Mathematics-for-Teachers (and Students)

Abstract

What mathematics do elementary teachers need and how might such mathematics be provided in a teacher education program? In this paper we discuss the development of a mathematics-for-teachers component for our elementary (K-8) preservice education program. Our mathematics-for-teachers program has evolved from an elective course for 20 preservice teachers, to 440 preservice teachers working in small groups in an auditorium setting, to a fully online component. The mathematics-for-teachers component immerses preservice teachers in mathematics experiences that many of them have never had, namely, experiences where they attend deeply to mathematical relationships and have opportunities to sense the pleasure of mathematical insight. As such, our primary goal is experiential therapy (authors, 2005) rather than content knowledge.

Full text

JOURNAL OF TEACHING AND LEARNING, 2007, VOL. 5, NO.1
Mathematics-for-Teachers (and Students)
George Gadanidis
University of Western Ontario
Immaculate K. Namukasa
University of Western Ontario
Abstract
What mathematics do elementary teachers need and how might such
mathematics be provided in a teacher education program? In this paper,
we discuss the development of a mathematics-for-teachers component
for our elementary (K-8) preservice education program. Our
mathematics-for-teachers program has evolved from an elective course
for 20 preservice teachers, to 440 preservice teachers working in small
groups in an auditorium setting, to a fully online component. The
mathematics-for-teachers component immerses preservice teachers in
mathematics experiences that many of them have never had, namely,
experiences where they attend deeply to mathematical relationships and
have opportunities to sense the pleasure of mathematical insight. As
such, our primary goal is experiential therapy (Gadanidis & Namukasa,
2005), rather than content knowledge.
Recently, at an orientation assembly, we asked our in-coming group of 440
elementary preservice teachers how they felt about mathematics. When asked if
they loved mathematics, 15-20 hands went up. When asked if they hated
mathematics, a sea of hands filled the auditorium. As one elementary preservice
teacher commented, “I hated math. I absolutely despised it. I still remember
sitting at my desk in grade one. I was sobbing quietly, because I was struggling a
bit, and I didn't finish my math on time. Thus I had to stay in at recess. Awful
isn't it!” Another preservice teacher said, “Math is like an iguana. As long as it
blends into its environment I don't mind it. But once I have to hold it I'm not so
fond of it.” Given that most elementary (K-8) teachers have to teach mathematics
(in the province of Ontario), we have a responsibility to try to help them change
their outlook towards the subject. We’re assuming that to do this, we need to
engage preservice teachers with doing mathematics, and not just learning about
pedagogy. We’re also assuming that engaging them with more school-like
mathematics—the type of mathematics that turned them off the subject in the
first place—would not be the most effective approach to take.
__________________________________________________________________
George Gadanidis is an Associate Professor at the University of Western Ontario. His current research interests
include: mathematics-for-teachers, students as performance mathematicians, learning and teaching in Web 2.0
environments, and mathematical learning objects.
Immaculate K. Namukasa is an Assistant Professor at the University of Western Ontario. Her current research
interests include: Mathematical thinking and activity; Mathematics teacher education, Non-routine problem
solving and learning environments, Complexity research framework, and Critical mathematics education.
Namukasa is also interested in globalization and internationalization of education.

14 Mathematics-for-Teachers (and Students)
In this paper, (1) we discuss the evolution of our mathematics-for-teachers
program, from an elective course for 20 preservice teachers, to 440 preservice
teachers working in small groups in an auditorium setting, to a fully online
component; (2) we define what we mean by mathematics-for-teachers and
distinguish it from pedagogical content knowledge (Shulman, 1987) and
specialized content knowledge (Ball, Bass, Sleep, & Thames, 2005); and (3) we
elaborate on our approach for offering mathematics-for-teachers in an online
environment.
Evolution of Our Mathematics-for-Teachers Program
Most Ontario preservice teacher education programs are composed of 5 months
of in-class instruction and 3 months of practicum experience. The program for
elementary (K-8) teachers must prepare them to teach all subject areas and,
consequently, this leaves little time for mathematics education. Accordingly,
Ontario elementary preservice teachers typically receive 25-30 hours of
mathematics education instruction.
Prior to 2001, our Faculty of Education mathematics education program for
elementary pre-service teachers consisted of 8 two-hour workshops (with
approximately 28 pre-service teachers per class) and 9 hours of lectures
(approximately 440 pre-service teachers in a large auditorium). The lectures gave
preservice teachers the mathematics education theory that they would then
experience and discuss in a more hands-on approach in the workshops. In
reviewing our program, we realized that both students and instructors valued the
workshops but viewed the large lectures as not very effective. Our preference
was to replace the large lectures with more small-group workshops. However, it
has been difficult to do this, given monetary and logistic constraints in our
Faculty of Education. Replacing the 9 lecture hours with 9 additional hours of
workshops would necessitate more instructional time and space. Our Faculty of
Education, whose ideal capacity is 650 preservice teachers, currently has about
850 preservice teachers (440 in K-8 and 410 in 9-12). In 2001, as a compromise
solution, we replaced the large lectures with 9 online modules accompanied by a
structured online discussion, where preservice teachers were organized in small
discussion groups. The online content came from the lecture notes. Unlike the
lectures, the online discussion offered preservice teachers the opportunity to
discuss the online content in small group settings. It also gave preservice teachers
a first-hand experience with online teacher education which is used with
increasing frequency in the school districts where they will be employed. The
online discussion was assessed which ensured participation by preservice
teachers. In the first year of implementation, instructors noticed that most
preservice teachers came to the workshops much better prepared in terms of
having read and thought about the course readings.
In our review of our mathematics education program, we felt that we needed
to add a component where pre-service teachers re-experienced mathematics. That
said, there was no time available to do this in the existing structure. In 2003, one
of the authors (Gadanidis, 2005) offered an elective Mathematics Course to
experiment with what a mathematics-for-teachers component for elementary
teachers might be like. The Mathematics Course consisted of nine 2-hour classes,
and it was offered to 20 pre-service teachers. In 2004, we were able to add 8

George Gadanidis & Immaculate Namukasa 15
hours of large group lectures without giving up the online content and discussion
that replaced the original lectures. That is, we in effect added 8 more hours of
large group contact time to our mathematics education program. We used this
time to offer a mathematics-for-teachers component, where 440 preservice
teachers worked on doing and discussing mathematics in (approximately 110)
small groups in an auditorium setting. Each hour-long session focused on one
mathematics task. Most of the 8 mathematics tasks employed came from the
elective Mathematics Course taught in the previous year. We purchased concrete
materials for each session and brought them into the auditorium in large
containers, and preservice teachers used paper plates to carry the materials they
needed to their groups.
The auditorium-based mathematics-for-teachers component had six
important characteristics. First, doing mathematics became the starting point.
Most elementary teachers have narrow views of what mathematics is and what it
means to do mathematics (Fosnot & Dolk 2001; McGowen & Davis 2001a;
McGowen & Davis 2001b). Fosnot and Dolk (2001, 159) suggest that “teachers
need to see themselves as mathematicians,” and towards this end we need to
foster environments where they engage with mathematics and construct
mathematical meaning. Second, the mathematics experiences for pre-service
teachers were designed to be interesting and challenging enough to capture their
interest and imagination and to offer the potential for mathematical insight and
surprise (Gadanidis, 2004). Third, a collaborative environment was fostered,
where pre-service teachers worked in small groups. Fourth, reflection was
fostered. In the last 5 minutes of each session, pre-service teachers took the time
to write about what they learned and what they felt during the class. Their ideas
were compiled into a single document under the headings of “learned” and “felt”
(anonymously), and this was distributed and briefly discussed at the beginning of
the next session. This helped preservice teachers see what others learned and how
they felt when doing mathematics. The learned/felt activity also served the
purpose of taking attendance. Fifth, between sessions, preservice teachers had
access to online interactive explorations of the activities they worked on in the
auditorium. They also had access to an online discussion where they could
collaborate to better understand and extend the mathematics of each activity.
Last, the culminating assessment activity of the math sessions was a Math Essay.
In the last workshop of the course, each preservice teacher randomly received
one of the math activities explored in the mathematics-for-teachers component
and had 30 minutes to ‘discuss’ one or two of the following: different solution
approaches, mathematical extensions, what they learned from the activity, or
pedagogical implications.
In 2005, we made a number of changes to the mathematics-for-teachers
component. First, it was converted to a fully online component. The decision to
do this was, in part, due to the challenge of running over 100 small groups in an
auditorium setting and, in part, due to our desire to explore what might be
possible in a fully online environment. We contemplated replacing the 8
auditorium mathematics sessions with in-class mathematics sessions, but this
would necessitate more instructional time and space, making it unfeasible due to
resource constraints. We also contemplated splitting into 4 smaller auditorium
groups, but still faced the limitations of space and human resources. With a large

16 Mathematics-for-Teachers (and Students)
online continuing teacher education program (approximately 5,000 online
students), we have ample online learning resources, experience and interest in
our Faculty of Education. Second, we reduced the number of mathematics tasks
from 8 to 4, to allow for a longer sustained focus on each task. Third, the online
mathematics activities were redesigned, becoming more comprehensive and
incorporating video as well as text, graphics and interactive content. Third, we
designed three different online Mathematics-for-Teachers courses: Measurement
and Geometry, Number, and Algebra. Organizing the activities into mathematics
curriculum strands allowed us to offer three distinct courses through our
Continuing Teacher Education Program to inservice elementary teachers and also
to parents of elementary school children. The four activities of the Algebra
course were the ones that were incorporated into the mathematics-for-teachers
component of our preservice program, and offered to all preservice teachers
during the regular teacher education program. The Measurement and Geometry
course was offered prior to their teacher education program, in August, as an
optional course that they would pay for and would appear as a quarter-credit on
their transcript. The course was offered to inservice as well as preservice
teachers, and thirty-six teachers enrolled in this course. The Number course will
be offered following their teacher education program as an optional credit. We
should note that all of our activities do cut across curriculum strands, thus an
activity whose main focus is Algebra may also integrate Measurement, Geometry
and Number concepts. We are not confident that the strand breakdown is the best
organization or the activities we use; however, the strands do correspond with the
curriculum strands used in sanctioned mathematics curriculum and assessment
documents in the province of Ontario and are also common to other jurisdictions.
Mathematics-for-Teachers
Two decades ago, Shulman (1987) suggested that teacher education (and
research) had “a blind spot with respect to content” and the emphasis was solely
“on how teachers manage classrooms, organize activities, allocate time and turns,
structure assignments, ascribe praise and blame, formulate the levels of their
questions, plan lessons, and judge general student understanding” (p. 8). Since
then, there is growing interest among mathematics educators in what
mathematical pedagogical content knowledge (MPCK) could encompass. While
we think that there needs to be a dialectical relationship between content and
pedagogy, the examples of MPCK that we have seen seem counterproductive as
they typically define what mathematics teachers need to learn by trivializing
what students need to learn. For example, MPCK tends to be defined by saying
that students need to know a mathematics concept like prime or multiplication in
two ways perhaps, but a teacher needs to know it in more ways. Likewise, Ball
(2003) suggests that “teachers need to know the same things that we would want
any educated member of our society to know, but much more (more
understanding of the insides of ideas, their roots and connections, their reasons
and ways of being represented)” (emphasis in original).
Many elementary school students and educated members of society, for
instance, think of multiplication only in terms of repeated addition or worse still
as times devoid of any deeper meaning. This is, unfortunately, the case even for
people who have been successful at mathematics (Ball & Bass, 2003). Educators

George Gadanidis & Immaculate Namukasa 17
and researchers acknowledge that this is a result of inadequate mathematics
teaching. Students who have experienced rich mathematics also do think about
multiplication in terms of areas or rows by columns; they can multiply fractions
by fractions and decimals by decimals meaningfully. To base conceptualization
of MPCK on the deficit of what students have not been taught is not very helpful.
Teachers and students can both have rich mathematical understandings and
attend to mathematics in deep and connected ways. The distinction made by the
proponents of MPCK is not as dramatic as they suggest, and consequently does
not warrant a “special” mathematics for beginning teachers as a starting point in
mathematics teacher education. We believe that the starting point of the
mathematics education for both students and teachers should be a sophisticated
and deep exploration of mathematics of which we will give an example in the
section on online mathematics.
We also do not agree with the conception of
Mathematical Knowledge for Teaching (MKT)
being developed by Ball et al. (2005). For
example, Ball et al. give the example of the
mathematical task, shown in Figure 1, and
suggest that “To teach, being able to perform
this calculation is necessary. This is common content knowledge. But being able
to carry out the procedure is not sufficient for teaching it.” They identify four
distinct domains of mathematical knowledge for teaching:
1. common content knowledge (calculating the answer to 307-168)
2. specialized content knowledge (analyzing calculation errors)
3. knowledge of students and content (identifying student thinking
that might have produced such errors)
4. knowledge of teaching and content (recognizing which
manipulatives would best highlight place-value features of the
algorithm)
The implication seems to be that the last three domains distinguish what
teachers need to know from what students need to know. But let us imagine a
classroom situation where a student is solving the problem in Figure 1 on the
blackboard and makes a mistake. We understand that the teacher is “analyzing
the calculation error,” “is identifying student thinking that might have produced
the error,” and is “thinking about which manipulatives (or other modeling tools)
would best highlight place-value features of the algorithm” so that the student
might realize the error made and be able to make sense of the formal procedure.
But what are the students doing? Are they thinking? What are they thinking
about? What should they be thinking about? We suggest that they should also be
invited to attend to the calculation error, making conjectures about the thinking
that might have produced the error, and they should be thinking about how they
might model all of this so as to communicate their thinking to their peers. In fact,
these types of thinking are expectations for students in many reform curricula
including the Ontario mathematics curriculum for K-8 (Ontario Ministry of
Education, 2005). It is not uncommon, for instance, to find a task in a textbook
that asks students to analyze an error that another student has made. Namukasa
(2005) argues that students should also be invited to attend to their own and to
307
- 168
Figure 1. A subtraction task

18 Mathematics-for-Teachers (and Students)
each other’s mathematical thinking processes. This is also what the
metacognitive, error, and interpretive analysis research is about.
Another problem with the example that Ball et al. use to illustrate their
conception of MKT is the nature of their focus on a traditional algorithm. Their
example is overly concerned with how the algorithm works and how students
should learn it, rather than also focusing on other procedures for subtracting the
two numbers, which would build on the personal knowledge and imagination of
students and not simply on the rigid and arbitrary rules of standard algorithms. In
the section that follows we share an example of the mathematics-for-teachers that
we have been focusing on, the aim of which is to immerse preservice teachers in
mathematics experiences that many of them have never had, namely, experiences
where they attend deeply to mathematical relationships and have opportunities to
sense the pleasure of mathematical insight.
Mathematics-for-Teachers Online
I felt lost at first as I struggled to remember math concepts from
childhood and adolescence. I felt confused. What did a poem have to
do with math? I was perplexed. Was there not only one answer to a
mathematical question? I felt apprehensive. How would I discuss a
mathematical concept that I did not fully understand? Then as I got into
the swing of things, I felt more confident with my opinions, my
answers and most importantly myself. I felt cheerful that I was
experiencing math as a student and that I would hopefully be able to
empathize with my future students. I felt happy that math instruction
could be made to be engaging. Finally, I was giddy that I was thinking
about math, actually thinking about math and not doing everything else
to avoid it.
The first problem explored in the Mathematics-for-Teachers Algebra Course was
Making 10. Preservice teachers are asked to find missing numbers in the equation
__ + __ = 10, and then plot them as ordered pairs on a coordinate grid. Pre-
service teachers expressed surprise that the ordered pairs lined up. “I had the
‘aha’ feeling when I saw the diagonal line pattern on the graph. That was my
favourite part.” Pre-service teachers also noticed that the graph of __ + __ = 10
(that is, x + y = 10) could be used as a visual proof of 12 + - 2 = 10 and 11 + - 1
= 10, since (12,-2) and (11,-1) line up with (10,0), (9,1), (8,2) and (7,3), thus
satisfying the equation x + y = 10. They also explored equations whose graphs
were parallel and then those that were not parallel to x + y =10. Such
mathematical connections appeared to be pleasing to pre-service teachers. “I
loved the adding/graphing we did and how you could take problems and branch
out … it really makes something in my mind click.” This problem was first used
with two classes of fourth grade students when one of the authors (Gadanidis,
2004) was invited to do a lesson on missing numbers (that is, solving equations
like _ + 3 = 7 and 5 + _ = 12). Using the above activity, students did solve a lot
of missing number problems. Typical classroom activities on this topic would
have students complete several unrelated missing number problems. By adding
the constraint that the sum of the numbers is constant we generate a mathematical
relationship among solutions. Students also explored ways of changing the

George Gadanidis & Immaculate Namukasa 19
equation x + y = 10 so that the pattern of plotted ordered pairs might slope in a
different direction or might be curved. Preservice teachers also used the online
activity shown in Figure 2 (Gadanidis, 2005), which was based on the fourth
grade activity, and which incorporated (1) a mathematical poem, (2) video
annotations that pose extension problems and offer pedagogical insights, and (3)
an interactive exploration of functions and their graphs. A mathematical poem is
Figure 2. The Pleasure of Making 10
used as the centrepiece because, as the poet Molly Peacock (1999) suggests,
poetry is screen-sized. A poem is compact enough and cohesive enough to be
held in one’s mind as a whole. Poetry also makes use of image and metaphor,
both of which help the reader sense deeper relationships to explore (Zwicky
2003).
Such mathematical experiences do offer preservice teachers opportunities to
learn mathematical concepts. However, our primary goal is not to increase their
mathematical content knowledge but to provide experiential therapy (Gadanidis
& Namukasa, 2005). That is, our intention is to provide experiences that
challenge and disrupt the mathematical discourse they have typically internalized
through past school experiences, which is characterized by such views as:
mathematics is a cold science—rather than an aesthetic, human experience
(Gadanidis & Hoogland, 2003); mathematics is about learning procedures for
getting correct answers—rather than attending to and gaining insights about the
complexity of mathematical ideas (Gadanidis 2004); a good teacher makes
learning easy—rather than creating situations where students have to think hard
(Jonassen 2000); and, teaching should start with what a child already knows and
understands—rather than with what a child can imagine (Egan 1997).
We have also developed an online discussion forum (Gadanidis, 2007) whose
features include (Wiki-style) editable postings, rich text postings, and a draw tool

20 Mathematics-for-Teachers (and Students)
with drawings embedded within postings. Such online tools help enhance and
enrich online mathematical communication. The drawing in Figure 3 was created
by a teacher in the online discussion of the Geometry Course offered in August
2005, to illustrate how her legs are positioned to form a triangle. In the text that
accompanied the drawing, the teacher discussed the three-dimensional figure she
imagined being ‘cut’ out of space as she twirls.
Figure 3. An online drawing
Looking Ahead
Pre-service teachers in the online Mathematics-for-Teachers:
Measurement/Geometry Course (offered in the summer of 2005) overwhelmingly
expressed that experiencing the Course helped change their view of mathematics
and what it means to teach and learn mathematics. Similar views have been
expressed by the preservice teachers in our other offerings of mathematics-for-
teachers. In fact, the concept of doing mathematics as a therapeutic experience
came from the original mathematics-for-teacher elective described above where
preservice referred to the problem solving sessions as “math therapy.” Statements
such as the one below were common in preservice teachers’ reflections on
mathematics-for-teachers and its effect on their views and beliefs.
Oh how the times have changed! In the few short months that have had
the pleasure of “exploring big ideas in elementary school
mathematics”, my mathematical mindset has been overhauled. Now I
feel empowered by math. I think mathematical experiences can change
you. I was initially frustrated when I found that my classmates and
myself were constantly being deprived of the solutions to mathematical
situations that we were instructed to work through. The purpose of this
type of an exercise soon became clear. It was the process of problem
solving rather than the accuracy of the response that was being focused
on. Soon I felt at ease in this pseudo-mathematical atmosphere. I started
to discuss my ideas more openly with my classmates than I had
initially. I had been programmed to withhold my ideas unless I was
convinced that I knew the right answer, but when we knew that we
were not going to be provided with the correct answer anyway, we
were more open to discussing our individual strategies.

George Gadanidis & Immaculate Namukasa 21
Despite such testimonials, however, a single course experience cannot create
comprehensive or permanent change in teachers’ perceptions of mathematics and
mathematics teaching. Neither can we assume that such an experience will
significantly affect teachers’ classroom practice. Teaching is also greatly affected
by accepted teaching practices in the wider school community (Buzeika, 1999;
Ensor, 1998) and by conflicting priorities (Skott, 1999). However, such
experiential therapy (Gadanidis & Namukasa, 2005) is an important starting
point for change in teachers’ perceptions and classroom practice (Gadanidis,
Hoogland & Hill, 2002a, 2002b).
As we look back on the short history of developing a mathematics course for
preservice teachers, we see three patterns emerging. First, our starting point has
always been to involve preservice teachers in doing mathematics—mathematics
where they have to attend deeply, mathematics that offers the potential of
experiencing the pleasure of mathematical insight, mathematics that engages
their imagination. Second, we have never viewed the mathematics we engage our
preservice teachers with as “mathematics only for teachers”—we have viewed it
as good mathematics, which is also good for students, and even parents. The
three Mathematics-for-Teachers courses we have developed are also now offered
through our Continuing Teacher Education Program to parents, as well as
inservice teachers. And, in our classroom-based research projects, they are used
with students. Last, we have been willing to experiment with doing mathematics
online. These patterns set the direction for our future development of
mathematics-for-teachers (and students!).
Acknowledgement
This research was supported by a Standard Research Grant from the Social Sciences and
Humanities Research Council of Canada.
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