By Barlow, Angela T; Reddish, Jill Mizell
Abstract
For more than 2 decades, professional, state, and federal agencies have adopted educational standards aimed at improving mathematics instruction. One way of measuring the success of these adopted standards is to examine their impact on the mathematics attitudes and beliefs of students who received their education during this reform period. How teachers approach and implement these mathematics standards is heavily influenced by what they believe about mathematics content and pedagogy. For those who seek to improve mathematics education, examining beliefs regarding mathematics and the factors that influence those beliefs is imperative. In this paper we explore the persistence of teacher candidates’ beliefs in myths despite changing educational standards. The beliefs of 76 current elementary teacher candidates were compared to beliefs of 131 elementary teacher candidates from 1990. The results confirm the pervasiveness and persistence of math myths among female elementary teacher candidates.
How teachers interpret and implement curricula is influenced significantly by their knowledge and beliefs (Cooney & Wiegel, 2003; Thompson, 1992). Beliefs about subject matter, including orientations to subject matter, have been found to contribute to the choices teachers make in their classroom instruction (Borko, 1992). Beliefs impact practices because beliefs affect how teachers see their students, how they view the practices of other teachers, and how they accept the ideas given to them to develop practice-whether those ideas are introduced through staff development, content courses, or pedagogy courses. In the discipline of mathematics, teacher practices are more influenced by beliefs about mathematical content than by beliefs about mathematical pedagogy (Raymond, 1997). Teachers “see what their beliefs allow them to see, that is, their beliefs act as a filter through which their observations are shaped” (Cooney & Wiegel, 2003, p. 800). Often, teachers’ content-related beliefs are influenced by their personal experiences as students.
Teacher Beliefs and Teacher Education
The potential impact of beliefs on teacher practices makes it a relevant topic of consideration for mathematics teacher educators (Chauvot, 2000; Grant, Hiebert, & Wearne, 1998; Skott, 2001). The current emphasis at both the national and state levels on preparing highly qualified professionals along with the ongoing efforts of the National Council of Teachers of Mathematics (NCTM) to shift the role of the mathematics teacher from one of a giver of information to one of a facilitator of learning make the importance of continually examining teacher beliefs about mathematics even more evident. Clearly, teacher preparation programs have been tasked with helping future teachers meet these expectations.
Understanding teacher candidates’ beliefs in myths about mathematics is vital for mathematics teacher educators striving to design and implement effective teacher education programs. Hart (2002) stated that in order to determine the effectiveness of a teacher education program, teacher candidate beliefs must be a primary consideration. Beliefs in myths about mathematics are of particular importance because beliefs in math myths can interfere with teacher candidates’ adoption of effective practices. Beliefs in myths about mathematics can lead to math anxiety and even math avoidance (Frank, 1990). Additionally, when false impressions or beliefs regarding mathematics are established, all other ideas regarding mathematics are accepted or rejected based on their alignment with these accepted impressions (Skott, 2001). For example, many teachers believe that students must master basic mathematical facts before engaging in problem solving, despite evidence to the contrary. Therefore, these teachers’ classrooms emphasize the memorization of basic facts and are void of rich mathematical tasks that engage students in problem solving.
Over 25 years of research on mathematics myths has revealed that teacher candidates have maintained stable myth beliefs despite active reform efforts in teacher training (Frank, 1990; Hart, 1999; Kogelman & Warren, 1978). The original study by Frank was conducted in the late 1980s and examined 12 mathematical myths identified by Kogelman and Warren. Our replication of Frank’s study is designed to explore cohort differences in myth beliefs. The timing of our study was important, given that the state and federal education governing bodies and the NCTM have implemented reform efforts since Frank’s study was conducted, allowing a direct examination of the level of concurrence in mathematical myths following 15 years of systematic reform efforts.
Myths About Mathematics
Kogelman and Warren (1978) described 12 mathematical myths, stating that these myths provided false impressions as to how mathematics is done. The presence of these myths or variations of these myths has been repeatedly demonstrated in the literature (Kenschaft, 1988; National Research Council, 1991; Op’t Eynde & DeCorte, 2003; Stanczuk, 2003). The paragraphs that follow provide each of the 12 myths, along with a brief description of Kogelman and Warren’s arguments.
Myth 1
Some people have a math mind and some don’t. This myth is based on the belief that people who have a math mind do mathematics easily and quickly with correct answers just popping in their heads. This belief is invalid because an individual’s ability to understand and do mathematics is impacted more by receiving instruction that supports his or her efforts to construct meaning than it is by the individual’s cognitive ability. When this belief is held, the inability to perform mathematics in such an effortless fashion leads to a lack of self-confidence, one of the most important determinants in mathematical performance.
Myth 2
Math requires logic, not intuition. Intuition is “the act or faculty of knowing without the use of rational processes” (Kogelman & Warren, 1978, p. 32). Mathematics incorporates a great deal of intuition. Number sense is an example of this. The role intuition plays in initiating and verifying solution processes is well noted. Many early mathematical ideas were initially based on intuitive notions.
Myth 3
You must always know how you got the answer. Arriving at a solution is not always the result of a conscious sequence of mathematical actions. Being able to reflect on one’s thinking or engage in metacognition is necessary in order to elaborate on how one arrives at an answer. Not all individuals are able to do this. Although it is the goal of mathematics education to develop this ability, not knowing how one arrived at a solution does not always indicate a lack of understanding. Often, it is more of an indicator of the need to develop the ability to reflect on one’s own thought processes than it is an indicator of a lack of mathematical understanding.
Myth 4
Math requires a good memory. The likely origins of this myth reside in the teaching practices of traditional mathematics teachers. If the goal of instruction is not conceptual understanding, the study of mathematics becomes the memorization of procedures and rules, and mastery of it necessitates a good memory. “Knowing math means that concepts make sense to you and rules and formulas seem natural. This kind of knowledge cannot be gained through rote memorization” (Kogelman & Warren, 1978, p. 40).
Myth 5
There is a best way to do a math problem. Many students are under the impression that the best way to solve a problem is the one presented either by their teacher or their textbook. In reality, math problems can be solved in a variety of ways. Being able to provide multiple ways to solve a problem is indicative of having a solid understanding of mathematics. While one way of solving a problem may be more efficient than another, the best way to solve a problem depends on the individual. No one way is necessarily better than another. When students believe one best way to solve a problem exists, they lack motivation to try to develop their own solution paths. The emphasis is on trying to remember what was presented by the teacher or text versus thinking through a problem.
Myth 6
Math is done by working intensely until the problem is solved. The idea that the manner in which math is done involves working nonstop until the problem is solved is probably rooted in classroom experiences. In schools, students are often rewarded for being the first one finished, and their mathematical experiences are often limited to the efficient application of algorithms. Solving mathematics problems or learning new concepts requires a process of alternating between working and resting. Periods of rest allow the mind to assimilate ideas and to develop new ones. Viewing math as the efficient application of algorithms discourages students from being persistent when faced with challenging problems that do not offer an immediate route to the solution. The true nature of mathematics as an exercise in thinking is replaced by a belief that mathematics is an exercise in recall.
Myth 7
Men are better in math than women. No evidence has been found to suggest that innate differences exist in mathematical ability between men and women.
Myth 8
It’s alway\s important to get the answer exactly right. In many circumstances, approximate answers are more appropriate than exact answers. This is true when a person is grocery shopping, estimating sales tax, calculating a tip, etc. Although problems have exact answers, as stressed in testing situations, there is value in estimating, conjecturing, and hypothesizing. These are important mathematical processes, none of which requires an exact calculation. An overemphasis on exact answers results in a depreciation of the importance of the process of mathematics.
Myth 9
Mathematicians do problems quickly in their heads. The process of solving a genuine mathematics problem, one for which there is no immediate route to a solution, is not a quick process. Time is needed to identify the question being asked by the problem as well as to plan the exploration of possible solution strategies. The only problems that mathematicians do quickly in their heads are ones they have previously solved. In the field of mathematics, it is not uncommon for a mathematician to spend months working on a single problem.
Myth 10
There is a magic key to doing math. There is no formula or rule that will demystify mathematics. Many students fail to develop a true understanding of the mathematics they are studying, relying instead on shortcuts for solving problems. The shortcuts are viewed as tricks because the mathematics is not understood and remains a mystery, despite the student’s ability to produce correct answers.
Myth 11
Math is not creative. Belief in this myth is most likely derived from schooling experiences that involve the modeling of one solution strategy that is to be mimicked by the students on a given assignment or exam. However, generating solution strategies, multiple representations, hypotheses, and conjectures are all mainstays of mathematics that require creativity. Mathematics is a result of a creative process. It is unfortunate that most people have not experienced the creative nature of mathematics, particularly when it is the essence of the discipline.
Myth 12
It’s bad to count on your fingers. Finger counting can be very useful in doing arithmetic and indicates an understanding of the mathematics being computed. Fingers are no less a mathematical manipulative than are base-10 blocks, two-sided counters, or other counting materials. The idea that students must compute answers in their heads fails to legitimize the understanding demonstrated through finger counting. In addition, prohibiting students from finger counting does not allow them to naturally progress through the developmental phases associated with counting.
Methods
The intent of the current study was to replicate a study conducted by Frank (1990) in which she examined beliefs in myths about mathematics held by a sample of elementary teacher candidates. Frank created a 12-item survey based on Kogelman and Warren’s (1978) mathematics myths on which participants noted whether they agreed or disagreed with each of the 12 statements. In addition, participants were instructed to select 1 of the statements with which they agreed and write a paragraph explaining why they agreed with that statement. No information was provided as to the validity or reliability of the survey.
The sample for Frank’s study included 131 preservice elementary teachers enrolled in a mathematics content course for teachers. No data were provided as to the makeup of this sample or the university from which it was drawn. Frank reported that 125 of the 131 participants agreed with at least one of the myths.
Subjects
In an effort to replicate Frank’s (1990) study, 138 teacher candidates enrolled in 4 sections of a mathematics content course for teachers completed a 12-item beliefs survey. Of these teacher candidates, 79 were early childhood majors seeking initial certification in elementary education. As only 3 of the early childhood majors were male, the decision was made not to include them. This small number of males could have resulted in misrepresentative generalizations regarding male elementary teachers. As a result, responses from the 76 female early childhood majors were used in this study.
The state university from which this sample was taken is located in northwest Georgia and has a current enrollment of over 10,000 students, over 7,600 of which are undergraduates. The average undergraduate age is 22 years, and 3.5% of students are from states other than Georgia. Approximately 8.5% of the undergraduates are early childhood majors.
Procedures
Participants in this study completed Frank’s (1990) beliefs survey during the 1st week of the semester in an effort to identify what beliefs teacher candidates held upon initially entering the teacher education program. The data were analyzed in three ways. First, each participant was assigned a total agreement number based on the number of myths with which she agreed. second, the percentage of teacher candidates agreeing with each myth was calculated. For each myth, participants’ responses in this sample were compared with Frank’s original findings.
The third analysis involved examing the participants’ written responses. First, we separated the responses according to which myth the participants addressed. For each myth, we identified the themes reported by preservice teachers regarding their adherence to a mathematical myth.
Limitations
Before presenting the results and implications from this research, limitations of the study must be noted. The first limitation is the unique sample and setting. The generalization of these results to other elementary teacher candidates is limited because of the makeup of the sample. The participants were all females attending the same state university. Therefore, the results may not necessarily generalize to larger universities or to male teacher candidates.
The second limitation is the survey instrument used in order to provide a direct replication of Frank’s (1990) study. The survey included only one item per myth, which limited the possible use of more sophisticated statistical analyses such as a factor analysis in drawing conclusions about the myths. In addition, without a Likert- type scale, conclusions regarding the level of agreement with each myth were not possible.
Results
In investigating the beliefs held by elementary teacher candidates, two questions were posed. First, what myths about mathematics do elementary teacher candidates hold? Second, how do the beliefs held by the current sample of elementary teacher candidates compare to the beliefs held by a previous sample of elementary teacher candidates?
Table 1
Teacher Candidates Agreement With Each Myth
Teacher Candidate Math Myths
Using the number of endorsed myths as a measure of myth presence, our results revealed teacher candidates agreed with 5 of the 12 myths on average (M= 5.0; SD = 1.78; range 1-10). In addition to participants’ total myth agreement, the percent of agreement with individual myths was calculated (see Table 1). Myths 1 through 4 received a higher percentage of agreement than the remaining 8 myths. The written responses provided by the teacher candidates offer insight into why the teacher candidates hold beliefs in these 4 myths. The myth, “Some people have a math mind and some people don’t,” received not only the highest percent of agreement (89%) but also was the subject of the highest percent of teacher candidates’ written responses. Many teacher candidates defined “having a math mind” as being able to do math easily:
I agreed to the statement “Some people have a math mind and some don’t” because I have seen this prove itself since my childhood. I believe some people can catch on more quickly to math and its concepts better than others; the same as in any other school subject.
I believe that some people can understand math better than others. I have never been one that math comes easily to; it just takes me a little longer to get it. I always had to get extra help from teachers.
I agree that some people have a math mind and some don’t because I am one who doesn’t, yet my sister does. Math has never come easily or naturally, and I’ve always had to work at it. On the other hand, my sister can pick up a problem and figure it out like it’s nothing. Math just comes a lot more naturally to her . . . she has the mind for it.
The myth, “Math requires a good memory,” received the second highest percentage of agreement with 76%. In the written responses, teacher candidates pointed towards the need to know formulas, equations, steps, and basic facts.
I agree with the statement that math requires a good memory. I feel that this is true b/c in math there are a lot of rules, theorms, and equations that you must remember in order to correctly complete a majority of problems.
My boyfriend does poorly in math and I think its partially due to his bad memory. He can’t ever remember the orders of steps in solving a problem.
Receiving the third highest percentage of agreement was, “You must always know how you got the answer” (75% agreed). Teacher candidates who chose to write about this statement seemed to view “not knowing” how you got the answer as a form of guessing. Many teacher candidates felt that a person needed to know how the correct answer was determined in order to be able to work similar problems in the future. Some teacher candidates also said it was needed so that one could judge the correctness of the answer, explain the process to someone else, or make sense of the mathematics.
I think that if you don’t show all of your work and know how you got through the one problem, you won’t understand the next one.
I believe that to fully understand a mathematical concept, you must always know how you get the answer. If you don’t, then you really do not know how to work the problem. Stumbling upon an answer will not help you in the future. You must learn and comprehend mathematical concepts!
Comparison With Fr\ank’s Sample
In comparing the beliefs held by the two samples, two similarities appear. First, for both samples, M1, M2, M3, and M4 received the highest percentages of agreement. second, both samples showed little agreement with 6 of the myths (M5, M7, M9, M10, M11, and M12). For each myth, a statistical comparison utilizing a z statistic was made between the proportions of participants agreeing in each sample (see Table 1). Using a p
Discussion
Our results indicate the elementary teacher candidates maintained beliefs in mathematical myths, many of which are in direct conflict with the beliefs about mathematics expressed in Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000). The incompatibility of these beliefs with those underlying improvement efforts “blocks reform and prolongs the use of a mathematics curriculum that is seriously damaging the mathematical health of our children” (Battista, 1994, p. 462). Therefore, close examination of these beliefs is imperative.
Three primary themes emerged from the analysis of the written responses. First, the majority of teacher candidates in this sample believe a person is either good in math-meaning that mathematical answers come quickly and easily-or they are not-meaning that learning mathematics is very difficult, if not impossible. When teacher candidates view themselves as not having a math mind, a lack of self-confidence in their mathematical ability may result (Kogelman & Warren, 1978). As practicing teachers, they may see their students as either having a math mind or not; they may fail to challenge all students to understand mathematics and to encourage students to develop the self-confidence needed to be successful in mathematics.
The second theme involved teacher candidates’ view of mathematics as a set of rules, procedures, or facts that must be memorized. Teachers with such beliefs will likely teach a concept as a procedure to be followed rather than as a process to be understood. This belief may lead to more class time being devoted to the memorization of mathematical rules and procedures and less time spent working to enable their students to understand mathematics and to make connections within and among the mathematical concepts.
The final theme revolved around teacher candidates’ lack of understanding of the role that intuition can play in doing mathematics. Participants’ perceptions of mathematics following a step-by-step procedure support a view that mathematics requires logic not intuition. Viewing intuition as merely blind guessing, these teacher candidates are likely to rely primarily on algorithmic approaches to math.
Unexpectedly, a comparison of the beliefs held by this sample with the beliefs of participants in Frank’s (1990) study revealed that mathematical myths have remained constant despite 15 years of reform in mathematics education. Based on this comparison, it appears as though mathematics education is experiencing a counter- productive cycle. Unless teacher education programs are designed to correct these beliefs and disrupt this cycle, these teacher candidates will return to the classroom and perpetuate these myths with their learners. Although reform efforts have addressed these issues, the results of this study indicate that this is not enough. We propose that teacher education programs hold the key to breaking the cycle, provided the programs are deliberate in debunking the standard myths held by students entering the teaching profession.
Implications for Teacher Educators
Given the goal of teacher preparation programs to prepare highly qualified mathematics teachers, this re-examination of elementary teacher candidates’ beliefs points toward critical recommendations for teacher educators. First, the literature and the findings of this study indicate the importance of recognizing the beliefs in myths teacher candidates bring with them into a teacher preparation program.
Teacher preparation programs must not only be concerned with the beliefs their teacher candidates have about mathematics, but also with how the program itself works to influence those beliefs. Research has shown that the experiences teachers have as learners can have a tremendous impact on the beliefs and attitudes they bring into their own classrooms (Chappell & Thompson, 1994). Thus, mathematics content courses are the most logical place within teacher education programs to combat beliefs in myths, calling into question the myths preservice teachers have and recognizing the power of these beliefs in altering the instructional processes of future math teachers.
The results of the written responses also revealed that teacher educators need to provide teacher candidates with experiences that require making sense of mathematics rather than viewing it as a set of rules. Teacher educators need to facilitate candidates’ examination of the issue of whether or not some people really have mathematical minds and encourage candidates’ exploration of the role of intuition in mathematics. Finally, teacher educators will need to dedicate time to guiding teacher candidates in reflecting back on these experiences and in exploring how these experiences contradict the beliefs that they hold.
References
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Angela T. Barlow
Mathematics, University of West Georgia
Jill Mizell Reddish
Curriculum and Instruction, University of West Georgia
Copyright Ball State University Teachers College Winter 2006
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