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A more parsimonious mathematics beliefs scales.

Abstract

Beliefs are mental representations of reality that guide peoples' thoughts and behaviors. Originally the Mathematics Beliefs Scales (MBS) was hypothesized to measure tour subscales. After administering it to 123 teachers and 54 preservice teachers, the factor analysis indicated a three-factor solution and a more parsimonious 18-item Revised Scale. This shortened version of the MBS should assist researchers in data collection by (a) shortening the time it takes to administer the scale, (b) removing seemingly redundant items, and (c) focusing on specific constructs contained within the instrument.

Introduction

Beliefs have been described and defined by different researchers in different ways. Beliefs are the bedrock and cornerstone at the heart of our actions (Corey, 1937). Beliefs are the best indicators of the decisions individuals make throughout their lives (Dewey, 1933). Beliefs are classified as instrumental and relational approaches to a situation (Carter & Yackel, 1989). Pajares (1992) proposed that beliefs are mental representations of reality that guide thought and behavior and are often initiated early in life and maintained in the lace of strong contradictions. These entrenched beliefs serve as a filter through which teachers view the world and interpret information. All teachers possess beliefs about their profession, their students, how learning takes place, and the subject areas they teach. It follows, therefore, that teacher practices should flow from these beliefs. Teacher beliefs are instrumental in defining teacher pedagogical and content tasks and for processing information relevant to those tasks (Nespor, 1987).

In the Principles and Standards for School Mathematics (NCTM, 2000), the National Council of Teachers of Mathematics state that "Effective teachers realize that the decisions they make shape students' mathematical dispositions and can create a rich setting for learning" (NCTM, 2000, p.18). These decisions are controlled and influenced by their beliefs. Thus beliefs are implicit in teacher discourse, teacher objectives, and teacher practices. Many researchers have studied teacher beliefs about mathematics. Teachers' beliefs and practices essentially mold classroom teaching, including discourse. "One's conception of what mathematics is affects one's conception of how it should be presented. One's manner of presenting it is an indication of what one believes to be the most essential in it ... The issue, then, is not, what is the best way to teach? But, what is mathematics really all about?" (Hersh, 1986, p. 13).

Researchers (Knapp & Peterson, 1995; Vacc, Bright, & Bowman, 1998) realized that changing beliefs took much time and support. Other researchers also found that substantial improvements occur in classroom achievement when teachers shift their beliefs along with their practices (Fennema, Franke, Carpenter, & Carey, 1993). Researchers (Carter & Norwood, 1997; Ford, 1994; Lubinski, 1993) have also compared teachers' and students' beliefs about mathematics. In addition researchers also revealed a definite relationship between teacher beliefs and actual classroom content, and how students learned in individual classrooms (Grant, Hiebert & Wearne, 1994; Hart, 2002). Clarke (1997) looked at how the beliefs held by teachers were reflected in the roles of the teachers--what the teachers did. Students in a research classroom studied by Carter and Norwood (1997) believed different factors such as task orientation, ego orientation, and extrinsic motivation scales were also important in mathematics success. Student responses possibly indicated that students were mirroring their teachers' beliefs. Battista (1994) stated that teachers who are presently teaching mathematics learned from teachers who used traditional curriculum depicting teachers as educators in a vicious cycle, teaching the same way that they were taught in school. These teachers held a view that was in direct contrast to the reform movement.

Constructivist teachers have beliefs and exhibit practices allowing students to construct their own knowledge through active investigation and meaningful discourse (Vacc, 1995). Beliefs are essential influences on how and whether teachers acquire constructivist knowledge in the first place, and on how and whether teachers would be inclined to implement constructivism in the classroom (Nespor, 1987). Agreement in constructivist beliefs and practices result in improved teaching. Conversely, when there is dissonance between beliefs and practices, ineffective teaching results (Pokalo, 1984; Thornton, 1985). Kloosterman et al. (1991) and Steele (1994) explored how implementing a constructivist approach in a mathematics methods class might change prospective teachers' conceptions about mathematics and mathematics teaching and learning. Steele administered the Mathematics Beliefs Scales (MBS). By the end of the standards-based course nearly all students in both had begun to talk differently about their own learning of mathematics and were willing to take risks, defending their own solutions to problems. They had a different image of teaching mathematics.

In conclusion, Carter and Norwood (1997) found high correlations between teachers' mathematical beliefs and their students' mathematical beliefs. This promotes the idea that teacher beliefs do impact upon student learning. This relationship, and instantiations through which these beliefs are communicated need exploration.

Methodology

For the purposes of this study, a Likert-type instrument entitled, Mathematics Beliefs Scales (MBS) (Fennema, Carpenter, & Loef, 1990), available upon request from the author, measured the mathematical beliefs of teachers. Teachers were asked to complete the MBS questionnaire that was adapted from Fennema, Carpenter and Peterson (1987). On a sample of 39 teachers, the internal consistency of teacher scores on the total scale was .93 using four subscales. The Likert-type instrument containing 48 statements, ranging from Strongly Agree (A) = 5 to Strongly Disagree (E) = 1, was coded as follows: positive items were left alone and the 24 negative items were coded in the opposite direction. The responses were added to get a total for each teacher, and a mean score was obtained by dividing by 48. Surveys were distributed to 4th- and 5th-grade teachers from 18 public schools in five school districts in a southeastern state. The MBS was either sent by mail or hand-delivered to 176 teachers, with 123 returned either in person or by mail.

Another administration was conducted at a large southwestern state public university during a senior methods block. The elementary preservice teachers (PTs) were traditional teacher education students, all female. Ethnicity was Caucasian (91.7%) and Hispanic (8.3%) with a mean age of 21.4 (SD=1.9). On the last day of mathematics methods class, (N=54) PTs were given the MBS. These PTs completed 52 full field-based days in classrooms, developed a weeklong integrated unit, wrote, and taught a minimum of four constructivist lessons. The PTs had also been involved in inquiry-type, hands-on, cooperative group activities involving the ten process and content standards (NCTM, 2000) during their mathematics block instruction. In addition, they maintained reflective journal of classroom activities and field experiences.

Data from both administrations of the surveys were analyzed using SPSS. Each of the responses to the 48 items of MBS was entered in 48 separate columns. Numerical data were entered from 5 to 1 based on the Likert scale responses of A to E respectively. Negative statements were then recoded. An average scale score on the total scale was obtained and on this score teachers were divided into two categories. Classroom teachers in the first study whose mean score was less than 2.5 were considered low constructivist in their beliefs and those whose mean score was greater than 3.5 were considered constructivist in their beliefs. Seven of these teachers placed in the low constructivist belief category and 25 in the constructivist belief category. In the second study, eleven of the PTs teachers could be considered constructivist in their beliefs, while only one would be categorized as having low constructivist beliefs. Those falling in the mid-range were not considered as either constructivist or low constructivist indicating that most teachers whether inservice or preservice are either undecided about their constructivist beliefs or both strongly agree and disagree on an even number of the scale items. Thus, each item or subscale needs to be examined more closely, individually.

Reliability

The coefficient-alpha reliability of the scores on the 48-item belief scale for the 123 classroom teachers was .68. This reliability is marginally acceptable according to Shavelson (1988). Reliability was lower than the published reliability of .93 obtained by Fennema, Carpenter, and Peterson (1987) using a sample of teachers in a Mid-western state. This difference possibly suggests this researcher used a more homogeneous sample of teachers rather than those used by the previously mentioned researchers because the heterogeneity of the items remained constant. The coefficient-alpha of the scores on the 48-item belief scale holding the heterogeneity of the items constant for the 54 PTs was .86 suggesting a more heterogeneous group than the classroom teachers. The combined reliability for both studies was .78 that was acceptable (Shavelson, 1988).

Factor Analysis

Factor analysis was employed in this study to determine a more parsimonious model of the MBS by condensing the information contained in the original variables into a smaller set of factors with a minimum loss of information (Hair, Anderson, Tathem, & Black, 1998) resulting in the number of factors being less than the number of original variables. The MBS contained 48 variables, and each time it was administered the participants in the study complained of its length and its repetitive nature. Therefore, the aim of this study was to reduce the number of variables and thus items on the scale but yet obtain roughly the same information. The original researchers designed the instrument to measure four subscales on the beliefs of teachers (a) about how children learn mathematics, (b) about how mathematics should be taught, (c) about the relationship between learning and concepts and procedures, and (d) about what should provide the basis for sequencing topics in addition and subtraction instruction. Due to the length of the scale and the repetitive nature of some of the statements, it was questioned whether this instrument actually measured four factors that the original authors claimed were measured.

To determine how many factors would emerge for the data in the present study, the techniques of factor analysis were used. Analysis of correlation matrix involving items from the original Beliefs Scales yielded 14 components with eigenvalues > 1, and the scree plot indicated at least six components. Because it was difficult to interpret this unrotated factor solution, a first-order principal components analysis with an orthogonal rotation method was employed on the 48 original items. Orthogonal rotation was used to obtain a more parsimonious and thus more replicable solution (Kieffer, 1999). Because the goal was to develop a shorter measure, criteria for omitting or retaining items were invoked. First, the last three of the six components had few items defining these constructs, and the first three components explained more variance and were deemed most noteworthy. Therefore, items primarily saturating the last three of the six components were omitted. Second, items that were "multivocal" (i.e., "spoke through two or more components, as reflected in pattern/structure coefficients on two more components > [.30] were omitted. Third, "univocal" items that most saturated the first three components were retained. In this manner 18 items were selected to constitute a short form of the measure.

These 18 variables explained 46.23% of the observed variance. The communality coefficients are the amount of variance in each item that was useful in all of the factors as a set. The arithmetic mean of these 18 values ranged from .192 to .638 with .462 of the variance being explained by the extracted factors. Although a little less than 50% of the variance was explained (3.5% more variance) when a four-factor model was employed, only two of the items correlated appreciably with factor IV. An examination of the results presented for the three-factor solution rotated to the varimax criterion is displayed in Table 1. See issue website http://rapidintellect.com/ AEQweb/fal2005.htm The rotated component matrix contains 18 variables, six correlating with each of the three components. The cutoff used for saliency was variables with pattern/structure coefficients greater than [.30].

The saturation of factors can be determined from the table. All of these items measure teacher beliefs concerning different areas of mathematics education, Factor I is most highly saturated with variables 22, 23, 29, 39, 42, and 47. All of these items deal with how children learn mathematics with a higher score, defining teachers who believe that children can construct their own knowledge. A lower score indicated that students receive most of their knowledge directly from the teacher. This factor can be named "Student Learning". Factor II is most highly saturated with variables 16, 25, 26, 36, 45, and 46. All of these items are measuring the sequence of learning, concentrating on which skills should be taught before what other skills and prerequisites for learning certain skills. A higher score indicates that students can solve real-world problems before knowing all their computational skills. A lower score designating a teacher who feels that it is necessary for all computational skills to be learned before a student can attempt to solve even simple word problems. This factor can be named "Stages of Learning". Factor III is most highly saturated with variables 2, 5, 9, 30, 32, and 37. All of these items measure teacher beliefs about how teachers should teach. This factor can be called "Teacher Practices". A higher score indicates that teachers facilitate student knowledge, while a lower score defines a teacher who feels that his/her practices need to be organized to direct student learning.

Results

The results of this study developed a revised Beliefs Scale consisting of 18 items shortened from 48. Six items measured each of the three factors. The analytic method produced a modified scale that still measures what the original authors intended for it to measure, namely the beliefs of teachers about how children learn (Factor 1: Student Learning), the role of the teacher in sequencing of teaching both computational and application skills (Factor 2: Stages of Learning), and the relationships between teaching computational skills and problem solving skills (Factor 3: Teacher Practices). The items by factor are:

Factor 1

* Recall of number facts should precede the development of an understanding of the related operation (addition, subtraction, multiplication, or division).

* Children will not understand an operation (addition, subtraction, multiplication, or division) until they have mastered some of the relevant number facts.

* Time should be spent practicing computational procedures before children are expected to understand the procedures.

* Children should not solve simple word problems until they have mastered some number facts.

* Time should be spent practicing computational procedures before children spend much time solving problems.

* Children should master computational procedures before they are expected to understand how those procedures work.

Factor 2

* Most young children have to be shown how to solve simple word problems.

* Children should understand computational procedures before they master them.

* Children learn math best by attending to the teacher's explanations.

* Most young children can figure out a way to solve many mathematics problems without any adult help.

* To be successful in mathematics, a child must be a good listener.

* Children need explicit instruction on how to solve word problems.

Factor 3

* Teachers should encourage children to find their own solutions to math problems even if they are inefficient.

* Teachers should teach exact procedures for solving word problems.

* Mathematics should be presented to children in such a way that they can discover relationships for themselves.

* The goals of instruction in mathematics are best achieved when students find their own methods for solving problems.

* Teachers should allow children who are having difficulty solving a word problem to continue to try to find a solution.

* Teachers should allow children to figure out their own ways to solve simple word problems.

The results presented here are a Revised Version of the MBS. The items were then reordered so that all of one factor was not all together. One negatively worded question was made positively worded matching the original scale that had an even number of negatively and positively worded statements. The factor analysis had eliminated more positively worded items. After completing those tasks, the scale was administered to an additional group of PTs to check the reliability of the scores with a similar sample obtaining a reliability of .86. This shortened version of the MBS should assist researchers in data collection by (a) shortening the time it takes to administer the scale, (b) removing seemingly redundant items, and (c) focusing on specific constructs contained within the instrument.

Discussion / Conclusion

It is important to understand teacher beliefs since ultimately these beliefs lead to student achievement. Researchers have investigated how teacher's knowledge of and beliefs about their students' thinking are closely related to student achievement (Carpenter, Fennema, Peterson, & Carey, 1988). Additionally, Fisher et al., (1980) found that teachers should help students construct their own knowledge and that their instruction should add to students' prior knowledge about a concept. These teachers did not see their role as providers of all knowledge and students as "sponges" waiting to absorb their knowledge.

Since beliefs are shaped through experience over time, pedagogical strategies and practices that support constructivist theory can be developed by involving preservice teachers and beginning teachers in constructivist experiences both in the learning and teaching of mathematics. Conceptions and beliefs teachers have about mathematical content can affect their ability to implement constructivist curriculum (Milsaps, 2002). These beliefs are important since they also influence teachers' choices of curriculum along with the choices they make concerning their instructional practices (Manouchehri & Goodman, 2000). Thus "as attention to beliefs becomes an increasingly common component of teacher preparation and professional development programs, it is important that the research community strives to ensure that representations of teacher's beliefs are as accurate as possible" (Speer, 2002, p.658). Hopefully this more parsimonious MBS will enable researchers to more quickly and accurately measure the beliefs of teachers. The next step is to measure whether the practices of these teachers match their beliefs. Long-lasting instructional changes results from essential modifications in what teachers believe, know, and practice (Putnam, Heaton, Pratt, & Remillard, 1992). The National Institute of Education (1975) in their document entitled, Teaching as Clinical Information Processing posited a close relationship between beliefs and practices stating "what teachers do is directed in no small measure by what they think" (p.1). Moreover, it is "necessary for any innovations in the context, practices and technology of teaching to be mediated through the minds and motives of teachers" (p. 1).

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Mary Margaret Capraro, Texas A & M University

Mary Margaret Capraro, Ph.D., is an Assistant ClinicaProfessor of Mathematics Education, in the Department of Teaching, Learning, and Culture..
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