____ THE
_____ MATHEMATICS ___
________ EDUCATOR _____
Volume 19 Number 2
Winter 2009/2010
MATHEMATICS EDUCATION STUDENT ASSOCIATION
THE UNIVERSITY OF GEORGIA
Editorial Staff
A Note from the Editors
Co-Editor
Allyson Hallman
Catherine Ulrich
Dear TME Readers,
We are pleased to present you with this concluding issue of The Mathematics Educator’s Volume 19.
This issue displays the variety of article types published in TME. Although all draw upon the work of
researchers, the first is a position paper, the second a more traditional research report, the third a literature
review focusing on theoretical frameworks of mathematics educators, and the last a research report
analyzing issues of methodology. Although we did not intend for this issue to have a particular theme, as
you read through the articles presented in Volume 19 Issue 2, you will see that the importance of the
mathematical community comes through in each article. In the first article, mathematics is argued to be an
inherently social endeavor. In the second article, the impact of student teachers’ interactions with mentors on
their attitudes towards technology in the classroom is underlined. In the third article, the implications of
viewing mathematics as a social construction are fleshed out. And in the fourth article, the effect of
communication through letter writing on both secondary students and preservice teachers is explored.
Associate Editors
Zandra deAraujo
Eric Gold
Erik D. Jacobson
Laura Singletary
Advisors
Dorothy Y. White
MESA Officers
2009-20010
President
Zandra deAraujo
Vice-President
Laura Singletary
Secretary
Laura Lowe
Treasurer
Anne Marie Marshall
NCTM
Representative
Allyson Hallman
Undergraduate
Representative
Hannah Channell
Derek Reeves
In lieu of a guest editorial, we have decided to open this issue with a position paper by Thomas Ricks.
He offers a challenge to return to the intellectual and social roots of mathematics activity in the classroom.
His article would make an excellent reading for preservice or inservice teachers who do not see the
importance of creativity and communication in school mathematics. The second article, by Asli ÖzgünKoca, Michael Meagher and Todd Edwards, is a report of research on the development of technological
pedagogical and content knowledge (TPACK). TPACK is an area of pedagogical content knowledge that is
sure to receive more attention in the coming years as increasingly sophisticated technology continues to offer
new possibilities and challenges in the classroom. The third article, by Kimberly White-Fredette, brings a
sense of the variety of theoretical frameworks used in mathematics education. An awareness, not only of
one’s own perspective, but also of the perspectives of others, is certainly important for all mathematics
educators. And, finally, Anderson Norton and Zachary Rutledge have written a follow-up article to
“Preservice Teachers’ Mathematical Task Posing: An Opportunity for Coordination of Perspectives,”
published in the first issue of Volume 18. Their original article focused on the theoretical perspectives of the
researchers when conducting a study on preservice teachers’ letter writing exchanges with secondary
students and on the results that these different perspectives helped them draw out of the data. In their followup article, they focus on the methodological obstacles when measuring student engagement.
We would like to thank our associate editors and authors for all their hard work and dedication. We
hope you enjoy reading this issue as much as we all have enjoyed working on it.
Catherine Ulrich & Allyson Hallman
TME Co-editors
105 Aderhold Hall
The University of Georgia
Athens, GA 30602-7124
tme@uga.edu
http://math.coe.uga.edu/TME/TMEonline.html
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
Winter 2009/2010
Volume 19 Number 2
Table of Contents
2 Mathematics Is Motivating
THOMAS E. RICKS
10 Preservice Teachers’ Emerging TPACK in a Technology-Rich Methods Class
S. ASLI ÖZGÜN-KOCA, MICHAEL MEAGHER, & MICHAEL TODD EDWARDS
21 Why Not Philosophy? Problematizing the Philosophy of Mathematics in a Time
of Curriculum Reform
KIMBERLY WHITE-FREDETTE
32 Measuring Task Posing Cycles: Mathematical Letter Writing Between Preservice
Teachers and Algebra Students
ANDERSON NORTON & ZACHARY RUTLEDGE
46 Upcoming conferences
47 Submissions information
48 Subscription form
© 2010 Mathematics Education Student Association
All Rights Reserved
The Mathematics Educator
2009/2010, Vol. 19, No. 2, 2–9
Mathematics Is Motivating
Thomas E. Ricks
Mathematics is motivating; at least, it should be. I argue that mathematical activity is an inherently attractive
enterprise for human beings because as intellectual organisms, we are naturally enticed by the intellectual
stimulation of mathematizing, and, as social beings, we are drawn to the socializing aspects of mathematical
activity. These two aspects make mathematics a motivating activity. Unfortunately, the subject that students
often encounter in school mathematics classes does not resemble authentic mathematical activity. School
mathematics is characterized by the memorization and regurgitation of rote procedures in isolation from peers.
It comes as no surprise that many students have little motivation to continue mathematics study because it lacks
intellectual and social appeal. I suggest several practical changes in school mathematics instruction that are
drawn from the literature. These changes will lead to instruction that more readily engages students with the
subject because they are rooted in the intellectually and socially appealing aspects of mathematics.
Mathematics is motivating. Or at least, it should
be. Mathematical activity is an inherently attractive
enterprise for human beings because of its intellectual
and social aspects. This may be difficult to believe,
especially when “so many people find mathematics
impossibly hard" (Devlin, 2000, p. 1) and many openly
admit strong dislike for the subject (Paulos, 1988).
Certainly, critics might argue, a few gifted individuals
might have a special inclination toward mathematical
study. But, can mathematics be appealing for
everyone? I claim that it can be; mathematics has the
potential to be interesting for everyone because it is an
intellectual and social endeavor. In the following
sections, I detail what is meant by authentic
mathematical activity, describing both its intellectual
and social aspects. Comparing authentic mathematical
activity to typical school mathematical activity, I
suggest ways that teachers can draw on the intellectual
and social aspects of mathematical activity to motivate
and engage students in the study of mathematics.
Mathematical Activity
Because
“mathematics
is
a
woefully
misunderstood subject” (Devlin, 2000, p. 3), for the
purposes of this article I define authentic mathematical
activity, or mathematical activity, to be what
Dr. Thomas E. Ricks earned his PhD in Mathematics Education
from the University of Georgia in 2007. He works as an Assistant
Professor in the Elementary Education division of the
Department of Educational Theory, Policy, and Practice at
Louisiana State University. Currently, he is collaborating with
LSU colleagues on a federal grant for the university’s preeminent
undergraduate technology-assisted mathematics program.
2
mathematicians do when they do mathematics. I use
examples from the life of Stanislaw Ulam, a Polish
mathematician, to describe authentic mathematical
activity.
Not as well-known as Euler, von Neumann or
Einstein, Ulam was a rather ordinary mathematician.
Ulam (1976) humbly stated, “after all these years, I
still do not feel much like an accomplished
professional mathematician” (p. 27). Much of Ulam’s
descriptions of his work focused on the people he met,
was inspired by, and worked with. Many of the
experiences he reported illustrate intellectual and social
aspects of mathematics.
Intellectual Aspect of Mathematics
Mathematical activity is the most intellectual
endeavor of all the sciences.
Mathematics is a creation of the brain...
Mathematicians …work …without any of the
equipment or props needed by other scientists….
mathematicians can work without chalk or pencil
and paper, and they can continue to think while
walking, eating, even talking. This may explain
why so many mathematicians appear turned inward
[which is] quite pronounced and quantitatively
different from the behavior of scientists in other
fields…. I have spent… on the average two to three
hours a day thinking…. Sometimes… I would
think about the same problem with incredible
intensity for several hours without using paper and
pencil. (Ulam, 1976, p. 53)
Since mathematics is “a thinking, flexible subject”
(Boaler, 1999, p. 264), mathematical activity is
Thomas E. Ricks
characterized by a variety of mental methods, such as
cognitive wrestle and creativity.
Cognitive Wrestle
Mathematicians
wrestle
with
cognitively
demanding problems that have no clear solution path.
Such
wrestle
is
invigorating,
and
“most
mathematicians begin to worry when there are no more
difficulties or obstacles” (Ulam, 1976, p. 54).
Mathematicians rarely make progress at steady rates;
rather, they struggle with little apparent progress, and
then major strides are made suddenly. The
mathematician Banach “worked in periods of great
intensity separated by stretches of apparent inactivity”
(Ulam, p. 33). Ulam described the importance of
“‘subconscious brewing’ (or pondering) [which]
produces better results than forced, systematic
thinking” (p. 54), being “a discontinuous process.
Nothing, nothing, at first, and suddenly one gets …it”
(p. 70). Part of this cognitive wrestle involves
exploring alternative possibilities through mental
experimentation.
I always preferred to try to imagine new
possibilities rather than merely to follow specific
lines of reasoning or make concrete calculations.
…Forcing oneself to persist in a logical exploration
becomes a habit after which it ceases to be forcing
since it comes automatically. (Ulam, 1976, p. 54)
Creativity
Mathematicians are creative by generating
mathematical content; they do this by posing and
solving problems. During this process they create the
very fabric of mathematics, weaving a thread that
connects to the work of others. For example, when
only 25, Ulam (1976) “established some results in
measure theory which soon became well known [by
solving] certain set theoretical problems attacked
earlier by Hausdorff, Banach, Kuratowski, and others”
(p. 55). In turn, the Russian Besicovitch solved a
problem posed many years earlier by Ulam. These
events also illustrate that problem posing is another
part of creative mathematical generation.
Devlin (2000) claimed that all people, everywhere,
have “a mind for mathematics" (p. 1), that every
human being with a functioning brain has “an innate
facility for mathematical thought” (p. xvi). The
intellectual aspects of mathematics, such as cognitive
wrestle and creative generation, are fundamental
attributes of our species: This helps explain why all
human cultures mathematize the world. The
predisposition for patterned thinking is even seen in
newborns (Dehaene, 1997).
Researchers know that “large parts of the brain are
active when a person is doing mathematics" (Devlin,
2000, p. 12). Because we are intellectual beings, the
intellectual appeal of mathematics makes it naturally
enjoyable; people’s brains like doing mathematics.
Mathematics is by far children's favorite subject in
school, at least well into the fourth grade (National
Council of Teachers of Mathematics [NCTM], 2000).
Social Aspect of Mathematics
Besides
its
intellectual
character,
doing
mathematics is also highly social. Every mathematician
works
within
a
mathematical
community.
Mathematicians are social on both a local and global
scale. Ulam, in particular, described communication
and collaboration.
Communicating
Mathematicians are engaged in constant
communication, in part to help them learn more
mathematics. This may entail reading about
mathematics, listening to lectures, or discussing
mathematical ideas with knowledgeable others. Ulam
was influenced by mathematicians’ books at an early
age, such as Sierpinski’s Theory of Sets, Steinhaus’s
What is and What is Not Mathematics, and Poincaré’s
La Science et l’Hypothèse, La Science et la Méthode,
La Valeur de la Science, and Dernières Pensées (1976,
p. 21). Kuratowski was an early teacher of Ulam who
made a formidable impression, and, in part, was
responsible for starting him on a career in mathematics.
Teachers and mentors play a significant role in
mathematicians’
development.
Many
famous
mathematicians bring to mind their mentors: Euler as a
student of Bernoulli; Ramanujan as a student of Hardy;
and Dedekind and Riemann as students of Gauss.
Mathematicians recognize the benefit that flows
from sharing and networking (Davis & Simmt, 2003).
Ulam (1976) described how he would engage in
mathematical discussions with friends and colleagues.
During a break while attending an International
Mathematical Congress, he got lost in the nearby
woods, and bumped “into Paul Alexandroff and Emmy
Noether [who were] walking together [among the
trees] and discussing mathematics” (p. 46).
The view of an isolated mathematician working
long hours alone in the office with little interaction is
almost everywhere false (Wiles’ work on the Fermat
theorem being a notable exception).
Much of the … historical development of
mathematics has taken place in specific centers [or]
a group in which mathematical activity flourished.
3
Mathematics Is Motivating
Such a group possesses more than just a
community of interests; it has a definite mood and
character in both the choice of interests and the
method of thought. Epistemologically this may
appear strange, since mathematical achievement,
whether a new definition or an involved proof of a
problem, may seem to be an entirely individual
effort, almost like a musical composition.
However, the choice of certain areas of interest is
frequently the result of a community of interests.
Such choices are often influenced by the interplay
of questions and answers, which evolves much
more naturally from the interplay of several minds.
(p. 38)
The work of a mathematician incorporates his or
her surrounding influences. For example, Ulam (1976)
also wrote that “most of my mathematical work was
really started in conversations with Mazur and
Banach” (p. 33, emphasis added). Even gatherings in a
local café provided opportunities for sharing and
discussing mathematics. A large notebook was
permanently kept in the café and brought out by a
waiter upon demand; it was the central repository of
the group’s ideas. Ulam later translated the notebook
and “distributed it to many mathematical friends in the
United States and abroad” (pp. 49–51).
Collaborating
Beyond communicating about mathematics, active
collaboration is also a natural part of mathematicians’
sociality. Ulam (1976) worked with many
distinguished mathematicians during his career;
“Collaboration [with Mazur and Banach] was on a
scale and with an intensity I have never seen surpassed,
equaled, or approximated anywhere—except perhaps
at Los Alamos during the war years” (p. 34). Ulam said
that upon his arrival in the United States, he and
Borsuk “started collaborating from the first… my first
publication in the United States ….was a joint paper
with Borsuk” (p. 41). Later, he said, “a whole series of
papers which we [Steinhaus and I] wrote jointly came
from …collaboration” (p. 43). Ulam also worked with
John von Neumann.
The joint work of mathematicians results in
mutually accepted definitions, terms, strategies,
methods, and algorithms. From parking lots to offices,
academic
lunchrooms
to
conference
halls,
mathematicians scribble and get stuck, share questions
and solution attempts, backtrack and refine, reattempt
and debate; they question, raise counterexamples,
reason, argue, collectively justify, and develop
communal metaphors (Polya, 1945). As such, many
believe mathematics, as a domain, transcends any
4
individual perspective (Boaler, 1999; Davis & Simmt,
2003; Devlin, 2000). It is not a static knowledge
domain––an external thing to be internalized by a
learner––but rather a socially created, culturally
dependent, fallible domain (Ernest, 1990).
Mathematics exists due to the collective actions of
many people over thousands of years. It belongs to no
one and yet is accessible to all; it is a constant,
communal, and humanistic creation (Romberg, 1994).
Great discoveries by many individuals and groups have
woven the tapestry of current mathematical thought;
people like the Pythagoreans, the Arab algebraists,
Cardano’s band, and Newton and Liebniz have all
contributed their threads. As Leopold Kronecker, a
nineteenth century mathematician, remarked, “God
made the integers, all else is the work of man" (quoted
in Devlin, 2000, p. 15).
All mathematics––
fundamental
axioms,
appropriate
terminology,
conventional representations, mathematically valid
propositions––is socially driven, “a cultural product”
(Ernest, 1990). From this perspective, mathematics is
much more than numbers or computations; it emerges
through correspondence, questions, and group
deliberations.
Inherent Mathematical Appeal
I am not alone in believing that mathematics can
be interesting for everyone. The authors of the NCTM
Standards (2000) opine that mathematics is a
meaningful, richly rewarding subject that all can learn
and enjoy. Additionally, when given the opportunity to
engage in meaningful mathematical tasks that maintain
their cognitive integrity, students not only tolerate
mathematical work, but report satisfaction and
enjoyment (e.g., Boaler, 1999). These findings are not
exclusive of any particular personality or culture. In
addition, I have seen ample evidence to suggest that
students, either high or low achieving, when allowed to
engage in mathematics, are drawn to the activity
(Ricks, 2007). There is something intrinsically
motivating in the subject.
School mathematics can share this attraction if
students are able to engage in authentic mathematical
activity. Unfortunately, “most people do not know
what mathematics is” (Devlin, 2000, p. xvii), perhaps
because they have not experienced authentic
mathematical activity and are thus dissuaded from
further mathematics study. School mathematics is
characterized by the memorization and regurgitation of
rote procedures in isolation from peers (Burton, 2004;
Stigler & Hiebert, 1999). Therefore, it comes as no
surprise that, devoid of its intellectual and social
Thomas E. Ricks
appeal, mathematics is not motivating for many
students and that many do not continue formal
mathematics study past high school. A corrective
possibility is to harness the intellectual and social
potential of mathematics activity; allowing students to
engage in mathematical activity in their own
classrooms affords simple, straightforward options to
improving mathematics instruction by returning to root
motivational aspects of the subject.
The Lack of Intellectual and Social Appeal in
School Mathematics
School mathematics is characterized by learning
definitions and practicing procedures (Stigler &
Hiebert, 1999), activities that lead to intellectual
boredom. The essential attributes of mathematical
activity—cognitive struggle and creative generation—
are absent. “The questions people [mathematicians]
worried about and the struggles they went through
trying to answer them almost never appear [in school
mathematics]; instead we see the results of the
struggles, neatly packaged into pieces of boxed text”
(Cuoco, 2001, p. 169). School mathematics is, quite
bluntly, an intellectual wasteland, a pseudomathematics. Richards (1991) describes the
intellectually stagnating initiation-reply-evaluation
sequence as the common form of classroom
interaction. No wonder students are confused; no
wonder they avoid further mathematics study!
Intellectual Lack in School Mathematics
Mathematics teachers often view their job as
showing “a few standard facts and algorithms” to
students, and, later, “supervis[ing] some drill and
practice” (Romberg, 1994, p. 314). Students are
expected only to memorize the various rules and
procedures the teacher demonstrates (Boaler, 1999;
Davis, 1994): Independent thought is not an
expectation. The intellectual possibilities for
“relatively sophisticated levels of mathematical
reasoning, well beyond what is typically thought of as
appropriate for primary school mathematics" (Yackel,
2000, p. 20) are rarely met. The current level of
intellectual engagement in learning school mathematics
pales in comparison to what could happen if children
were allowed to think things through for themselves
(Davis, 1994).
School mathematics is also not viewed as a
creative endeavor. The curriculum is set, the teacher
and textbook are the authorities in classrooms. There is
no room for questioning, no room for exploration, no
room for experimentation. “In many schools,
mathematics is perceived as an established body of
knowledge that is passed on from one generation to the
next. Instead of seeing [theorems, formulas, and
methods of mathematics] as the products of doing
mathematics, these artifacts are seen as the
mathematics” (Cuoco, 2001, p. 169). Said Burton
(2004):
It has long been my opinion that the mathematics
experienced by students in formal education, and
the ways in which it is encountered, offer
explanations for [the] decline in interest. Public
interest books about mathematics are readily
bought so it cannot be that people have no wish to
engage with mathematics…. once students make a
choice to study mathematics, many of them report
experiences that are not conducive to holding them
in the discipline. The same pattern holds whether
they are studying mathematics at school, as a preuniversity choice, at university as undergraduates
or even at doctoral level. (p. 4, emphasis added)
Contrasting this intellectually diluted school
mathematics with the work of mathematicians is
enlightening: “as a result of such limited experiences,
many students are prejudiced against the broader, more
interesting aspects of mathematics” (Romberg, 1994, p.
290).
Social Lack in School Mathematics
Similarly, school mathematics deprives students of
the natural socializing appeal of mathematical activity.
Students are expected to sit quietly and listen to the
teacher with little to no interaction with others (Davis,
1994). The necessary mathematical interactions needed
for full mathematical activity are absent. This severely
curtails children’s abilities to make “judgments about
what is acceptable mathematically, for example, with
respect to mathematical difference, mathematical
sophistication,
mathematical
inefficiency,
mathematical elegance, and mathematical explanation
and justification,” and it deprives them of autonomous
“mathematical power” (Yackel, 2000, p. 21).
In school mathematics, students usually do
problem sets alone, do homework alone, and take
quizzes and tests in isolation. When do they have a
chance to engage in social mathematical work? School
mathematics perpetuates beliefs that heterogeneous
class makeup is an “obstacle to effective teaching” and
that “the tutoring situation is best, academically,
because instruction can be tailored specifically for each
student” (Stigler & Hiebert, 1999, p. 9). When they do
work in groups, students usually do only superficial
5
Mathematics Is Motivating
computational exercises, even though the group could
enable individual students to overcome personal
barriers in the problem-solving process. Rarely is
mathematical understanding created by the group as a
whole.
School mathematics neglects the social aspects that
make mathematics so appealing—the ability to
participate in larger mathematizing collectives working
toward shared meanings and common objectives.
Classroom dialogue is characterized by univocal
“number talk” rather than socially intertwined,
mutually specified, dialogic functioning (Davis &
Simmt, 2003; Richards, 1990, Wertsch & Toma,
1995). Such absences of mathematical activity in
school mathematics classrooms led one researcher
studying U.S. mathematics lessons to bemoan, “I have
trouble finding the mathematics [in these lessons]”
(quoted in Stigler & Hiebert, 1999, p. 26).
The great ironic tragedy is that most students who
claim to have little motivation to study mathematics
have never really experienced authentic mathematics.
To deal with a lack of motivation, non-mathematical
strategies are often employed to maintain students’
attention in mathematics classes. However, such
strategies do not work. Some examples are: (i)
interrupting mathematics instruction to talk about a
more interesting subject, (ii) using candy or prizes to
excite students, (iii) presenting the lesson in the context
of competitive games, or (iv) letting students work
together on projects where the focus often shifts from
mathematical ideas to creating attractive displays
(Stigler & Hiebert, 1999).
Making School Mathematics Authentic
Mathematics learning does not require games,
dramatic teacher presentations, external motivators, or
even connections to real world activities, all common
suggestions to motivate students in traditional
mathematics classrooms. It requires instead a return to
the core components of mathematics. The reason
motivation is an issue at all is that current school
mathematics is neither intellectual nor social; by
focusing on “habitual, unreflective, arithmetic
problems” (Richards, 1991, p. 16), school mathematics
strips from the subject the very constituents that
provide for meaningful mathematical experiences.
Slight, subtle changes in the way mathematics is taught
can significantly increase students’ motivation to learn
mathematics.
For example, teachers can engage students in (1)
cognitively challenging (Stein, Smith, Henningsen, &
Silver, 2000) and (2) socially oriented activities in
6
mathematics classrooms (Stein & Brown, 1990).
Students can then be involved in the genesis of
mathematical ideas in a group setting. These two
components make mathematics a motivating activity:
When a class follows an inquiry tradition of
instruction, many of the ‘tasks’ that children
engage in are tasks that they set for themselves as
they attempt to reason about the dynamic
interactions that occurs in small group interactions
and in whole class discussions. In a real sense, by
choosing what they reflect on, students
individualize instruction for themselves in ways
that only they can do. (Yackel, 2000, p. 20)
We can see how this understanding of what makes
mathematics motivating is reflected in current trends in
mathematics education. For example, the common
recommendation to structure lessons around central
challenging tasks (Stigler & Hiebert, 1999) would
support the intellectual and social requirements of
mathematical activity. A teacher’s ability to recognize,
modify, or develop a central activity that is cognitively
demanding, while, at the same time, maintaining the
intellectual integrity of the task as students struggle,
allows the mathematics to retain its intellectual vitality.
Students’ mathematical experience would be less likely
to degrade into mimicry, repetition, and boredom.
Jointly working on a central task also provides for
more robust whole-class discussions; the class shares
the common foundations necessary to truly collaborate
on mathematical work. The class can begin to emerge
as a mathematical community through developing a
common vocabulary and engaging in collective sense
making. Ball & Bass (2003) write:
Making mathematics reasonable is more than
individual sense making. Making sense refers to
making mathematical ideas sensible [and]
comprises a set of practices and norms that are
collective, not merely individual or idiosyncratic…
That an idea makes sense to me is not the same as
reasoning toward understandings that are shared by
others with whom I discuss and critically examine
that idea toward a shared conviction. (p. 29)
A second trend in mathematics education—
relinquishing ‘mathematical authority’ (Cobb, Yackel,
& Wood, 1992; Smith, 1996)—also respects the
intellectual and social dimensions of mathematical
activity. By purposely removing herself or himself as
the source of mathematical truth, the teacher enables
students to collectively develop mathematical
knowledge.
In fact, most current trends in mathematics
education respect and enable the intellectual or social
Thomas E. Ricks
dimension of mathematics. For instance, establishing
sociomathematical
norms—establishing
an
environment for shared ways of mathematical sensemaking and making explicit appropriate means of
questioning, justifying, and reasoning—enables the
social aspect of professional mathematicians’ work in
the classwork (Cobb, 1994; Cobb, Wood, & Yackel,
1990;Yackel, 2000; Yackel, Cobb, & Wood, 1999).
The trend toward whole-class discussions, where the
teacher orchestrates a respectful space for students to
discuss and question each others’ thinking, (Cobb,
Wood, & Yackel, 1990; Yackel, Cobb, & Wood,
1999);
joint
mathematizing
or
encouraging
collaboration, where students combine their
mathematical efforts (Grossman, et al., 2001; Ricks,
2007); and equalizing participation of students so
particular students do not dominate class discussions
(Noddings, 1989; Webb, 1995) are recommendations
that attempt to catalyze the types of social interactions
characteristic of mathematicians’ work. Eliciting
students’ mathematical diversity, the teacher selection
of different manners of student approaches and
solutions to mathematical problems (Bennie, Olivier,
& Linchevski, 1999; Borba, 1992; Linchevski,
Kutscher, & Olivier, 1999; Linchevski, Kutscher,
Olivier, & Bennie, 2000a, 2000b; Smith, 1992), helps
students experience the range of creativity that is a
hallmark
of
mathematical
problem
solving.
Emphasizing dialogic functioning (Wertsch & Toma,
1995), when students think about each others’ thinking,
also contributes to the intellectual work needed to
make sense of others’ ways and means of operating
mathematically.
The ultimate goal of mathematics instruction
should be for students to become “lifelong
mathematics learners” (Cuoco, 2001, p. 169). What
might this look like in a classroom? Although this
article is not the place for delving into the specifics of
intellectual and social rejuvenation of mathematics
classes, I do offer three categories of examples at the
levels of task, lesson, and overall curriculum to whet
the reader’s appetite (e.g., similar to Usiskin, 1998).
Mathematical Tasks
The most obvious place for change is at the level
of mathematical activities. Stein et al. (2000) detailed a
task framework for measuring a task’s cognitive
demand, ordered from lowest to highest:
memorization, procedures without connections,
procedures with connections, and doing mathematics.
For them, doing mathematics is a cognitively
demanding activity “requiring complex and non-
algorithmic thinking” where no rehearsed approach is
used (p. 16), as opposed to memorization, defined as
the recall or reproduction of previously learned
material, or procedures, defined as emphasizing
algorithms to produce correct answers with little
explanation of thinking. They consider doing
mathematics tasks to be the most beneficial student
activities, and their study details several classroom
case studies of teachers iteratively attempting to setup
and implement doing mathematics tasks appropriately.
Definitions
can
be
developed
through
investigating, rather than having definitions presented
by the teacher at the beginning of lessons as though
they were axiomatic. For example, developing a
definition for trapezoid could lead to intriguing
intellectual and social possibilities. As there is no
accepted standard definition for trapezoid (Wolfram,
2010). Are trapezoids quadrilaterals with at least one
pair or only one pair of parallel sides? There are
tantalizing ramifications of this choice, such as
implications for trapezoidal classification as a subset or
superset of parallelograms, how the trapezoid and
parallelogram area formulas relate, etc. Students can
explore how the taxonomy of other shapes change with
similar definition modification. The teacher can
mediate a class discussion about which definition is
better and why, and the class can then adopt this
specific definition in their future work.
Mathematical Lessons
Teachers can also structure their lessons to
accentuate the intellectual and social aspects of
mathematics. Instead of a lesson structured around
teacher presentation, demonstration, and modeling of
pre-packaged mathematical procedures (Cuoco, 2001),
the teacher can pose challenging tasks, and then allow
individual and/or group work followed by whole class
discussions (Yackel, 2000). There are many examples
of such lessons available for teachers, e.g., videos from
the Annenberg collection (WGBH Boston, 1995).
Another excellent example is the released 1995 Trends
in Mathematics and Science Study (TIMSS) videos
(NCES, 2003), including one of an eighth grade
Japanese geometry lesson that revolved around a single
task of dividing land equally. This lesson’s unified
structure is particularly powerful when juxtaposed
against the piecemeal problem review and teacher
lecture of an eighth grade U.S. geometry lesson in
another TIMSS video.
Lessons from Deborah Ball’s 1989 third grade
classroom offer further examples of rich intellectual
and social mathematics lessons (Ball, 1991). In the
7
Mathematics Is Motivating
lesson known as Shea’s Numbers, a student claims that
the number six is both an even and an odd number.
Rather than telling Shea that he is mistaken about the
definitions (a typical teacher’s response), Ball instead
allows him to fully explain his reasoning. Doing so
allows others to value his argument, and to recognize
that certain even numbers (2, 6, 10,…, 2 + 4n) have an
odd number of twos; these the class calls Shea
numbers.
Mathematical Curriculum
Mathematics teachers can also structure their units,
courses, and curriculum to more accurately mimic the
work of mathematicians. Examples of this abound,
such as Moses’ (2001) Algebra Project where students
learn algebra from real-life experiences, the Moore
Method (Corry, 2007) of building mathematical
structure from a small set of teacher-provided axioms,
and Anderson’s (1990) mathematics courses
“emphasizing that ordinary people create mathematical
ideas and ‘do’ mathematics” (p. 354). Deborah Ball’s
(1991) year-long third-grade class offers another
glimpse at such curriculum innovation because she
allows students to reason through their thinking in
whole class discussions.
Conclusion
Current school mathematics strips from the subject
the very constituents that provide for meaningful
mathematical experiences. A solution to the crisis may
be far easier than some think and this solution would
not require more rigorous standards, more standardized
testing, more funding for smaller classes, or more
content training—only a return to the fundamental
aspects that make mathematics so intriguing.
The primary way to engage students in mathematics
classrooms is to allow them to experience
mathematical activity. Mathematical activity is a
motivating activity because it connects the ubiquitous
human capabilities of intellectualizing and socializing
(Devlin, 2000). More specifically, mathematical
activity welds together the intellectual and social
dimensions of human beings as they collaboratively
wrestle with and jointly create mathematical terrain in
a process of social mathematizing.
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9
The Mathematics Educator
2009/2010, Vol. 19, No. 2, 10–20
Preservice Teachers’ Emerging TPACK in a Technology-Rich
Methods Class
S. Asli Özgün-Koca, Michael Meagher, & Michael Todd Edwards
There is a dearth of research on the mechanisms for preservice teachers' development of the pedagogical
knowledge necessary for effective use of such technologies. We explored the emergent Technological
Pedagogical and Content Knowledge (TPACK) (Niess 2005, 2006, 2007) of a group of secondary mathematics
preservice teachers in a methods course as they designed and implemented technology-rich teaching materials
in field settings. Participant surveys and collected assignments were analyzed through the lens of the TPACK
framework. The data were also analyzed to examine the trajectory of the participants’ beliefs about the
appropriate role of advanced digital technologies in mathematics. The results indicate that the participants’
understanding of technology shifted from viewing technology as a tool for reinforcement into viewing
technology as a tool for developing student understanding. Collected data supports the notion that preservice
teacher TPACK development is closely related to a shift in identity from learners of mathematics to teachers of
mathematics. In a class where advanced digital technologies were used extensively as a catalyst for promoting
inquiry-based learning, preservice teachers retained a great deal of skepticism about the appropriateness of
using technology in concept development roles, despite their confidence that they can incorporate technology
into their future teaching.
To me, it's [the use of calculators in mathematics
instruction] more about where kids are at
developmentally. The methods are influenced by
this. When kids are younger and inexperienced,
they need to be taught the basics using direct
instruction like I was. Now that I know some
things, I can use the calculator to learn more. But I
have a good foundation in the basics FIRST.
In the above quote, a preservice teacher shares his
views on the use of technology to teach mathematics.
Based on his strong views on this issue, we might not
expect him to use a great deal of advanced digital
technologies in his classroom nor to employ discovery
activities (using technology or otherwise). Many
researchers have highlighted the important influence of
S. Asli Ozgun-Koca teaches mathematics and secondary
mathematics education courses at Wayne State University,
Detroit, MI. Her research interests focus on the use of technology
in mathematics instruction and understanding mathematics
teachers' views about and knowledge on the technology use in
teaching and learning of mathematics.
Michael Meagher is an Assistant Professor of Mathematics
Education at Brooklyn College - CUNY. His research interests
include the use of advanced digital technologies in teaching and
learning mathematics, and the role alternative certification
programs in training mathematics teachers for teaching in urban
schools.
M. Todd Edwards teaches mathematics and secondary
mathematics education courses at Miami University, Oxford, OH.
His research interests include the use of technology in the
learning and teaching of school mathematics. In his spare time,
he enjoys exploring the joys of dynamic geometry software with
his three children, Cassady, Ian, and Dylan.
10
teachers' beliefs and views on instructional decisionmaking and classroom practice (Ball, Lubienski, &
Mewborn, 2001; Borko & Putnam, 1996; National
Council of Teachers of Mathematics [NCTM], 1991;
Richardson, 1996; Stipek, Givvin, Salmon, &
MacGyvers, 2001; Thompson, 1984, 1992).
Furthermore, Peressini, Borko, Romagnano, Knuth,
and Willis-Yorker (2004) argue that none of the
experiences (mathematics and teacher preparation
courses, preservice field experiences, and employment)
in learning to teach are independent of one another,
which ensures a complicated collection of influences
on a prospective teacher's learning trajectory.
A growing body of research indicates that digital
technologies, including graphing and Computer
Algebra System (CAS)-enabled calculators, can
enhance young students' conceptual and procedural
knowledge of mathematics (Dunham, 2000; Thompson
& Senk, 2001). As teachers decide whether and how to
use advanced digital technologies in their teaching,
they need to consider the mathematical content that
they will teach, the technology that they will use, and
the pedagogical methods that they will employ.
Moreover, they need to reflect on the critical
relationships between these concepts: content,
technology, and pedagogy. Drawing on a series of case
studies, Zbiek (2002) suggests some direction for the
development of a model of effective teaching using
CAS. This model stresses the importance of many of
the influences discussed by Peressini et al. (2004),
including conceptions of school mathematics and how
available curriculum materials intersect with
S. Asli Özgün-Koca, Michael Meagher, & Michael Todd Edwards
technology. Zbiek concludes that such a model could
be a useful analytic tool for describing and facilitating
teachers' evolution in teaching with CAS. As will be
discussed in more detail in the next section, Niess
(2005, 2006, 2007) developed the Technological
Pedagogical and Content Knowledge (TPACK)
framework to provide such a tool for using advanced
digital technologies in teaching in general. However,
empirical studies of preservice teachers' emerging
TPACK remain in short supply.
Theoretical Context
Shulman (1986) provided an analysis of teachers'
knowledge as a complex structure including content
knowledge, pedagogical knowledge, and pedagogical
content knowledge (PCK). Ensuing research on teacher
knowledge is grounded in his framework. With Mishra
and Koehler's (2006; Koehler & Mishra, 2005) and
Niess' (2005, 2006, 2007) conception of TPACK, the
field has additionally gained “an analytic lens for
studying the development of teacher knowledge about
educational technology” (Mishra & Koehler, 2006, p.
1041).
TPACK involves the content knowledge (CK),
pedagogical knowledge (PK), and technological
knowledge (TK) required to teach in technology-rich
environments (see Figure 1). In our case, content
knowledge is high school mathematics. Pedagogical
knowledge includes learning theories and instructional
methods. Technological knowledge includes the
knowledge of how to operate technology-oriented tools
(such as Geometer's Sketchpad or TI-Nspire) and the
ability to adapt to ever-changing, novel technologies.
Figure 1. Re-creation of Mishra and Koehler's TPACK
model.
Shulman's (1986) discussion of PCK focuses on
the two-way relationship between content and
pedagogy, for instance, how particular pedagogical
methods might help (or hinder) students' learning of
specific content. Niess's (2005, 2006, 2007) TPACK
model extends this relationship to include relationships
with other constructs, including technological content
knowledge (TCK) and technological pedagogical
knowledge (TPK). TCK is viewed as the intersection
of the technology and the content wherein a wholly
different perspective on content may arise. For
example, technology can be used to explore the fact
that a quadratic with integer coefficients is highly
unlikely to be factorable, drawing attention to and
questioning the traditional content of school
mathematics. With respect to TCK, Mishra & Koehler
(2006) say, “teachers need to know not just the subject
matter they teach but also the manner in which the
subject matter can be changed by the application of
technology” (p.1028). On the other hand,
“technological pedagogical knowledge (TPK) is
knowledge of the existence, components, and
capabilities of various technologies as they are used in
teaching and learning settings, and conversely,
knowing how teaching might change as the result of
using particular technologies” (p. 1028).
Research Question
There is much to consider when studying pre- and
inservice teachers' knowledge, views, beliefs, attitudes,
and decisions about the use of technology in their
classrooms. Whereas Niess (2006, 2007) discussed
how teachers' beliefs and views about teaching
mathematics with technology play a crucial role in the
development of TPACK, our research question was:
How does preservice teachers' TPACK emerge during
their methods classes and field placement? Therefore,
in a methods course intended specifically for
preservice secondary mathematics teachers, we
examined teachers' emerging TPACK (Niess 2005,
2006, 2007) as manifested in their use of advanced
digital technologies in the design and implementation
of technology-rich teaching materials in field
placements. Moreover, through written responses
regarding the use of the TI-Nspire (Texas Instruments,
2007) and other advanced digital technologies, we
studied their views about the use of technology to teach
mathematics.
Data Collection
We studied a group of 20 preservice teachers
enrolled in a first-semester mathematics teaching
11
Preservice Teachers’ Emerging TPACK
methods course at a small Midwestern university. The
sample is one of convenience (Lodico, Spaulding, &
Voegtle, 2006). The participants were students in one
of the researchers’ classes, which was designed to
introduce participants to inquiry-based learning with
open-ended questioning. In past research, we have
found that technologies such as the TI-Nspire
calculator, virtual manipulatives, and dynamic
geometry software (DGS) open up new possibilities for
teachers
to
promote
connections
between
representations, encourage students to explore dynamic
mathematics environments, develop students' skills of
inquiry, and support students’ construction of
knowledge (Özgün-Koca, Meagher, & Edwards, in
press). Based on this result, the instructor placed
considerable emphasis on the use of such technologies
in the teaching and learning of mathematics, with
particularly extensive use of the TI-Nspire. The TINspire is a handheld device that incorporates
functionalities such as graphing, manipulating
algebraic expressions, and constructing geometric
figures and analyzing data in a dynamic environment,
while dynamically linking all of these representations.
Activities in the course focused primarily on
pedagogical tasks (e.g. constructing lesson plans and
grading rubrics, creating technology-oriented math
activities) and content-related activities (solving
mathematics problems, analyzing mathematical
accuracy of student work). For example, participants
completed problem sets designed to give them the
opportunity to explore (and extend) content and
pedagogical knowledge of secondary school
mathematics. As part of their field experience,
participants completed two reports in which they
researched, developed, and implemented mathematics
lessons. In addition, they submitted five secondarylevel mathematics activities constructed and/or
modified for use with the TI-Nspire. They were
encouraged to use these materials in their field
teaching whenever possible. Finally, participants
conducted original research dealing with the teaching
of a secondary mathematics problem (or set of related
problems) using the TI-Nspire. The field experiences
varied in the extent to which technology was used,
from almost none in some classrooms to extensive and
skilled implementation in others.
At both the beginning and end of the course, the
participants completed a mathematics technology
attitudes survey (MTAS), which included questions
rated on the Likert scale. Additionally, they
participated in three short surveys administered
electronically in weeks 4, 8, and 13 of the course. Each
12
of these surveys consisted of a combination of Likert
scale and open-ended items. Finally, participants
completed an open-ended exit survey at the end of the
course with questions that were more general than
those asked in the earlier surveys. Likert scale
questions from the MTAS and short surveys included:
1. Graphing calculators help me understand
mathematics.
2. Graphing calculators are a useful support for
discovering algebraic rules.
3. Students shouldn't use calculators until they
have thoroughly mastered the required skills
by-hand.
4. Graphing calculators help people who have
difficulties with algebra to still be able to do
mathematics.
5. I have been thinking and working a lot on the
technology of the course we are designing.
6. Our group has been considering how course
pedagogy and technology influence one
another.
Some of the examples of open-ended questions
were:
1. What kind of technology skills that you can
use later in your profession are you learning?
Describe how you intend to use those skills in
your future teaching.
2. Discuss the extent to which you have been
thinking and working with pedagogical issues
in the student activities you have been
designing in our class. While recently
observing classroom instruction in a local high
school, a mathematics teacher made the
following comment to me: "Content and
pedagogy influence one another, especially
when I use technology with kids in my
classroom." Discuss your thoughts regarding
this statement.
3. Similarly, a student in the aforementioned
classroom noted that "technology changes the
way our teacher teaches mathematics and the
way I learn mathematics." Discuss your
thoughts regarding this statement
4. Lastly, the classroom teacher noted that
"technology changes the mathematics content
that I teach." Discuss your thoughts regarding
this statement.
S. Asli Özgün-Koca, Michael Meagher, & Michael Todd Edwards
Data Analysis
Our analysis focused on the data collected through
the five secondary-level mathematics activities, field
experience reports, and surveys. The first level of
analysis focused on survey responses. We used
descriptive statistics for quantitative data and searched
for emerging codes and themes in the qualitative data.
The initial themes arising from the analysis were (i) a
shift in thinking of technology as a reinforcement tool
to thinking of technology as a tool for developing
mathematical concepts, and (ii) a change in
relationship to technology predicated on a shift of the
participants’ own identity from learner of mathematics
to teacher of mathematics. Once these themes had been
identified, we re-analyzed changes between pre- and
post-survey data and found the themes to validly
reflect key characteristics of the data. We analyzed the
activities and field experience reports through this lens
and found further evidence to support the conclusions.
We used the TPACK framework to guide the
qualitative data analysis for the open-ended survey
questions, the secondary-level mathematics activity
write-ups, and the field experience reports. The first
level of analysis involved coding the data for
instantiations of the participants' attitudes towards,
skill in using, and deployment of, TK, CK, and PCK.
For instance, if a participant were to say that
calculators should not be used until students master the
skills by hand, or if a participant were to discuss how
using technology in their activity write-ups affected
instructional planning, then we would code this as
TPK. While analyzing the activity write-ups, we
focused on three key issues: implementation of
technology, implementation of inquiry-based methods,
and quality of problem solving. Our interest in the
interactions among the various domains of the model
fueled our second level of analysis. We developed
codes for each of the possible interactions between TK,
CK, and PCK. We then analyzed the data for important
aspects of these interactions. For example, a
participant’s statement suggesting that the use of
calculators means that certain topics should be deemphasized would be coded as how technological
knowledge influences content knowledge. We feel that
the multiple data sources and use of different lenses in
the analysis provided sufficient data and researcher
triangulation to ensure trustworthiness of our findings
in this study (Miles & Huberman, 1994).
Results and Discussion
Two major themes emerged from the data analysis.
Firstly, the participants’ understanding of technology
showed perceptible shifts and mutations from thinking
of technology as a reinforcement tool to a tool for
developing mathematical concepts. Glimpses of this
evolving relationship to technology, which we see as a
positive development in their TPACK, are reflected in
candidate comments throughout the semester.
Secondly, we saw an interesting change in participants’
relationship to technology as they shifted their identity
from being a learner of mathematics to being a teacher
of mathematics. This also represents a positive step in
developing TPACK, specifically in TPK. The course
was the first methods course for these teacher
participants and, therefore, their first opportunity to
give serious thought to the use of technology from a
teacher's perspective.
Reinforce or Develop: The Use of Technology in
Lesson Plan Development
The development, or lack thereof, of TPACK in
teacher participants is reflected in the learning
activities they design for students. Developing good
tasks that incorporate technology presents a challenge
for the preservice teachers since they have to mix CK
of the topic they wish to address, TK of the technology
they choose to use, and PK in designing an inquirybased task for their students. Their intersection,
TPACK, proved particularly interesting in this
challenge. In the quotes below, we see two
participants’ reflections on how they started to think
about technology as an instructional tool to build
conceptual understanding:
I am using technology because we are required to
do so. However, the second activity write-up used
the TI-Nspire extensively because I thought it
would be really neat to see if I could use it for my
idea.
At first, the activity seemed to me that we had to
use and had to incorporate technology in our
activity. Now is seems that technology is more of a
tool to help us design a really good hands on,
visual activity.
TPACK was evident in the content-specific ways
that preservice teachers took advantage of the
functionalities and affordances of the technology to
engage students in inquiry-based tasks. The examples
discussed below illustrate that participants moved
beyond a naïve use of the technology into a more
sophisticated incorporation of technology into the
mathematics of their tasks.
When developing activities and creating lesson
plans using technology, preservice teachers often
incorporated technology into lessons through a
13
Preservice Teachers’ Emerging TPACK
superficial use of the available tools rather than taking
advantage of those capabilities that were specific to the
technology at hand (e.g. drawing shapes within a DGS
environment while ignoring dynamic construction
capabilities of the software, calculating simple results
or using graphing functions while ignoring linkages
between different representations), thus showing some
lack of TK. This also showed an underdeveloped sense
of TPK since the technology was not being used in a
sophisticated way to help provide inquiry-based
experiences for the students to develop understanding.
In many of the participants’ lessons, technology use
was not tied to acquisition of the mathematics
content—technology and content were envisaged as
separate constructs rather than as intertwined entities.
Therefore, although we found little evidence of TPK,
there was an increasing trend throughout the study. For
example, early work samples of student-generated
activities revealed naïve understandings of various TINspire tools and had low cognitive demand (see Figure
2). The activity in Figure 2 does not use the dynamic
capabilities of TI-Nspire, focusing strictly on its
drawing and measurement capabilities. Utilized in this
manner, the technology contributed few, if any,
advantages over use of traditional paper and pencil.
answer the question, determining the length of each
side of the triangle and verifying that the square of the
side opposite theta was strictly less than the sum of the
squares of the other two side lengths. While theta is
clearly an acute angle when viewed statically (as is the
case on the printed page), when vertices of the triangle
are dragged, theta may also assume values larger than
90 degrees. Hence, viewed dynamically, it is
impossible to determine if the triangle is right or not.
Therefore, in addition to being a very low-level
identification task (identify the type of angle denoted
by theta), the prompt makes no sense in a dynamic
context.
Below we can see a participant’s first activity (see
Figure 3) in which he used a real-world problem and
both the tabular and graphing capabilities of the
technology to find a point of intersection of two
graphs. Here the use of technology was helpful, but not
essential; the problem could have been solved just as
easily with pencil and paper.
Figure 3. Participant 3’s Activity, highlighting weak
use of technology
Figure 2. An example of a student-generated activity
of low cognitive demand.
In this activity, asking the student to “say whether
this triangle is a right triangle or not” implies that the
angle theta remains fixed. Although it is not clear from
the prompt alone, the participant intended the student
to use the converse of the Pythagorean Theorem to
14
However, when we analyzed the second activity
that he created later in the semester (see Figure 4), we
saw that he constructed the cross-sections of various
polyhedra using CABRI 3D to determine that crosssections of a cone can form an ellipse, a circle, and
parabola, and that cross-sections of a regular
tetrahedron can form a scalene triangle, an isosceles
triangle, an equilateral triangle, an isosceles trapezoid,
and a quadrilateral. Without the technology, such
constructions are impractical and not readily available
to teachers or students. In this second example, the
software is arguably an instructional necessity,
indicating a more mature utilization of technology.
S. Asli Özgün-Koca, Michael Meagher, & Michael Todd Edwards
Figure 4. Participant 3’s Second Activity with
Effective Use of Technology
While there was a general improvement in the
quality of the activities and lesson plans written by
participants as the semester progressed, the activities
written by those students with field placements in
technology-rich
environments
showed
more
sophistication, not just in the use of technology, but
also in terms of implementing inquiry-based and openended instructional approaches. Not only did both their
technological and pedagogical knowledge develop, but
the intersection of these two constructs, their TPK, also
developed during those experiences. When participants
did not have a rich technology experience in the field,
they typically indicated that technology was not an
instructional must:
I didn't really see any technology in schools when I
had field, and I'm not convinced that kids need it.
They need the basics first in order to actually
UNDERSTAND what the calculator is doing. This
is like reading. Kids need to know letters of the
alphabet first before they can read.
Change in Identity: Technology for Them and
Technology for Their Students
On the pre-and post-surveys, a large percentage of
participants agreed or strongly agreed that graphing
calculators helped them better understand mathematics.
In addition, a clear majority of the subjects agreed or
strongly agreed that graphing calculators increased
their desire to do mathematics (73% in the pre-survey
and 65% in the post-survey). Based on these
observations, it seems that the participants had a wellestablished understanding of how they themselves can
use advanced technologies in doing mathematics. On
the other hand, when it came to the issue of teaching
with calculators, the participants had mixed views. In
the pre-survey, 82% agreed that calculators help people
who have difficulties with algebra to still be able to do
mathematics. However, this percentage decreased to 70
in the post-survey. We, therefore, conclude that their
perspective changed somewhat when putting
themselves in the position of teachers of mathematics.
This could be an indication that the participants, based
on their experiences in their field placements, remained
attached to the idea that students' can be overly
dependent on calculators and that calculators can
interfere with students' learning of basic concepts.
In addition, during the fourth week, two of the
preservice teachers discussed their concern that they
were learning about technology to which they were
unlikely to have access as classroom teachers. Again,
this shows that they had started to reflect on the issues
as prospective teachers. That week the preservice
teachers also discussed SMARTTM Boards (n=8),
websites (n=4), and Geometer’s Sketchpad (n=1) as
possible tools to use in their future teaching. However,
by the eighth week, after some field experience, no one
discussed TI-Nspire calculators, although many
discussed the limited access that they had to advanced
technologies in actual school settings: “After going out
in the field, I believe more than I did before that the
technology I am learning to base my lessons off of,
though, is far too advanced.” After their field
experiences, many participants reported that they found
internet-based resources, such as interactive web
applets,more practical—both in terms of their
accessibility in classroom situations (e.g. most
classrooms were equipped with one demonstration
computer with internet access) and their low cost
(unlike the TI-Nspire, most applets were freely
available).
The TPACK Model and Advanced Digital
Technologies
In this section, we discuss how the TPACK model
helped us to reach the two main conclusions discussed
above.
Technological knowledge. Participants mentioned
a variety of technologies when discussing which of the
technological skills they were learning would be useful
in their future teaching. In the fourth week, eight
preservice teachers mentioned that they liked TINspire calculators. Some mentioned the technological
skills that they were learning, such as how to operate
the TI-Nspire:
I've not had a lot of practice in using calculators
besides the TI-84 and with that only the basic
functions. The technological advanced TI-Nspire
on the other hand, as I'm learning, is very user
friendly, with menus you can go to find out more
about what is available.
On the other hand, some preservice teachers had
technical difficulties learning how to use some of the
technology: “I found the TI-Nspire to be too
15
Preservice Teachers’ Emerging TPACK
complicated and not worth the hassle figuring it all out.
I spent more time trying to figure out how to use it than
I did learning about math.” Another mentioned that,
“one of the issues I've struggled with is the extent [to
which] we would use technology. A number of cases in
using
technology
have
required
extensive
knowledge/experience with the technology.” Clearly a
lack of TK for a particular technology could be an
important factor in preservice teachers’ consideration
of whether to use that technology in their future
classrooms.
Content and pedagogical content knowledge. The
CK required by these teachers is, minimally, the high
school mathematics content they will teach.
Participants in the methods course mostly agreed that
both their university class and field placement required
them to work extensively with high school
mathematics content as they designed teaching
activities. Even though many said that they were not
learning about mathematics content as they designed
activities, quite a few mentioned they were
“remembering” and that the activities were “refreshing
us” on the high school mathematics while looking at
content from a teacher's perspective:
So far, we have covered many of the content areas
including algebra and geometry. These were
important for my growth because I was unaware of
the severity of my 'rustiness' when it came to basic
algebraic and geometric principles.
Quite a few of the activities we have done in class
have made use of mathematics that I have not used
since high school. These activities have reminded
me how to do a number of problems. Overall, the
course is requiring me to look at mathematics from
a teachers’ perspective and not a students’ view of
question and answer.
Moreover, one preservice teacher mentioned that he or
she was focusing on the “why” question more:
I feel as though I am not learning new mathematics
content, but instead, I am thinking of what I
already know in a different light. The class has
caused me to think more about the why than the
how, and to me, that is, the most important element
of being a mathematics teacher.
We see here the participants’ transition from
thinking of themselves as learners of mathematics to
thinking of themselves as teachers of mathematics. The
participants' CK provided a basis for the development
of their PCK. Another participant mentioned that:
Both the problem sets and our activities have
required us to investigate mathematics content. The
16
Folded Paper problem is a perfect example. This
problem could be solved with a table, via a graph
or using calculus. Exploring each of these is
valuable in understanding the content of the math
and understanding multiple representations.
Preservice teachers started thinking about
pedagogical issues together with content. Moreover,
another preservice teacher discussed how designing
lesson plans helped him learn, not only about the
content, but also about incorporating technology:
I have been trying to figure out how to make
lessons based on certain content. This class has
been helping me to identify how to design lessons
based on content which is something I had no
experience with. I now have a better understanding
of how to incorporate things like technology into
the lesson as well.
Pedagogical knowledge. PK includes teaching
strategies appropriate for student learning. When
preservice teachers were asked to discuss pedagogical
issues as they designed activities during the fourth
week, only two mentioned the use of technology.
Specifically, one mentioned TI-Nspire and one
mentioned websites. Other than that, participants
focused their discussion primarily on the use of
manipulatives,
inquiry,
problem
solving,
differentiation, and other pedagogical issues.
The class has made me realize the importance of
manipulatives and hands-on activities in the
classroom. These types of activities help students to
become active learners, and thus, cause them to
retain more of the material.
In the quote above, the candidate did not make an
explicit connection between the use of technology and
the use of manipulatives in an active/hands-on learning
style.
I feel in my activity write-ups, I am constantly
considering pedagogical issues. The one main
issue is using an inquiry method for problem
solving. I try to have my students explore a topic,
such as finding the length of the diagonal of a
square through investigation rather than lecture.
Once again, the candidate does not explicitly link the
use of digital technologies to the use of an inquirybased pedagogical approach.
The interactions among pedagogy, content and
technology. In a survey given in the thirteenth week,
three open-ended questions asked the participants to
discuss the relationships between content and
pedagogy, content and technology, and technology and
pedagogy. Figure 5 illustrates the interactions between
S. Asli Özgün-Koca, Michael Meagher, & Michael Todd Edwards
content, technology, and pedagogy reflected in the
data. The direction and thickness of the arrows
represent the relationships articulated by the preservice
teachers. If many preservice teachers articulated a
relationship, the arrow representing that relationship is
bolded. Dashed arrows signify relationships that were
not mentioned or are non-existent.
Even though no one mentioned how the content
influenced the technology directly, several preservice
teachers mentioned “appropriate uses” of technology
when discussing the relationship between content and
pedagogy. In these cases, they were not focusing on
technological skills, but on the use of technology as
one of many possible teaching strategies:
Some things that we taught kids were naturally
studied with the Nspire. For instance graphing
functions was a natural with the “Graphs and
Geometry” application.
The math you are teaching totally influences the
way you teach it. Some topics are well suited for
use with technology. Like when you are learning
about transformations, then Sketchpad is a natural
tool to use. When you're studying graphs, then
calculators are a natural learning tool. Other
topics, such as trigonometric proofs and identities,
aren't as obviously hands-on. I feel more limited
teaching these topics using methods other than
lecture.
Figure 5. Interactions between the content, the
technology and the pedagogy.
Many preservice teachers mentioned that they
could see the influence of content on pedagogy. For
instance, they acknowledged that the complexity of the
high school content affects the teaching method to be
employed:
I totally think that content influences how topics
are taught. I'm not sure if pedagogy should
influence content.
The methods you use depend on the kids you teach
and the topics you are teaching.
The content you teach is related to the way you
teach it. When I taught kids about area and
perimeter in the field, we used plastic tiles to study
the topics in a hands-on manner. But we didn't use
hands-on materials when we studied cross
multiplying. There just wasn't a good way to do
this hands-on.
Even though a few said that they were not sure if
pedagogy should influence the content, the examples
that they gave showed that they were considering the
pedagogy to content direction in Figure 5. One
mentioned, “in my field placement, I saw how dynamic
geometry software influences what content is taught
and how it is taught.”
Thus, in Figure 5, we connected content and
technology through pedagogy with one arrow.
When discussing the influences of technology on
content, participants focused on the capabilities of
technology. Due to the capabilities of advanced
technologies, the content (curriculum) might have been
affected or some contents might have become more
accessible:
I have been using more technology now with the
problems than I ever thought I would. I remember
from problem set two that using technology made
doing the problem much easier to do.
Technology does change the content. There are
things that I studied in school that don't seem
relevant in algebra class anymore. For instance,
factoring. We did an assignment on the Nspire that
showed that 99% (or more) of quadratics aren't
factorable. So why do we spend all of this time
factoring?
I think technology changes the content sometimes,
but I don't think it should. I think math should
influence the technology, not the other way around.
In regard to the relationship between technology
and pedagogy, some preservice teachers discussed
technology as a pedagogical tool as opposed to
focusing on needed technical skills. Therefore
technology was embedded in pedagogy. They noted
how the use of technology might impact how a task
develops, which in turn could influence student
learning. In these instances, the participants are
thinking through how technology is deployed rather
17
Preservice Teachers’ Emerging TPACK
than, as discussed earlier, which topics might naturally
lend themselves to technology use:
Technology can make it easier to test conjectures.
For instance, with sketchpad, we can test countless
conjectures much more quickly than possible when
using pencil and paper. When we observe
behaviors in sketchpad (or with the TI-Nspire),
students are more motivated to ask “does this
always happen?”
I have been able to incorporate things such as the
TI Nspire and GSP into my activity write-ups, and I
think that incorporating these types of technology
into lessons helps to make them (the activities)
more multifaceted and thus easier for a larger
percentage of the students in a classroom to
understand.
Not all preservice teachers thought that technology
influenced students’ learning. In particular, one
candidate noted:
Technology doesn't change the way kids learn
math. They have to learn it the way I learned it, by
repetition and practice. It's like learning how to
read. You have to do some memorizing and
repetition before you can get to the good stuff.
Other preservice teachers noted the importance of
first using paper and pencil experiences in students’
learning. In these cases, it appeared that their beliefs
about how learning occurs affected the extent to which
they would use technology in their teaching. During
the eighth week, after having some field experiences,
one preservice teacher felt that students were
dependent on calculators for computation: “The main
issue that I dealt with in my classroom (i.e. methods
field experience) was the severe dependency on
calculators that students seemed to have.” In the exit
survey, approximately 70% of the preservice teachers
agreed that they would specifically like to use the
Nspire when they become a full-time teacher.
However, in the fourth week, on a more general
question about the place of technology in the
mathematics classroom, approximately 68% of the
preservice teachers agreed that students should not use
calculators until they have thoroughly mastered the
required skills by hand. This percentage decreased to
44% at the end of the study. We should point out that
only 9 participants completed the survey in the
thirteenth week, as opposed to the 20 that completed
the survey in the fourth-week survey.
18
One of the preservice teachers’ main messages was
that content should be a teacher’s first priority. As one
preservice teacher put it:
Good teachers think about content first and ways
to better deliver content to students. Putting
pedagogy first (and even WORSE, putting
technology first) is irresponsible. We should
always be thinking about WHAT we [want] our
kids to know MATHEMATICALLY . . . then figure
out how (or if) technology or various teaching
methods support THAT . . . not the other way
around.
Eventually, they started to look at the content from
a teacher's perspective, thinking about issues related to
teaching and learning Later, technology came into play
as a pedagogical tool with novel capabilities that paper
and pencil (or chalk and blackboard) cannot provide.
At the end of the course, participants thought
themselves better prepared to use technology in their
teaching. At the beginning of the course, only 40% of
them considered themselves at least fairly prepared to
have students use technology to explore new concepts.
This percentage increased to 84% at the end of the
course. Similarly, 64% of them felt fairly or very well
prepared to have students use appropriate educational
technology to learn mathematics at the beginning of
the year. This percentage also increased to 84% in the
exit survey.
Conclusions
This study sought to examine teachers’ emerging
TPACK as manifested in their use of advanced
technologies in the design and implementation of
technology-rich activities in their student teaching. We
did this through an examination of their views on the
use of advanced technologies, such as the TI-Nspire.
Our major conclusions are that (1) preservice teachers’
development of TPACK is related to their shift in
identity from being learners of mathematics to teachers
of mathematics; and that, even in a class where
advanced digital technologies are used extensively as a
catalyst for promoting inquiry-based learning, (2)
preservice teachers retain a great deal of skepticism
about the role of technology in mathematics education
even though they felt much better prepared to
incorporate technology into their teaching.
We have shown above that, often, a preservice
teacher’s first use of advanced technologies is naïve
and incorporates technology superficially. We believe
this results from a combined intial lack of PCK and
TK. These two deficits, in tandem, make it hard to
S. Asli Özgün-Koca, Michael Meagher, & Michael Todd Edwards
design tasks that allow students to explore
mathematical concepts. The data show that, initially,
preservice teachers’ mathematical focus is on content
and their own ability to solve problems. By and large,
preservice teachers have been successful in doing
mathematics in traditional environments. As they make
the shift to being mathematics teachers, they begin to
develop pedagogical knowledge and become interested
in hands-on activities and inquiry-based learning.
However, for many, there may remain a feeling that
advanced technologies “do too much.” The data shows
that they do not see advanced technologies as part of
an inquiry-based approach.
Preservice teachers can certainly develop their
technological, pedagogical, and content knowledge
separately, but integrating these types of knowledge
through the development of their TPK, TCK and
TPACK gives them a more holistic view of their
teaching and helps them transition from learners of
mathematics to teachers of mathematics. Our data
show that close attention must be paid to the
relationship between the university classroom and the
field placement; ideally, every preservice teacher
would see that what they learn in the university
classroom has an impact on their work in the field.
Field placements are where preservice teachers face the
reality of a classroom and experience first-hand that
how they design tasks affects student learning.
Our conclusions suggest several directions for
further research. Perhaps the most obvious of these is
the need for further investigation of what happens
when participants complete their preservice training
and become full time teachers: What are the crucial
influences on the development of TPACK? Our past
research (Özgün-Koca, Meagher, & Edwards, in press)
suggests that experiencing success in the classroom
and reflection, through journal writing or interviews,
are vital elements in continuing the development of
TPACK. Other potential influences to consider include
access to technology, availability of materials to
support inquiry-based instruction, and the existence of
a supportive professional environment.
Another area for further research is studying the
effect, on preservice teachers’ attitudes and practices,
of seeing exemplary inquiry-based instruction in a
technology-rich environment. There is not enough data
in this study to support strong claims, but our data does
suggest that students found it difficult to appreciate the
possibilities of advanced technologies in instruction
without experiencing exemplary use in an authentic
classroom situation. Such experience is highly
dependent on field placement, although use of remote
video technology could be employed to make an
exemplary experience available to an entire class.
Using advanced technologies in methods classes
puts preservice teachers in the position of being
learners. This allows them to pay explicit attention to
developing their TCK, which in turn encourages them
to reflect on their PCK and CK. Thinking about, and
engaging with, advanced technologies gives preservice
teachers a vantage point from which to examine their
beliefs about, and attitudes towards, what it means for
their students to be successful.
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20
The Mathematics Educator
2009/2010, Vol. 19, No. 2, 21–31
Why Not Philosophy? Problematizing the Philosophy of
Mathematics in a Time of Curriculum Reform
Kimberly White-Fredette
This article argues that, as teachers struggle to implement curriculum reform in mathematics, an explicit
discussion of philosophy of mathematics is missing from the conversation. Building on the work of Ernest
(1988, 1991, 1994, 1998, 1999, 2004), Lerman (1990, 1998, 1999), the National Council of Teachers of
Mathematics (1989, 1991, 2000), Davis and Hersh (1981), Hersh (1997), Lakatos (1945/1976), Kitcher (1984),
and others, the author draws parallels between social constructivism and a humanism philosophy of
mathematics. While practicing mathematicians may be entrenched in a traditional, Platonic philosophy of
mathematics, and mathematics education researchers have embraced the fallibilist, humanist philosophy of
mathematics (Sfard, 1998), the teachers of school mathematics are caught somewhere in the middle.
Mathematics teachers too often hold true to the traditional view of mathematics as an absolute truth independent
of human subjectivity. At the same time, they are pushed to teach mathematics as a social construction, an
activity that makes sense only through its usefulness. Given these dichotomous views of mathematics, without
an explicit conversation about and exploration of the philosophy of mathematics, reform in the teaching and
learning of mathematics may be certain to fail.
The teaching and learning of mathematics is
going through tremendous changes. The National
Council of Teachers of Mathematics’ (NCTM, 2000)
Principles and Standards for School Mathematics
calls for reforms to both curriculum and classroom
instruction. Constructivist learning, student-centered
classrooms, worthwhile tasks, and reflective teaching
are all a part of NCTM’s vision of school
mathematics in the 21st century. Along with calls for
changes in how mathematics is taught, there are
numerous calls for changes in who engages in higher
level mathematics courses. NCTM’s Equity Principle
calls for high expectations, challenging curriculum,
and high-quality instructional practices for all
students. In addition, recent publications from the
National Research Council (NRC, 2001, 2005) have,
in many ways, redefined the teaching and learning of
mathematics. These documents call for a move away
from the teaching of isolated skills and procedures
towards a more problem solving, sense-making
instructional mode. This changing vision of school
mathematics—student-centered
pedagogy,
constructivist learning in our classrooms, focus on
the problem-solving aspects of mathematics, and
mathematics success for all— cannot come about
Kimberly White-Fredette has taught elementary, middle, and
high school mathematics, and is currently a K-12 math
consultant with the Griffin Regional Educational Service Agency.
She recently completed her doctorate at Georgia State
University. Her work focuses on supporting teachers as they
implement statewide curriculum reform.
without a radical change in instructional practices
and an equally radical change in teachers’ views of
mathematics teaching and learning, as well as the
discipline of mathematics itself.
As state curricula, assessment practices, and
teaching expectations are revamped, a discernable
theoretical framework is essential to the reform
process (Brown, 1998). This theoretical framework
must include a re-examination of teachers’ views of
mathematics as a subject of learning. What are
teachers’ beliefs about mathematics as a field of
knowledge? Do teachers believe in mathematics as a
problem-solving discipline with an emphasis on
reasoning and critical thinking, or as a discipline of
procedures and rules? Do teachers believe
mathematics should be accessible for all students or
is mathematics only meant for the privileged few?
Recent studies examining teacher beliefs and
mathematical reform have primarily focused on
teachers’ views of mathematics instruction (see e.g.,
Bibby, 1999; Cooney, Shealy, & Arvold, 1998; Hart,
2002; Mewborn, 2002; Sztajn, 2003). Further
research is required to understand how teachers view
not only mathematics teaching and learning, but
mathematics itself. Unlike many previous studies,
this research should examine teachers’ philosophies,
not simply their beliefs, regarding mathematics.
Philosophy and beliefs, although similar, are not
identical. Beswick (2007) asserted that there is no
agreed upon definition of the term beliefs, but that it
can refer to “anything that an individual regards as
21
Why Not Philosophy?
true” (p. 96). Pajares (1992) affirmed the importance
of researching teacher beliefs, although he
acknowledged that “defining beliefs is at best a game
of player’s choice” (p. 309). Not only is any
definition of beliefs tenuous, but distinguishing
beliefs from knowledge is also a difficult process
(Pajares, 1992). I argue that a study of philosophy
moves beyond the tenuousness of beliefs, in that
philosophy is a creative process. “Philosophy is not a
simple art of forming, inventing, or fabricating
concepts, because concepts are not necessarily forms,
discoveries, or products. More rigorously,
philosophy is the discipline that involves creating
concepts” (Deleuze & Guattari, 1991/1994, p. 5).
Why philosophy?
Current calls for reform in mathematics
education are not without controversy (Schoenfeld,
2004). This controversy, and the reluctance towards
change, may well be rooted in philosophical
considerations (Davis & Mitchell, 2008). Webster’s
Dictionary (2003) defines philosophy as “the critical
study of the basic principles and concepts of a
particular branch of knowledge, especially with a
view to improving or reconstituting them” (p. 1455).
A study examining philosophy, therefore, seeks to
better understand those basic principles and concepts
that teachers’ hold regarding the field of
mathematics. Philosophy, not just philosophy of
mathematics teaching and learning, but the
philosophy of mathematics, is rarely examined
explicitly: “Is it possible that teachers’ conceptions
of mathematics need to undergo significant revisions
before the teaching of mathematics can be revised?”
(Davis & Mitchell, p. 146). That is a question not yet
answered by the current research on teacher change
and mathematics education.
Researchers seldom ask teachers to explore their
philosophies of the mathematics they teach. But such
a study is in keeping with the writings of Davis and
Hersch (1981) and Kitcher (1984) who sought to
problematize the concept of mathematics. If we are
to change the nature of mathematics teaching and
learning, we have to look beyond the traditional view
of mathematics as a fixed subject of absolute truths,
what Ernest (1991) and Lerman (1990) termed an
absolutist view. Constructivist teaching and inquirybased learning demand a new view of mathematics,
the fallibilist view that envisions “mathematical
knowledge [as a] library of accumulated experience,
to be drawn upon and used by those who have access
to it” (Lerman, p. 56) and “focuses attention on the
22
context and meaning of mathematics for the
individual, and on problem-solving processes”
(Lerman, p. 56).
Sfard (1998) argued that mathematicians are
entrenched in an absolutist view of mathematics
while researchers in mathematics education are
deeply immersed in the fallibilist view:
On the one hand, there is the paradigm of
mathematics itself where there are simple,
unquestionable criteria for distinguishing
right from wrong and correct from false. On
the other hand, there is the paradigm of
social sciences where there is no absolute
truth any longer; where the idea of
objectivity is replaced with the concept of
intersubjectivity, and where the question
about correctness is replaced by the concern
for usefulness. (p. 491)
The teachers of school mathematics are caught
between these two opposing groups, yet are rarely
asked to explore their philosophies of mathematics.
The very existence of philosophies of mathematics is
often unknown to them. Yet the question, what is
mathematics, is as important to the work of K–12
mathematics teachers as it is to the mathematics
education researcher and the mathematician. In the
following sections, I will outline the recent
explorations that researchers and mathematicians
have undertaken in the areas of philosophy of
mathematics
and
mathematics
education.
Unfortunately, few researchers have engaged
teachers of mathematics in this important discussion.
Changing Views of Education and Mathematics
An investigation of philosophy of mathematics is
rooted in three areas. Postmodern views of
mathematics, 20th century explorations in the
philosophy
of
mathematics,
and
social
constructivism have contributed to discussions
regarding the philosophy of mathematics. I begin by
describing the emergence of social constructivism,
which in many ways is the driving force behind
mathematical reform in the United States and other
nations (Forman, 2003).
Social Constructivism
Forman and others (e.g., Restivo & Bauchspies,
2006; Toumasis, 1997) have argued that NCTM’s
Professional Standards for Teaching Mathematics
(1991) and the later Principles and Standards for
School Mathematics (2000) clearly build upon a
social constructivist model of learning. But Ernest
Kimberly White-Fredette
(1991, 1994, 1998, 1999) argues that social
constructivism is more than just a learning theory
applicable to the teaching and learning of
mathematics.
According to Ernest, social
constructivism is a philosophy of mathematics that
views mathematics as a social construction. Social
constructivism focuses on the community of the
mathematics classroom and the communication that
takes place there (Noddings, 1990), and grew out of
Vygotsky’s (1978) work in social learning theory. It
has been further developed in mathematics teaching
and learning through the work of Confrey (1990),
Lerman (1990, 1998, 1999), and Damarin (1999).
This theory is in keeping with NCTM’s (2000)
emphasis on the social interplay in mathematics
instruction:
Students’ understanding of mathematical
ideas can be built throughout their school
years if they actively engage in tasks and
experiences designed to deepen and connect
their
knowledge.
Learning
with
understanding can be further enhanced by
classroom interactions, as students propose
mathematical ideas and conjectures, learn to
evaluate their own thinking and that of
others, and develop mathematical reasoning
skills. Classroom discourse and social
interaction can be used to promote the
recognition of connections among ideas and
the reorganization of knowledge. (p. 21)
Overall, social constructivists advocate that
educators form a view of mathematical learning as
something people do rather than as something people
gain (Forman, 2003).
It is upon this foundation that Ernest (1998) built
his theory of social constructivism as a philosophy of
mathematics. He argued that the teaching and
learning of mathematics is indelibly linked to the
philosophy of mathematics:
Thus the role of the philosophy of
mathematics is to reflect on, and give an
account of, the nature of mathematics. From
a philosophical perspective, the nature of
mathematical knowledge is perhaps the
central feature which the philosophy of
mathematics needs to account for and reflect
on. (p. 50)
Without that link, Ernest argued, we cannot truly
understand the aims of mathematics education.
Ernest (2004) emphasized the need for researchers,
educators, and curriculum planners to ask “what is
the purpose of teaching and learning mathematics?”
(p. 1). But, in order to answer that, both mathematics
and its role and purpose in society must be explored.
Dossey (1992) also placed an emphasis on the
philosophy of mathematics: “Perceptions of the
nature and role of mathematics held by our society
have a major influence on the development of school
mathematics curriculum, instruction, and research”
(p. 39). Yet, in the educational sphere, there is a lack
of conversation about and exploration of philosophy
that “has serious ramifications for both the practice
and teaching of mathematics” (Dossey, p. 39).
Without a direct focus on philosophy, the
consequences of differing views of mathematics are
not being explored.
Ernest (1991, 1998) described two dichotomist
philosophical views of mathematics—the absolutist
and the fallibilist. The Platonist and formalist
philosophies both stem from an absolutist view of
mathematics as a divine gift or a consistent,
formalized language without error or contradiction.
Both of these schools of thought believe mathematics
to be infallible, due either to its existence beyond
humanity, waiting to be discovered (the Platonist
school), or to its creation as a logical, closed set of
rules and procedures (the formalist school). The
fallibilist philosophy, what Hersh (1997) termed a
philosophy of humanism, views mathematics as a
human construction and, therefore, fallible and
corrigible. One important implication of the fallibilist
philosophy of mathematics is that if mathematics is a
human construct then so must be the learning of
mathematics. In the fallibist philosophy, mathematics
is no longer knowledge that is simply memorized in
a rote fashion. It is societal knowledge that must be
interpreted in a manner that holds meaning for the
individual. The constructivist approach to learning,
therefore, aligns well with the fallibilist philosophy
of mathematics.
Ernest (1991, 1998) characterized a cycle of
subjective and objective knowledge to support his
view of the social constructivist foundations of
mathematical knowledge. In this cycle, new
knowledge begins as subjective knowledge, the
mathematical thoughts of an individual. This new
thought becomes objective knowledge, knowledge
that may appear to exist independent of humanity,
through a social vetting process. This objective
knowledge then enters the public domain where
individuals test, reformulate, and refine the
knowledge. The individuals then internalize and
23
Why Not Philosophy?
interpret the objective knowledge, once again
transforming it to subjective knowledge. The social
process of learning mathematics is intricately linked
to society’s ideas of what is and is not mathematics.
Thus, Ernest was able to connect a learning theory,
social constructivism, with a philosophy of
mathematics.
A Postmodern View of Mathematics
During the past 50 years, there has been a
growing discussion of the historical and
philosophical foundations of mathematics. What was
once seen as objective is now viewed by some as a
historical and social construction, changing and
malleable, as subjective as any social creation.
Aligned with these changing views of mathematics
are new ideas about mathematics instruction. The
absolutist view of mathematics is associated with a
behaviorist approach, utilizing drill and practice of
discrete skills, individual activity, and an emphasis
on procedures. The fallibilist view of mathematics
aligns itself with pedagogy consistent with
constructivist theories, utilizing problem-based
learning, real world application, collaborative
learning, and an emphasis on process (Threlfall,
1996). Although there have been numerous calls to
change and adapt the teaching of mathematics
through the embracing of a constructivist
epistemology, little has been done to challenge
teachers’ conceptions of mathematics. The push
towards student-centered instructional practices and
the current challenges to traditional views of
mathematics teaching have been brought together
through Ernest’s work over the past 20 years:
“Teaching reforms cannot take place unless teachers’
deeply held beliefs about mathematics and its
teaching and learning change” (Ernest, 1988).
Ernest’s (2004) more recent work advocates a
postmodern view of mathematics. He seeks to break
down the influence of what he terms the “narratives
of certainty” that have resulted in “popular
understandings of mathematics as an unquestionable
certain body of knowledge” (Ernest, 2004, p. 16).
Certainly, this understanding still predominates in
mathematics classrooms today (see e.g., Bishop,
2002; Brown, Jones, & Bibby, 2004; Davison &
Mitchell, 2008; Handel & Herrington, 2003).
However, Ernest draws upon postmodern
philosophers such as Lyotard, Wittgenstein,
Foucault, Lacan, and Derrida, to challenge traditional
views of mathematics and mathematics education.
He embraces the postmodern view because it rejects
24
the certainty of Cartesian thought and places
mathematics in the social realm, a human activity
influenced by time and place. Others have joined
Ernest in exploring mathematics and mathematics
instruction through the postmodern perspective (see
e.g. Brown, 1994; Walkerdine, 1994; Walshaw,
2004). Neyland (2004) calls for a postmodern
perspective in mathematics education to “address
mathematics as something that is enchanting, worthy
of our esteem, and evocative of wonder” (p. 69). In
so doing, Neyland hopes for a movement away from
mathematics instruction emphasizing procedural
compliance and onto a more ethical relationship
between teacher and student, one that stresses not
just enchantment in mathematics education but
complexity as well.
Walshaw (2004) ties sociocultural theories of
learning to postmodern ideas of knowledge and
power, drawing, as Ernest does, on the writings of
Foucault and Lacan: “Knowledge, in postmodern
thinking, is not neutral or politically innocent” (p. 4).
For example, issues of equity in mathematics can be
seen in ways other than who can and cannot do
mathematics. Indeed, societal issues of power and
reproduction must be considered. A postmodern
analysis forces a questioning of mathematics as
value-free, objective, and apolitical (Walshaw,
2002). Why are the privileged mathematical
experiences of the few held up as the needed (but
never attained) mathematical experiences of all?
Furthering a postmodern view of mathematics,
Fleener (2004) draws on Deleuze and Guattari’s idea
of the rhizome1 in order to question the role of
mathematics as lending order to our world: “By
pursuing the bumps and irregularities, rather than
ignoring them or ‘smoothing them out,’ introducing
complexity, challenging status quo, and questioning
assumptions, the smoothness of mathematics is
disrupted” (p. 209). The traditional view of
mathematics has ignored the bumps and
irregularities, forcing a vision of mathematics as
smooth, neat, and orderly.
Another postmodern view is that our
representations of mathematics cannot be divorced
from the language we use to describe those
representations:
Any act of mathematics can be seen as an
act of construction where I simultaneously
construct in language mathematics notions
and the world around me. Meaning is
produced as I get to know my relationships
to these things. This process is the source of
Kimberly White-Fredette
the post-structuralist notion of the human
subject being constructed in language.
(Brown, 1994, p. 156)
Brown used Derrida’s ideas on deconstructing
language to examine how the social necessity of
mathematical learning means that mathematics is
always, in some way, constructed. And, in examining
new mathematical ideas, the learners cannot help but
bring their entire mathematical and personal history
to the process.
Brown’s
(1994)
postmodern
view
of
mathematics strengthens Ernest’s (1998) own
contention of the philosophical basis of social
constructivism. Mathematical ideas begin as social
constructions but “become so embedded within the
fabric of our culture that it is hard for us to see them
as anything other than givens” (Brown, p. 154).
Thus, the establishment of mathematical metanarratives2 camouflages the social/cultural roots of
mathematical knowledge. As a result, mathematics
continues to be viewed primarily as something
discovered, not constructed. Siegel and Borasi
(1994) described the pervasive cultural myths that
continue to represent mathematics as the discipline
of certainty. In order to confront this idealized
certainty, they state a need for an inquiry
epistemology that “challenges popular myths about
the truth of mathematical results and the way in
which they are achieved, and suggests, instead, that:
mathematical knowledge is fallible… (and)
mathematical knowledge is a social process that
occurs within a community of practice” (p. 205).
This demystifying process is necessary, argued
Siegel and Borasi, if teachers are to engage students
in doing mathematics, not simply memorizing rote
procedures and discrete skills.
New Ideas in the Philosophy of Mathematics
A world of ideas exists, created by human
beings,
existing
in
their
shared
consciousness. These ideas have objective
properties, in the same sense that material
objects have objective properties. The
construction of proof and counterexample is
the method of discovering the properties of
these ideas. This branch of knowledge is
called mathematics. (Hersh, 1997, p. 19)
A reawakening of the philosophy of mathematics
occurred during the last part of the 20th century
(Hersh, 1997). In their landmark book The
Mathematical Experience, Davis and Hersh (1981)
explored ideas of mathematics as a human invention,
a fallibilist construct. Davis and Hersh described
several schools of philosophical thought regarding
mathematics—including Platonism and formalism.
In the Platonist view, mathematics “has evolved
precisely as a symbolic counterpart of the universe. It
is no wonder, then, that mathematics works; that is
exactly its reason for existence. The universe has
imposed mathematics upon humanity” (p. 68). The
Platonist not only accepts, but embraces, God’s place
in mathematics. For what is mathematics but God’s
gift to us mortals? (Plato, trans. 1956). The Platonist,
forever linking God and mathematics, sees the
perfection of mathematics. If there are errors made in
our mathematical discoveries (and, of course, they
are discoveries not inventions because they come
from a higher power), then the errors are ours as
flawed humanity, not inherent to the mathematics.
And because mathematics is this higher knowledge it
follows that some will succeed at mathematics while
many others fail. Mathematics, in the Platonic view,
becomes a proving ground, a place where those who
are specially blessed can understand mathematics’
truths (and perhaps even discover further truths)
while the vast numbers are left behind.
Euclid’s Elements was (and still is) the bible of
belief for mathematical Platonists (Hersh, 1997). As
Davis and Hersh (1981) pointed out, “the appearance
a century and a half ago of non-Euclidean geometries
was accompanied by considerable shock and
disbelief” (p. 217). The creation of non-Euclidean
geometries—systems in which Euclid’s fifth
postulate (commonly known as the parallel
postulate) no longer held true—momentarily shook
the very foundations of mathematical knowledge.
The loss of certainty in geometry was
philosophically intolerable, because it
implied the loss of all certainty in human
knowledge. Geometry had served, from the
time of Plato, as the supreme exemplar of
the possibility of certainty in human
knowledge. (Davis & Hersh, p. 331)
A result of the uncertainty brought on by the
formation of non-Euclidean geometries was the
development
of
formalism.
In
formalism,
mathematics is the science of rigorous proofs, a
language for other sciences (Davis & Hersh, 1981).
“The formalist says mathematics isn’t about
anything, it just is” (Hersh, 1997, p. 212). In the
early part of the 20th century, Frege, Russell, and
Hilbert, among others, each attempted to formalize
25
Why Not Philosophy?
€
all of mathematics through the use of the symbols of
logic and set theory. Russell and Whitehead’s
“unreadable masterpiece” (Davis & Hersh, p. 138),
Principia Mathematica, attempted the complete
logical formalization of mathematics. But the
attempts to complete the logical formalization of
mathematics were doomed to failure as demonstrated
by Gödel’s Incompleteness Theorem that proved any
formal system of mathematics would remain
incomplete, not provable within its own system
(Goldstein, 2005).
Proofs and Refutations: The Logic of
Mathematical Discovery, a beautifully written
exploration of the philosophy of mathematics penned
by Imre Lakatos (1976), offered an alternative
philosophy of mathematics to those of Platonism and
formalism, termed the humanist philosophy. In
Proofs and Refutations, Lakatos used the history of
mathematics, as well as the structure of an inquirybased mathematics classroom, to explore ideas about
proof. Through a lively Socratic discussion between
a fictional teacher and students, Euler’s formula
( V − E + F = 2 ) is dissected, investigated, built upon,
improved, and, finally, made nearly unrecognizable.
Lakatos used the classroom dialogue to challenge
accepted ideas about proof. He forced the reader to
question if proofs are ever complete or if
mathematicians simply agree to ignore the nonexamples, which Lakatos’s students termed
monsters, that contradict the proof. Through this
analogy, Lakatos demonstrated that in mathematics
there are many monsters, most of which are ignored,
as though the mathematical community has made a
tacit agreement to turn away from that which makes
it uncomfortable.
Ernest built much of his philosophy of
mathematics and mathematics education on the
writings of Lakatos. Like Lakatos, Ernest (1998) saw
mathematics as indubitably tied to its creator—
humankind: “Both the creation and justification of
mathematical knowledge, including the scrutiny of
mathematical warrants and proofs, are bound to their
human and historical context” (p. 44). Hersh (1997),
in his book, What is Mathematics, Really?, included
both Lakatos and Ernest on his list of “mavericks”—
thinkers who see mathematics as a human activity
and, in so doing, influenced the philosophy of
mathematics. Others are included as well:
philosophers Charles Sanders Peirce (Siegel &
Borasi, 1994) and Ludwig Wittgenstein (Ernest,
1991, 1998b), psychologists Jean Piaget and Lev
26
Vygotsky (Confrey, 1990, and Lerman, 1994), and
mathematicians George Polya and Philip Kitcher.
Polya’s (1945/1973) classic, How to Solve It: A
New Aspect of Mathematical Method, revived the
study of the methods and rules of problem solving—
called heuristics—in mathematics. Although he
eschewed philosophy, Polya saw mathematics as a
human endeavor. He described the messiness of the
mathematician’s work:
Mathematics in the making resembles any
other human knowledge in the making. You
have to guess a mathematical theorem
before you prove it; you have to guess the
idea of the proof before you carry through
the details. You have to combine
observations and follow analogies; you have
to try and try again. (Polya, 1954/1998, pp.
99)
Both Polya and Lakatos led mathematicians into
new areas that questioned the very basis of
mathematical knowledge. Their combined impact on
the philosophy of mathematics was as important as
the development of non-Euclidean geometries and
Gödel’s Incompleteness Theorem (Davis & Hersh,
1981). By defining mathematics as a social construct,
they opened up the field to new interpretations.
Polya’s heuristic emphasized the accessibility of
problem solving. Lakatos, by using dialogue to trace
the evolving knowledge of mathematics—the proofs
and refutations—stressed the social aspects of
mathematical learning as well as the fallibility of
mathematical knowledge, and defined mathematics
as quasi-empirical. No longer was mathematics a
subject for the elite. Ernest (1998) credited Lakatos
with a synthesis of epistemology, history, and
methodology in his philosophy of mathematics—a
synthesis that influenced the sociological,
psychological, and educational practices of
mathematics.
Ernest (1998) and Hersh (1997) also referred to
Kitcher as a maverick, in that he stressed the
importance of both the history of mathematics and
the philosophy of mathematics. Kitcher (1983/1998)
underscored the concept of change in mathematics:
“Why do mathematicians propound different
statements at different times? Why do certain
questions wax and wane in importance? Why are
standards and styles of proof modified?” (p. 217).
His conclusion was that mathematics changes in
practice, not just in theory. Kitcher identified five
components of mathematical practice—language,
Kimberly White-Fredette
metamathematical views, accepted questions,
accepted statements, and accepted reasoning—that
are developmentally compatible: As one component
changes, others must change as well (Hersh, 1997).
Kitcher’s five components emphasized the social
aspect of mathematics as a community activity with
agreed upon norms and practices. Kitcher’s view of
mathematics mirrors Ernest’s cycle of subjective
knowledge → objective knowledge → subjective
knowledge and Lakatos’ idea of proofs and
refutations in that each generation simultaneously
critiques, internalizes, and builds upon the
mathematics of the previous generation (Hersh,
1997).
Conclusion
Few studies have addressed the issue of teachers’
philosophies of mathematics. Too often, those that
have relied on surveys and questionnaires to define
the complexity that is a teacher’s philosophy (see
e.g., Ambrose, 2004; Szydlik, Szydlik, & Benson,
2003; Wilkins & Brand, 2004). Although studies
conducted by Lerman (1990), Wiersma and
Weinstein (2001), and Lloyd (2005) briefly
examined their participants’ expressed perceptions of
mathematics, none of these studies specifically
examined the results of an exploration of philosophy.
What remains to be investigated is what happens
when teachers are presented with non-traditional
views of mathematics and explore philosophical
writings about mathematics. At a university in
Greece, Toumasis (1993) developed a course for
preservice secondary school mathematics teachers
that centered on readings about the history and
philosophy of Western mathematics, as well as
“discussion and an exchange of views” (p. 248). The
purpose of the course was to develop a reflective
mathematics teacher because:
To be a mathematics teacher requires that
one know what mathematics is. This means
knowing what its history, its social context
and its philosophical problems and issues
are. . . . The goal is to humanize
mathematics, to teach tolerance and
understanding of the ideas and opinions of
others, and thus to learn something of our
own heritage of ideas, how we came to think
the way we do (p. 255).
According to Toumasis, teacher preparation
programs continue to shortchange mathematics
teachers by focusing only on coursework in higher
level mathematics, e.g., Linear Algebra, Discrete
Mathematics, and Analysis. Knowledge of
mathematics, especially if one is to teach
mathematics, must include a reflexive study of
mathematics.
Toumasis (1997) argued that the philosophical and
epistemological beliefs about the nature of
mathematics are intrinsically bound with the
pedagogy of mathematics. In his examination of the
philosophical underpinnings of NCTM’s Curriculum
and Evaluation Standards for School Mathematics
(1989), Toumasis identified a clear fallibilist point of
view; mathematics is “a dialogue between people
tackling mathematical problems” (p. 320). Yet in
current attempts to reform mathematics based on
both the 1989 Standards and the later Principles and
Standards for School Mathematics (NCTM, 2000),
an investigation of philosophy is rarely undertaken.
The teaching and learning of mathematics is a
politically charged arena. Strong feelings exist in the
debate on how “best” to teach mathematics in K–12
schools, feelings that are linked to varying
perceptions about the nature of mathematics
(Dossey, 1992). Is mathematics an abstract body of
ideal knowledge, existing independently of human
activity, or is it a human-construct, fallible and everchanging? These perceptions of mathematics then
drive beliefs about the appropriateness of
instructional
practices
in
mathematics.
Is
mathematics a body of knowledge that must be
memorized and unquestionably mastered, or do we
engage the learners of mathematics in personal
sense-making,
in
constructing
their
own
mathematical knowledge? That we are still in the
midst of “math wars” is indisputable (Schoenfeld,
2004). What it means to teach mathematics and the
very nature of mathematics is at the center of these
wars:
Traditionalists or back-to-basics proponents
argue that the aim of mathematics education
should be mastery of a set body of
mathematical knowledge and skills. The
philosophical complement to this version of
the teaching and learning of mathematics is
mathematical absolutism. Reform-oriented
mathematics educators, on the other hand,
tend to see understanding as a primary aim
of school mathematics. Constructivism is
often the philosophical foundation for those
espousing this version of mathematics
education. (Stemhagen, 2008, p. 63)
27
Why Not Philosophy?
I agree with Schoenfeld (2004), Greer and
Mukhopadhyay (2003), and others (e.g., Davison &
Mitchell, 2008) that the math wars are based on
philosophical differences. It has therefore been my
intent to inject philosophy into the discussion of
mathematics educational reform and research.
Research is needed that focuses on what we teach as
mathematics and, more importantly, how teachers
view the mathematics that they teach. Is mathematics
transcendental and pure, something that exists
outside of humanity, or is it a social activity, a social
construction whose rules and procedures are defined
by humanity (Restivo & Bauchspies, 2006)? An
extensive review of the literature found no studies
that led teachers to explore their philosophies of
mathematics. Yet Restivo and Bauchspies
recognized the need to push teachers’ understanding
of mathematics beyond the debate of mathematics as
a social construction. To understand mathematics
(and thus to teach mathematics) is to understand the
social, cultural, and historical worlds of mathematics
(Restivo & Bauchspies). Should we not then explore
mathematics in a philosophical sense, its “basic
principles and concepts…with a view to improving
or reconstituting them” (Webster’s Dictionary, 2003,
p. 1455)? Change in classroom practices may not be
possible without first “improving or reconstituting”
teachers’ philosophies of mathematics.
Many studies have addressed the need to engage
both preservice and inservice teachers in
constructivist learning in order to change their
instructional practices (see, e.g., Hart, 2002;
Mewborn, 2003; Thompson, 1992). Yet little has
been done to engage teachers in a philosophical
discussion of mathematics: “Teachers, as well,
should be encouraged to develop professionally
through philosophical discourse with their peers”
(Davison & Mitchell, 2008, p. 151). Philosophy and
mathematics have a long-standing connection, going
back to the ancient Greeks (Davis & Hersh, 1981).
Mathematics teachers are seldom asked to explore
philosophy beyond an introductory Philosophy of
Education course. If one is going to teach
mathematics, one should ask “Why?” What is the
purpose of teaching mathematics? What is the
purpose of mathematics in society at large? Should
not mathematics’ purpose be tied to how we then
teach it? These questions come back to teachers’
perception of mathematics, and more specifically,
their philosophies of mathematics.
28
I contend that discussions of philosophy,
particularly philosophy of mathematics, should be
brought to the forefront of mathematics education
reform. If teachers are never asked to explore the
philosophical basis of their perceptions of
mathematics, then they will continue to resist
change, to teach the way they were taught. The
growing philosophical investigations of mathematics
(see, e.g., Davis & Hersh, 1981; Hersh, 1997;
Restivo, Van Bendegen, & Fischer, 1993;
Tymoczko, 1998) in the past 30 years have not often
been addressed in mathematics education research.
We seem afraid to raise issues of philosophy as we
implement curriculum reform and study teacher
change. But philosophy too often lies hidden, an
unspoken obstacle in the attempt to change
mathematics education (Ernest, 2004). Researchers
can bring the hidden obstacle to light, should engage
both policymakers and educators in a conversation
about philosophy, not with the intent of enforcing the
“right” philosophy but with the acknowledgement
that, without a continued dialogue about philosophy,
the curriculum reform they research may continue to
fall short.
I end this article by revisiting a definition of
philosophy of mathematics: “The philosophy of
mathematics is basically concerned with systematic
reflection about the nature of mathematics, its
methodological problems, its relations to reality, and
its applicability” (Rav, 1993, p. 81). If our goal in
mathematics education reform is to make
mathematics more accessible and more applicable to
real-world learning, we should then help guide
today’s teachers of mathematics to delve into this
realm of systematic reflection and to ask themselves,
“What is mathematics?”
1
Building from the botanical definition of rhizome,
Deleuze and Guattari (1980/1987) used the analogy of the
rhizome to represent the chaotic, non-linear, postmodern
world. Like the tubers of a canna or the burrows of a mole,
rhizomes lead us in many directions simultaneously.
Deleuze and Guattari described the rhizome as having no
beginning or end; it is always in the middle.
2
A meta-narrative, wrote Kincheloe and Steinberg
(1996), “analyzes the body of ideas and insights of social
theories that attempt to understand a complex diversity of
phenomena and their interrelations” (p. 171). It is, in other
words, a story about a story; a meta-narrative seeks to
provide a unified certainty of knowledge and experience,
removed from its historic or personalized significance.
Kimberly White-Fredette
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31
The Mathematics Educator
2006, Vol. 19, No. 1, 32–45
Measuring Task Posing Cycles: Mathematical Letter Writing
Between Algebra Students and Preservice Teachers
Anderson Norton
Zachary Rutledge
In a secondary school mathematics teaching methods course, a research team engaged 22 preservice secondary
teachers (PSTs) in designing and posing tasks to algebra students through weekly letter writing. The goal of the
tasks was for PSTs to elicit responses that would indicate student engagement in the mathematical processes
described by NCTM (2000) and Bloom’s taxonomy (Bloom, Englehart, Furst, Hill, & Krathwohl, 1956), as
well as student engagement in the highest levels of cognitive activity described by Stein, Smith, Henningsen,
and Silver (2000). This paper describes our efforts to design reliable measures that assess student engagement in
those processes as a product of the evolving relationship within letter-writing pairs. Results indicate that some
processes are easier to elicit and assess than others, but that the letter-writing pairs demonstrated significant
growth in terms of elicited processes. Although it is impossible to disentangle student factors from teacher
factors that contributed to that growth, we find value in the authenticity of assessing PSTs’ tasks in terms of
student engagement rather than student-independent task analysis.
Designing and posing tasks plays a central role for
mathematics teaching (Krainer, 1993; NCTM, 2000).
However, research indicates that preservice teachers
lack ability to pose appropriately challenging
mathematical tasks for students (e.g., Silver, MamonaDowns, Leung, & Kenney, 1996). This article
addresses the development of such ability by engaging
preservice secondary teachers (PSTs) in posing
mathematical tasks to high school algebra students
through mathematical letter writing. We consider our
approach an extension of the kind of letter-writing
study performed by Crespo (2000; 2003). In a previous
article (Rutledge & Norton, 2008), we reported results
from this project related to the letter-writing
interactions between PSTs and students. That article
focused on comparing cognitive constructivist and
socio-cultural lenses for examining the interactions.
The purpose of this article is to investigate the
mathematical processes that PSTs’ tasks elicited from
students.
Crespo (2003) engaged preservice elementary
school teachers in posing mathematical tasks to fourthgrade students through letter writing. The purpose of
her study was to elicit and assess students’
mathematical thinking. She found the tasks preservice
Anderson Norton is an Assistant Professor in the Department of
Mathematics at Virginia Tech. He teaches math courses for
future secondary school teachers and conducts research on
students' mathematical development.
Zachary Rutledge is completing a PhD in Mathematics Education
at Indiana University. He works as an actuary in Salem, Oregon.
32
teachers wrote became more open-ended and
cognitively complex over the weeks of letter writing.
This result affirmed her key hypothesis that the
preservice teachers’ extended and reflective
interactions with an “authentic audience” (p. 243)
would provide opportunities for them to learn how to
pose appropriately challenging tasks. Crespo’s work
informed our approach to studying the development of
task-posing ability among PSTs, and we too used letter
writing with PSTs to foster such development. Rather
than focusing on the tasks PSTs posed, as Crespo did,
we specifically examined elicited student responses as
a product of the evolving relationship within letterwriting pairs.
During a secondary methods course, 22 PSTs were
paired with high school algebra students; the PSTs
posed tasks to their student partners and assessed the
responses. As researchers, we independently examined
the responses from the algebra students to make
inferences about their cognitive activity. Considering
this study to be an extension of Crespo’s work, we
introduce a method for measuring PSTs’ progress in
learning to design and pose individualized
mathematical tasks through letter writing. We
measured the effectiveness of PSTs’ tasks by assessing
the cognitive activities those tasks elicited from
students (as indicated by student responses), and we
hypothesized that such measurements would
demonstrate growth over the course of letter-writing
exchanges between the pairs.
We report on our design of measurements for the
effectiveness of the letter-writing pairs, in addition to
Anderson Norton & Zachary Rutledge
the results of applying that design. In particular, we
relied on descriptions of cognitive activities described
in three main sources: Bloom’s taxonomy (Bloom et
al., 1956; Kastberg, 2003), Principles and Standards of
School Mathematics (NCTM, 2000), and a chapter on
“cognitively complex tasks” by Stein, Smith,
Henningsen, and Silver (2000). We chose these sources
because they are common readings in the PSTs’
methods courses, and they provide potential metrics for
assessing the quality of tasks. We collectively modified
them to form a comprehensive and complementary
framework for assessing students’ responses to the
tasks.
In the following section we summarize the original
authors’ descriptions of these processes. We then
describe how we operationalized the processes to
assess the cognitive activities indicated by each student
response. In the final two sections, we report on the
reliability of our measures and the evolution of
cognitive activity elicited by the PSTs’ tasks over the
course of 12 weeks. Findings from this study inform
the following research questions: How can we reliably
measure the effectiveness of the letter-writing
exchanges in terms of elicited cognitive activity from
the high school students? And, using the measurements
we develop, in what ways do the PSTs demonstrate
progress in designing and posing appropriately
challenging mathematical tasks for students?
We wanted PSTs to learn to pose more engaging
mathematical tasks and to assess students’ thinking
based on their written responses. We hypothesized that
over the 12 weeks the PSTs’ tasks would elicit more of
NCTM’s Process Standards (2000) and the four highest
levels of reasoning in Bloom’s taxonomy (i.e.
application, analysis, synthesis, and evaluation). We
also expected a general progression toward responses
that indicated students were using Procedures with
Connections and Doing Mathematics, moving away
from responses reliant upon Memorization or
Procedures without Connections (Stein et al., 2000).
Theoretical Orientation
Task Posing
Since Brown and Walter’s (1990) seminal work on
problem posing (read as “task posing”), many
subsequent publications focused on teachers engaging
students in posing problems (e.g., Gonzales, 1996;
Goldenberg, 2003). Whereas these publications have
implications for teacher education, they do not
examine teachers’ abilities to design appropriately
challenging tasks for their students. Research
investigating teachers’ abilities to design such tasks has
typically focused on student-independent attributes of
the tasks, such as whether they introduce new implicit
assumptions, initial conditions, or goals (Silver et al.,
1996). Similarly, Prestage and Perks (2007) engaged
PSTs in modifying givens and analyzing mathematical
demand of tasks in order for these future teachers to
develop fluency in creating ad hoc tasks in the
classroom. However, Prestage and Perks noted, “the
analysis of the mathematics within a task can only
offer a description of potential for learning” (p. 385).
Understanding the actual cognitive demand of a task
depends upon the learner. “Today, there is general
agreement that problem difficulty is not so much a
function of various task variables, as it is of
characteristics of the problem solver” (Lester & Kehle,
2003, p. 507). One such characteristic that has received
insufficient attention during task posing is the students’
understanding of mathematics content (NCTM, 2000,
p. 5). As Crespo (2000; 2003) demonstrated, letter
writing can provide a rich context for PSTs to develop
task-posing ability through mathematical interactions
with students and help PSTs better attend to student
understanding of content.
Liljedhal, Chernoff, and Zazkis (2007) described
another important component of task-posing for PSTs:
“predicting the affordances that the task may access”
(p. 241) as PSTs attempt to elicit particular
mathematical concepts or processes from students.
However, within letter writing such analyses no longer
determine whether the task is ‘good’ because PSTs can
rely on students’ actual responses for making that
determination. Crespo (2003) described letter writing
as an opportunity for “an authentic experience in that it
paralleled and simulated three important aspects of
mathematics teaching practice: posing tasks, analyzing
pupils’ work, and responding to pupils’ ideas” (p. 246).
The authenticity of PST-student interactions is highly
desirable because the PSTs can assess the effectiveness
of their tasks without relying on the authority of a
teacher educator. The benefits of this kind of
authenticity might be analogous to students’
experiences when they view their own mathematical
reasoning as an authority, rather than relying on the
text or a teacher for validation. The PSTs’ problem
becomes one of “witnessing the development of the
activities provoked by the task, and comparing it to the
ones they predicted and to the initial task” (Horoks &
Robert, 2007, p. 285). This development allows PSTs
to use these comparisons as they modify their initial
tasks and design new tasks.
33
Measuring Task Posing Cycles
Cognitive Measures (in Theory)
In order to measure PSTs’ progress in eliciting
mathematical activity from students through task
posing, we looked to three sources: Bloom’s taxonomy
(Bloom et al., 1956), the NCTM Process Standards
(2000), and the levels of cognitive demand designed by
Stein et al. (2000). PSTs’ familiarity with these sources
was important to us for the following reason: Often
these sources (or others that describe a hierarchy for
analyzing student thinking) are introduced to PSTs as
useful ideas to adapt into their future teaching. Teacher
educators should move beyond introduction of these
sources and instead facilitate opportunities for PSTs to
investigate ways that they prove beneficial in working
with students. Therefore, we asked PSTs to assess their
students’ responses to tasks using these sources. This
mirrors the way that we used them in this study to
assess the PST’s task-posing ability.
34
Table 1 presents the measures created for this study
based on these frameworks. The short definition
provides a summary of the different measures as
described by the original authors. The first four
measures come from Bloom’s taxonomy of educational
objectives (Bloom et al., 1956), the next five measures
come from NCTM’s Process Standards (2000), and the
last four measures come from Stein et al. (2000). It is
important to note that we chose to disregard the first
two levels of Bloom’s taxonomy, Knowledge and
Comprehension, because we felt that these measures
were too low-level and would likely be elicited with
great frequency. On the other hand, we kept all four of
Stein’s levels of cognitive demand because they
provide a necessary hierarchy for ranking tasks and
measuring growth, as we describe in the following
section.
Anderson Norton & Zachary Rutledge
Methodology
Setting
The 22 PSTs who participated in this study were
enrolled in the first of two mathematics methods
courses that precede student teaching at a large
midwestern university. Mrs. Rae, a local high school
mathematics teacher, was interested in finding ways to
challenge her students by individualizing instruction.
When the PSTs’ methods instructor (first author)
approached her about task-posing through letter
writing, Mrs. Rae agreed that such an activity would
serve the educational interests of her students, as well
as the PSTs. Each PST was assigned to one student
from Mrs. Rae’s Algebra I class, and wrote letters back
and forth to her or his assigned student, once per week
for seven weeks. The PSTs were given no guidelines
on the type of problems to pose; instead they were
instructed to focus on building students’ mathematical
engagement. The high school term (trimester) ended
after the seventh week and students were assigned to
new classes, so the PSTs began writing letters to a new
group of students in the eighth week. They wrote to
Mrs. Rae’s Algebra II students the final five weeks of
the project. Each week, the methods instructor and
Mrs. Rae collected the letters and responses,
respectively, and exchanged them.
In the title of this article, we use the term cycle to
refer to the PSTs’ iterative task design. After posing an
initial task, we expected PSTs to use student responses
to construct models of students’ mathematical thinking.
That is, we expected PSTs to try to “understand the
way children build up their mathematical reality and
the operations by means of which they try to move
within that reality” (von Glasersfeld & Steffe, 1991, p.
92). Using this knowledge, the PSTs could design tasks
more attuned with their students’ mathematics—the
students’ particular mental actions and ways of
applying those actions to problem-solving situations. In
turn, we hypothesized that the well-designed tasks
would presumably increase student engagement and
cognitive activity. By focusing PSTs’ attention on the
cognitive activities described by NCTM, Bloom et al.,
and Stein et al., we hoped to provide a framework for
PSTs to begin building models.
Whereas PSTs’ goals for student learning often
revert to mastery of procedural knowledge (Eisenhart
et al., 1993), we promoted goals for conceptual
learning among the PSTs through class readings and
discussions. We encouraged PSTs to use open-ended
tasks (i.e. tasks that invite more than one particular
response) so student responses would be rich enough
for PSTs to make inferences about the students’
thinking. We hoped the opportunity to make inferences
about the students’ mathematical thinking would lead
the PSTs to construct models of students’ mathematics.
We also encouraged the PSTs to rely on their models
to imagine how students’ mathematics might be
reorganized in order to become more powerful,
allowing the students to engage with a broader range of
mathematical situations.
Data Analysis
Data consisted of PSTs’ letters and students’
responses. PSTs complied these documents into their
notebooks, and we collected them at the end of their
methods course. After removing 31 letters that were
not matched with task responses, 233 tasks/response
pairs remained to be analyzed.
Data
analysis
had
four
phases:
(a)
operationalization of our cognitive measures, (b) the
raters’ individual coding, (c) reconciliation of our
individual coding, and (d) interpretation of the final
codes. The operationalization concerns the way in
which we transformed the theoretical processes given
in the previous section into heuristics that allowed us to
identify cognitive activity. Individual coding relied on
this operationalization while continuing to inform
further operationalization of the cognitive measures.
As not to distort inter-rater reliability scores, in the
interim we met only to discuss clarifications of the
cognitive activities, without sharing notes or discussing
particular responses. At the end of the letter-writing
project, we computed the inter-rater reliability of our
coding for the cognitive measures. Following this
analysis, we reconciled our codes by arguing points of
view regarding scoring differences until we reached
consensus. Finally, we could interpret the reconciled
codes, graphically and statistically.
Graphs of the relative frequency of each cognitive
activity, as measured week-by-week, provide an
indication of growth among the PST-student pairs. We
use the graphs to describe patterns in elicited activity
over time. Although the two different groups of
students involved in letter writing (Algebra I students
in the first seven weeks, and Algebra II students in the
final five weeks) render a 12-week longitudinal
analysis untenable, data from the two groups do
provide opportunity for us to consider differences in
PSTs’ success in working across the groups. Finally,
we performed linear regressions on aggregate results to
provide a statistical analysis of progress.
35
Measuring Task Posing Cycles
Cognitive Measures (in Practice)
The operationalization occurred mostly during the
individual coding phase with minor adjustments
required during the reconciliation phase. That is to say,
we essentially transformed the 13 measures into a
system allowing a researcher or a practitioner to
categorize his or her inferences of students’ cognitive
activity. To achieve this transformation, we began with
the previously discussed definitions for the various
measures and then made adjustments throughout the
individual coding phase. Whenever one of the raters
(authors) encountered difficulty in assessing a student
response, he would approach the other to discuss the
difficulty, in a general way, without referring to a
particular student response. This interaction would
allow the raters to decide how to resolve the difficulty
and individually reassess previous ratings to ensure
consistent use of the newly operationalized measure.
Table 2 describes the most fundamental changes
that we made to the measures. The adjustments are the
results of the following two goals: (1) to ensure that
measures could be consistently applied from task to
task and (2) to ensure that no two measures were
redundant.
With regard to redundancy, we had to differentiate
Connections, Procedures with Connections, and
36
Application. We used Connections to refer to
connections among disparate mathematical concepts;
we reserved Procedures with Connections to describe
connections between mathematical procedures and
concepts; and we reserved Application to describe
connections among mathematical concepts and other
domains. With regard to our ability to consistently
apply a measure, we had difficulty with Problem
Solving and Doing Mathematics. As defined in Table
1, Problem Solving requires a struggle toward a novel
solution, rather than the application of an existing
procedure or concept. Lester and Kehle (2003) further
characterized problem solving as “an activity requiring
the individual to engage in a variety of cognitive
actions” (p. 510).
Lester and Kehle (2003) described a tension
between what is known and what is unknown:
Successful problem solving involves coordinating
previous
experiences,
knowledge,
familiar
representations and patterns of inference, and
intuition in an effort to generate new
representations and patterns of inference that
resolve the tension or ambiguity (i.e., lack of
meaningful representations and supporting
inferential moves) that prompted the original
problem solving activity. (p. 510)
Anderson Norton & Zachary Rutledge
Other cognitive activities, such as representation
and inference (possibly involving reasoning and proof),
may support a resolution to this tension. For us to infer
that a student engaged in Problem Solving, we needed
to identify indications of this perceived tension, and we
needed to infer a new construction through a
coordination of cognitive actions. This meant
responses labeled as involving Problem Solving were
often labeled as involving other cognitive activities as
well, such as Analysis and Reasoning & Proof.
While Stein et al.’s (2000) definition of Doing
Mathematics provided some orientation for our work,
we found it to be too vague for our purposes. So, we
relied on Schifter’s (1996) definition for further
clarification; she defined Doing Mathematics as
conjecturing. To infer a student had engaged in Doing
Mathematics, we needed to infer the student had
engaged in making and testing conjectures. For
example, if we inferred from student work that the
student designed a mathematical formula to describe a
situation and then appropriately tested this formula,
then we would consider this to be Doing Mathematics.
We recognize that our restriction precludes assessment
of other activities that Stein et al. (2000) would
consider to be “doing mathematics,” but this restriction
provided a workable resolution to assessing students’
written responses.
As a final and general modification of the original
cognitive measures, we required that each process
(other than the lowest two levels of cognitive demand)
produce a novelty. For example, assessing Synthesis
required some indication that the student had produced
a new whole from existing constituent parts.
Furthermore, we needed indication that the student
generated the cognitive activity as part of their
reasoning. If a PST explicitly asked for a bar graph, the
students’ production of it would not constitute
Representation, because it would not indicate
reasoning. Such a response would probably indicate a
Procedure without Connections.
Results
A Letter-Writing Exchange
To illustrate the exchanges between letter-writing
pairs, and to clarify the manner of our assessments, we
provide the following sample exchange. A complete
record of the exchange can be found in Rutledge and
Norton (2008). Figure 1 shows the task posed by a
PST, Ellen, in her initial letter to her student partner,
Jacques. The task is similar to other introductory tasks
posed by PSTs and seems to fit the kinds of tasks to
which they had become accustomed from their own
experiences as students. However, there is evidence
(i.e. the “why” questions at the task’s end) that Ellen,
like fellow PSTs, attempted to engage the student in
responding with more than a computational answer.
37
Measuring Task Posing Cycles
Jacques’ response (Figure 2) indicates that he did
not meaningfully engage in the task of finding
equations for lines meeting the specified geometric
conditions. However, he was able to assimilate (make
sense of) the situation as one involving solutions to
systems of equations. From his activity of
manipulating two linear equations and their graph, we
inferred that the task elicited only procedural
knowledge from Jacques. It is possible that Jacques
may have had a more connected understanding of the
concepts underlying the procedure, but there was no
clear indication from his response that allowed us to
infer this. Therefore, we coded the elicited activity as
Procedures without Connections.
provoked Jacques to struggle through finding equations
for the lines. In addition, Ellen seemed to detect an
overall trend that Jacques engaged more readily with
familiar procedures. She adapted to the trend and
began to frame future tasks around a procedure with
which she felt Jacques was likely familiar. This kind of
adaptation to the student indicates that Ellen began to
model the student’s thinking.
Other codes assigned to Jacques’ response
included Application and Communication. The former
was based on our inference that Jacques used existing
ideas in a novel situation. He effectively applied an
algebraic procedure to a new domain when he applied
his knowledge of systems of equations to a situation
involving finding equations of intersecting lines, When
coding for Communication, we inferred that Jacques’
written language intended to convey a mathematical
idea involving the use of systems of equations to find
points of intersection.
Subsequent tasks and responses indicate Ellen
began to model Jacques’ mathematical thinking. Using
these models, she designed tasks that successfully
engaged Jacques in additional cognitive activities, such
as problem solving. For example, Ellen asked
questions to focus Jacques’ attention on the angles
formed in the drawing on Figure 1. Her questions
reconcile discrepancies. To understand inter-rater
reliability, we considered three measures as shown in
Tables 4 and 5. These measures were Cohen’s Kappa,
Percent Agreement, and Effective Percent Agreement.
As Table 4 indicates, percent agreement was high on
all four measures from Bloom’s taxonomy, but this
result is because of the rarity of either rater identifying
the measures. Effective percent agreement provides
further confirmation of this outcome by considering
agreement among those items positively identified by
at least one of the raters.
Moving from left to right, the Kappa scores in
Table 3 show decreasing inter-rater reliability as we
progress to higher levels of Bloom’s taxonomy. Sim
and Wright (2005) cite Landis and Koch who suggest
the following delineations for interpreting Kappa
scores: less than or equal to 0 poor, 0.01-0.20 slight,
0.21-0.40 fair, 0.41-0.60 moderate, 0.61-0.80
38
Inter-Rater Reliability Results
When we finished individually coding all of the
task responses for the letter-writing pairs, we met to
compile the results into a spreadsheet, for this process
allowed us to measure inter-rater reliability and
Anderson Norton & Zachary Rutledge
substantial, and 0.81-1 almost perfect. With this in
mind, we see Application has a substantial agreement,
Analysis has moderate agreement, Synthesis has slight
agreement, and Evaluation did not have agreement
distinguishable from random.
The high percent agreement in combination with a
low effective percent agreement for Synthesis and
Evaluation highlight the fact the raters rarely identified
these two constructs. In the few times one rater
identified such a construct when the other did not, this
resulted in a low Kappa score. For further support of
this assessment, we note that the 95% confidence
interval for these two measures includes 0; thus, there
is no support for inter-rater reliability with these two
measures.
Table 4 displays the data for the NCTM Process
Standards. In a similar analysis, Communication and
Problem Solving have moderate agreement.
Representation is on the border between fair and
moderate; Reasoning & Proof is on the border between
slight and fair; and Connections has a poor inter-rater
reliability. Connections and Reasoning & Proof were
quite rare, hence the high levels of percent agreement
and the lower level of Kappa, as was the case with
Synthesis and Evaluation. In fact, the 95%
confidence interval for each of these measures includes
0, indicating no reliability on these measures.
Table 5 displays the raters’ responses for the levels
of cognitive demand described by Stein et al. (2000).
The Kappa was .55—described as moderately
reliable—with a 95% confidence interval of .47 to .64
(note that we report only one Kappa because we could
choose only one categorization for each task response).
We see that although 164 of the 233 items are on the
main diagonal (showing agreement), there is definite
spread away from the diagonal as well. We determined
this was partly attributable to a shift in how
conservatively the raters interpreted student responses.
Specifically, one rater tended to identify items as
eliciting lower cognitive demand than the other rater.
This is seen in the total column and row, where one
rater identified 178 items as either Memorization or
Procedures without Connections and only 45 items as
Procedures with Connections. Alternately, the other
rater found 157 items to be either Memorization or
Procedures without Connections and 70 items to be
Procedures with Connections.
39
Measuring Task Posing Cycles
design in that these measures were more commonly
identified in the coding.
Elicited-Response Results
To summarize, Evaluation was the measure with
the weakest reliability, for it had a negative Kappa
associated with it. Although having a positive Kappa,
Synthesis, Connections, and Reasoning & Proof had
confidence intervals that contained 0; this result
indicates we cannot be certain whether it was above 0
randomly. It is also important to note these measures
were some of the least-often identified. Conversely, the
most reliable measures were Communication, Problem
Solving, and Application. In addition, our assessments
of levels of cognitive demand were reliable to a similar
degree. Again, this conclusion is supported by our
40
After measuring inter-rater reliability, we
reconciled our scores by arguing for or against each
discrepant score. For example, the second author
assessed Task 1 (Figure 1 and Figure 2) as having
elicited Connections. However, the first author
successfully argued that the evidence was stronger for
connections to concrete situations, and according to
our operationalization that should be coded as
Application. We report on the reconciled scores in
Figures 3, 4, and 5. Each of those figures illustrates the
percentages of responses that satisfied our negotiated
measures over the course of the twelve weeks. We
excluded missing responses from all calculations.
Although we included letters from week 1, we note
many of the introductory letters did not include tasks,
presumably because the PSTs were becoming familiar
with the students and the format of the activity. We
also note the first seven letters were written between
PSTs and Algebra I students, whereas the final five
letters were written between PSTs and Algebra II
students. Once again, the letters written in week 8 were
introductory letters, though these included many more
tasks. In Figures 3, 4, and 5, the dark vertical line
Anderson Norton & Zachary Rutledge
between weeks 7 and 8 marks the transition from
Algebra I letters to Algebra II letters.
Ignoring the introductory letters from week 1, the
general trends illustrated in Figure 3 indicate the
frequency of PSTs eliciting Application from students
decreased over the duration of letter writing, even
across correspondences with Algebra I and Algebra II
students. Analysis was elicited more frequently, with a
pronounced spike among correspondence between
PSTs and Algebra II students in the final weeks. As
previously mentioned in the reliability results, we
found both Synthesis and Evaluation were rarely
elicited in correspondence with either group of
students.
These patterns indicate the levels of cognition
described by Bloom’s taxonomy—at least in our
operationalization of them—were heavily dependent
on the PSTs and the tasks they posed. Interestingly,
these patterns show little apparent dependence on the
groups of students (i.e. Algebra I and Algebra II). This
outcome may be because many of the posed tasks
inherently required application and analysis to resolve
them, with PSTs gaining a greater appreciation for
students’ use of analysis over the course of the
semester. Application tasks tended to describe new
situations where the PSTs inferred, often correctly, that
the students could use existing knowledge. For
example, we characterized Jacques’ response in Figure
2 as indicating an Application, for he applied his
knowledge about solving linear equations to a concrete
situation that required finding the point of intersection
of two lines. Analysis tasks often involved equations
whose components needed to be examined. For
example, in a subsequent exchange with Ellen, Jacques
broke down the triangle (Figure 1) into three lines and
correctly identified the sign of the slopes of these lines.
Using this knowledge, he attempted to formulate the
equations of these lines. It seems that either PSTs were
less familiar with the kinds of tasks that might elicit
Synthesis and Evaluation, or students did not readily
engage in such activity.
In Figure 4, we begin to see some differences
between the elicited responses of the two groups of
students. Whereas Connections, Representation, and
Reasoning & Proof were rarely elicited from either
group of students, there is a pronounced increase in
Problem Solving among correspondence with the
Algebra II students as compared to the Algebra I
students. Communication also increased among
correspondences with Algebra II students, but seemed
to be elicited in a pattern that was similar across the
two groups.
We see mathematical communication from both
groups of students increased to a peak during the
41
Measuring Task Posing Cycles
middle weeks, and then decreased toward the end of
the correspondence between letter writing pairs. This
trend may be due to the PSTs’ initial interest in the
students’ thinking, which was replaced by more goaldirected tasks, once the PSTs determined a particular
trajectory along which to direct the students. We found
Communication dropped among both groups of
students after their fourth week. This could be due to a
lack of enthusiasm among the students once the
novelty of letter writing had faded. In fact, Mrs. Rae
noticed the Algebra I students began to tire of writing
responses and wrote less in later weeks.
Figure 5 illustrates a general trend away from tasks
eliciting Memorization. It seems the PSTs used
students’ recall of facts in order to gauge where the
students were developmentally, both at the beginning
and end of their correspondence with the students.
Procedures without Connections dominated the
elicited responses from students, whereas Procedures
with Connections seemed to play a significantly lesser
role. It is also interesting to note that the few instances
identified as Doing Mathematics occurred among
correspondence with Algebra II students. Along with
the previous observation about Problem Solving
(namely, that problem solving was elicited much more
with Algebra II students), these results lead us to one
42
of two conclusions: (1) the PSTs held higher
expectations for Algebra II students (in terms of
cognitive activity, and not just content) and were,
therefore, more inclined to challenge them with higherlevel tasks, or (2) the Algebra II students were better
prepared (either from previous learning or accepted
social norms) to engage in these higher levels of
cognitive activity.
A Statistical Analysis
In addition to considering the measures
individually, we performed a linear regression on
aggregate results over time. The first column in Table
6 lists the average number of processes elicited weekby-week, among the five NCTM Process Standards
and the four highest levels of Bloom’s taxonomy. For
example, the PSTs elicited, on average, two of the nine
processes during Week 10. The second column in
Table 6 lists the average ranking of the levels of
cognitive demand elicited week-by-week. We ranked
Memorization as 0, Procedures without Connections as
1, Procedures with Connections as 2, and Doing
Mathematics as 3. If we accept this simple form of
ranking, the student responses from Week 7 indicated,
on average, as a 1, Procedures without Connections.
Anderson Norton & Zachary Rutledge
measurements. We intend to improve the
measurements in terms of their reliability and their
value as assessments of professional growth. First, we
recognize areas of weakness in reliable uses of the
measures, as well as areas of weakness in elicited
responses. These areas coincide because cognitive
activities that were least assessed were assessed least
reliably; they include Synthesis, Evaluation,
Connections, Reasoning & Proof, and Representation.
Reliability of Measures as Operationalized
Table 7 reports the slopes and r-squared values for
each column over each of the following time periods:
the first seven weeks (interactions with Algebra I
students), the final five weeks (interactions with
Algebra II students), and the entire 12 weeks (across
the two groups of students). In addition, Table 7
includes the corresponding p-values to indicate
whether the slopes are statistically significant. We
calculated these values using rank coefficients. The
slopes provide indications of the group’s growth from
week to week to the degree that the r-squared values
approach 1 and p-values approach 0.05. It is interesting
to note that the slope, r-squared value, and p-value for
the final five weeks of letter writing suggest
considerable growth in the level of mathematical
engagement during interactions between PSTs and
Algebra II students.
Discussion of Findings and Implications
Having operationalized measures of cognitive
activity and having applied them to a cohort of letterwriting pairs, we are now prepared to evaluate the
When we reconciled our independent assessments
of task responses, common themes emerged
concerning the least assessed cognitive activities. Some
of these involved highly subjective judgments, such as
the novelty of the activity for the student and the
student’s familiarity with particular concepts and
procedures. This subjectivity highlights the need for us
to make our assessments based on inferences about the
student’s mathematical activity, just as we asked the
PSTs to design their tasks based on such inferences.
For example, one student assimilated information from
a story problem in order to produce a simple linear
equation. During coding, this response would typically
be evaluated as Synthesis; however, one of the raters
inferred that the student was so familiar with the
mathematical material that her actions indicated a
procedural exercise.
For our reconciliations of all measures, we agreed
each cognitive activity needed to produce a
mathematical novelty, such as a tabular representation
a student produced to organize data in resolving the
task. If the PST were to request the table, then the
student’s production of it would not be considered
novel. Therefore, this response would not be labeled as
a Representation. Likewise, we decided Connections
should be used to refer to a novel connection between
two mathematical concepts, such as a connection a
student might make between his or her concepts of
function and reflection in resolving a task involving
transformational geometry. The need to make
inferences about the novelty of students’ activities
introduced ambiguity in assessing student responses,
43
Measuring Task Posing Cycles
particularly because we assessed responses week-byweek without considering the history of each student’s
responses.
We also realized we introduced some reliability
issues through our selection of cognitive activities.
Whereas we were pleased with the diversity of
measures offered by the three categorizations (Bloom’s
taxonomy, NCTM’s Process Standards, and Stein’s
levels of cognitive demand), these are not mutually
exclusive. Despite our efforts to operationalize the
measures in a way that would reduce overlap, we
realized, for example, Connections would always
implicate Procedures with Connections or Doing
Mathematics. Additionally, we recognize that Doing
Mathematics would always implicate Problem Solving,
and Reasoning & Proof would always implicate
Communication. Recognizing such implications might
reduce ambiguity and increase reliability by
eliminating some of the perceived need for raters to
choose one measure over another. Finally, because
frequently elicited cognitive activities were measured
reliably, we conclude that supporting PSTs’ attempts to
elicit all cognitive activities can increase the reliability
of each measurement. This support would also promote
our goals for PSTs to design more engaging tasks.
Eliciting Cognitive Activity
We originally hypothesized that PSTs’ tasks would
elicit more cognitive processes over time, and we
anticipated a general progression toward the highest
levels of cognitive demand. Such findings would
indicate growth in the evolving problemposing/problem-solving relationships between PSTs
and students. Our hypothesis is confirmed to the degree
that r-squared values indicate the positive slopes
reported in Table 7. Those values indicate that the
relationships were particularly productive between
PSTs and Algebra II students. There are many reasons
students’ content level might have influenced the
relationship, and we cannot discern the main
contributors. Possible contributors include the
following: (1) PSTs wrote to the Algebra II students
second and for a shorter duration so the students’
remained motivated throughout the project; (2) greater
content knowledge of Algebra II students contributed
to greater process knowledge as well by allowing the
students to engage in more problem solving or make
more connections; (3) PSTs were more familiar with
the content knowledge of Algebra II students so the
PSTs were better prepared to design more challenging
tasks; (4) social norms in the two classes differed and
affected students’ levels of engagement. In any case,
44
our findings do indicate that PSTs—as a whole and
over the course of the entire twelve weeks—became
more successful in eliciting cognitive activity through
their letter writing relationships.
Our findings also indicate which cognitive
activities seem most difficult to elicit through letter
writing, and we have suggested classroom social norms
play a role in students’ reluctance to engage in some of
those activities, such as Problem Solving and Doing
Mathematics. However, we also found that PSTs were
able to engage students in some cognitive activities,
such as Application and Communication, which
affirms, “prospective teachers have some personal
capacity for mathematical problem posing” (Silver et
al., 1996, p. 293). Moreover, PSTs demonstrated
increased proficiency at engaging their student partners
in additional higher-level cognitive activities, such as
Analysis.
Silver et al. found, “the frequency of inadequately
stated problems is quite disappointing” (1996, p. 305).
Although, like Silver et al., our expectations for our
PSTs’ task design were not met, we found students
accepted nearly all of the tasks as personally
meaningful and engaged in some kind of mathematical
activity as a result. The disparity of this finding with
that of Silver et al. (1996) might be attributed to our
disparate approaches in studying problem posing. Most
notably, the PSTs in our study designed tasks with
particular students in mind and used student responses
to assess the effectiveness of those tasks and to model
students’ thinking. We believe such experiences are
essential to making methods courses personally
meaningful to future teachers.
References
Bloom, B., Englehart, M. Furst, E., Hill, W., & Krathwohl, D.
(1956). Taxonomy of educational objectives: The
classification of educational goals. Handbook I: Cognitive
domain. New York: Longmans.
Brown, S. I., & Walter, M. I. (1990). The Art of Problem Posing.
Hillsdale, NJ: Erlbaum.
Crespo, S. (2000). Seeing more than right and wrong answers:
Prospective teachers’ interpretations of students’ mathematical
work. Journal of Mathematics Teacher Education, 3, 155–
181.
Crespo, S. (2003). Learning to pose mathematical problems:
Exploring changes in preservice teachers' practices.
Educational Studies in Mathematics, 52, 243–270.
Eisenhart, M., Borko, H., Underhill, R., Brown, C., Jones, D., &
Agard, P. (1993). Conceptual knowledge falls through the
cracks: Complexities of learning to teach mathematics for
understanding. Journal for Research in Mathematics
Education, 24, 8–40.
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Goldenberg, E. P. (2003). Problem posing as a tool for teaching
mathematics. In H. L. Schoen (Ed.), Teaching mathematics
through problem solving: Grades 6-12 (pp. 69–84). Reston,
VA: National Council of Teachers of Mathematics.
Gonzales, N. A. (1996). A blueprint for problem posing. School
Science and Mathematics, 98, 448–456.
Horoks, J., & Robert, A. (2007). Tasks designed to highlight taskactivity relationships. Journal of Mathematics Teacher
Education, 10, 279–287.
Kastberg, S. (2003). Using Bloom’s taxonomy as a framework for
classroom assessment. The Mathematics Teacher, 96, 402–
405.
Krainer, K. (1993). Powerful tasks: A contribution to a high level
of acting and reflecting in mathematics instruction.
Educational Studies in Mathematics, 24, 65–93.
Lester, F. K., & Kehle, P. E. (2003). From problem solving to
modeling: The evolution of thinking about research on
complex mathematical activity. In R. Lesh & H. M. Doerr
(Eds.), Beyond constructivism (pp. 501–518). Mahwah, NJ:
Lawrence Erlbaum.
Liljedahl, P., Chernoff, E., & Zazkis, R. (2007). Interweaving
mathematics and pedagogy in task design: A tale of one task.
Journal of Mathematics Teacher Education, 10, 239–249.
Prestage, S., & Perks, P. (2007). Developing teacher knowledge
using a tool for creating tasks for the classroom. Journal of
Mathematics Teacher Education, 10, 381–390.
Rutledge, Z., & Norton, A. (2008). Preservice teachers'
mathematical task posing: An opportunity for coordination of
perspectives. The Mathematics Educator, 18(1), 31–40.
Schifter, D. (1996). A constructivist perspective on teaching and
learning mathematics. In C. T. Fosnot (Ed.), Constructivism:
Theory, perspectives, and practice (pp. 73–80). New York,:
Teachers College Press.
Silver, E. A., Mamona-Downs, J., Leung, S. S., & Kenney, P. A.
(1996). Posing mathematical problems: An exploratory study.
Journal for Research in Mathematics Education, 27, 293–309.
Sim, J., & Wright, C. C. (2005). The Kappa Statistic in reliability
studies: Use, interpretation, and sample size requirements.
Physical Therapy, 85, 257–268.
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A.
(2000). Implementing standards-based mathematics
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von Glasersfeld, E., & Steffe, L. P. (1991). Conceptual models in
educational research and practice. The Journal of Educational
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and standards for school mathematics. Reston, VA: Author.
45
UPCOMING CONFERENCES …
AMESA
Sixteenth Annual National Congress
Durban, South Africa
March 28–April
1, 2010
Seoul, Korea
July 8–15, 2012
Belo Horizonte, Brazil
July 18–23,
2010
Vancouver, British
Columbia, Canada
July 31–August
5, 2010
Grahamstown, South
Africa
September 11–
17, 2011
GCTM
Georgia Council of Teachers of Mathematics Annual Conference
http://www.gctm.org/
Rock Eagle, GA
October 2011
SSMA
School Science and Mathematics Association
Ft. Meyers, FL
November 4–6,
2010
Denver, CO
April 30–May 4,
2010
Columbus, OH
October 28–31,
2010
New Orleans, LA
January 5–8,
2011
Irvine, CA
January 28–29,
2011
RCML
Research Council on Mathematics Learning
http://www.unlv.edu/RCML/conference2007.html
Conway, AR
March 11–13,
2010
NCSM
National Council of Supervisors of Mathematics
SanDiego, CA
April 19–21,
2010
SanDiego, CA
April 21–24,
2010
http://www.amesa.org.za/AMESA2010/
ICME12
International Congress on Mathematical Education
http://www.icme12.org
PME-34
International Group for the Psychology of Mathematics Education
http://pme34.lcc.ufmg.br/
JSM of the ASA
Joint Statistical Meetings of the American Statistical Association
http://www.amstat.org/meetings/jsm/2010/
The 11th International Conference of the Mathematics Education into the 21st Century
Project
http://math.unipa.it/~grim/21project.htm
http://www.ssma.org
AERA
American Educational Research Association
http://www.aera.net
PME-NA
North American Chapter: International Group for the Psychology of Mathematics
Education
http://pmena.org/2010
MAA-AMS
Joint Meeting of the Mathematical Association of America and the American
Mathematical Society
http://www.ams.org/meetings/national_meetings.html
AMTE
Association of Mathematics Teacher Educators
http://amte.net/conferences
http://www.ncsonline.org/
NCTM
National Council of Teachers of Mathematics
http://www.nctm.org
46
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49
In this Issue,
Mathematics Is Motivating
THOMAS E. RICKS
Preservice Teachers’ Emerging TPACK in a Technology-Rich Methods Class
S. ASLI ÖZGÜN-KOCA, MICHAEL MEAGHER, & MICHAEL TODD EDWARDS
Why Not Philosophy? Problematizing the Philosophy of Mathematics in a Time of
Curriculum Reform
KIMBERLY FREDETTE-WHITE
Measuring Task Posing Cycles: Mathematical Letter Writing Between Preservice
Teachers and Algebra Students
ANDERSON NORTON & ZACHARY RUTLEDGE
The Mathematics Education Student Association is an official affiliate of the
National Council of Teachers of Mathematics. MESA is an integral part of
The University of Georgia’s mathematics education community and is
dedicated to serving all students. Membership is open to all UGA students,
as well as other members of the mathematics education community.
Visit MESA online at http://www.coe.uga.edu/mesa