Document Sample
					Vol. 5                                                                                                        363


                                              Megan Burton
                                        University of South Carolina

Elementary teacher candidates enrolled in a mathematics methods course were asked to “draw
math” at the beginning and end of the semester. Findings display the vision of mathematics that
teacher candidates have before and after exploring teaching methods and implementing these
methods with elementary students. In addition, it examines the specifics of the changes that
occurred during the semester of methods and field placement experience.

    What is math? Often people reflect back to their vision of school mathematics, while others
reflect upon its relevance to the real world when this question is asked. Vinter (1999) found that
teachers often struggle to find the application of much of the math they teach. This can be due to
the lack of meaningful experience with the content taught in the elementary grades (Ball & Bass,
2000). In addition to lack of experience with math, many elementary teacher candidates have
high levels of mathematics anxiety (Swars, 2006). These factors can affect the impression of
mathematics that teachers give to their students. Through examining their own perceptions of
mathematics, teachers and teacher candidates can begin to explore how to deepen their own
understanding, overcome anxiety, and connect the content to elementary students.
This article documents an elementary mathematics methods course, which begins by asking
teacher candidates to draw math and write a few sentences describing the drawing. The drawings
often involved students communicating and reflecting upon their emotions and past experiences
associated with the content, including mathematics anxiety. In addition it became a theme of the
course throughout the semester, inviting students to revisit their perceptions as various methods
and content were introduced. This simple task provided insight into the perceptions teacher
candidates bring to their teacher preparation programs and the impact that positive experiences
with students and content can make upon these perceptions.
    Student drawings have been used to examine students’ perceptions about various content
areas for years. In literacy, drawings of reading and writing have been used to understand their
perceptions of the subject areas (McKay & Kendrick, 2001). Students’ impressions and attitudes
of scientists and science have been studied in many elementary and teacher candidate classrooms
in order to understand student perceptions (Thompson et al., 2002). Drawing images before
writing or verbalizing ideas can foster more creative responses and help generate ideas, because
often language can slow down the creative process (Caldwell & Moore, 1991). Ideas can be
explored through drawing without the cognitive demands often found when using language.
Art is often used in therapy, because thoughts and emotions can be expressed vividly through
images (Lusebrink, 2004). These same techniques can be useful in supporting meta-cognition
and addressing negative emotions often tied to mathematics by elementary teacher candidates.
Drawings by teacher candidates of various subject areas can reveal dispositions, attitudes, and
experiences related to a subject area. These drawings allow the artist to establish and reflect
upon these attitudes and experiences in a non-threatening way (Rule & Harrell, 2006).

Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
Vol. 5                                                                                                        364

By acknowledging and giving voice to negative emotions and experiences, such as mathematics
anxiety, through drawings, one is better able to deal with and move beyond those negative
emotions and experiences (Rule & Harrell, 2006). This study relies on the theoretical
understanding that the relationship between the conscious and unconscious mind can be
expressed through images and thus given a voice where it otherwise might be ignored (Hillman,
1992). Often the negative emotions surrounding a concept, such as mathematics, develop in ways
that distort original experiences. Instead reflection upon past experiences tends to reflect the
current emotional attachment that has evolved over time and repeated experiences with that
concept (Hillman, 1992). Sometimes images used to express the larger concept, such as
mathematics, can be literature snapshots of a particular event and at other times they are more
representational. However, both reveal a deep insight into the true relationship of the artist to the
subject, such as mathematics (Rule & Harrell, 2006). By examining his/her own understanding
and perception of a subject, the artists are better able to improve the negative emotions related to
the concept and this in turn allows them to focus on the learning, without the obstacles associated
with their negative experiential baggage. Watkins (1984) suggests that by investigating images
and discussing feelings related to the images, the artists become empowered to engage actively
in changing the negative perceptions related to the subject.
Mathematics Anxiety
    Mathematics anxiety often begins in elementary school when students have negative
interactions with the content and are taught by procedural, rather than conceptual teaching
methods (Harper & Daane, 1998). Tying instruction to the exact procedures in the textbook,
timed tests, hostile teacher behavior, embarrassing students in front of peers, only accepting one
method of solving a problem, and lack of differentiation based on student needs are all factors
that can contribute to mathematics anxiety (Swars, 2006). Hembree (1990) found the highest
level of mathematics anxiety among college students came from elementary teacher candidates.
To resolve mathematics anxiety, teachers need to have positive experiences with mathematics
and see the purpose behind the mathematics they are teaching and have mathematical
experiences with manipulatives and working in groups (Harper & Daane, 1998; Swars, 2006).
Vinson (2001) found that exploring the conceptual content in meaningful ways with
manipulatives before learning the procedural aspects of mathematics reduced the mathematics
anxiety among teacher candidates. This impacts the way they teach their students.
When teachers are confident in their mathematics ability they spend 50% more time teaching
mathematics than those who have mathematics anxiety (Schmidt & Buchmann, 1983). In
addition teachers with math anxiety spend less time implementing standards based instruction
and more time teaching to the whole class and assigning seat work (Bursal & Paznokas, 2006;
Bush 1989). These activities perpetuate the notion that mathematics lacks real-world meaning.
Perception of the Use of Mathematics
    Vinter (1999) found that many teacher candidates lack an applied understanding of
mathematics and this in turn affects their ability to make the content meaningful for their
students. Resnick (1987) suggests that many times teachers prepare students to do school math,
but this is not the same as mathematics beyond the classroom. NCTM (2000) stresses the
importance of problem solving, communication, and connecting math content, which requires
teachers to have a deeper understanding in order to support these connections. Chappell and
Thompson (1994) expressed the importance of the mathematical courses that teacher candidates
experience in their preservice programs. The preservice program is crucial to the development of

Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
Vol. 5                                                                                                        365

teacher candidate’s beliefs, content knowledge and attitudes about the way math should be taught
at various grades and their effectiveness as educators.

                                     Research Questions
    1. What is the perception of mathematics held by elementary teacher candidates at the
       beginning of a mathematics methods course?
    2. How do elementary teacher candidates perceptions of mathematics change during a
       mathematics methods course?

     This study examined the symbolic representations of math drawn by teacher candidates at the
beginning and end of a mathematics methods course. The drawings were analysed and
categorized to explore the initial impressions of mathematics that teacher candidates bring to
their teacher preparation courses and the changes experienced through opportunities to discuss
anxieties, work with students, and explore various pedagogical theories for mathematics.
     Sixty-two teacher candidates were enrolled in a mathematics methods course for elementary
teachers at a midsized university in the southeast. The study took place over a period of two
years with the same mathematics instructor, but four different sections all offered in the spring of
their junior year. There were fifty-nine females and three males, fifty-five Caucasian, four
African-American, and three Latino teacher candidates.
     During the first class meeting, participants were asked to draw pictures of math. Teacher
candidates were told to draw whatever came to mind and not to filter images. When they asked
for further details about what to draw, the instructor simple advised them to draw what comes to
mind about math. It was emphasised that the grades in the course would in no way be influenced
by the drawings. Teacher candidates were asked to put the last four digits of their student
identification number on the paper in order to compare pre-test and post-test results. Then they
were asked to write a few sentences related to their drawing on the back of their paper.
     During the semester teacher candidates spent one day a week in a practicum experience. In
addition during the weekly three-hour mathematics methods course, each teacher candidate spent
30 minutes working with two fourth graders exploring various mathematics content. A portion of
the methods course involved discussing the lessons learned from the fourth graders. In addition,
teacher candidates explored various theories about teaching and learning mathematics, effective
use of technology in mathematics, and the role of manipulatives in mathematics.
     Throughout the class, there were informal conversations about the impressions of math that
teachers bring to the classroom. Teacher candidates were able to evaluate the content and their
experiences with their students with a conscious understanding of their lens that developed from
their personal experiences as a learner. For example, those who understood algorithms were able
to listen to students with misconceptions or invented algorithms. Teacher candidates who entered
the methods course with hesitation due to their own struggles with math often found their
struggles could help them relate and better explain the content to their students.
     The same drawing activity was conducted the final day of class. Teacher candidates were
asked to draw math and write a few sentences about it. In addition they were asked to write 2-3
sentences describing if and how the course changed their impressions. These drawings, and
sentences were used to investigate the mathematics views held by the teacher candidates and the
changes experienced over the course of the semester.

Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
Vol. 5                                                                                                        366

Data Analysis
    The writing was used as needed for clarification of the meaning behind the drawings.
Drawings were categorized in three ways: positive, neutral, and negative emotions; particular
experiences and general meanings, and classroom, abstract or real world connection. Positive,
negative and neutral were based on if the drawings or descriptions had specific emotional
prompts, such as faces with smiles or tears. Next, the pictures were grouped by particular
experiences or general meaning. Particular experiences were drawings in which one point in
time was displayed, such as drawing on the board. When pictures simply had mathematical
images such as a fraction, this was classified as general meaning, Finally, the drawings were
categorized based on if the pictures showed images connected to the classroom, were abstract, or
were connected to the real world. Pictures that were connected to the classroom displayed
images such as books, teachers, or a white board. Abstract images were images such as fractions,
multiplication problems and algorithms. Real world connections, had images such as shopping or
cooking. This type of classification for analysis is based on the evaluation technique used by
Rule and Harrell (2006). These categories were confirmed by a member of the mathematics
education faculty. This faculty member evaluated and supported the initial findings, categories,
groupings, and count. This confirmation provided validity of the interpretation. Then the analysis
to determine the change was charted by grouping pre and post drawings and then analysing them
for positive and negative changes. These findings were also confirmed by the aforementioned
mathematics education faculty member.

Initial Drawings
    Through the drawings and writings, teacher candidates expressed a variety of experiences
and impressions of mathematics. A majority (32) of the experiences were negative. However,
there were nine that were positive and twenty-one were neutral. All the positive drawings related
mathematics to the content, real world examples, and puzzle. Most negative drawings related to
the teacher candidates’ emotions and experiences in school (see figure 1 and 2). For example,
three people drew themselves at the board with question marks. Question marks seemed to be a
common expression for teacher candidates to show their feelings of confusion. Also, many drew
textbooks and jumbled ideas in their drawings.

Figure 1                                            Figure 2

Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
Vol. 5                                                                                                        367

    Table 1 shows the positive, neutral, and negative emotions connected to the drawings. A
common thread for many of the negative drawings was the struggle that teacher candidates felt in
school mathematics. One teacher candidate explained, “I never understand it. I always feel
stupid and like what the teacher says is a foreign language.” Several wrote sentences expressing
the desire to change these emotions in order to avoid negatively impacting future students.

                 Table 1
                 Categories of Drawings
                                                      Pre-test                 Post-test
                         Positive                        9                       38
                         Neutral                        21                       24
                        Negative                        32                        0
                  Particular Experience                 33                       19
                         General                        29                       43
                   Classroom Setting                    28                       22
                   Read World Setting                    7                       11
                         Abstract                       27                       29

    After analysing the emotions in the drawings, were categorized based on if they referred to
particular experiences or were more general (see table 1). Thirty-three drawings displayed a
particular point in time. For example one teacher candidate drew tears and question marks
around an illustration of herself. In her writing she explained that she remembers her fifth grade
math teacher being angry with her. She went on to describe a time she was at the board in fifth
grade and had no idea how to solve the problem. She described this situation as representative of
her feelings of math. Twenty-nine drawings were math symbols or items related to math, rather
than a specific memory related to math.
    The final category evaluated if the images related math to the classroom setting, real world
setting, or were more abstract (see table 1). Many teacher candidates drew content images such
as shapes, numbers, and equations. The abstract images had a mix of written responses varying
between positive, negative, and neutral emotions. The teacher candidates who drew themselves
cooking, shopping, or building all expressed a passion and conceptual understanding of
connections, real world meaning, and reasoning essential to mathematics.
Drawings of Change
    The experiences of teaching mathematics to fourth grade students during the methods course
as well as exploring content in the methods course from a Standards-based approach (NCTM
2000) were positive for the teacher candidates. Fifty-eight teacher candidates reflected in the
final writing, that they now saw the connections between the real world and the content they
were teaching. While five were still hesitant and concerned about their own understanding, they
expressed growth and a more positive perception of math. Thirty-eight of the final drawings and
writings showed a new perception that math is fun, meaningful, and makes sense. Teacher
candidates expressed more confidence in teaching mathematics, but also more confidence in
their own personal mathematical abilities.
                I always thought I was bad at math and dreaded this course,
                but now I see that math isn’t just memorizing stuff the
                teacher says. It is talking about stuff, exploring different
                ways of doing things, and thinking about what makes sense.
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
Vol. 5                                                                                                        368

               I am actually good at it, now that I understand that.
    When the emotions attached to the final images were compared to the initial drawings a
positive growth in emotional affect and reduction of anxiety was seen (see Table 1). The
drawings that showed positive emotions in mathematics contained images of collaboration,
manipulatives, real world connections, and discussions. There were no changes towards
negative emotional connections to mathematics in the drawings. Those that initially drew
positive drawings kept these, but several included more images of collaboration and simplified
the mathematics in the drawings to match the mathematics that they will use in the elementary
classroom. In the final drawings the images of specific classroom experiences were positive,
rather than the negative images initially drawn. In addition, drawings displayed more
collaboration and meaningful learning (see figure 3).

Figure 3

    These findings contribute to the body of research on perceptions of mathematics held by
elementary teacher candidates. Several key findings provide insight into teacher candidates’
perceptions and have implications for teacher education programs.
    This study provided further evidence of the negative experiences teacher candidates bring to
the classroom (Swars, 2006). It goes deeper than simply recognizing these negative emotions and
experiences. It examines how teacher candidates view the concept of math. The details in the
images provide visual understanding of the anxiety experienced by many. One interesting
finding was the discovery that the negative experiences are often related to the classroom rather
than real world math. This aligns with the work of Nicol, who suggests this lack of real world
connection negatively impacts their students (2002). When teacher candidates connected math
with real world experiences, they viewed math in a positive light and displayed confidence in the
    When teacher candidates changed their depiction of mathematic to a more positive image,
this included images of discussions, manipulatives, understanding, and connections to the real
world beyond the classroom. Even when classroom images were the focus of the drawing, the
writings on the back referred to the importance of making connections. Those who expressed
math in a positive light at either the beginning or the end did not make references to textbooks,
isolation, or working problems on the board. Instead meaningful, connected, and engaging math
was the focus. By exploring the methods for teaching elementary content within a context

Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
Vol. 5                                                                                                        369

connected to working with students, teacher candidates expressed a change in beliefs about the
concept of mathematics as displayed in their drawings.
    The connection between teaching children and reduction of mathematics anxiety aligns with
Harper and Daane's study (1998). Because conscious reflections on attitudes can alter negative
complexes (Hillman, 1992), the attitudes may have been changed through the symbolic analysis,
which was both a part of the unique methodology of the study and a conscious reflective process.
This is one of the reasons that the images can be analyzed, but assumptions about why they
changed cannot be made. By having teacher candidates draw their perception of math initially,
they were more aware of this throughout the course. Their emotions attached to math were a
natural part of the conversations about methods and their experiences with students. These
teacher candidates were challenged to draw their own perceptions of math and many used this
opportunity to express their own problems they encountered in math as a student. This in turn
made them more aware of problems students might have throughout the semester.
    Teacher educators need to allow time for teacher candidates to reflect upon the perceptions
and beliefs they bring to the classroom. This reflection allows teacher candidates to acknowledge
their biases and begin to explore how to create more meaningful experiences for students (Rule
and Harrell 2006). In addition time for teacher candidates to make real world connections
between math concepts should be an essential portion of elementary methods courses (Chappell
& Thompson, 1994).
    While this study offers insight into the perceptions that elementary teacher candidates hold,
further studies are needed. Continuing this activity by having teacher candidates at the end of
internship and inservice teachers draw math could show how these perceptions change and are
refined as they progress in their development as teachers as well as the sustainability of the
newfound positive perceptions. In addition, it would be interesting to compare the images of
inservice teachers with the students they teach in order to see the correlation of perceptions.
Asking teacher candidates to draw, write, and explain their conceptions of math can further
understanding into the mathematics anxiety that many experience, the pedagogical stances they
bring into teacher preparation courses, and their understanding of connections between concepts
and real world uses of school mathematics. These understandings can provide insight into
supporting teacher candidates as they develop positive affect, effective pedagogical strategies,
and content knowledge for teaching mathematics. These are all critical areas of development to
increase student achievement and end the cycle of mathematics anxiety.

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Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
Vol. 5                                                                                                        370

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Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.