Mathematics Autobiographies: A Window into Beliefs, Values, and Past Mathematics Experiences of Preservice Teachers.

Abstract

This paper explores the usage of a Mathematics Autobiography to investigate the beliefs, attitudes, and experiences that preservice teachers have had while learning mathematics in order to inform instruction. The following were found to be important in students' experiences in learning mathematics: the role of the teacher; issues of fear, failure and avoidance; learning strategies and content issues; support and influence of family; and challenge. This paper will discuss in depth three of these themes. Implications for instruction are also discussed.

Introduction

Affective variables related to the learning of mathematics play an important role in the development of preservice teachers. Throughout the process of learning mathematics, preservice teachers collect a wide range of experiences. Both positive and negative, these experiences have led to the development of their beliefs and attitudes about mathematics. Recent research has found that teachers' beliefs about mathematics and the teaching of mathematics are significantly influenced by their mathematical experiences as students (Brown & Borko, 1992; Brown, Cooney & Jones, 1990; Raymond, 1997). Furthermore, research has found that success in solving mathematics problems is not based solely on one's knowledge of mathematics. It is also based on metacognitive processes related to mathematics strategy usage, the emotions an individual feels when doing a problem, and personal beliefs about one's mathematical abilities (Garafalo & Lester, 1985; Schoenfeld, 1985; McLeod, 1988).

As mathematics educators preparing students to teach mathematics, we have often thought that the beliefs, attitudes, and experiences our students have had learning mathematics may have an impact on their learning and performance in mathematics, which ultimately may affect how they teach mathematics. This belief is supported by McLeod's (1992) assertion that affective issues play a central role in mathematics learning. Beliefs that students hold about mathematics and their abilities to perform mathematically are critical in the development of their responses in mathematical situations.

For the purposes of this study beliefs will be defined as the personal assumptions from which individuals make decisions about the actions they will undertake. This notion is consistent with research that indicates that actions are motivated by what individuals perceive are the outcomes of their actions (Kloosterman, 1996). From "I hate math" to "Math is my favorite subject," these statements speak to the range of feelings and beliefs student have about mathematics. Students, in the process of learning mathematics, experience both positive and negative emotions, which influence the development of their attitude towards mathematics as a whole. These beliefs about mathematics, about what they need it for and how strong they are as mathematics students, are related to learning and can significantly affect what students do in a mathematics classroom (Kloosterman, 1996).

Teacher educators are charged with the daunting task of shaping and reshaping the attitudes, beliefs, and content knowledge of preservice teachers. Teacher education programs must sometimes help participants to deconstruct and reconstruct their views on teaching and learning (Brown & Borko, 1992; Wilson & Ball, 1996). It is important for mathematics to be learned in a supportive community of learners (Brown & Campione, 1994). This environment can also provide an arena for discussion of these important affective issues.

The authors' primary purpose in this research was to investigate the beliefs, attitudes, and experiences that preservice teachers have had while learning mathematics in order to inform instruction and facilitate the development of a supportive learning environment. This article will describe students' perceived experiences of learning mathematics in grades K-14 and discuss the implications of these experiences for college level instruction.

Methodology

The participants in this study were preparing to teach at either the elementary, middle, or secondary level, and were enrolled in a mathematics or mathematics education course required for students seeking teaching certification in their respective level. Data were gathered in the form of written responses called "Mathematics Autobiographies" during the first week of the semester. We requested that students respond in writing about any experience in mathematics that they felt contributed to their mathematical development. A total of 72 mathematics autobiographies were gathered from three different levels of college mathematics courses and two secondary mathematics education courses. In the second session of each of these courses students were asked to share with the class any experiences they were comfortable sharing, and then as a group we discussed the implications for mathematics instruction.

Analysis of Results

Data were analyzed in two ways: a holistic examination of each student's response, and a comparison across chosen teaching levels: elementary, middle, and secondary. Each individual mathematics autobiography was summarized and coded according to emerging themes. These themes were then compared within and across teaching levels with the help of conceptually ordered displays (Miles & Huberman, 1994).

After analysis, several themes emerged. The following were found to be important in students' experiences in learning mathematics: the role of the teacher; issues of fear, failure and avoidance; learning strategies and content issues; support and influence of family; and challenge. This paper will discuss in depth three of these themes. All of these issues mentioned are in some fashion tied to one variable, the teacher. Since the teacher generally guides classroom instruction, many of these issues focus on a teacher's approach to classroom instruction. Each of these experiences is stated from the student's perspective and thus are perceived experiences and beliefs about their learning of mathematics.

The Role of the Teacher

Virtually all students mentioned the importance of the role of the teacher in the development of their understanding of mathematics. This finding is consistent with previous research (Jackson, & Leffingwell, 1999). The description of the teachers seemed to fall on a continuum, with one end of the continuum indicating teachers with "enabling" traits, and the other end indicating teachers with "disabling" traits. The "enabling" teacher was often portrayed as one who was patient and understanding, who would always fully answer student questions and explain a concept until all students understood. Further, this type of teacher would not embarrass a student, and in fact, would often single out a student independently to provide additional assistance and encourage him or her to continue developing his or her mathematical skills. Teachers classified as "enablers" were typically remembered for the ability to make learning mathematics fun. In contrast, the "disabling" teacher was depicted as one who would often ridicule students if they did not know an answer, intimidate students, not fully explain concepts or homework problems when asked, and who was often inconsiderate of students' feelings.

Generally, an experience with an enabling teacher left students, even those who were not as successful in the class as they would have liked, feeling as if they had been successful in the class, and could be successful in future mathematics classes if they worked hard. Commenting about such an experience Corrine states, "My high school mathematics teacher had a positive attitude toward me which gave me the confidence to do well in math and not feel stupid any longer." Scan recalls his trigonometry teacher in this manner" He gave me confidence and I still have bad math anxiety but it goes away when I think of that experience (in Trig Class) and how it made me believe in myself and how I could learn mathematics."

Students clearly articulated their feelings about learning in a class with a disabling teacher. Rich recalls

   I was always nervous about going to my Calculus class because the teacher
   made me feel dumb. If I would raise my hand and ask a question, he would
   give me the answer but would give it in a way that made me feel I was
   wasting his time by asking a question. I felt so dumb. I ended up barely
   passing the class. I didn't want to ever have to take anymore math.

Jessica wrote about her fifth grade teacher

   who wrote continuously on the board, not once turning around to see if we
   were understanding the material. He would go over it once and if we didn't
   understand it we would have to depend on our friends to help us. By this
   time I lost all interest in math and began to loose my self-esteem, I had
   no desire to continue with math. The only thing standing in my way was that
   I had to complete three years of math to go to a good college.

Bernadette also had a difficult experience learning her multiplication tables:

   I had a teacher who would make us recite them (multiplication tables) in
   front of the class and often made me feel very stupid and embarrassed when
   I made a mistake. I was a very shy child who didn't have a lot of friends
   and I was very insecure about myself, which made this experience especially
   painful for me. For the years that followed I decided that I was not good
   in math.

A bad experience with a teacher who made students feel uncomfortable with their mathematical abilities tended to deter students from wanting to pursue further mathematics study. Many students felt that they were just "not math students."

The teacher was a pivotal experience for each of the students. A good experience with an "enabling" teacher promoted the students' positive outlook on their mathematical abilities and the subject. In contrast, an experience with a "disabling" teacher discouraged positive student beliefs of mathematical ability and also dampened outlook on future mathematical experiences.

Fear, Failure and Avoidance

A general trend that emerged from the data is fear, failure, and then avoidance. First a student experiences fear of mathematics, often followed by failure which can then lead to avoidance of the subject. Avoidance of mathematics is often overcome only after a positive experience. This cycle may repeat several times throughout a student's education. Students may also experience fear, failure and avoidance in exclusion of each other. Our findings are supported by previous research, which has found traditional mathematics instruction to cause fear, anxiety, and avoidance of mathematics by students (Tobias, 1981).

Bruce's experiences in geometry class were representative of this cycle:

   As a result of my shortcomings in geometry I began to fear math for the
   first time in my life. So I spent the next few years shying away from the
   subject for fear that I may fail at something that I was once so in love
   with. This continued until I took my first physics class....Fortunately
   physics intrigued me so much that I simply knew I had to stand up to my
   fears of math and overcome them in order to proceed in that area of study.

A majority of students at all levels spoke of fear of mathematics. These are some of their feelings,

   * "By this time, I internalized and truly believed that I hated math, was
   stupid and couldn't do it";

   * "From the age of 15 until the age of 41, I avoided math like the plague."

These comments highlight the important component that fear played in students' learning of mathematics. Students whose last memory of learning mathematics resulted in fear or failure will likely have quite different perceptions about their learning of mathematics than students who recall a more positive resolution to a fearful situation in a mathematics class.

Learning Strategies and Content

Two topics closely related that were consistently mentioned were learning strategies used to learn mathematics and several specific mathematics topics, which included number sense (basic counting), multiplication, and geometry. One of the major approaches to the successful learning of mathematics is the use of learning strategies. The National Council of Teachers of Mathematics (NCTM, 2000) states that to meet new challenges in the work place students will have to be able to adapt and extend whatever mathematics they know. This requires problem solving which is dependent upon students' knowledge of strategies. Learning strategies mentioned consistently throughout the mathematics autobiographies were concrete manipulatives, learning aides, and outside help.

In an environment that supports children's development of personal strategies, these will come naturally. (Carpenter, Fennema, Franke, Levi, & Empson, 1999) However, many of our preservice teachers had difficulty with this topic at a young age. For example, Bernadette had difficulties learning how to count; her mother helped her learn math by "counting socks and lima beans." Nicole's third grade teacher brought in candies and nuts to help the class with their mathematics lessons. Use of manipulatives has been found to be an effective tool to facilitate the learning of mathematics (Lambert, 1996).

Several students like Michelle used flash cards to help them learn. Michelle says her sister "helped me create flash cards and helped me learn my multiplication facts. I remember I was so proud of myself. I know I need visual aids, that's when I learn best." Help from family members, roommates, and tutors was often used as a strategy to learn mathematics.

Mathematics content was a frequently discussed topic, mostly the difficulty of learning the specific topic. Multiplication, which is typically taught in the third grade, was an area where these students had a mixture of experiences. Though the majority of these students had difficulty learning multiplication facts, others expressed they did not. Those who had difficulty learning their facts were very upset by teaching strategies their teachers used to facilitate their learning. These techniques usually included some form of timed assessment to test a student's knowledge, and/or creating a competitive environment to assess learning. These assessment strategies again elicited opposing feelings from students. Those students who understood their multiplication facts were excited and felt proud of their achievements in these assessments. On the other hand, students who did not perform well on these types of assessments were embarrassed and left feeling that they "could not do mathematics" at a very early age.

Discussion

The major difference we found in each of the three groups, elementary, middle and high school, was with affective variables. Typically preservice elementary teachers had more negative feelings and attitudes and less confidence toward learning mathematics. Middle school preservice teachers felt some of the same anxiety as their elementary teacher counterparts. However, this anxiety was mixed with enough successful experiences in mathematics to encourage a greater commitment to the teaching and learning of mathematics. High school preservice teachers' memories of their mathematics classes were the least tainted by frustration and failure; most had always been good at math and never had to struggle. Thus those preservice teachers who experienced less failure had more confidence in their mathematics abilities.

Many of our students spoke of perceived good experiences in learning mathematics, even if they had not achieved high grades, when they had a teacher that gave them the impression that they could succeed in mathematics. Others spoke of bad experiences when they had the opposite, a teacher who did not express an interest or a belief that the student could learn mathematics. Also of concern is whether or not a student would continue to pursue mathematics if they never had a good experience learning mathematics. Further, since all of our students are working toward a teaching license with the goal of teaching as a career we are concerned that if these students do not have good experiences learning mathematics, how will they then know how to create good mathematical learning experiences for their students? Will they express a love of the subject or exude a dislike of the subject, which their students could then also adopt? Specifically, could the negative fears and experiences these preservice teachers have transfer to their students, thus continuing the cycle of negative feelings and beliefs about mathematics? This is a major issue for all students, especially those students who may have already had a negative learning experience in mathematics.

Implications for Teacher Educators

Mathematics autobiographies can be used as a tool for teacher educators because they can provide insight into the mathematical learning experiences of their students. Specifically, this information can be useful for instructional planning and curriculum development as it enables an instructor to link a student's past learning experience to instructional techniques which may facilitate the student understanding of the topic in a deeper manner. However, it is essential that a discussion occur that highlights the differences and similarities between past learning experiences and present instructional strategies used to learn the same content.

Research has supported our findings that some students believe they cannot do mathematics (National Research Council, 1989). Kloosterman, Raymond, and Emenaker (1996) found that students pick up teachers' unintended as well as intended messages about what it means to know and do mathematics. Thus, it is essential that appropriate modeling be used. This is especially important in preservice courses since the teaching for which teacher educators are preparing their students is different from most of what students recall from their own learning experiences (Wilson & Ball, 1997). For example, traditionally mathematics students have learned mathematics from a "sage on the stage." Now the approach to teaching mathematics has shifted to a student-centered constructivist approach to learning where students are often given problems that require them to work in considerable depth and often, struggle with confusion rather than be told the "one method" of solving the problem. In addition, there are shifts in teacher interactions with students and the expectation of high achievement for all students. This deconstruction of past views on instruction and teaching is often a difficult task for teacher educators, but a necessary one (Wilson & Ball, 1997). It is imperative that preservice teachers experience current approaches to teaching and learning advocated by current reforms in an explicit manner.

Mathematics autobiographies can also be used to facilitate a positive classroom environment, one in which all students feel that their experiences and background are valued. Sharing autobiographies in a classroom discussion can help students identify with others' similar experiences as well as enlighten students on the diversity of student backgrounds. For example, as our students shared their experiences, a common discussion that developed focused on difficulties learning mathematics. The discussion progressed into exploring a variety of different learning strategies used to learn mathematics. Shared experiences help students connect to one another and often realize they are not alone with their feelings, beliefs, experiences, and approaches toward learning mathematics. In addition, such discussion facilitates the development of a supportive mathematical community, one in which students can construct their own ideas, find their own representations, and connect mathematical ideas in their own ways while doing mathematics together. Students can share their work in a comfortable atmosphere where discourse and collaboration are valued (Brown & Campione, 1994; Bruner, 1996).

Finally, since prior experiences appear to have shaped these preservice teachers' attitudes and self-efficacy about math and math learning, it is important to consider how theft experiences may affect their future teaching of math. Though our students did not explicitly mention the impact their learning experiences might have on their future teaching, we believe there is an impact. There is the possibility that these students may teach just as they were taught. Conversely, there is the possibility that preservice teachers, when taught using appropriate methodologies, may embrace these methods and utilize them in their own teaching. The impact that a positive or negative experience learning mathematics has on future teaching is an area, which needs to be explored further. There is the possibility that these students may teach just as they were taught (Wilson & Ball, 1997).

References

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Julie is an Assistant Professor of Mathematics Education. Her research interests include affective factors related to the teaching and learning of mathematics, and best practices for teaching special needs learners mathematics. Cheryl is an Assistant Professor of Mathematics Education. Her current research interests include affective factors related to the teaching and learning of mathematics, calculus reform, and the issue of transfer related to mathematical understanding.

Julie A. Sliva, San Jose University, CA

Cheryl Roddick, San Jose University, CA