Teaching Fellows and traditionally prepared teachers

in grades 3 to 8 (Boyd, Grossman, Lankford, Loeb,

Michelli, & Wyckoff , 2006; Kane et al., 2006), but,

after several years of teaching experience, the students

of Teaching Fellows outperform the students of

traditionally prepared teachers in academic

achievement (Boyd, Grossman, Lankford, Loeb,

Michelli, & Wyckoff , 2006). However, very few

studies have focused on Teaching Fellows who teach

mathematics in particular, and an emphasis on

secondary mathematics Teaching Fellows is needed

because much of the existing research has focused on

teachers in elementary schools only.

Teacher quality typically addresses content and

pedagogical knowledge, but examining teacher

attitudes is also important. Previous studies have

shown that attitudes in mathematics have a positive

relationship with achievement in mathematics for

students (Aiken, 1970, 1974, 1976; Ma & Kishor,

1997), which may translate to teachers as well.

Attitudes toward mathematics are defined for this study

as the sum of positive and negative feelings toward

mathematics in terms of self-confidence, value,

enjoyment, and motivation held by teachers. Amato

(2004) found that negative teacher attitudes can affect

student attitudes. Trice and Ogden (1986) found that

teachers who had negative attitudes toward

mathematics often avoided planning mathematics

lessons. Charalambous, Panaoura, and Philippou

(2009) called for teacher educators to actively work to

improve teachers’ attitudes.

Like teacher attitudes, teacher efficacy is a strong

indicator of quality teaching (Bandura, 1986; Ernest,

1989). Teachers with high efficacy, defined as a

teacher’s belief in his or her ability to teach well and

belief in the ability to affect student learning outcomes

(Bandura, 1986), are more student-centered,

innovative, and exhibit more effort in their teaching

(Angle & Moseley, 2009). Additionally, teachers with

high efficacy are more likely to teach from an inquiry

and student-centered perspective (Czerniak & Schriver,

1994), devote more time to instruction (Gibson &

Dembo, 1984; Soodak & Podell, 1997), and are more

likely to foster student success and motivation (Angle

& Moseley, 2009; Ashton & Webb, 1986; Haney,

Lumpe, Czerniak, & Egan, 2002). Mathematics anxiety

is one hurdle in building efficacy in teachers: Teachers

with higher mathematics anxiety were found to believe

themselves to be less effective (Swars, Daane, &

Giesen, 2006).

Research in Alternative Certification

Concern about alternative teacher certification

programs has led to an interest in studying the effects

of these programs in U.S. classrooms, particularly in

terms of teacher quality issues (Darling-Hammond,

1994, 1997; Darling-Hammond et al., 2005; Evans,

2009, in press; Humphrey & Wechsler, 2007; Laczko-

Kerr & Berliner, 2002; Raymond, Fletcher, & Luque,

Mathematics Teacher Differences

26

2001; Xu, Hannaway, & Taylor, 2008). Many recent

studies examining the Teaching Fellows in New York

schools focus on teacher retention and student

achievement as variables to determine success. Though

these variables are important (Boyd, Grossman,

Lankford, Loeb, Michelli, & Wyckoff, 2006; Boyd,

Grossman, Lankford, Loeb, & Wyckoff, 2006; Boyd et

al., 2007; Kane, et al., 2006; Stein, 2002), there is also

a need to investigate other variables related to success,

such as teacher content knowledge, attitudes toward

mathematics, and teacher efficacy because these

variables can affect student learning outcomes (Angle

& Moseley, 2009; Ball et al., 2005; Bandura, 1986;

Ernest, 1989). Few studies have examined the

relationship between mathematical content knowledge

and teacher efficacy. Those that exist have examined

preservice teacher content knowledge and efficacy for

traditionally prepared teachers (i.e. Swars et al., 2006;

Swars, Hart, Smith, Smith, & Tolar, 2007).

Researchers have called for a strong academic

coursework component for alternative certification

teachers (Suell & Piotrowski, 2007), yet little is known

about the knowledge and skills that these teachers

already possess on entering the program. In order to

most effectively use limited teacher training resources,

policymakers need more research in this area.

Humphrey and Wechsler (2007) noted, “Clearly, much

more needs to be known about alternative certification

participants and programs and about how alternative

certification can best prepare highly effective teachers”

(p. 512).

Theoretical Framework

The theoretical framework of this study is based

upon the positive relationship between mathematical

achievement and attitudes found in students (Aiken,

1970, 1974, 1976; Ma & Kishor, 1997), the need for

strong teacher content knowledge (Ball et al., 2005),

and teaching efficacy theory (Bandura, 1986). Bandura

found that teacher efficacy can be subdivided into a

teacher’s belief in his or her ability to teach well and

his or her belief in a student’s capacity to learn well

from the teacher. Teachers who feel that they cannot

effectively teach mathematics and affect student

learning are more likely to avoid teaching from an

inquiry and student-centered approach (Angle &

Moseley, 2009; Swars et al., 2006).

Purpose of the Study and Research Questions

This study is a continuation of a previous study

(Evans, in press) that examined changes in content

knowledge, attitudes toward mathematics, and the

teacher efficacy over time of new teachers in the

Teaching Fellows program. The previous study found

that Teaching Fellows increased their mathematical

content knowledge and attitudes over the course of the

semester-long mathematics methods course while

teaching in their own classroom. They also held

positive attitudes toward mathematics and had high

teacher efficacy both in terms of their ability to teach

well and their ability to positively affect student

outcomes. The focus of the present study is finding

differences in the various categories of Teaching

Fellows across these three variables.

Teacher quality is an important concern in teacher

preparation (Eide et al., 2004), and particularly for

mathematics teachers of high-need urban students (Ball

et al., 2005). The purpose of this study was to

determine differences in these variables among

different categories of alternative certification teachers

in New York City. Determining these differences is

important for two reasons. First, it is important for

teacher recruitment. If policy makers, administrators,

and teacher educators know which teacher

characteristics lead to the highest levels of content

knowledge, attitudes, and efficacy, recruitment can be

better focused. Second, it is important for teacher

preparation. Knowing which teachers need the most

support, and in which areas, can lead to increased

teacher quality through better preparation and focused

professional development. This study addresses the

following research questions:

1. Are there differences in mathematical content

knowledge, attitudes toward mathematics, and

teacher efficacy between middle and high school

Teaching Fellows?

2. Are there differences in mathematical content

knowledge, attitudes toward mathematics, and

teacher efficacy between Mathematics and

Mathematics Immersion Teaching Fellows?

3. Are there differences in mathematical content

knowledge, attitudes toward mathematics, and

teacher efficacy between undergraduate college

majors among the Teaching Fellows?

4. Is mathematical content knowledge related to

attitudes toward mathematics and teacher efficacy?

The first three research questions addressed the

differences that existed among types of teachers in

content knowledge, attitudes toward mathematics, and

teacher efficacy. These questions are important

because it is imperative that policy makers,

administrators, and teacher educators determine

teacher quality for those who will be teaching mostly

Brian R. Evans

27

high-need urban students. In this study “high-need”

refers to urban schools in which students are of lower

socio-economic status, have low teacher retention, and

lack adequate resources. The fourth research question

involved synthesizing the results of the first three

questions to generate further implications.

Methodology

This study employed a quantitative methodology.

The sample consisted of 42 new teachers in the

Teaching Fellows program (N = 30 Mathematics

Immersion and N = 12 Mathematics Teaching Fellows)

with approximately one third of the participants male

and two thirds of the participants female. The teachers

in this study were selected due to availability and thus

represented a convenience sample with limited

generalizability. The Teaching Fellows in this study

were enrolled in two sections of a mathematics

methods course, which involved both pedagogical and

content instruction in the first semester of their

program. These sections, taught by the author, focused

on constructivist methods with an emphasis on

problem solving and real-world connections in line

with NCTM Standards (2000).

Teaching Fellows completed a mathematics

content test and two questionnaires at the beginning

and end of the semester. The mathematics content test

consisted of 25 free-response items ranging from

algebra to calculus and was designed to measure

general content knowledge. The mathematics content

test taken at the end of the semester was similar in

form and content to the one taken at the beginning.

Prior to their coursework and teaching, the Teaching

Fellows take the Content Specialty Test (CST). CST

scores were recorded as another measure of

mathematical content knowledge. The scores range

from 100 to 300, with a minimum state-mandated

passing score of 220. The CST consists of multiple-

choice items and a written assignment and has six sub-

areas: Mathematical Reasoning and Communication;

Algebra; Trigonometry and Calculus; Measurement

and Geometry; Data Analysis, Probability, Statistics

and Discrete Mathematics; and Algebra Constructed

Response. Data from the CST were analyzed to

validate findings suggested by the mathematics content

test.

Attitudes toward mathematics were measured by a

questionnaire designed by Tapia (1996) that has 40

items measuring characteristics such as self-

confidence, value, enjoyment, and motivation in

mathematics. The instrument uses a 5-point Likert

scale of strongly agree, agree, neutral, disagree, to

strongly disagree. Teacher efficacy was measured by a

questionnaire adapted from the Mathematics Teaching

Efficacy Beliefs Instrument (MTEBI) developed by

Enochs, Smith, and Huinker (2000). The MTEBI is a

21-item 5-point Likert scale instrument with the same

choices as the attitudinal questionnaire. It is grounded

in the theoretical framework of Bandura’s efficacy

theory (1986). Based on the Science Teaching Efficacy

Belief Instrument (STEBI-B) developed by Enochs and

Riggs (1990), the MTEBI contains two subscales:

Personal Mathematics Teaching Efficacy (PMTE) and

Mathematics Teaching Outcome Expectancy (MTOE)

with 13 and 8 items, respectively. Possible scores

range from 13 to 65 on the PMTE, and 8 to 40 on the

MTOE. Higher scores indicated better teacher efficacy.

The PMTE specifically measures a teacher’s concept

of his or her ability to effectively teach mathematics.

The MTOE specifically measures a teacher’s belief in

his or her ability to directly affect student-learning

outcomes. Enochs et al. (2000) found the PMTE and

MTOE had Cronbach α coefficients of 0.88 and 0.77,

respectively.

Research questions one and two were answered

using independent samples t-tests on data collected

from the 25-item mathematics content test, CST, 40-

item attitudinal test, and 21-item MTEBI with two

subscales. Research question three was answered using

one-way ANOVA on data also collected from the same

instruments. In this study there was a mix of middle

school and high school teachers in the Mathematics

and Mathematics Immersion programs. For the third

research question Teaching Fellows were divided into

three categories based upon their undergraduate

college majors: liberal arts, business, and mathematics

and science majors. Liberal arts majors consisted of

majors such as English, history, Italian, philosophy,

political science, psychology, sociology, Spanish, and

women studies. Business majors consisted of majors

such as accounting, business administration and

management, commerce, economics, and finance.

Mathematics and science majors consisted of majors

such as mathematics, engineering, and the sciences

(biology and chemistry). Research question four was

answered through Pearson correlations with the same

instruments used in the other research questions.

The data were analyzed using the Statistical

Package for the Social Sciences (SPSS), and all

significance levels were at the 0.05 level. Teachers

were separated by teaching level (middle and high

school), mathematics credits earned (Mathematics and

Mathematics Immersion), and undergraduate major

(liberal arts, business, and mathematics and science

majors) in order to determine differences between the

Mathematics Teacher Differences

28

different types of mathematics teachers sampled to

determine teacher quality.

Results

To determine internal reliability of the attitudinal

instruments, it was found that the Cronbach α

coefficient was 0.93 on the pretest and 0.94 on the

posttest for the 40-item attitudinal test. For the efficacy

pretest, α = 0.80 for the PMTE α = 0.77 for the MTOE.

For the efficacy posttest, α = 0.82 for the PMTE and α

= 0.83 for the MTOE, respectively. These values are

fairly consistent with the literature (Enochs et al.,

2000; Tapia, 1996).

The first research question was answered using

independent samples t-tests comparing middle and

high school teacher data using responses on the

mathematics content test, CST, attitudinal test, and

MTEBI with two subscales: PMTE and MTOE. There

was a statistically significant difference between

middle school teacher scores and high school teacher

scores for the mathematics content pretest, posttest,

and CST (see Table 1). Thus, high school teachers had

higher content test scores than middle school teachers,

and the effect sizes were large. There were no

statistically significant differences found between

middle and high school teachers on both pre- and

posttests measuring attitudes toward mathematics and

teacher efficacy beliefs.

Table 1

Independent Samples t-Test Results on Mathematics Content Tests by Level

Assessment Mean SD t-value d-value

Mathematics Content

Pre-Test

Middle School (N = 26)

High School (N = 16)

68.42

85.13

15.600

16.041

-3.334**

1.056

Mathematics Content

Post-Test

Middle School (N = 26)

High School (N = 16)

79.46

92.63

15.402

6.582

-3.230**

1.112

Mathematics CST

Middle School (N = 26)

High School (N = 16)

255.31

269.25

20.372

17.133

-2.283*

0.741

N = 42, df = 40, two-tailed

* p < 0.05

** p < 0.01

The second research question was answered using

independent samples t-tests comparing Mathematics

Immersion and Mathematics Teaching Fellows data

using the mathematics content test, CST, attitudinal

test, and MTEBI with two subscales: PMTE and

MTOE. There was a statistically significant difference

between Mathematics Immersion Teaching Fellows’

scores and Mathematics Teaching Fellows’ scores for

the mathematics content pretest, posttest, and CST (see

Table 2). Thus, Mathematics Teaching Fellows had

higher content test scores than Mathematics Immersion

Teaching Fellows, and the effect sizes were large.

There were no statistically significant differences

found between Mathematics and Mathematics

Immersion Teaching Fellows on both pre- and posttests

measuring attitudes toward mathematics and teacher

efficacy beliefs.

Table 2

Independent Samples t-Test Results on Mathematics Content Tests by Background

Assessment Mean SD t-value d-value

Mathematics Content

Pre-Test

Mathematics Teaching

Fellows (N = 12)

Mathematics

Immersion (N = 30)

89.50

68.90

7.868

17.008

-4.005**

1.555

Mathematics Content

Post-Test

Mathematics Teaching

Fellows (N = 12)

Mathematics

Immersion (N = 30)

94.33

80.53

7.390

14.460

-3.130**

1.202

Mathematics CST

Mathematics Teaching

Fellows (N = 12)

Mathematics

Immersion (N = 30)

276.33

254.33

16.104

18.291

-3.636**

1.277

N = 42, df = 40, two-tailed

** p < 0.01

The third research question was answered using

one-way ANOVA comparing different undergraduate

college majors using the mathematics content test,

CST, attitudinal test, and MTEBI with two subscales:

PMTE and MTOE. Teaching Fellows were grouped

according to their undergraduate college major. Three

categories were used to group teachers: liberal arts (N

= 16), business (N = 11), and mathematics and science

(N = 15) majors. The results of the one-way ANOVA

revealed statistically significant differences between

Brian R. Evans

29

undergraduate major area for the mathematics content

pretest, posttest, and CST, with large effect sizes in

each case (see Tables 3, 4, 5, and 6). A post hoc test

(Tukey HSD) revealed that mathematics and science

majors had significantly higher content knowledge

than business majors with p < 0.01 (pretest, posttest,

and CST) and liberal arts majors with p < 0.01 (pretest)

and p < 0.05 (posttest and CST). There were no other

statistically significant differences. In summary, in this

study mathematics and science majors had higher

content knowledge scores than non-mathematics and

non-science majors. No statistically significant

differences were found between the undergraduate

college majors on both pre- and posttests in attitudes

toward mathematics and teacher efficacy.

Table 3 Means and Standard Deviations on Content Knowledge for Major

Pre-, Post-, and CST Tests Mean Standard Deviation

Content Knowledge Pre Test; Total (N = 42)

Liberal Arts (N = 16)

Business (N = 11)

Math/Science (N = 15)

74.79

70.13

64.45

87.33

17.605

16.382

15.820

12.804

Content Knowledge Post Test; Total (N = 42)

Liberal Arts (N = 16)

Business (N = 11)

Math/Science (N = 15)

84.48

81.19

76.82

93.60

14.225

15.132

14.034

7.679

CST Content Knowledge; Total (N = 42)

Liberal Arts (N = 16)

Business (N = 11)

Math/Science (N = 15)

260.62

255.81

249.64

273.80

20.184

18.784

18.943

15.857

Table 4

ANOVA Results on Mathematics Content Pretest for Major

Variation Sum of Squares df Mean Square F η2

Between Groups 3883.261 2 1941.630 8.582** 0.31

Within Groups 8823.811 39 226.252

Total 12707.071 41

** p < 0.01

Table 5 ANOVA Results on Mathematics Content Posttest for Major

Variation Sum of Squares df Mean Square F η2

Between Groups 2066.802 2 1033.401 6.469** 0.25

Within Groups 6229.674 39 159.735

Total 8296.476 41

** p < 0.01

Mathematics Teacher Differences

30

Table 6 ANOVA Results on Mathematics Content Specialty Test (CST) for Major

Variation Sum of Squares df Mean Square F η2

Between Groups 4302.522 2 2151.261 6.765** 0.26

Within Groups 12401.383 39 317.984

Total 16703.905 41

** p < 0.01

Research question four was analyzed using

Pearson correlations to determine if there were any

relationships between content knowledge and attitudes

toward mathematics or efficacy. No significant

relationships were found. This suggests that Teaching

Fellows’ attitudes toward mathematics and efficacy are

unrelated to how much content knowledge they

possess.

Discussion and Implications

The results of the analyses on the data collected

from this particular group of Teaching Fellows

revealed that high school teachers had higher

mathematics content knowledge than middle school

teachers, Mathematics Teaching Fellows had higher

mathematics content knowledge than Mathematics

Immersion Teaching Fellows, and mathematics and

science majors had higher mathematics content

knowledge than non-mathematics and non-science

majors. The sample size in this study was small, but

effect sizes were found to be quite large. Moreover, no

differences in attitudes toward mathematics and

teacher efficacy were found between middle and high

school teachers; between Mathematics and

Mathematics Immersion Teaching Fellows; or among

liberal arts, business, and mathematics and science

majors. Surprisingly, no relationships were found

between mathematical content knowledge and attitudes

toward mathematics and teacher efficacy. The

statistically significant differences in content

knowledge found in this study led to further analysis to

determine if there were differences in gain scores for

content knowledge on the mathematics content test

over the course of the semester for any group;

however, no significant differences were found in gain

scores between middle and high school teachers,

between Mathematics Teaching Fellows and

Mathematics Immersion Teaching Fellows, or among

the different undergraduate college majors.

In the first study (Evans, in press) the sampled

teachers had positive attitudes toward mathematics and

high teacher efficacy. The present study revealed that

there were no differences between the different

categories (teaching level, immersion status, and

major) of Teaching Fellows in attitudes toward

mathematics and efficacy, and that content knowledge

was unrelated to attitudes toward mathematics and

efficacy. Combining the results of the first study

(Evans, in press) with the results found in this present

study, an interesting finding emerged. Teachers in this

study had the same high level of positive attitudes

toward mathematics and the same high level of teacher

efficacy regardless of content ability. Thus, some of

the teachers in this study believed they were just as

effective at teaching mathematics, despite not having

the high level of content knowledge that some of their

colleagues possessed. This finding is significant

because high content knowledge is a necessary

condition for quality teaching (Ball et al., 2005). This

finding also contradicts other research conducted that

found a positive relationship between content

knowledge and attitudes (Aiken, 1970, 1974, 1976; Ma

& Kishor, 1997). It is possible that the unique sample

of alternative certification teachers may have

contributed to this difference, and this possibility

should be further investigated. It should also be noted

that the instructor in the mathematics methods course

was also the researcher. Thus, consideration must be

given for possible bias in participant reporting since

the participants in this study knew that the instructor

would be conducting the research. Participants were

assured that their responses would not be used as an

assessment measure in the methods course.

Although New York State requires a minimum of

30 mathematics credits for both middle and high

school teachers, high school teachers had higher

content knowledge than middle school teachers. This

may be due to their experience working with higher

level mathematics in their teaching. However, this does

not explain the reason that sampled high school

teachers scored better on the CST and content pretest

instruments: this study began at the beginning of their

teaching careers, and the teachers did not yet have

significant classroom experience. It is possible that

teachers with stronger content knowledge may be

drawn more to high school teaching, rather than middle

Brian R. Evans

31

school teaching, and the more rigorous content that

comes with teaching high school mathematics. Because

the participants in this study represent a convenience

sample due to availability, which restricts the

generalizability of this study, further research should

extend to larger sample sizes.

Many alternative certification teachers, such as the

Teaching Fellows, teach in high-need urban schools in

New York City (Boyd, Grossman, Lankford, Loeb, &

Wyckoff, 2006) and throughout the United States.

Therefore, it is imperative that policy makers,

administrators, and teacher educators continually

evaluate teacher quality in alternative certification

programs. NCTM (2005) stated, “Every student has the

right to be taught mathematics by a highly qualified

teacher—a teacher who knows mathematics well and

who can guide students' understanding and learning”

(p. 1). New York State holds the same high standards

for both high school and middle school teachers. Thus,

educational stakeholders should investigate and

implement strategies to better middle school teachers’

content knowledge. Based on the results of this study it

is recommended that middle school teachers be given

strong professional development in mathematics

content knowledge by both the schools in which they

teach and the schools of education in which they are

enrolled. Future studies should examine this issue with

larger samples of Teaching Fellows and teachers from

other alternative certification programs to increase

generalizability. It is imperative that future research

address whether or not there are differences in actual

teaching ability among the Mathematics and

Mathematics Immersion Teaching Fellows and

different college majors held by the teachers. One way

to determine this would be to measure students’

mathematics performance to identify differences in

student achievement among the variables examined in

this study.

As earlier stated, Teaching Fellows currently

account for one-fourth of all New York mathematics

teachers (NYCTF, 2008), and increasingly alternative

certification programs account for more teachers

coming to the profession throughout the United States

(Humphrey & Wechsler, 2007). For the sake of

students who have teachers in alternative certification

programs, the certification of high quality teachers

must continually be a priority for policy makers,

administrators, and teacher educators. Considering the

call for high quality teachers, high stakes examinations,

and accountability, now more than ever we need to

ensure that the teachers we certify are fully prepared in

both content knowledge and dispositions to best teach

our high-need students.

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The Mathematics Educator

2011, Vol. 20, No. 2, 33–43

33

Sense Making as Motivation in Doing Mathematics: Results from Two Studies

Mary Mueller, Dina Yankelewitz, & Carolyn Maher

In this article, we present episodes from two qualitative research studies. The studies focus on students of

different ages and populations and their work on different mathematical tasks. We examine the commonalities

in environment, tools, and teacher-student interactions that are key influences on the positive dispositions

engendered in the students and their interest and engagement in mathematics. In addition, we hypothesize that

these positive dispositions in mathematics lead to student reasoning and, thus, mathematical understanding. The

resulting framework is supported by other educational research and suggests ways that the standards can be

implemented in diverse classrooms in order to achieve optimal student engagement and learning.

The National Council of Teachers of Mathematics

(NCTM, 2000) describes a vision for mathematics

education focusing on conceptual understanding. This

vision includes students engaged in hands-on activities

that incorporate problem solving, reasoning and proof,

real-world connections, multiple representations, and

mathematical communication. NCTM and others have

prepared multiple documents and resources (e.g.,

Chambers, 2002; Germain-McCarthy, 2001; NCTM,

2000; Stiff & Curcio, 1999) to support teachers in

achieving this vision and putting the standards into

practice. However, differences in age, gender,

ethnicity, and school culture often impede the

implementation of successful teaching practice in

mathematics classrooms and prevent students from

taking ownership of mathematical ideas in the ways

that have been envisioned.

While NCTM addresses factors such as classroom

environment and mathematical tasks, this provides an

incomplete picture of how to build students’

conceptual understanding. For example, motivation to

learn is pivotal in students’ attainment of

understanding in all content areas (Middleton &

Spanias, 1999), but the NCTM vision does not

explicate how to help students experience motivation

as they learn mathematics. We have developed a

framework for mathematics teaching and learning that

provides this missing link. It provides teachers and

researchers with a conceptual tool that explains how

students build the positive attitudes (motivation,

autonomy, self-efficacy, and positive dispositions)

towards mathematics that are necessary to engage in

mathematical reasoning. We believe that this approach

that can be implemented across the spectrum of

mathematics classrooms in the US.

Our research focuses on students who are working

collaboratively as they engage in mathematical

problem solving. We videotaped students as they

engaged in mathematical tasks and then analyzed the

reasoning that occurred as they worked to formulate

strategies and defend their solutions. We have found

that, although the demographics of the groups of

students and the tasks may be different, the reasoning

and subsequent understanding that occurs is quite

similar.

In this article, we present two episodes from our

research, focusing on students of different ages and

populations as they work on different mathematical

tasks. We then examine the commonalities in

environment, tools, and teacher-student interactions

that are key influences both on the positive attitudes

towards mathematics engendered in the students and

on their engagement in mathematics. We hypothesize

that these positive attitudes towards mathematics lead

to student reasoning and, thus, mathematical

understanding. Based on our research, we created a

Mary Mueller is an Associate Professor in the Department of

Educational Studies at Seton Hall University. Her research

interests include the development of mathematical ideas and

reasoning over time.

Dina Yankelewitz is an Assistant Professor in the School of

General Studies at the Richard Stockton College of New Jersey.

Her research interests include the development of mathematical

thinking and the identification and development of mathematical

reasoning in students and teachers of mathematics.

Carolyn Maher is a Professor of mathematics education in the

Graduate School of Education at Rutgers University. Her research

interests include the development of mathematical thinking in

students, mathematical reasoning, justification and proof making in

mathematics, and the development of a model for analyzing

videotape data.

Sense Making as Motivation

34

framework for teaching and learning that identifies the

key factors in encouraging positive attitudes in the

mathematics classroom as well as their role in enabling

student reasoning and understanding. We support this

framework using the extensive literature base centering

on students’ motivation in the mathematics classroom.

The resulting framework suggests ways that the

standards can be implemented in diverse classrooms in

order to achieve optimal student engagement and

learning. Although the development of our framework

began with our data and then was supported by the

literature, we begin by presenting the supporting

literature in order to give the readers a background for

the framework.

The Role of Intrinsic Motivation

In our framework, there are four factors that

mediate between elements in the classroom

environment, such as tasks, and the development of

conceptual understanding through mathematical

reasoning. These four factors are autonomy, instrinsic

motivation, self-efficacy and positive dispositions

towards mathematics. Because the literature

concerning all four of these factors is interrelated, we

have picked one factor, intrinsic motivation, to

organize our discussion around.

All students must be motivated in some way to

engage in mathematical activity, however, the nature of

that motivation largely determines the success of their

endeavor. In particular, students’ motivations can be

divided into two distinct types: extrinsic motivation

and intrinsic motivation. Extrinsically motivated

students engage in learning for external rewards, such

as teacher and peer approval and good grades. These

students do not necessarily acquire a sense of

ownership of the mathematics that they study; instead

they focus on praise from teachers, parents and peers

and avoiding punishment or negative feedback

(Middleton & Spanias, 1999). In contrast, students who

are intrinsically motivated to learn mathematics are

driven by their own pursuit of knowledge and

understanding (Middleton & Spanias, 1999). They

engage in tasks due to a sense of accomplishment and

enjoyment and view learning as impacting their self-

images (Middleton, 1995). Intrinsically motivated

students, therefore, focus on understanding concepts.

Thus, intrinsic, rather than extrinsic, motivation

benefits students in the process and results of

mathematical activities.

Sources of Intrinsic Motivation

Researchers (Deci & Ryan, 1985; Hidi, 2000;

Renninger, 2000) have found that sources of intrinsic

motivation include perceptions of autonomy, interests

in given tasks, and the need for competence. Brophy

(1999) concurs and notes that a supportive social

context, challenging activities, and student interest and

value in learning are crucial to the development of

intrinsic motivation.

Autonomous students, in attending to problem

situations mathematically, rely on their own

mathematical facilities and use their own resources to

make decisions and make sense of their strategies

(Kamii,1985; Yackel & Cobb, 1996). Autonomy

promotes persistence on tasks and thus leads to higher

levels of intrinsic motivation (Deci, Nezdik, &

Sheinman, 1981; Deci & Ryan, 1987; Stefanou,

Perencevich, DiCinti, & Turner, 2004). Furthermore,

through participation in classroom activities,

mathematically autonomous students begin to rely on

their own reasoning rather than on that of the teacher

(Cobb, Stephan, McClain, & Gravemeijer, 2001;

Forman, 2003) and thus become arbitrators of what

makes sense.

Studies show that teacher support and classroom

environments play a crucial role in the development of

another source of intrinsic motivation, namely, positive

(or negative) dispositions toward mathematics

(Bransford, Hasselbring, Barron, Kulewicz, Littlefield,

& Goin, 1988; Cobb, Wood, Yackel, & Perlwitz, 1992;

Middleton, 1995; Middleton & Spanias, 1999).

According to NCTM (2000), “More than just a

physical setting … the classroom environment

communicates subtle messages about what is valued in

learning and doing mathematics (p. 18). The document

then describes the implementation of challenging tasks

that challenge students intellectually and motivate

them through real-world connections and multiple

solution paths (NCTM, 2000). Stein, Smith,

Henningsen, and Silver (2000) stress that teachers need

be thoughtful about the tasks that they present to

students and use care to present and sustain cognitively

complex tasks. They suggest that during the problem

solving implementation phase, teachers often reduce

the cognitive complexity of tasks. Overall, when

students are presented with meaningful, relevant, and

challenging tasks; offered the opportunity to act

autonomously and develop self-control over learning;

encouraged to focus on the process rather than the

product; and provided with constructive feedback, they

become intrinsically motivated to succeed (Urdan &

Turner, 2005).

Effects of Intrinsic Motivation

Intrinsic motivation leads to self-efficacy, an

individual’s beliefs about their own ability to perform

Mary Mueller, Dina Yankelewitz, & Carolyn Maher

35

specific tasks in specific situations (Bandura, 1986;

Pajares, 1996). Students’ self-efficacy beliefs often

predict their ability to succeed in a particular situation

(Bandura, 1986). Specifically, in mathematics, research

has shown that self-efficacy is a clear predictor of

students’ academic performance (Mousoulides &

Philippou, 2005; Pintrich & De Groot, 1990).

Furthermore, studies suggest that students with highly

developed self-efficacy beliefs utilize cognitive and

metacognitive learning strategies more vigorously

while being more aware of their own motivational

beliefs (Mousoulides & Philippou, 2005; Pintrich,

1999).

Unlike sources of extrinsic motivation, which need

to be constantly reinforced, research shows that the

common sources of intrinsic motivation are reinforced

when students are encouraged to develop their self-

efficacy (Urdan & Turner, 2005), For example,

intrinsic motivation helps students succeed at a given

learning objective, thereby further developing students’

self-efficacy.In general, students are more likely to

engage and persist in tasks when they believe they

have the ability to succeed (Urdan & Turner, 2005).

Therefore, intrinsic motivation can lead to an increased

willingness to engage in reasoning activities.

In summary, research shows that when students are

intrinsically motivated to learn mathematics, they

spend more time on-task, tend to be more persistent,

and are confident in using different, or more

challenging, strategies to solve mathematical problems

(Lepper, 1988; Lepper & Henderlong, 2000). These

qualities of mathematical learners better enable them to

actualize the recommendations put forth by NCTM

(2000) and to master key mathematical processes in

their pursuit of understanding mathematics. Intrinsic

motivation, then, is correlated with self-efficacy and

positive dispositions towards a conceptual

understanding of mathematics, whereas extrinsic

motivation results in merely a superficial grasp of the

information presented.

Results from Two Studies

Through a combination of cross-cultural and

longitudinal studies we have observed that a mixture of

factors contribute to students’ motivation to participate

in mathematics and their dispositions towards

mathematics (for details on our methodologies, see

Mueller, 2007; Mueller & Maher, 2010; Mueller,

Yankelewitz, & Maher, 2010; Yankelewitz, 2009;

Yankelewitz, Mueller, & Maher, 2010). These include

classroom environment, teacher questioning that

evokes meaningful support of conjectures, and well-

designed tasks. Together, these factors positively

influence the establishment of favorable dispositions

towards learning mathematics. In their quest to make

sense of appropriately challenging tasks, students enjoy

the pursuit of meaning and thereby become

intrinsically motivated to engage in mathematics.

In this paper we present results from two research

studies investigating students’ mathematics learning. In

particular, we present specific examples of elementary

and middle school students who demonstrated sense

making and higher order reasoning when working on

mathematical tasks. In these episodes, the students

were engaged, motivated, and, importantly, confident

in their ability to offer and defend mathematical

solutions; they demonstrated positive dispositions

towards mathematics. We identified student behaviors

that indicated confidence in mathematics and a high

level of engagement. These behaviors include

perseverance; the ability to consider more challenging,

alternative solutions; and the length of time spent of

the task. In the discussion that follows, we analyze the

commonalities in the two teaching experiments, and

consider how these commonalities may have positively

influenced the level of motivation and confidence that

students exhibited as they worked on mathematical

tasks. In the discussion, we use our findings to define a

framework that can be used to inform a teaching

practice that will motivate students and encourage

student engagement and mathematical understanding.

Data Analysis and Results

The episodes presented below come from two data

sets. Data from the first study is drawn from sessions

during an informal after-school mathematics program

in which 24 sixth-grade students from a low

socioeconomic urban community worked on open-

ended tasks involving fractions. The students

represented a wide range of abilities and thus their

mathematical levels ranged from those who were

enrolled in remedial mathematics to those who were

successful in regular mathematics classrooms. The

present discussion focuses on one table of four

students, two boys and two girls.

The second source of data includes segments from

sessions in which fourth and fifth grade students from a

suburban school investigated problems in counting and

combinatorics. This data is drawn from a longitudinal

study of children’s mathematical thinking. As part of

the students’ regular school day, researchers led the

students in exploring open-ended tasks during which

students were expected to justify their solutions to the

satisfaction of their peers. These strands of tasks were

separate from the school-mandated curriculum.

Because of space limitations, we give examples of one

Sense Making as Motivation

36

task from each data set, one involving fractions and the

other focusing on combinatorics.

Episode 1: Reasoning about Fractions in the Sixth

Grade

The students in the first study worked

collaboratively on tasks involving fraction

relationships. Cuisenaire® rods (see Figure 1) were

available and students were encouraged to build

models. A set of Cuisenaire rods contains 10 colored

wooden or plastic rods that increase in length by

increments of one centimeter. For these activities, the

rods have variable number names and fixed color

names. The colors increased incrementally as follows:

white, light green, purple, yellow dark green, black

brown, blue, and orange.

Figure 1. “Staircase” Model of Cuisenaire Rods.

Students were encouraged to build models to

represent fraction tasks. For example, in one task, the

blue rod was given the number name one and students

initially worked on naming the red rod (two-ninths)

and the light green rod (three-ninths or one-third).

When the group completed this task, they initiated their

own task of naming all of the rods in the set, given that

the blue rod was named one.

Chanel used the staircase model (shown in Figure

1) to incrementally name the remainder of the rods

beginning with naming the white rod one-ninth. As she

was working, she said the names of all of the rods,

“One-ninth, two-ninths, three-ninths, four-ninths, five-

ninths, six-ninths, seven-ninths, eight-ninths, nine

ninths, ten..– wow, oh, I gotta think about that one …..

nine-tenths”.

Disequilibrium. The teacher/researcher

encouraged Chanel to share her problem with Dante.

Chanel showed Dante her strategy of using the

staircase to name the rods and explained the dilemma

of naming the orange rod, “See this is One-ninth, two-

ninths, three-ninths, four-ninths, five ninths, six-ninths,

seven-ninths, eight-ninths, nine-ninths - what’s this

one?” Dante replied, “That would be ten-ninths.

Actually that should be one. That would start the new

one (one-tenth)”. Chanel and Michael then named the

blue rod “a whole”. The students worked for a few

more minutes and then Dante explained that he had

overhead another table naming the rods.

Dante: Why are they calling it ten-ninths and [it]

ends at ninths?

Michael: Not the orange one. The orange one’s a

whole.

Dante: But I’m hearing from the other group

from over here, they calling it ten-ninths.

Michael: Don’t listen to them! The orange one is a

whole because it takes ten of these to

make one.

Dante: I’m hearing it because they speaking out

loud. They’re calling it ten-ninths

Michael: They might be wrong! …

Chanel: Let me tell you something, how can they

call it ten-ninths if the denominator is

smaller than the numerator?

Dante: Yeah, how is the numerator bigger than

the denominator? It ends at the

denominator and starts a new one. See

you making me lose my brain.

A teacher/researcher joined the group and asked

what the students were working on. Dante presented

his argument of naming the orange rod one-tenth and

explained that “it starts a new one”. The

teacher/researcher reminded him that the white rod was

named one-ninth and that this fact could not change.

Again she asked him for the name of the blue rod and

he stated, “It would probably be ten-ninths”. When

prompted, Dante explained that the length of ten white

rods was equivalent to the length of an orange rod.

The teacher/researcher asked Dante to convince his

partners that this was true.

Chanel: No, because I don’t believe you because–

Michael: I thought it was a whole.

Dante: But how can the numerator be bigger

than the denominator?

T/R: It can. It is. This is an example of where

the numerator is bigger than the

denominator.

Mary Mueller, Dina Yankelewitz, & Carolyn Maher

37

Chanel: But the denominator can’t be bigger

than the numerator, I thought.

Michael: That’s the law of facts.

T/R: Who told you that?

Chanel: My teacher.

Dante: One of our teachers.

Direct reasoning. The students continued working

on the task. At the end of the session students were

asked to share their work. Another student explained

that she named the orange rod using a model of two

yellow rods, “We found out the denominator doesn’t

have to be larger than the numerator because we found

out that two yellows [each] equal five-ninths so five-

ninths plus five-ninths equals ten-ninths.” Another

student explained that the orange rod could also be

named one and one-ninth and used a model of a train

of a blue rod and a white rod lined up next to the

orange rod to explain (see Figure 2), “If you put them

together then this means that it’s ten-ninths also known

as one and one-ninth.”

Figure 2. A train of rods to show that 910

91

99

=+ or

911

Finally, Dante came to the front of the class and

explained that he found a different way to name the

orange rod. Building a model of an orange rod lined up

next to two purple rods and a red rod (see Figure 3), he

explained that the purple rods were each named four-

ninths and therefore together they were eight-ninths;

the red rod was named two-ninths and therefore the

total was eight-ninths plus two-ninths or ten-ninths,

“four and four are eight so which will make it eight-

ninths right here and then plus two to make it ten-

ninths.”

Figure 3. A train of rods to show

that910

92

94

94

=++ .

In the beginning of the session described above,

Dante and his partners were convinced that a fraction’s

numerator could not be greater than its denominator.

At some point it seems that they were taught about

improper fractions and may have internalized this to

mean incorrect fractions. The children referred to this

rule as “the law of facts” and, when presented with the

task, although they visually saw that the orange rod

was equivalent to ten white rods (or ten-ninths), they

resisted using this nomenclature. We highlight this

episode to show that the students did not simply accept

the rule that they recalled and move on to the next task.

Instead they heard another group naming the orange

rod ten-ninths and grappled with the discrepancy

between this name and their rule. Remaining engaged

in the task, the students focused on sense making; they

were motivated to make sense of the models they built

and in doing so exhibited confidence in their solutions.

For over an hour, Dante attempted to make sense of his

solution by building alternative models, sharing his

ideas, conjectures, and solutions, questioning the

teacher, and revisiting the problem. When faced with a

discrepancy between what he had previously learned

and the concrete model that he built, Dante relied on

reasoning, rather than memorized facts, to convince

himself and others of what made sense. In particular,

he relied on his understanding of the model that he had

constructed to make sense of the fraction relationships.

This quest for sense making triggered the use of a

variety of strategies, and the success of meaning-

building led to persistence and flexibility in thinking,

which, as described by Lepper and Henderlong (2000),

are positively correlated with self-efficacy. Dante’s

self-efficacy gave him the confidence and autonomy to

move beyond his erroneous understanding that was

based on previous memorized facts. Similarly to

discussions about autonomy from Kamii (1985) and

Yackel and Cobb (1996), this autonomy encouraged

Dante to believe in his own mathematical ability and

use his own resources to make sense of his model. This

autonomy, coupled with his positive dispositions

toward mathematics, allowed him to use reasoning to

make sense of and fully understand the mathematics

inherent in the problem.

Episode 2: Reasoning about Combinatorics in the

Fourth and Fifth Grades

In the second study, fourth- and fifth-grade

students were introduced to combinatorial tasks. The

students were given Unifix cubes and were asked to

find all combinations of towers that were four tall

when selecting from cubes of two colors. Over the

course of the two years, students revisited the task in

Sense Making as Motivation

38

various settings. This provided multiple opportunities

for them to think about and refine their thinking about

the problem.

Stephanie, along with her partner, Dana, first

constructed all possible towers four cubes tall by

finding patterns of towers and searching for duplicates.

After her first attempt to find all possible towers,

Stephanie organized her groups of towers according to

color categories (e.g., exactly one of a color and

exactly two of a color adjacent to each other) in order

to justify her count of 16 towers, thus she organized the

towers by cases (see Figure 4). Stephanie then used

this organization by cases to find all possible towers of

heights three cubes tall, two cubes tall and one cube

tall when selecting from two colors.

Figure 4. Stephanie’s organization of towers by cases.

During further investigation, Stephanie noticed a

pattern in the sequence of total number of towers for

each height classification: “Two, four, eight, sixteen…

that’s weird! Look! Two times 2 is 4, 4 times 2 is 8,

and 8 times 2 is 16. It goes like a pattern! You have the

2 times 2 equals the 4, the 4 times 2 equals the 8 and

the 8 times 2 equals the 16.” A few minutes later,

Stephanie gave a rule to describe a method for

generating towers, “all you have to do is take the last

number that you had and multiply by two.”

Stephanie’s persistent attempts to make sense

of the problem enabled her to think about the problem

in flexible, yet durable, ways. She used multiple forms

of reasoning to examine the problem from different

angles and was confident in her findings. She was

motivated by her own discoveries and the chance to

create and share her own conjectures.

Milin also used cases to organize towers five cubes

tall. He then went back to the problem and used

simpler problems of towers four cubes tall and three

cubes tall to build on to towers five cubes tall. While

his partners based their arguments on number patterns

and cases, Milin explained his solution using an

inductive argument. Milin’s explanation in each

instance was based on adding on to a shorter tower to

form exactly two towers that were one cube taller (see

Figure 5). For example, when asked to explain why he

created four towers from two towers, Milin explained:

Milin: [pointing to his towers that were one cube

high] Because – for each one of them, you could

add … two more – because there’s … a blue, and a

red- … for red you put a black on top and a red on

top – I mean a blue on top instead of a black. And

blue – you put a blue on top and a red on top – and

you keep doing that.

Figure 5. Milin’s inductive method of generating and organizing towers.

Blue Red

Mary Mueller, Dina Yankelewitz, & Carolyn Maher

39

Later in the year, four students participated in a

group session, during which Stephanie and Milin

presented their solutions to the towers problem. The

next year, in the fifth grade, when the students again

thought about this problem, Stephanie worked with

Matt to find all tower combinations. Initially, they used

trial and error to find as many combinations as they

could. However, they only found twelve combinations.

Stephanie remembered the pattern that she had

discovered the year before.

Stephanie: Well a couple of us figured out a

theory because we used to see a

pattern forming. If you multiply the

last problem by two, you get the

answer for the next problem. But you

have to get all the answers. See, this

didn't work out because we don't have

all the answers here.

Matt: I thought we did.

Stephanie: No. I mean all the answers, all the

answers we can get . . . I don't know

what happened! Because I am

positive it works. I have my papers at

home that say it works.

Persistence. Stephanie and Matt worked to find

more tower combinations, but their search proved

unsuccessful. Stephanie insisted that there were more

combinations.

Stephanie: I don’t know how it worked. I know

it worked. I just don’t know how to

prove it because I’m stumped.

Matt: Steph! Maybe it didn’t work!

Stephanie: Oh no. No. Because I’m pretty sure it

would… I think we goofed because

I’m still sticking with my two thing.

I’m convinced that I goofed, that I

messed up because I know that…

Flexible Thinking. The teacher/researcher

encouraged Stephanie and Matt to discuss the problem

with other students. Stephanie and Matt approached

other groups to see how they had solved the problem.

They visited Milin and Michelle, who had been

discussing the inductive method of finding all tower

combinations. After hearing Michelle’s explanation of

Milin’s method, Matt adopted that method and told

other students about it. Stephanie attempted to explain

Milin’s strategy to others, and, after the

teacher/researcher questioned Stephanie about her

explanation, she returned to her seat to work on

refining her justification. Later in the session, the

teacher/researcher again asked Stephanie to explain her

original prediction of the number of four-tall towers

using the inductive method. This time she

demonstrated a newfound understanding and

enthusiastically presented the solution to the class.

The motivation to make sense of the mathematical

task and the confidence in the power of their own

reasoning exhibited by this young group of students is

evident from the transcripts and narrative above. In

addition, the students exhibited characteristics that are

correlated with intrinsic motivation (e.g., Lepper &

Henderlong, 2000), including perseverance, the length

of time spent on the task, and the students’ flexibility

of thought as they considered and adopted the ideas of

others. Stephanie’s investigations are especially

interesting. Although she had previously solved the

problem and was certain of her previous solution,

Stephanie’s autonomy motivated her to continue to

work on the problem until she was convinced that her

strategy made sense. Rather than accept the solutions

of her classmates, Stephanie persisted in verifying her

model in order to make sense of the mathematics. The

episode described took a full class period, during

which the students were actively engaged in solving

the task. Stephanie insisted on rethinking the problem,

eventually learning from Milin’s explanation, and then

she used her newfound knowledge to reason correctly

about the task and verify her solution. Similar to Dante,

she persisted in understanding why her solution

worked and insisted on reasoning about the problem,

thereby successfully solving and understanding the

mathematical task.

Discussion

Both highlighted tasks, one dealing with fraction

ideas and the other with combinatorics, engaged

students in sense making. The students described in the

above episodes demonstrated confidence in their own

understanding as they justified their solutions in the

presence of their peers, even as their partners offered

alternate representations. It is important to note that the

episodes described above are exemplars of numerous

similar incidents involving many of the students.

Students developed this confidence as they were

encouraged to defend their solutions first in their small

groups and then in the whole class setting. They relied

on their own models and justifications and did not seek

approval from an authority or guidance from the

teacher/researchers for validation of their ideas. These

findings correspond with Francisco and Maher’s

(2005) findings that certain classroom factors promote

mathematical reasoning. The factors identified by

Francisco and Maher include the posing of strands of

challenging, open-ended tasks, establishing student

Sense Making as Motivation

40

ownership of their ideas and mathematical activity,

inviting collaboration, and requiring justification of

solutions to problems, all of which were present in the

episodes above.

On the surface, the two classroom episodes seem

quite different from one another. Specifically, the two

classrooms were comprised of students of different

ages and demographics. In addition, one of the

highlighted tasks focuses on fractional relationships

and the other on combinatorics. However, despite these

dissimilarities, they share many characteristics that

encouraged students to be intrinsically motivated by

the mathematics that they learned.

In both episodes, an environment was created that

facilitated an active, responsible, and engaged

community of learners: Students were encouraged to

share ideas and representations and to listen to,

question, and convince one another of their solutions.

The teacher/researchers facilitated learning while

affording students the opportunity to create and defend

their own justifications. The teacher/researchers

employed careful questioning and support when

needed, but the students were the arbitrators of what

made sense, giving them a sense of autonomy.

Students had opportunities to be successful in building

understanding and in communicating that

understanding through the arguments they constructed

to support their solutions. The resulting discussions

also required students to develop representations of

their thinking in order to express their ideas with

others.

In both groups, students used rich and varied forms

of direct and indirect reasoning. The reasoning that

emerged during these tasks may be explained, at least

in part, by the open-ended nature of the two tasks: The

tasks lent themselves to multiple strategies, and, hence,

they elicited various forms of reasoning. The behaviors

that were observed and the depth of reasoning

exhibited can also be explained as a byproduct of

intrinsic motivation. The students in both groups strove

for conceptual understanding, were persistent in their

endeavors, and displayed confidence in their final

solutions.

Perhaps most importantly, in both episodes

described, the students gained ownership of new

mathematical ideas after being confronted with other

students’ differing understanding of challenging tasks.

In accordance with other research (Deci, Nezdik, &

Sheinman, 1981; Deci & Ryan, 1987; Stefanou et al.,

2004), the students’ autonomy led to their perseverance

to find or defend their solutions and further increased

their intrinsic motivation to make sense of the tasks at

hand. Rather than accept the solutions of their

classmates, both Stephanie and Dante verified their

own strategies using the models they built and, thus,

relied on their own reasoning to gain mathematical

understanding. Dante and Stephanie were both

motivated to rethink their understanding and justify

their solutions after being exposed to the ideas of

others and being challenged by the researchers to make

sense of the task. Dante and Stephanie are

representative of the other students we worked with,

who displayed the ability to think about the solutions

of others and use their own models to make sense of

and acquire these solutions as their own. The

consistency of these behaviors among our diverse

sample suggests that, given the correct environment, all

students can reason mathematically and succeed in

engaging in mathematics.

Based on our analysis, we hypothesize that

motivation and positive dispositions toward

mathematics lead to mathematical reasoning, which, in

turn, leads to understanding. Furthermore, we

constructed a framework to show the relationship

between contextual factors and the chain of events

leading to conceptual understanding (Figure 6). Our

framework begins with the posing of an open-ended,

engaging, and challenging task that the students have

the ability to solve. The task is supported by a carefully

crafted learning environment, carefully planned

facilitator roles and interventions, student

collaboration, and the availability of mathematical

tools.

In the episodes described above, both challenging

tasks allowed students to deploy their own, personal

solution strategy. Both tasks encouraged students to

work collaboratively and utilize mathematical tools. In

addition, the teacher/researcher adopted the role of

facilitator and allowed the students to grapple with

their own strategies as they listened to the strategies of

their peers. Stephanie was given the opportunity to

work on the problem independently and with a partner.

She then listened to the strategies of others before

refining her own solution strategy. Likewise, Dante

was given the space and time to work through his

misconception that the numerator of a fraction could

not be larger than the denominator.

Due to the nature of the task and the environment,

Dante and his peers were motivated to resolve the

discrepancy and find a solution. As with Stephanie,

after listening to the ideas of others, Dante worked to

make sense of the problem himself and create his own

justification. Both students spent over an hour

developing their solutions. Their positive dispositions,

Mary Mueller, Dina Yankelewitz, & Carolyn Maher

41

coupled with intrinsic motivation, gave them the

confidence and desire to find a solution. This is

apparent in the amount of time that they spent

developing their solutions. Both students persevered

even after a classmate had offered a viable solution. In

both episodes, the students’ motivation to succeed at

the tasks at hand led to feelings of self-efficacy and

autonomy. Both Stephanie and Dante took the

initiative to build several models and justifications in

order to justify their solutions, first to themselves and

then to the larger community.

The students relied on reasoning, rather than

memorized facts or the solutions of others, to convince

themselves and others of what made sense. This

reasoning led to their mathematical understanding. In

Figure 6. The relationship between contextual factors, motivation and other events leading to conceptual

understanding.

Autonomy

Tasks

Open-ended

Engaging

Challenging yet attainable

Self-efficacy

Positive

Dispositions

Toward Math

Teacher

Variables

Student

Collaboration

Mathematical

Tools

Environment

Understanding

Mathematical

Reasoning

Intrinsic

Motivation

Sense Making as Motivation

42

particular, Dante proved to himself that 10/9 was a

reasonable fraction and Stephanie was able to defend

her doubling rule.

In summary, in such a learning environment,

students are encouraged to communicate their

understandings of the task, and their ideas are valued

and respected. This respect engenders students’

positive self-concepts in mathematics. At the same

time, students become intrinsically motivated to

succeed at mathematics. Intrinsic motivation fosters

positive dispositions toward mathematics, which, in

turn, encourage students to develop self-efficacy and

mathematical autonomy as they discuss and share their

understandings with their classmates. At the same time,

students enjoy doing mathematics and develop

ownership of their ideas. In such an environment and

with such dispositions, students are more likely to

engage in mathematical reasoning and, thus, acquire

conceptual understanding.

Our framework and research suggest that with

careful attention to developing appropriate and

engaging tasks, a supportive mathematical

environment, and timely teacher questioning, students

can be encouraged to build positive dispositions

towards mathematics in all mathematics classrooms.

These positive dispositions towards mathematics, in

turn, form the ideal conditions for achieving

conceptual understandings of mathematics.

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1 This work was supported in part by grant REC0309062

(directed by Carolyn A. Maher, Arthur Powell and Keith

Weber) from the National Science Foundation. The opinions

expressed are not necessarily those of the sponsoring agency

and no endorsements should be inferred. 2 The research was supported, in part, by National Science

Foundation grants MDR9053597 and REC-9814846. The

opinions expressed are not necessarily of the sponsoring

agency and no endorsement should be inferred.

The Mathematics Educator

2011, Vol. 20, No. 2, 44–50

44

An Alternative Method to Gauss-Jordan Elimination: Minimizing Fraction Arithmetic

Luke Smith & Joan Powell

When solving systems of equations by using matrices, many teachers present a Gauss-Jordan elimination

approach to row reducing matrices that can involve painfully tedious operations with fractions (which I will call

the traditional method). In this essay, I present an alternative method to row reduce matrices that does not

introduce additional fractions until the very last steps. The students in my classes seemed to appreciate the

efficiency and accuracy that the alternative method offered. Freed from unnecessary computational demands,

students were instead able to spend more time focusing on designing an appropriate system of equations for a

given problem and interpreting the results of their calculations. I found that these students made relatively few

arithmetic mistakes as compared to students I tutored in the traditional method, and many of these students who

saw both approaches preferred the alternative method.

When solving systems of equations by using

matrices, many teachers present a Gauss-Jordan

elimination approach to row reducing matrices that can

involve painfully tedious operations with fractions

(which I will call the traditional method). In this essay,

I present an alternative method to row reduce matrices

that does not introduce additional fractions until the

very last steps. As both a teacher using this alternative

method and a tutor working with students instructed in

the traditional method, I have some anecdotal

experience with both. The students in my classes

seemed to appreciate the efficiency and accuracy that

the alternative method offered them. Since they were

freed from unnecessary computational demands, they

were instead able to spend more time focusing on

designing an appropriate system of equations for a

given problem and interpreting the results of their

calculations. I found that these students made relatively

few arithmetic mistakes as compared to students I

tutored in the traditional method, and many of these

students who saw both approaches preferred the

alternative method. I find (and it is likely true for

students) that it takes significantly less time to row

reduce a matrix using the alternative approach than the

traditional approach. Teachers are free to choose a

preferred method (some may want to emphasize

practice with fractions), but I believe this alternative

method to be a strong alternative to the traditional

method since students will perform significantly fewer

computations and teachers can extend the technique to

finding the inverse of matrices.

Many students are not proficient at solving

problems involving fractions, and this lack of

proficiency is not restricted to any one grade band. For

example, when Brown and Quinn (2006) studied 143

ninth graders enrolled in an elementary algebra course

at an upper middle-class school, they found that many

of the students had a lack of experience with both

fraction concepts and computations. In their study,

52% of the students could not find the sum of 5/12 and

3/8, and 58% of the students could not find the product

of 1/2 and 1/4. Unfortunately, students’ difficulty with

fractions can persist into postsecondary education.

When studying elementary education majors at the

University of Arizona, Larson and Choroszy (1985)

found that roughly 25% of the 391 college students

incorrectly added and subtracted mixed numbers when

regrouping was involved. Hanson and Hogan (2000)

studied the computational estimation skills of 77

college students who were majoring in a variety of

disciplines; many of the students in their study

struggled with problems that involved fractions and

became frustrated with the process of finding common

denominators. They noted that a few students in the

lower performing groups added (or subtracted) the

numerators and denominators and did not find common

denominators. Commenting on the lack of

understanding commonly associated with fractions,

Steen (2007) observed that even many adults become

confused if a problem requires anything but the

simplest of fractions.

Luke Smith has several years of experience teaching high school

mathematics. He currently manages a math and science tutoring

lab at Auburn University Montgomery.

Joan Powell is a veteran professor with over 26 years of college

teaching experience.

Luke Smith & Joan Powell

45

The use of matrices to solve systems of equations

has long been a topic in high school and college

advanced algebra and precalculus algebra courses. An

increasing number of colleges and high schools teach

Finite Mathematics, sometimes as a core course option.

This means that increasing numbers of college and

college-bound students are introduced to solving

systems of equations by converting them into matrices

and then row reducing them. For example, at the

university where I teach, childhood education majors

see this topic in a required core course. Fraction skills

may be a reasonable requirement for all of these

students, but I believe this is not the best context for

practicing numerous fraction computations,

particularly for students who are not typically math or

science majors. Indeed, students’ difficulties with

fractions lead many instructors to carefully pick

matrices that do not involve fractions during the

intermediate steps of the traditional approach to row-

reducing a matrix. However, the alternative method

discussed below is similar to traditional Gauss-Jordan

elimination but allows instructors to use any system of

linear equations over the rational numbers because it

prevents new fractions from appearing until the very

last steps. Furthermore, the alternative method

involves a similar number of computations as the

traditional method, which decreases the likelihood of

arithmetic mistakes.

When deciding which approach students should

learn in order to row reduce matrices, teachers need to

consider their motivation for showing students how to

row reduce matrices. Typically, we want our students

to be able to solve resource allocation problems,

geometric problems, or other types of applications by

finding the values of the variables in a system of

equations and then correctly interpreting the results of

their findings. In other words, we are interested in

showing our students how to solve problems where

row reduction of matrices is an appropriate strategy.

Therefore, if we have two mathematically sound

approaches for finding the values of the variables, one

whose computational demands may distract from the

main concept and the other that involves fewer

computations and is less distracting, it seems

reasonable to show students the method that will free

them to focus on setting up the problem and

interpreting the results rather than being immersed in

the intermediate calculations. Such an instructional

decision aligns with the National Council of Teachers

of Mathematics (2000) teaching principle (2000) that

advocates the skillful selection of teaching strategies to

communicate mathematics.

The alternative method is not a new approach, but

after reviewing many Finite Mathematics and Linear

Algebra textbooks from a variety of publishers, I found

that the vast majority of the texts do not clearly present

to students with a method of solving a system of

equations without incurring fractions in the

intermediate steps (Goldstein, Schneider, & Siegel,

1998; Poole, 2003; Rolf, 2002; Uhlig, 2002; Young,

Lee, & Long, 2004). Even the texts used at my

university (Barnett, Ziegler, & Byleen, 2005; Lay,

2006) do not demonstrate the alternative method.

Warner and Costernoble (2007), Shifrin and Adams,

(2002), and Lial, Greenwell, and Ritchey (2008) were

the only texts that I found that clearly presented the

alternative method. In all of the aforementioned books

no characteristics seemed to predict whether or not the

alternative method was presented and they all covered

roughly the same concepts that are traditionally

presented in Finite Mathematics and Linear Algebra

courses. For the benefit of students and teachers who

have only been exposed to the traditional Gaussian

methods of row-reduction, the remaining portion of the

article develops the alternative technique. The

following paragraphs describe operations with matrices

of the type provided below (Figure 1).

33,32,31,3

23,22,21,2

13,12,11,1

kaaa

kaaa

kaaa

Figure 1. A typical 33× augmented matrix.

The most common method that students are taught

Gauss-Jordan-elimination for solving systems of

equations is first to establish a 1 in position a1,1 and

then secondly to create 0s in the entries in the rest of

the first column. The student then performs the same

process in column 2, but first a 1 is established in

position a2,2 followed secondly by creating 0s in the

entries above and below. The process is repeated until

the coefficient matrix (Figure 1) is transformed into the

identity matrix, where 1s are along the main diagonal

and 0s are in all other entries (Barnett, Ziegler &

Byleen, 2005). Some teachers use a variation of Gauss-

Jordan elimination called back-substitution that

simplifies the process somewhat for solving systems of

equations; however, back-substitution can not be used

to find inverses of matrices.

The traditional approach of finding first the 1s for

each of the diagonal entries and secondly finding the 0s

for the remaining elements in each corresponding

column becomes extremely cumbersome when

Gauss-Jordan Elimination

46

fractions are involved. Students who are not

comfortable or proficient with fractions may become

frustrated with these types of problems. Asking

instructors to teach students a method that they are

only able to use to solve a limited class of problems

does those students a disservice. The alternative

Gaussian approach where 1s on the diagonal are not

obtained until the very end of the problem is a nice

alternative to the traditional method. In my opinion, the

strength of this approach is that (a) no new fractions

are introduced until the very last steps and (b) this

process can still be implemented to find the inverse of

a matrix (in contrast to the back-substitution method).

To set up this method, I review an approach for

solving a system of two equations in two variables. For

this smaller system, teachers commonly teach the

addition method, which relies on multiplying each

equation by the (sometimes oppositely signed)

coefficients in the other equation and then adding the

two equations to eliminate the target variable. Consider

the following problem (Example 1 in Figure 2).

Step 1: We can choose to eliminate either the x or y variable. For this example, we

will eliminate the x variable. 152

8 23

−=−

=+

yx

yx

Step 2: To eliminate the x variable, we will multiply the top row (R1) by 2 and the

bottom row (R2) by -3. Then we will add the two equations together to create a new

equation.

Note: We know that we are proceeding in the correct direction because we

successfully eliminated the x variables when we added the equations together.

152

8 23

−=−

=+

yx

yx

( )( )32

−

3

16

15

4

6

6

=

=

+

+

− y

y

x

x

1919 =y

Step 3: At this point, we simply solve for y and substitute our solution back into

either equation to solve for x, checking both in the other equation. 2

1

=

=

x

y

Figure 2. Solving Example 1, a 2×2 linear system.

The process of eliminating the x variable in the

above problem (Figure 2) by producing opposite

coefficients of x is used in the alternative method for

row-reducing matrices. Next, I show how to use the

above idea to solve a typical system of n equations

with n variables without incurring any fractions

(Example 2 in Figures 3a and b).

Step 1: Recopy from the original system of equations into augmented matrix form.

12

7

5

2

2

3

4

4

2

3

−

=

=

=

−

+

+

+

−

+

z

z

z

y

y

y

x

x

x

−

−

−

12

7

5

2

1

2

1

3

4

4

2

3

Step 2: Multiply R1 and R2 in such a way that you create oppositely signed common

multiples in entries a1,1 and a2,1 as shown below.

)3(7132

)2(5243

−−

−

Adding and then substituting the sum for row 2 results in a 0 in entry a2,1.

311170

21396

10486

−−−

−−+

−−−−

−

−

−−

12

31

5

2

1

2

1

17

4

4

0

3

Figure 3a. Step-by-step process for solving Example 2 using the alternative Gaussian approach. Note: The process in the left column produces the matrix in the right column for each step.

Luke Smith & Joan Powell

47

Step 3: Multiply R1 and R3 in such a way that you create oppositely signed common

multiples in entries a1,1 and a3,1.

)3(12214

)4(5243

−

−

Adding and then substituting the sum for row 3 results in a 0 in entry a3,1.

1614130

366312

2081612

−−

−+

−−−−

−

−

−

−

−

16

31

5

14

1

2

13

17

4

0

0

3

Figure 3b. Step-by-step process for solving Example 2 using the alternative Gaussian approach.

It is not important what values are produced on

the main diagonal until the last step of this

process. So, I will not divide the top row by 3 to

get a value of 1 in position a1,1 which would

produce fractions in this intermediate step. Now, I

will must establish 0s in the entries above and

below a2,2 (Figure 4).

Step 4: Multiply R1 and R2 in such a way that you create oppositely signed common

multiples in entries a1,2 and a2,2.

)4(311170

)17(5243

−−−

Adding and then substituting the sum for row 1 results in a 0 in entry a1,2.

3930051

1244680

85346851

−

−−−+

−−

−−−

−

1614130

311170

3930051

Step 5: Multiply R2 and R3 in such a way that you create oppositely signed common

multiples in entries a2,2 and a3,2.

)17(1614130

)13(`31170

−−

−−−−

Adding and then substituting the sum for row 3 results in a 0 in entry a3,2.

393000

12442210

85132210

−

−−−+

−

−−−

−

67522500

311170

3930051

Figure 4. A continuation of the solution of Example 2 using the alternative Gaussian approach.

Having now established 0s in the appropriate

positions in columns 1 and 2 (Figure 4), we repeat the

process to establish 0s in column 3. However, it would

be useful at this point to reduce the numbers in row 3

before we establish the last set of 0s (See optional step

in Figure 5).

Gauss-Jordan Elimination

48

Optional Step: Since 675 is a multiple of -225, simplifying R3 by dividing the entire

row by “-225” (or multiplying by the reciprocal) will make the arithmetic easier from

this point on:

)225(67522500 1−− → 3100 −

Note: Dividing a row by a common factor simplifies the arithmetic by producing

smaller values for each entry.

−

−−−

−

3100

311170

3930051

Step 6: Multiply R1 and R3 in such a way that you create oppositely signed common

multiples in entries a1,3 and a3,3.

)30(3100

)1(3930051

−−

−

Adding and then substituting the sum for R1 results in a 0 in entry a1,3.

510051

903000

3930051

−+

−

−

−−−

3100

311170

510051

Step 7: Multiply R2 and R3 in such a way that you create oppositely signed common

multiples in entries a2,3 and a3,3

)1(3100

)1(311170

−

−−−

. And then substituting the answer in for R2 results in a 0 in entry a2,3.

340170

3100

311170

−−

−+

−−−

−

−−

3100

340170

510051

Final Step: The last step in this process is to divide each row by its first non-zero

entry (multiply by its reciprocal), in this case the values on the main diagonal.

)1(3100

)17(340170

)51(5100511

1

−

−−− −

−

Thus, x = 1, y = 2, z = -3.

−

−−

3100

340170

510051

Figure 5. Concluding steps for solving Example 2 using the alternative Gaussian approach.

Showing students how to solve systems of linear

equations using the alternative version of Gaussian

elimination allows them to avoid becoming inundated

with fraction computations. For Example 2, if the

operation between any two integers counts as one

computation, then using the traditional method to solve

the system of equations results in 58 computations; the

alternative method results in 46 computations. Because

the alternative method produced 21% fewer

computations than the traditional method, students are

less likely to get lost in the intermediate computations

and are more able to focus on the overall purpose of

the method.

Note again that the alternative method can be used

for systems of rational equations and can be followed

fairly mechanically for rational systems containing n

equations with n variables. In the event that the system

of equations has infinitely many solutions or no

solution, the idea behind the alternative method is the

same: get 0’s for entries above and below the leading

non-zero entry in each row, then divide each row by

the value of this non-zero entry. The following

example illustrates this point (Example 3 in Figure 6).

Luke Smith & Joan Powell

49

Step 1: Recopy from the original system of equations into augmented matrix form.

12

7

513

8

6

4

4

9

6

=

=

=

−

+

+

+

+

z

z

y

y

y

x

x

x

− 121812

7069

51346

Step 2: Multiply R1 and R2 in such a way that you create oppositely signed common

multiples in entries a1,1 and a2,1 as shown below.

)2(7069

)3(51346 −

Adding and then substituting the answer for R2 results in a 0 in entry a2,1.

13900

1401218

15391218

−−

+

−−−−

−

−−

121812

13900

51346

Step 3: Multiply R1 and R3 in such a way that you create oppositely signed common

multiples in positions a1,1 and a3,1.

)1(121812

)2(51346

−

−

Adding and then substituting the answer for R3 results in a 0 in entry a3,1.

22700

121812

1026812

−

−+

−−−−

−

−−

22700

13900

51346

Figure 6. Beginning steps of solution for Example 3.

Looking at the preceding matrix, we have a 0 in

position a2,2, so I cannot use it to eliminate the 4 in

position a1,2; and since I have a 0 in position a3,2, I do

not benefit from switching row 2 and row 3. Thus, I

can focus our attention on -39 in position a2,3. (I could

also focus our attention on -27, but the end result

would not change). The objective is still the same: get

“0’s” in the entries above and below -39 (Figures 7a

and 7b).

Step 4: Multiply R1 and R2 in such a way that you create oppositely signed common

multiples in positions a1,3 and a2,3.

)1(13900

)3(51346

−−

Adding and then substituting the answer in for R1 results in a 0 in position a1,3.

1401218

13900

15391218

−−+

−

−−

22700

13900

1401218

Figure 7a. Continuation of solution for Example 3.

Gauss-Jordan Elimination

50

Step 5: Multiply R2 and R3 in such a way that you create oppositely signed common

multiples in positions a2,3 and a3,3.

)39(22700

)27(13900

−−

−−

Adding and then substituting the answer in for R3 results in a 0 in position a3,3..

105000

78105300

27105300

−

−+

−−

−

−−

105000

13900

1401218

Figure 7b. Continuation of solution for Example 3.

Based on the previous matrix (Figures 7a and 7b)

we can see that the system of equations does not have a

solution since row 3 states that 0 = -105 (clearly a false

statement). If we wanted to finish simplifying the

matrix, we would divide rows 1 and 2 by the values of

their leading non-zero entries to get the following

(Figure 8).

Final Step:

R1 →÷18 R1

R2 →−÷ 39 R2

−105000

100

01

391

97

32

Figure 8. Final steps of solution for Example 3.

I hope that those who have not considered this

alternative method will see the possible advantages for

themselves and their students. First, this method may

increase the accessibility of matrix material for

students with weaknesses in fractions. Next, the

method has the potential to increase the speed and

accuracy of computations for students and teachers

alike by the substitution of integer computations for

rational number computations. I have found that some

students avoid fractions by using decimal

approximations, sacrificing precision. However, with

this method, teachers can still require the precision of

fractional solutions without the excessive mire of

fractions, potentially encouraging more student effort

and success. Finally, teachers who are wary of

requiring extensive fractional computations may be

freed by this method to have a greater flexibility in

problem selection.

REFERENCES

Barnett, R., Ziegler, M., & Byleen, K. (2005). Applied mathematics

for business and economics, life sciences, and social sciences.

Upper Saddle River, NJ: Pearson Prentice Hall.

Brown, G., & Quinn, R. (2006). Algebra students’ difficulty with

fractions. Australian Mathematics Teacher, 62(4), 28–40.

Goldstein, L., Schneider, D., & Siegel, M. (1998). Finite

mathematics and its applications, 6th ed. Upper Saddle River,

NJ: Prentice Hall.

Hanson, S., & Hogan, T. (2000). Computational estimation skill of

college students. Journal for Research in Mathematics

Education, 31, 483–499.

Larson, C., & Choroszy, M. (1985). Elementary education majors’

performance on a basic mathematics test. Retrieved from

http://www.eric.ed.gov/.

Lay, D. (2006). Linear algebra and its applications, 3rd ed.

Boston, MA: Pearson Education.

Lial, M., Greenwell, R., & Ritchey, N. (2008). Finite mathematics,

9th ed. Boston: Pearson Education.

National Council of Teachers of Mathematics (2000). Principles

and standards for school mathematics. Reston, VA.: Author.

Poole, D. (2003). Linear algebra: A modern introduction. Pacific

Grove, CA: Thompson Learning.

Rolf, H. (2002). Finite mathematics, 5th ed. Toronto: Thompson

Learning.

Shifrin, T., & Adams, M. (2002). Linear algebra: A geometric

approach. New York, NY: W. H. Freeman and Company.

Steen, L. (2007). How mathematics counts. Educational

Leadership, 65(3), 8–14.

Warner, S., & Costernoble, S. (2007). Finite mathematics, 4th ed.

Pacific Grove, CA: Thompson Learning.

Young, P., Lee, T., Long, P., & Graening, J. (2004). Finite

mathematics: An applied approach, 3rd ed. New York, NY:

Pearson Education.

52

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In this Issue,

Guest Editorial… From the Common Core to a Community of All Mathematics Teachers SYBILLA BECKMANN You Asked Open-Ended Questions, Now What? Understanding the Nature of Stumbling Blocks in Teaching Inquiry Lessons NORIYUKI INOUE & SANDY BUCZYNSKI

Secondary Mathematics Teacher Differences: Teacher Quality and Preparation in a New York City Alternative Certification Program BRIAN R. EVANS

Sense Making as Motivation in Doing Mathematics: Results From Two Studies MARY MUELLER, DINA YANKELEWITZ, & CAROLYN MAHER An Alternative Method to Gauss-Jordan Elimination: Minimizing Fraction Artihmetic LUKE SMITH & JOAN POWELL

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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.