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____ 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 [email protected]
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.
References
Bryk, A. S., Sebring, P. B., Allensworth, E., Luppescu, S., &
Easton, J. Q. (2010). Organizing schools for improvement:
Lessons from Chicago. Chicago: The University of Chicago
Press.
Collins, A. (2010). The Science of Teacher Development.
Education Week, 30(13), 27, 36.
Colvin, G. (2008). Talent is overrated. New York, NY: Penguin.
Conference Board of the Mathematical Sciences (2001). The
mathematical education of teachers (In “Issues in
Mathematics Education” series, Vol. 11). Washington, DC:
Author. (Available at
http://www.cbmsweb.org/MET_Document/)
Corey, D. L., Peterson, B. E., Lewis, B. M., & Bukarau, J. (2010).
Are there any places that students use their heads? Principles
of high-quality Japanese mathematics instruction. Journal for
Research in Mathematics Education, 41, 438-478.
Cross, C. T., Woods, T. A., & Schweingruber, H. (Eds.). (2009).
Mathematics learning in early childhood, paths toward
excellence and equity. Washington, DC: The National
Academies Press.
Darling-Hammond, L., Wei, R. C., Andree, A., Richardson, N., &
Orphanos, S. (2009). Professional Learning in the Learning
Profession: A Status Report on Teacher Development in the
United States and Abroad. Dallas, TX: National Staff
Development Council and The School Redesign Network at
Stanford University.
Deci, E. L., & Ryan, R. M. (2008a). Facilitating optimal motivation
and psychological well-being across life's domains. Canadian
Psychology, 49, 14-23.
Deci, E. L., & Ryan, R. M. (2008b). Self-determination theory: A
macrotheory of human motivation, development, and health.
Canadian Psychology, 49, 182-185.
Duncan, A. (June, 2009). Robust data gives us the roadmap to
reform. Address by the Secretary of Education to The Fourth
Annual Institute of Education Sciences Research Conference,
Washington, DC. (Available at
http://www2.ed.gov/news/speeches/2009/06/06082009.pdf)
Ericsson, K. A., Krampe, R. T., & Tesch-Roemer, C. (1993). The
role of deliberate practice in the acquisition of expert
performance. Psychological Review, 100, 363-406.
Greenberg, J., & Walsh, K. (2008). No common denominator: The
preparation of elementary teachers in mathematics by
America’s education schools Washington, DC: National
Council on Teacher Quality.
Greene, D., & Lepper, M. R. (1974). Effects of extrinsic rewards
on children's subsequent intrinsic interest. Child Development,
45, 1141-1111.
Hearing on FY 2011 Dept. of Education Budget: Hearing before
the Subcommitee on Labor, Health and Human Services,
Education, and Related Agencies, of the Senate Committee on
Appropriations, 111th Cong. (2010) (testimony of Arne
Duncan). (Available at http://appropriations.senate.gov/sc-
labor.cfm)
Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B.,
Hollingsworth, H., Jacobs, J., … Stigler, J. (2003). Teaching
mathematics in seven countries: Results from the TIMSS 1999
Video Study. Washington, DC: U.S. Department of Education,
National Center for Education Statistics.
Sybilla Beckmann
9
Jacobs, J. K., & Morita, E. (2002). Japanese and American
Teachers' Evaluation of Videotaped Mathematics Lessons.
Journal for Research in Mathematics Education, 33, 154-175.
Lewis, C. C. (2002). Lesson study: A handbook of teacher-led
instructional change. Philadelphia, PA: Research for Better
Schools.
Lewis, C. C. (2010). A public proving ground for standards-based
practice: Why we need it, what it might look like. Education
Week, 30(3), 28–30.
Lutzer, D. J., Rodi, S. B., Kirkman, E. E., & Maxwell, J. W.
(2007). Statistical abstract of undergraduate programs in the
mathematicalsSciences in the United States, Fall 2005 CBMS
Survey. Washington, DC: Conference Board of the
Mathematical Sciences .
National Commission on Excellence in Education. (1983). A nation
at risk: The imperative educational reform (Report No. 065-
000-00177-2). Washington, DC: U.S. Department of
Education.
National Commission on Mathematics and Science Teaching for
the 21st Century. (2000). Before it’s too late: A report to the
nation from the National Commission on Mathematics and
Science Teaching for the 21st Century. Washington, DC: U.S.
Department of Education.
National Council of Teachers of Mathematics. (2000). Principles
and standards for school mathematics. Reston, VA: Author.
National Governors Association Center for Best Practices and the
Council of Chief State School Officers. (2010). Common core
state standards for mathematics. Retrieved July, 2010, from
http://www.corestandards.org/the-standards/mathematics
National Mathematics Advisory Panel. (2008). Foundations for
success: The final report of the National Mathematics
Advisory Panel. Washington, DC: U.S. Department of
Education.
O.C.G.A. § 43-10-9 (LexisNexis, 2011).
Penuel, W. R., Riel, M., Krause, A. E., & Frank, K. A. (2009).
Analyzing teachers' professional interactions in a school as
social capital: A social network approach. Teachers College
Record, 111, 124–163.
Pink, D. H. (2009). Drive, the surprising truth about what
motivates us. New York, NY: Penguin.
Seymour, E., & Hewitt, N. M. (1997). Talking about leaving: Why
undergraduates leave the sciences. Boulder, CO: Westview
Press.
Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas
from the world’s teachers for improving education in the
classroom. New York, NY: Free Press.
Undergraduate science, math, and engineering education: What's
working?, House of Representatives, 109th Cong. 14 (2006)
(testimony of Dr. Elaine Seymore).
1 Additional references can be found at the website
http://www.psych.rochester.edu/SDT/index.php.
The Mathematics Educator 2011, Vol. 20, No. 2, 10–23
10
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
Noriyuki Inoue & Sandy Buczynski
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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.
References
Ball, D. L., Hill, H. C, & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29(3), 14–46.
Berliner, D. C. (1994). Expertise: The wonder of exemplary performances. In J. N. Mangieri & C. C. Block (Eds.), Creating powerful thinking in teachers and students: Diverse
perspectives (pp.161–186). Fort Worth, TX: Harcourt Brace.
Bramwell-Rejskind, F. G., Halliday, F., & McBride, J. B. (2008). Creating change: Teachers’ reflections on introducing inquiry teaching strategies. In B. M. Shore, M. W. Aulls, M. A. B., Delcourt (Eds.), Inquiry in education, volume 2: Overcoming barriers to successful implementation (pp. 207–234). New York: Erlbaum.
Buczynski, S., & Hansen, B. (2010). Impact of professional development on teacher practice: Uncovering connections. Teaching and Teacher Education, 26, 599–607.
Chen, A., & Ennis, C. D. (1995). Content knowledge transformation: An examination of the relationship between content knowledge and curricula. Teaching and Teacher Education, 4, 389–401.
Clements, D. H. (1997). In my opinion: (Mis?) Constructing constructivism. Teaching Children Mathematics, 4, 198–200.
Cobb, P., & Bausersfeld, H. (1995). Introduction: The coordination of psychological and sociological perspectives in mathematics education. In P. Cobb & H. Bausersfeld (Eds.), The emergence of mathematical meaning (pp.1-16). Hillsdale, NJ: Erlbaum.
Cobb, P., Stephan, M., McClain, K., & Gravemijer, K. (2001). Participating in classroom mathematical practices. The Journal of the Learning Sciences, 10, 113–163.
Cobb, P., & Yackel, E. (1998). A constructivist perspective on the culture of the mathematics classroom. In F. Seeger, J. Voigt, & U. Waschescio (Eds.), The culture of the mathematics classroom. City Needed: Cambridge University Press.
Cooper, B. (1994). Authentic testing in mathematics? The boundary between everyday and mathematical knowledge in national curriculum testing in English schools. Assessment in Education: Principles, Policy & Practice, 11, 143–166.
Cooper, B. (1998). Using Bernstein and Bourdieu to understand children's difficulties with “realistic” mathematics testing: An exploratory study. International Journal of Qualitative Studies in Education, 11, 511–532.
Davis, R. B., Maher, C. A., & Noddings, N. (1990) Constructivist views on the teaching and learning of mathematics. Journal for Research in Mathematics Education, 1, 1–6.
Fernandez, C. & Yoshida, M. (2004). Lesson study: A Japanese approach to improving mathematics teaching and learning. Mahwah, NJ: Lawrence Erlbaum.
Foss, D., & Kleinsasser, R. (1996) Preservice elementary teachers’ views of pedagogical and mathematical content knowledge, Teaching and Teacher Education, 12, 429–442.
Greer, B. (1997). Modeling reality in mathematics classrooms: The case of word problems. Learning & Instruction, 7, 293–307.
Green, S. K., & Gredler, M. E. (2002). A review and analysis of constructivism for school-based practice. School Psychology Review, 31, 53–71.
Inoue, N. (2005). The realistic reasons behind unrealistic solutions: The role of interpretive activity in word problem solving. Learning and Instruction, 15, 69– 83.
Inoue, N. (2009). Rehearsing to teach: Content-specific deconstruction of instructional explanations in preservice teacher trainings. Journal of Education for Teaching, 35, 47–60.
Inoue, N. (2011). Consensus building for negotiation of meaning: Zen and the art of neriage in mathematical inquiry lessons through lesson study. Journal of Mathematics Teacher Education, 14, 5–23.
Noriyuki Inoue & Sandy Buczynski
23
Kinach, B. M. (2002a). A cognitive strategy for developing pedagogical content knowledge in the secondary mathematics methods course: Toward a model of effective practice. Teaching and Teacher Education, 18, 51–71.
Kinach, B. M. (2002b). Understanding and learning-to-explain by representing mathematics: Epistemological dilemmas facing teacher educators in the secondary mathematics “methods” course. Journal of Mathematics Teacher Education, 5, 153–186.
Klein, M. (1997) Looking again at the ‘supportive’ environment of constructivist pedagogy. Journal of Education for Teaching, 23, 277–292.
Klein, M. (2004). The premise and promise of inquiry based mathematics in preservice teacher education: A poststructuralist analysis. Asia-Pacific Journal of Teacher Education, 32, 35–47.
Knuth, E. J. (2002). Teachers’ conceptions of proof in the context of secondary school mathematics. Journal of Mathematics Teacher Education, 5, 61–88.
Leinhardt, G. (1989). Math lessons: A contrast of novice and expert competence. Journal for Research in Mathematics Education, 20, 52–75.
Leinhardt, G. (2001). Instructional explanations: A commonplace for teaching and location for contrast. In V. Richardson (Ed.), Handbook of research on teaching (4th ed., pp.333–357). Washington, DC: American Educational Research Association.
Lubienski, S. (2007). What we can do about achievement disparities? Educational Leadership, 65, 54–59.
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in
China and the United States. Mahwah, NJ: Lawrence Erlbaum.
Mapolelo, D. C. (1998). Do pre-service primary teachers who excel in mathematics become good mathematics teachers? Teaching and Teacher Education, 15, 715–725.
Manconi, L., Aulls, M. W., & Shore, B. M. (2008). Teachers’ use and understanding of strategy in inquiry instruction. In B. M. Shore, M. W. Aulls, & M. A. B., Delcourt (Eds.), Inquiry in education, volume 2: Overcoming barriers to successful
implementation (pp. 247–269). New York: Lawrence Erlbaum.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
National Research Council (NRC). (2000). Inquiry and the national science education standards: A guide for teaching and
learning. Washington, D.C.: National Academy Press.
Ormrod, J. R. (2003). Educational psychology: Developing learners (5th Ed.). Upper Saddle River, NJ: Merrill.
Parsons, R., Hinson, S. L., & Sardo-Brown, D. (2000). Educational psychology: A practitioner-researcher model for teaching. Belmont, CA: Wadsworth.
Robinson, A., & Hall, J. (2008). Teacher models of teaching inquiry. In B. M. Shore, M. W. Aulls, & M. A. B., Delcourt (Eds.), Inquiry in education, volume 2: Overcoming barriers to successful implementation (pp. 235–246). New York: Lawrence Erlbaum.
Stonewater, J. K, (2005). Inquiry teaching and learning: The best math class study. School Science and Mathematics, 105, 36–47.
Segall, A. (2004). Revisiting pedagogical content knowledge: The pedagogy of content/the content of pedagogy. Teaching and Teacher Education, 20, 489–504.
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15, 4–14.
Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57, 1–22.
Tillema, M., & Knol, W. (1997). Collaborative planning by teacher educators to promote belief changes in their students. Teachers and Teaching: Theory and Practice, 3, 29–46.
Van de Walle, J. (2004). Elementary and middle school mathematics: Teaching developmentally. Boston, MA: Pearson.
Verschaffel, L., Greer, B., & De Corte, E. (2000). Making sense of word problems. Lisse, The Netherlands: Swets & Seitlinger.
Voigt, J. (1996). Negotiation of mathematical meaning in classroom processes: Social interaction and learning mathematics. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin & B. Greer. (Eds.), Theories of mathematical learning. Mahwah, NJ: Erlbaum.
Wallace C. S., & Kang, N. H. (2004). An investigation of experienced secondary science teachers’ beliefs about inquiry: An examination of competing belief sets. Journal of Research in Science Teaching, 41, 936–960.
Wang, J-R, & Lin, S-W. (2008). Examining reflective thinking: A study of changes in methods students’ conceptions and understandings of inquiry teaching. International Journal of Science and Mathematics Education, 6, 459–479.
Warfield, J., Wood, T., & Lehman, J. D. (2005). Autonomy beliefs and the learning of elementary mathematics teachers. Teaching and Teacher Education, 21, 439–456.
Wiggins, G., & McTighe, J. (2005). Understanding by design. Alexandria, VA: Association for Supervision and Curriculum Development.
Woolfolk, A. E. (2006). Educational psychology (10th Ed.). Boston, MA: Allyn & Bacon.
Yin, R. (2003). Case study research: Design and methods (3rd ed.). Thousand Oaks, CA: Sage. Mathematics Curricula (pp. 457-468). Mahwah, NJ: Erlbaum.
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.
The Mathematics Educator
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.
References
Aiken, L. R. (1970). Attitudes toward mathematics. Review of
Educational Research, 40, 551–596.
Aiken, L. R. (1974). Two scales of attitude toward mathematics.
Journal for Research in Mathematics Education, 5, 67–71.
Aiken, L. R. (1976). Update on attitudes and other affective
variables in learning mathematics. Review of Educational
Research, 46, 293–311.
Amato, S. A. (2004). Improving student teachers’ attitudes to
mathematics. Proceedings of the 28th Annual Meeting of the
International Group for the Psychology of Mathematics
Education (IGPME), 2, 25–32. Bergen, Norway: IGPME.
Angle, J., & Moseley, C. (2009). Science teacher efficacy and
outcome expectancy as predictors of students’ End-of-
Instruction (EOI) Biology I test scores. School Science and
Mathematics, 109, 473–483.
Ashton, P., & Webb, R. (1986). Making a difference: Teachers’
sense of efficacy and student achievement. New York:
Longman.
Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics
for teaching: Who knows mathematics well enough to teach
third grade, and how can we decide? American Educator, 14–
17, 20–22, & 43–46.
Bandura, A. (1986). Social foundations of thought and action: A
social cognitive theory. Englewood Cliffs, NJ: Prentice Hall.
Boyd, D. J., Grossman, P., Lankford, H., Loeb, S., Michelli, N. M.,
& Wyckoff, J. (2006). Complex by design: Investigating
pathways into teaching in New York City schools. Journal of
Teacher Education, 57, 155–166.
Boyd, D., Grossman, P., Lankford, H., Loeb, S., & Wyckoff, J.
(2006). How changes in entry requirements alter the teacher
workforce and affect student achievement. Education Finance
and Policy, 1, 176–216.
Boyd, D., Lankford, S., Loeb, S., Rockoff, J., & Wyckoff, J.
(2007). The narrowing gap in New York City qualifications
and its implications for student achievement in high poverty
schools (CALDER Working Paper 10). Washington, DC:
National Center for Analysis of Longitudinal Data in
Education Research. Retrieved August 26, 2008, from
http://www.caldercenter.org/PDF/1001103_Narrowing_Gap.p
df.
Charalambous, C. Y., Panaoura, A., & Philippou, G. (2009). Using
the history of mathematics to induce changes in preservice
teachers’ beliefs and attitudes: Insights from evaluating a
teacher education program. Educational Studies in
Mathematics, 71, 161–180.
Czerniak, C. M., & Schriver, M. (1994). An examination of
preservice science teachers’ beliefs. Journal of Science
Teacher Education, 5, 77–86.
Darling-Hammond, L. (1994). Who will speak for the children?
How "Teach for America" hurts urban schools and students.
Phi Delta Kappan, 76(1), 21–34.
Darling-Hammond, L. (1997). The right to learn: A blueprint for
creating schools that work. San Francisco, CA: Jossey-Bass.
Darling-Hammond, L., Holtzman, D. J., Gatlin, S. J., & Heilig, J.
V. (2005). Does teacher preparation matter? Evidence about
Mathematics Teacher Differences
32
teacher certification, Teach for America, and teacher
effectiveness. Education Policy Analysis Archives, 13(42), 1–
32.
Eide, E., Goldhaber, D., & Brewer, D. (2004). The teacher labour
market and teacher quality. Oxford Review of Economic
Policy, 20, 230–244.
Enochs, L. G., & Riggs, I. M. (1990). Further development of an
elementary science teaching efficacy belief instrument: A
preservice elementary scale. School Science and Mathematics,
90, 695-706.
Enochs, L. G., Smith, P. L., & Huinker, D. (2000). Establishing
factorial validity of the Mathematics Teaching Efficacy
Beliefs Instrument. School Science and Mathematics, 100,
194–202.
Ernest, P. (1989). The knowledge, beliefs and attitudes of the
mathematics teacher: A model. Journal of Education for
Teaching, 15, 13–33.
Evans, B. R. (2009). First year middle and high school teachers’
mathematical content proficiency and attitudes: Alternative
certification in the Teach for America (TFA) program. Journal
of the National Association for Alternative Certification
(JNAAC), 4(1), 3–17.
Evans, B. R. (in press). Content knowledge, attitudes, and self-
efficacy in the mathematics New York City Teaching Fellows
(NYCTF) program. School Science and Mathematics Journal.
Gibson, S., & Dembo, M. H. (1984). Teacher efficacy: A construct
validation. Journal of Educational Psychology, 76, 569–582.
Haney, J. J., Lumpe, A.T., Czerniak, C.M., & Egan, V. (2002).
From beliefs to actions: The beliefs and actions of teachers
implementing change. Journal of Science Teacher Education,
13, 171–187.
Humphrey, D. C., & Wechsler, M. E. (2007). Insights into
alternative certification: Initial findings from a national study.
Teachers College Record, 109, 483–530.
Kane, T. J., Rockoff, J. E., & Staiger, D. O. (2006). What does
certification tell us about teacher effectiveness? Evidence from
New York City. Working Paper No. 12155, National Bureau
of Economic Research.
Laczko-Kerr, I., & Berliner, D. C. (2002). The effectiveness of
“Teach for America” and other under-certified teachers on
student academic achievement: A case of harmful public
policy. Education Policy Analysis Archives, 10(37). Retrieved
August 26, 2008, from http://epaa.asu.edu/epaa/v10n37/.
Ma, L. (1999). Knowing and teaching elementary mathematics.
Mahwah, NJ: Lawrence Erlbaum Associates Publishers.
Ma, X., & Kishor, N. (1997). Assessing the relationship between
attitude toward mathematics and achievement in mathematics:
A meta-analysis. Journal for Research in Mathematics
Education, 28, 26–47.
National Council of Teachers of Mathematics. (2000). Principles
and standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (2005). Highly
qualified teachers. NCTM Position Statement. Retrieved
February 18, 2009, from
http://www.nctm.org/about/content.aspx?id=6364.
New York City Teaching Fellows. (2008). Retrieved August 26,
2008, from http://www.nyctf.org/.
New York City Teaching Fellows. (2010). Retrieved May 25, 2010,
from http://www.nyctf.org/.
Raymond, M., Fletcher, S. H., & Luque, J. (2001). Teach for
America: An evaluation of teacher differences and student
outcomes in Houston, Texas. Stanford, CA: The Hoover
Institution, Center for Research on Education Outcomes.
Soodak, L. C., & Podell, D. M. (1997). Efficacy and experience:
Perceptions of efficacy among preservice and practicing
teachers. Journal of Research and Development in Education,
30, 214–221.
Stein, J. (2002). Evaluation of the NYCTF program as an
alternative certification program. New York: New York City
Board of Education.
Suell, J. L., & Piotrowski, C. (2007). Alternative teacher education
programs: A review of the literature and outcome studies.
Journal of Instructional Psychology, 34, 54–58.
Swars, S. L., Daane, C. J., & Giesen, J. (2006). Mathematics
anxiety and mathematics teacher efficacy: What is the
relationship in elementary preservice teachers? School Science
and Mathematics, 106, 306–315.
Swars, S., Hart, L. C., Smith, S. Z., Smith, M. E., & Tolar, T.
(2007). A longitudinal study of elementary pre-service
teachers’ mathematics beliefs and content knowledge. School
Science and Mathematics, 107, 325–335.
Tapia, M. (1996). The attitudes toward mathematics instrument.
Paper presented at the Annual Meeting of the Mid-South
Educational Research Association, Tuscaloosa, AL.
Trice, A. D., & Ogden, E. D. (1986). Correlates of mathematics
anxiety in first-year elementary school teachers. Educational
Research Quarterly, 11(3), 3–4.
Xu, Z., Hannaway, J., & Taylor, C. (2008). Making a difference?
The effects of Teach for America in high school. Retrieved
April 22, 2008, from
http://www.urban.org/url.cfm?ID=411642.
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.
References
Ames, C. A. (1992). Classrooms: Goals, structures and student
motivation. Journal of Educational Psychology, 84, 261–271.
Bandura, A. (1986). Social foundation of thought and action: A
social cognitive theory. Englewood Cliffs, NJ: Prentice Hall.
Bransford, J., Hasselbring, T., Barron, B., Kulewicz, S., Littlefield,
J., & Goin, L. (1988). Uses of macro-contexts to facilitate
mathematical thinking. In R. I. Charles & E. A. Silver (Eds.),
The teaching and assessing of mathematical problem solving
(pp. 125–147). Reston, VA: NCTM and Hillsdale, NJ:
Erlbaum.
Brophy, J. (1999). Toward a model of the value aspects of
motivation in education: Developing appreciation for
particular learning domains and activities. Educational
Psychologist, 34, 75–85.
Chambers, D. L. (Ed.). (2002). Putting research into practice in the
elementary grades. Reston, VA: NCTM.
Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001).
Participating in classroom mathematical practices. The
Journal of the Learning Sciences, 10, 113–163.
Cobb, P., Wood, T., Yackel, E., & Perlwitz, M. (1992). A follow-
up assessment of a second-grade problem-centered
mathematics project. Educational Studies in Mathematics, 23,
483–504.
Deci, E. L., Koestner, R., & Ryan, R. M. (1999). A meta-analytic
review of experiments examining the effects of extrinsic
rewards on intrinsic motivation. Psychological Bulletin, 125,
627–668.
Deci, E. L., Nezlek, J., & Sheinman, L. (1981). Characteristics of
the rewarder and
intrinsic motivation of the rewardee. Journal of Personality and
Social Psychology, 40, 1–10.
Deci, E. L., & Ryan, R. M. (1980). The empirical exploration of
intrinsic motivational processes. In L. Berkowitz (Ed.),
Advances in experimental social psychology (Vol. 13, pp. 39-
80). New York, NY: Academic Press.
Deci, E. L., & Ryan, R. M. (1985). Intrinsic motivation and self
determination in human behavior. New York, NY: Plenum.
Deci, E. L., & Ryan, R. M. (1987). The support of autonomy and
the control of behavior. Journal of Personality and Social
Psychology, 53, 1024–1037.
Eccles, J., Wigfield, A., & Reuman, D. (1987, April). Changes in
self-perceptions and values at early adolescence. Paper
presented at the annual meeting of the American Educational
Research Association, San Francisco, CA.
Forman, E. A. (2003). A sociocultural approach to mathematics
reform: Speaking, inscribing, and doing mathematics within
communities of practice. In J. Kilpatrick, W. G. Martin, &
D.Schifter (Eds.), A research companion to principles and
standards for school mathematics (pp. 333-352). Reston, VA:
National Council of Teachers of Mathematics.
Francisco, J. M. & Maher, C. A. (2005). Conditions for promoting
reasoning in problem solving: Insights from a longitudinal
study. Journal of Mathematical Behavior, 24, 361–372.
Germain-McCarthy, Y. (2001). Bringing the NCTM standards to
life. Poughkeepsie, NY: Eye on Education.
Hidi, S. (2000). An interest researcher’s perspective: The effects of
extrinsic and intrinsic factors on motivation. In C. Sansone &
J. Harackiewicz (Eds.), Intrinsic and extrinsic motivation: The
search for optimal motivation and performance (pp. 309-
339). New York, NY: Academic Press.
Kamii, C. (1985). Young children reinvent arithmetic: Implications
of Piaget's theory. New York, NY: Teacher College Press.
Lepper, M. R. (1988). A whole much less than the sum of its parts.
American Psychologist, 53, 675–676.
Lepper, M. R., & Henderlong, J. (2000). Turning “play” into
“work” and “work” into “play”: 25 years of research on
intrinsic versus extrinsic motivation. In C. Sansone & J.
Harackiewicz (Eds.), Intrinsic and extrinsic motivation: The
search for optimal motivation and performance (pp. 257–307).
New York, NY: Academic Press.
Middleton, J. A. (1995). A study of intrinsic motivation in the
mathematics classroom: A personal constructs approach.
Journal for Research in Mathematics Education, 26, 254–279.
Middleton, J. A. & Spanias, P. A. (1999). Motivation for
achievement in mathematics: Findings, generalizations, and
criticisms of the research. Journal for Research in
Mathematics Education, 30, 65–88.
Midgley, C., Feldlaufer, H., & Eccles, J. S. (1989). Student/teacher
relations and attitudes toward mathematics before and after
transition to junior high school. Child Development, 60, 981–
992.
Mousoulides, N., & Philippou, G. (2005). Students’ motivational
beliefs, self-regulation strategies and mathematics
achievement. In H. L. Chick & J. L. Vincent (Eds.),
Mary Mueller, Dina Yankelewitz, & Carolyn Maher
43
Proceedings of the 29th Conference of the International
Group for the Psychology of Mathematics Education (PME)
(pp. 321-328). Melbourne, Australia: PME.
Mueller, M. F. (2007). A study of the development of reasoning in
sixth grade students (Unpublished doctoral dissertation).
Rutgers, The State University of New Jersey, New Brunswick.
Mueller, M., & Maher, C. A. (2009). Learning to reason in an
informal math after-school program. Mathematics Education
Research Journal, 21(3), 7–35.
Mueller, M., Yankelewitz, D., & Maher, C. (2010). Promoting
student reasoning through careful task design: A comparison
of three studies. International Journal for Studies in
Mathematics Education, 3(1), 135–156.
National Council of Teachers of Mathematics. (2000). Principles
and standards for school mathematics. Reston, VA: Author.
Pajares, F. (1996). Self-efficacy beliefs in achievement settings.
Review of Educational Research, 66, 543–578.
Pintrich, P. R. (1999). The role of motivation in promoting and
sustaining self regulated learning. International Journal of
Educational Research, 31, 459–470.
Pintrich, P. R., & De Groot, E. (1990). Motivational and self-
regulated learning: Components of classroom academic
performance. Journal of Educational Psychology, 82, 33–50.
Renninger, K. A. (2000). Individual interest and its implications for
understanding intrinsic motivation. In C. Sansone & J.
Harackiewicz (Eds.), Intrinsic and extrinsic motivation: The
search for optimal motivation and performance (pp. 373–404).
New York, NY: Academic Press.
Sansone, C., & Harackiewicz, J. M. (2000). Looking beyond
extrinsic rewards: The problem and promise of intrinsic
motivation. In C. Sansone & J. Harackiewicz (Eds.), Intrinsic
and extrinsic motivation: The search for optimal motivation
and performance (pp. 1–9). New York, NY: Academic Press.
Stiff, L. V., & Curcio, F. R. (Eds.). (1999). Developing
mathematical reasoning in grades K-12. Reston, VA: National
Council of Teachers of Mathematics.
Stefanou, C. R., Perencevich, K. C., DiCintio, M., & Turner, J. C.
(2004). Supporting autonomy in the classroom: Ways teachers
encourage students’ decision making and ownership.
Educational Psychologist, 39, 97–110.
Stein, M. K., Smith, M.S., Henningsen, M. A., & Silver, E. A.
(2000). Implementing standards-based mathematics
instruction: A casebook for professional development. New
York, NY: Teacher College Press.
Urdan, T., & Turner, J. C. (2005). Competence motivation in the
classroom. In A. J. Elliot & C. S. Dweck (Eds.), Handbook of
competence and motivation (pp. 297–317). New York, NY:
Guilford.
Wigfield, A., & Eccles, J. S. (2002). The development of
competence beliefs, expectancies for success, and
achievement values from childhood through adolescence. In
A. Wigfield & J. S. Eccles (Eds.), Development of
achievement motivation (pp. 91–120). San Diego, CA:
Academic Press.
Yackel, E., & Cobb, P. (1996). Sociomathematical norms,
argumentation, and autonomy in mathematics. Journal for
Research in Mathematics Education, 27 , 458–477.
Yankelewitz, D. (2009). The development of mathematical
reasoning in elementary school students’ exploration of
fraction ideas. Unpublished doctoral dissertation, Rutgers, The
State University of New Jersey, New Brunswick.
Yankelewitz, D., Mueller, M., & Maher, C. (2010). Tasks that elicit
reasoning: A dual analysis. Journal of Mathematical Behavior,
29, 76–85.
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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National Council of Teachers of Mathematics. MESA is an integral part of
The University of Georgia’s mathematics education community and is
dedicated to serving all students. Membership is open to all UGA students,
as well as other members of the mathematics education community.