Kruskal Wallis

IMPLICATIONS OF TEACHERS’ BELIEFS ABOUT MATHEMATICS FOR CLASSROOM AND TEACHER EDUCATION REFORM Catherine P. Vistro-Yu

Department of Mathematics, Ateneo de Manila University, Quezon City

Abstract This article reports partial results from a study that investigated secondary school mathematics teachers’ beliefs and attitudes about mathematics. Using Ernest’s (1988) model, it describes what teachers believe about the nature of mathematics and about the teaching and learning of mathematics consistent with the instrumentalist, Platonist, and problem solving views of mathematics. A 47-item Mathematics Attitude Survey and a demographic questionnaire were administered to 57 high school mathematics teachers. Data analyses used the non-parametric tests of Friedman, Mann-Whitney, and Kruskal-Wallis and a non-parametric correlation test. Results revealed that teachers hold strong beliefs about mathematics that are consistent with the problem solving view. There is a significant difference in the intensity with which public and private school teachers hold the instrumentalist view of mathematics. There is also a significant difference in the intensity with which relatively low- and high-scorers in the Licensure Examination for Teachers hold the instrumentalist view. Implications on how mathematics teacher education programs can be improved are discussed. Keywords: teachers’ beliefs, mathematics for classroom, teacher education reform, instrumentalist, Platonist, problem solving Introduction Ernest noted that the practice of teaching mathematics depends on a number of key elements (1): • • •

The teacher’s mental contents or schemas, particularly the system of beliefs concerning mathematics and its teaching and learning; The social context of the teaching situation, particularly the constraints and opportunities it provides; and The teacher’s level of thought processes and reflection.

These factors determine the autonomy of the mathematics teacher and subsequently, also the outcome of teaching innovations, like problem solving that depends on teacher autonomy for their successful implementation (1). The present study addresses the first key element – the teacher’s mental contents or schemas including the system of beliefs concerning mathematics and its teaching and learning. Teachers’ mental contents or schemas include a) mathematics content knowledge, b) beliefs and attitudes displayed towards an object or a group of objects, and c) belief systems about the nature of mathematics and the teaching and learning of it. Although mathematics content knowledge is important, it is not enough to determine teachers’ readiness or capacity to shift to more desirable teaching approaches. Teaching reforms cannot take place unless teachers’ deeply held beliefs about mathematics and its teaching and learning change (1). Theoretical Framework Teachers’ conceptions of mathematics are almost always interpreted in light of a philosophy or several philosophies of mathematics because these constitute their views, attitudes, beliefs, and

preferences about the nature of mathematics as well as the teaching and learning of mathematics. Ernest distinguished three possible conceptions of mathematics (1, p.2) that relate significantly to a philosophy of mathematics (2): First of all, there is the instrumentalist view that mathematics is an accumulation of facts, rules and skills to be used in the pursuance of some external end. Thus, mathematics is a set of unrelated but utilitarian rules and facts. Secondly, there is the Platonist view of mathematics as a static, but unified body of certain knowledge. Mathematics is discovered not created. Thirdly, there is the problem solving view of mathematics as a dynamic, continually expanding field of human creation and invention, a cultural product. Mathematics is a process of enquiry and coming to know, not a finished product, for its results remain open to revision. Ernest further associated the above views to corresponding models of teaching in the aspects of the teacher’s role, the intended outcome of teaching, and the teacher’s use of curricular materials (1). All these further reflect a corresponding learning model implicitly adhered to by the teacher. These aspects are embedded in a diagram (cf. 1, p.3) illustrating the specific relationships between teachers’ beliefs about the nature of mathematics and their models of teaching and learning. Table 1 shows a likely set of significant associations and implications of one’s view about mathematics based on Ernest’s model (1). This matrix, where the instrumentalist view of mathematics is at the bottom, appears to model a hierarchy. As shown by Table 1, a teacher who follows the instrumentalist view of mathematics will tend to take an instructor’s role in teaching where the main objective is for students to master the skills needed in mathematics. Performance is important because that is how an instructor can determine if mastery has been achieved. To ensure this, teachers of this view would strictly follow the prescribed curriculum that is broken down into a hierarchy of specific skills to be learned. The basis of knowledge here is rules, not necessarily with understanding. There is an implicit belief that the curriculum and the corresponding instructional materials offer the best formula for mastering skills; thus, instructions would tend to be very rigid. However, teachers who follow the Platonist view of mathematics would tend to take on an explainer’s role in teaching. They tend to lecture and explain concepts, focusing on mathematical content. They emphasize students’ understanding of ideas and processes, particularly students’ understanding of the logical relationships of mathematical concepts. The objective of instruction is for students to have a unified concept of mathematics and a consistency of ideas. The problem-solving view of mathematics is the highest in the hierarchy because such a view encourages learning by active construction of one’s knowledge. Teachers who subscribe to this view see themselves as facilitators of learning. As such, they would prefer to construct or develop their own materials that would suit the needs and interests of their students. Their instructional objective is to develop more confident and better problem-solvers. This view of mathematics encourages creativity and multiple approaches to learning a concept or skill.

Table 1. Teachers’ views about mathematics and their implications based on Ernest’s model (1, pp. 2-4).

Related Research Various studies on mathematics teachers’ beliefs and conceptions have consistently confirmed one major idea: that mathematics teachers’ beliefs and conceptions, particularly about the nature of mathematics and about the teaching and learning of mathematics, have an impact on the type of mathematics instruction they deliver in the classroom. Thompson identified several studies in mathematics education (3-12) that have indicated that beliefs about mathematics and its teaching play a significant role in shaping the teachers’ patterns of instructional behavior (2). In particular, Thompson concluded that the relationship between teachers’ conceptions and their instructional decisions and behavior is a complex one (8). In her study of three junior high school teachers, she found that the quality and level of these teachers’ reflection about their beliefs contributed to the presence or lack of congruence between their beliefs and their instructional behavior, making the said relationship complex. Additional conceptions about their students and the social and emotional make-up of the classroom also affect their instructional behavior patterns, perhaps, much more for some teachers. These conceptions likewise affect teachers’ view about mathematics and its teaching. Many studies (13-15) have strongly suggested that teachers’ cognitive knowledge of mathematics directly influence what they do in the classroom and ultimately what their students learn (16). A critical study of the literature related to teachers’ cognitive knowledge of mathematics, also reveals that the linkage between teachers’ mathematical knowledge and their instructional practices is complex. Fennema & Franke proposed a working framework that provides a more complete picture of how teachers’ mathematical knowledge influences their pedagogical practice (16). Rather than focusing on teachers’ mathematical knowledge alone, the framework illustrated how teachers’ mathematical knowledge along with teachers’ knowledge of the learning contexts, of their students’ understanding of mathematics, and of specific pedagogical techniques, including mathematics teachers’ beliefs contribute to the way they teach mathematics. For instance, a classroom teacher knows that a teaching methodology would not elicit the same response from two different classes principally because of the varying personalities that make up the two classes. Therefore, the teacher would try to use different methodologies that would suit the classes. Thompson also reported varying degrees of consistency between mathematics teachers’ espoused beliefs about mathematics and their instructional practices based on studies done by other researchers (2). For example, (6) reported that strong relationships exist between the knowledge base of novice teachers and their instructional practice. McGalliard (10) observed a high degree of consistency between the mathematics conceptions of four senior high school teachers and their instructional practices in geometry (2). In the same study, McGalliard also reported inconsistencies between teachers’ professed beliefs and their instructional practice. This cautions researchers in interpreting results of studies involving teachers’ professed beliefs. It further suggests that meaningful studies on teachers’ beliefs should include observable data on teachers’ teaching practices. Purpose of the study The purpose of the study was to investigate high school mathematics teachers’ beliefs about the nature of mathematics and about the teaching and learning of mathematics. Following Ernest’s framework (1), the study sought to identify the beliefs that teachers hold consistent with the instrumentalist, Platonist, and the problem solving views of mathematics and to determine the intensity with which they hold these beliefs. Related to this is the importance of the relationship between the beliefs held and certain grouping variables used to describe the sample of teachers for the study. In sum, the study attempted to answer the following questions: 1. What do high school mathematics teachers believe about the nature of mathematics and about the teaching and learning of mathematics? 2. Do teachers hold much more strongly beliefs that are consistent with one type of view of mathematics according to Ernest’s framework?

3. Do certain subgroups of teachers hold much more strongly beliefs consistent with the types of view of mathematics according to Ernest’s framework than other subgroups do? 4. Is there a relationship between any of the grouping variables used to describe the sample of teachers for the study and the types of view of mathematics according to Ernest’s framework? Method Participants. Fifty-seven high school mathematics teachers participated in the study. Of these, 26 teach in public schools spread over 14 regions in the country. The remaining 31 teach in private sectarian schools that belong to one geographical cluster in Metro Manila. Tables 2a and 2b summarize the demographic variables used to describe the teachers. Table 2a. Mean and standard deviation of age, years of teaching, and Licensure Exam score Score in Years of Age Licensure Teaching Examination Number of teachers who a 53 51 37 reported (n) Mean (x) 32.6 9.4 78.9% Standard Deviation (σ) a

7.4

6.8

4.3

One teacher simply reported “passed” for this item.

Table 2b. Frequency and percentage for gender, post-baccalaureate studies, awards, non-teaching jobs, and in-service training Variables Frequency % GENDER Female 37 70 Male 16 30 Total 53 100 POST-BACCALAUREATE STUDIES Yes 4 8 No 47 92 Total 51 100 HONORS/AWARDS RECEIVED Yes 5 10 No 46 90 Total 51 100 NON-TEACHING JOB EXPERIENCE Yes 9 17 No 43 83 Total 52 100 IN-SERVICE TRAINING Yes 25 48 No 27 52 Total 52 100 Instrument. In addition to the demographic questionnaire, the main instrument used to determine teachers’ beliefs about mathematics was a 47-item Mathematics Attitude Survey containing a mixture of belief and attitude statements. This instrument was used in the Second International Mathematics Study (SIMS) that was completed in 1982 as part of the effort to learn about the 12-year-old

participants from a cross-cultural sample of 18 countries (17). The same instrument was used to obtain information about middle school mathematics teachers in a NSF-funded Middle School Mathematics Project at the University of Georgia in 1989 (18). Thirteen of the items in the survey have been classified as pertaining to beliefs about the nature of mathematics while 12 of these refer to beliefs about the teaching and learning of mathematics. Two items were classified as pertaining to both. The rest of the items are a combination of belief and attitude statements pertaining to other aspects of mathematics, which will not be discussed in this paper. The classification followed the one used for SIMS. The survey uses a 5-scale response: strongly disagree, disagree, undecided, agree, and strongly agree. Numerical values of 1, 2, 3, 4, and 5, respectively, were assigned to the responses to facilitate quantitative analyses. Data Collection and Analyses. The two instruments were administered to the private school teachers at the beginning of the aforementioned half-day in-service workshop while they were administered to the public school teachers at the end of a class session. The 27 items in the Mathematics Attitude Survey were classified further according to the instrumentalist (I), Platonist (P), and problem-solving (PS) views of mathematics. The researcher and a mathematics education professor who had done extensive studies on the philosophy of mathematics education independently classified the items. Initially, the two agreed on the classification of only 25 items. A 100% agreement was achieved after some discussion. It is important to note, however, that this categorization forced one to place items in one and only one category if only to simplify the analysis. The reality is that some of these items may be classified in two or three categories. For example, the item on logic may be considered important in the problem solving view. However, because the Platonist view emphasizes logic, it is classified under the Platonist view of mathematics. Furthermore, since a hierarchical structure has been assumed, classifying the item at a lower level in the hierarchy does not diminish its importance in the higher level. Table 3 indicates the classified items. Table 3. Classification of items according to the type of belief and the type of view of mathematics InstrumentProblemPlatonist alist solving Nature of Items 5, 7, 8, Items 3, 6, Items 1, 2, 4, Mathematics 27, 40 12, 14 9, 10, 11 Teaching and Learning of Mathematics

Items 27, 40, Item 44 43

Items 13, 23, 34, 36, 37, 41, 42, 45, 46, 47

For every item, a weighted average was obtained based on the responses of all 57 teachers. A weighted average of 3.5 and above is considered as manifesting the belief statement. A weighted average of 2.5 and below is considered as not manifesting the belief statement. A weighted average between 2.5 and 3.5 is considered as manifesting indecision about the belief statement. For every teacher, one score for each of the three views of mathematics was obtained by taking the average of the responses to the items classified under each type of view. Thus, associated with each teacher are three scores. Hence, there are 57 scores reported for each type of view of mathematics. The mean of the 57 scores for each type of view was then obtained. Each of the demographic variables type of school, age, gender, years of teaching experience, postbaccalaureate studies, awards, in-service training, non-teaching job, and Licensure Examination score was assigned a nominal value according to the subgroups determined by the researcher. Some of the subgroups were natural groupings (e.g. female vs. male, private vs. public school, or have vs. have not) while the rest were ascertained based either on the available data or on the researcher’s perception of critical periods or points in the life of or about a teacher. For example, the following

were grouped based on perceived critical periods or points: AGE – (x < 30, 30 ≤ x < 40, x > 40) and YEARS OF TEACHING – (1 ≤ x ≤ 3, 4 ≤ x ≤ 7, 8 ≤ x ≤ 15, x > 15). Since the variables involved were discrete, distribution-free (non-parametric) tests were used to check for significant differences between the means, and to determine significant correlations between variables (19). The following non-parametric tests were used from the SPSS software: the Friedman Test, the Mann-Whitney Test, and the Kruskal-Wallis Test for comparison of mean ranks. Finally, correlations were obtained using the Spearman ρ coefficient. A non-parametric test for the Spearman ρ coefficient was also used from the SPSS software to check for any significant correlation between each demographic variable and each type of view of mathematics. Results Beliefs about the Nature of Mathematics. The weighted averages showed that teachers believe in the following aspects of mathematics: • The importance of rules • The importance of logic • The importance of proofs • That mathematics can be solved in different ways • That mathematics offers opportunities for creativity • That mathematics is a dynamic field • That mathematics allows for the use of trial and error in solving problems The weighted averages also showed that teachers do not seem to believe in the following aspects of mathematics: • That mathematics is made up of unrelated topics • That new discoveries are not being made in mathematics It appears that teachers are undecided about the following aspects of mathematics: • The role of memorization in mathematics. • That there is little place for originality in mathematics • That problems can be solved without using rules Beliefs about the Teaching and Learning of Mathematics. The weighted averages showed that teachers believe in the following with respect to the teaching and learning of mathematics: • The importance of memorization in learning mathematics • The role of tests in learning mathematics • The importance of teachers’ explanations in the learning of mathematics • The need for estimation skills • The importance of the problem-solving process • The value of using manipulative objects to learn mathematics • The role of play • The role of problem solving in the teaching of mathematics • The value of computers in teaching mathematics • The value of group work • The value of writing about mathematics There are no belief statements about the teaching and learning of mathematics that the teachers do not believe in or are undecided about. Comparison Tests on the Type of View of Mathematics. To determine whether teachers tend toward one type of view of mathematics more than the other two, the Friedman Test was used. In this test, a

teacher’s scores for each of the three views are converted to ranks and a mean rank for each of the three views is then obtained (19). Table 4 shows the results from this test. Table 4. Mean, standard deviation, and mean rank of each type of view of mathematics based on Friedman Test Instrument- Platonist Problem p-value alist Solving Nature of X = 3.20; X = 3.57; X = 3.87; mathematics σ = 0.67 σ = 0.48 σ = 0.54 Teaching and X = 3.53; X = 4.02; X = 4.39; learning of σ = 0.81 σ = 0.94 σ = 0.30 mathematics Overall X = 3.36; X = 3.66; X = 4.20; σ = 0.61 σ = 0.45 σ = 0.32 Mean Rank 1.40 1.75 2.84 0.000* *significant at p < 0.02

The p-value obtained from this test is 0.000, which is significant (p < 0.02). Hence, it implies that teachers significantly tend to hold much more strongly beliefs consistent with the problem solving view (highest mean rank) than those consistent with the instrumentalist and the Platonist views of mathematics. Comparison Tests on Subgroups within Variables. To determine whether subgroups within each variable differ in the way they view mathematics, the Mann-Whitney Test for 2-independent samples and the Kruskal-Wallis Test for k-independent samples (k > 2) were performed. Both tests compare the mean ranks of the subgroups. The Mann-Whitney Test is the non-parametric equivalent of the t-test while the Kruskal-Wallis Test is the non-parametric equivalent of the one-way ANOVA (20). Table 5 shows that the tests yielded only 3 significant results. These are for the variables school and Licensure Exam Score with the instrumentalist view and for the variable years of teaching with the problem solving view. Table 5. Significant test results for 3 variables. Variable Sub-group N Mean Rank School (for Public 26 34.94 Instrumentalist view) Private 31 24.02 Passed to 22 23.18 Licensure Exam Score 79.5% (for X > 79.5% 16 14.44 Instrumentalist view) 5 18.10 Years of 1 ≤ x ≤ 3 Teaching (for Problem 20 23.40 4≤x≤7 Solving view) 19 33.82 8 ≤ x ≤ 15 x > 15 7 17.86 *significant at p < 0.05 level

Test MannWhitney

p-value (2-tailed) .013*

MannWhitney

.016*

KruskalWallis

.024*

Results from Table 5 can then be stated as follows: 1. Public school teachers significantly tend to hold much more strongly beliefs consistent with the instrumentalist view of mathematics compared to private school teachers. 2. Teachers who obtained relatively low scores in the Licensure Examination significantly tend to hold much more strongly beliefs consistent with the instrumentalist view of mathematics compared to teachers who obtained relatively high scores in the said exam. 3. There is a significant difference in the intensity with which different subgroups for the variable years of teaching hold beliefs consistent with the problem solving view. Discussion If beliefs are to be interpreted as expressions of values, then teachers may value many of the important aspects of mathematics, foremost of which are logic, proof, and the dynamic aspect of mathematics. Rules seem to play an important role in mathematics. However, research has shown that placing too much emphasis on rules may limit higher-order thinking skills, such as creative thinking (21) and cognitive elaboration (22). Results seem to indicate that teachers generally agree that mathematics is a coherent body of knowledge and a dynamic field of endeavor. On the average, teachers seem to be undecided about the role of memorization in mathematics, the place of originality in mathematics, and the tolerance for problem solving without rules. The second point appears to be inconsistent with the beliefs that mathematics is a good field for creative people and that mathematics is a dynamic field. A possible explanation for this lies in the way teachers perceive mathematics – first, as a school subject and second, as a non-school endeavor. Research studies have shown these two separate perceptions to exist among children (23) and, possibly, among practicing teachers (8). The teachers’ beliefs indicate that they value multiple approaches to the teaching and learning of mathematics, and that these approaches are wide-range. They include the use of technology, writing, play, group work, and manipulative objects for the high school level. These beliefs indicate that teachers have very positive views about using creative and fun approaches to learning mathematics. Comparisons and Correlations. The Friedman Test showed that teachers significantly tend to hold much more strongly the problem solving view of mathematics compared to the instrumentalist and Platonist views. This is encouraging because the problem solving view is the highest in the hierarchy of Ernest’s framework. This is a good starting point and, if they persist in these beliefs and carry them on to actual practice, genuine reforms in mathematics teaching can begin to take shape. The Mann-Whitney Test, the Kruskal-Wallis Test, and the non-parametric correlation test for Spearman ρ coefficient indicate crucial areas that need further investigation. The Mann-Whitney Test results supported by the correlation test showed that public school teachers significantly tend to hold more strongly the instrumentalist view of mathematics compared to private school teachers. This result might partly explain why public schools have not been successful in mathematics, as revealed by results of the Third International Mathematics and Science Study-Repeat (TIMSS-R) that included 114 public schools out of a total sample of 150 schools all over the Philippines (24). Teachers who hold more strongly the instrumentalist view tend to teach mathematics as a set of unrelated topics, focusing more on rules and skills (procedural understanding), rather than on a deep conceptual understanding of the subject matter. Previous studies on teachers (25) have shown that procedural understanding without conceptual understanding of mathematics does not last. Procedural understanding encourages a fragmented type of learning mathematics. Conceptual understanding, on the other hand, promotes connectivity and networking of concepts, which is much more useful and needed in mathematics. The result bolsters the need for training programs aimed at developing deeper understanding of mathematical concepts among public school teachers. One can deduce from here that part of helping public school teachers is to help them “weaken” their instrumentalist beliefs about

mathematics, it being the lowest level in the hierarchy. The aim is to help them learn to teach mathematics much more effectively and correctly, thereby leading them to turn their “problem solving” beliefs into meaningful actions. The Mann-Whitney Test results, again supported by the correlation test, also showed that teachers with relatively low scores in the Licensure Examination significantly hold more strongly beliefs consistent with the instrumentalist view compared to those who got relatively higher scores in the same exam. The results have implications for pre-service teacher education programs. Studies by Ibe (2628) have shown that mean scores obtained by prospective high school mathematics teachers in the Professional Board Examination for Teachers (PBET) (given from 1978 to 1995) and the Licensure Examination (given beginning in 1996), both in the General Education and Major Subject components of the exams, fall below the desired score of 50%. This pattern of performance in the exams reflects the level of competence of mathematics teachers from the cognitive standpoint. The results from the current study add to these data. Performance in the Licensure Exam also reflects the affective tendencies of teachers, in this case through their espoused beliefs. Because relatively lowscorers in the Licensure Exam tend to be more instrumentalist in the way they view mathematics, they would be less effective teachers of mathematics. Research has shown, e.g. (1), that the instrumentalist approach may fail to engage students in genuine learning. Teacher educators must continue to be vigilant and ensure that teacher preparation programs do address the cognitive and affective needs of future teachers. The Kruskal-Wallis Test calls attention to seemingly critical periods in teachers’ professional lives. By looking at the mean ranks of the four subgroups, the result from this test may be interpreted in the following way: teachers who have taught for 8 to 15 years tend most toward the problem solving view followed by teachers who have taught for 4 to 7 years. There does not seem to be a significant difference between the remaining two subgroups. This may mean that it is the first two subgroups of teachers who have the potential to initiate reforms toward a more problem solving approach to teaching. One possible explanation for this is the fact that these are teachers who have been in the profession beginning in the last half of the 1980s to the first two-thirds of the 1990s. Those years comprised a vibrant period of reform efforts and innovations in mathematics education locally (the advent of the SEDP curriculum and the nationwide Engineering and Science Education Project) and internationally (the launching of the NCTM Curriculum and Evaluation Standards in 1989 and the Third International Mathematics and Science Study in 1995). These would be teachers who may have been exposed to these efforts, not necessarily through in-service training (a significance test on the data reveal no significant differences involving in-service training and years of teaching variables), but, perhaps, through involvements in the reform efforts themselves. These results need further investigation. What is it about the 8-15 years and 4-7 years of teaching experience that caused the significant difference? What other explanations can be offered? What alternative groupings can be used? Conclusions and Recommendations The study showed that high school mathematics teachers hold certain beliefs and views about mathematics, which have both desirable and undesirable consequences in the way they teach in the classroom. These beliefs cannot be ignored because of the impact they have on classroom teaching (8). This study also showed that there are variables that significantly relate to the intensity with which teachers view mathematics from the instrumentalist perspective. There is a need, therefore, to pay attention to the variables type of school and Licensure Examination score. For instance, in the Licensure Exam, the cut-off point of 80% was based on the available data – the scores ranged from 71% to 87%. Would the test yield the same result if a different kind of grouping were used? Further studies on this variable have to be done.

The study has also shown that teachers significantly tend to hold more strongly the problem solving view of mathematics over the other two types of views. Therefore, teacher educators should assist and support teachers in concretizing these beliefs by undertaking genuine reforms at both the pre-service and in-service education levels. Teacher education programs must pay attention to both affective and cognitive issues that surround prospective and practicing teachers. The study has its limitations that have to be noted. First, the sample of 57 teachers may be too small especially since non-parametric methods were used for the analyses. Replicate studies over a larger sample of teachers are highly recommended to confirm the results that have been obtained. In the Mann-Whitney Test, one pair of variables yielded a (2-tailed) p-value of 0.052, which is slightly over the largest acceptable p-value of 0.10. One cannot help but wonder if a larger sample would have yielded significant results. Second, the Kruskal-Wallis test performed on the years of teaching variable in relation to the problem solving view offered limited information. Perhaps, more detailed investigations using pairs of subgroups within the same variable or tests using different subgroups would be very enlightening. Third, Ernest’s framework (1) is not the only perspective that can be used to investigate teachers’ beliefs. Ernest’s categorizations alone can be too limiting for the complex nature of our school system. The political, socioeconomic, and socio-cultural factors that greatly affect schools and students may not be addressed by Ernest’s framework. There is a need, therefore, to use other frameworks that might enrich the data that have been gathered from this study. An alternative framework may be a cross between Ernest’s levels (1) and Green’s (29) three dimensions of belief systems that describe the way beliefs relate to one another in the system. Green used the dimensions of primary and derivative beliefs, central or peripheral beliefs, and clustered beliefs. Fourth, what have been looked at are general views and group results. It would be interesting to know what certain individuals really believe in and how consistent they are in their expression of beliefs across different contexts. Case studies of individual teachers are recommended to follow up on the results here. Future studies can also move toward more general affective issues influencing the teacher. Fifth, it must be remembered that what have been gathered are teachers’ espoused beliefs, which may be translated differently into action. Consistent with McGalliard’s recommendation (10), studies on how teachers translate these beliefs into their teaching are also recommended. In what ways can we help teachers so that their actions are consistent with their espoused beliefs? Studies that answer this question are very useful in the teacher education field. References 1. P. Ernest, “The impact of beliefs on the teaching of mathematics”, paper prepared for ICME VI, Budapest, Hungary, July 1988, pp.1-4. 2. A. G. Thompson, “Teachers’ beliefs and conceptions: A synthesis of the research”, in Handbook of Research on Mathematics Teaching and Learning, D. A. Grouws, Ed., (Macmillan, New York, 1992), pp. 127 - 146. 3. B. J. Dougherty, “Influences of teacher cognitive/conceptual levels on problem-solving instruction”, in G. Booker et al., Eds., Proceedings of the Fourth International Conference on the Psychology of Mathematics Education, (International Group for the Psychology of Mathematics Education, Oaxtepec, Mexico, 1990), pp. 119 – 126. 4. R. Marks, Those who appreciate: The mathematician as secondary teacher. A case study of Joe, a beginning mathematics teacher, (Stanford University School of Education, Stanford, CA, 1987). 5. R. Kesler, Jr., doctoral dissertation, University of Georgia, Athens (1985). 6. R. Steinberg, J. Haymore, R. Marks, paper presented at the annual meeting of the American Educational Research Association, San Francisco, April 1985. 7. C. E. Grant, doctoral dissertation, University of North Dakota (1984); Dissertation Abstracts International, 46, DA8507627 (1984). 8. A. Thompson, Educational Studies in Mathematics 15, 105 – 127 (1984).

9. S. Lerman, International Journal of Mathematical Education in Science and Technology 14 (1), 69 –66 (1983). 10. W. A. McGalliard, Jr., doctoral dissertation, University of Georgia (1982); Dissertation Abstracts International, 44, 1364A (1983). 11. T. Kuhs, doctoral dissertation, Michigan State University, East Lansing (1980). 12. J. C. Shroyer, paper presented at the annual meeting of the American Educational Research Association, Toronto (1978, March). 13. D. L. Ball, Unlearning to teach mathematics (Issue Paper 88-1), (Michigan State University, National Center for Research on Teacher Education, East Lansing, 1988). 14. T. R. Post, G. Harel, M. J., Behr, R. Lesh, “Intermediate teachers’ knowledge of rational number concepts”, in Integrating research on teaching and learning mathematics, E. Fennema, T. P. Carpenter, S. J. Lamon, Eds., (SUNY Press, Albany, NY, 1991), pp. 177 – 198. 15. S. I. Brown, T. J. Cooney, D. Jones, “Mathematics teacher education”, in Handbook of Research on Teacher Education, W. R. Houston (Ed.), (Macmillan, New York, 1990), pp. 639 – 656. 16. E. Fennema, M. L. Franke, “Teachers’ knowledge and its impact”, in Handbook of Research on Mathematics Teaching and Learning, D. A. Grouws, Ed., (Macmillan, New York, 1992), pp. 147 – 164. 17. D. F. Robitaille, R. A. Garden, Eds., The IEA Study of Mathematics II: Contexts and Outcomes of School Mathematics, (Pergamon Press, Oxford, 1989). 18. D.E. Barnes, T. J. Cooney, I. Coronel, C. P. Vistro, “Evaluation of the Middle School Project”, (The University of Georgia, Atlanta, GA, 1989). 19. M. Hollander, D. A. Wolfe, Nonparametric Statistical Methods, (John Wiley & Sons, NY, 1973). 20. M. F. Triola, Elementary Statistics, (Addison-Wesley Publishing Company, Massachusetts, ed. 6, 1995). 21. P. Zeitz, The Art and Craft of Problem Solving, (John Wiley & Sons, NY, 1999). 22. E. A. Silver, “Knowledge organization and mathematical problem solving”, in Mathematical Problem Solving Issues in Research, F. K. Lester, J. Garofalo, Eds., Franklin Institute Press, Pennsylvania, 1982), pp.14 – 24. 23. D. W. Carraher, T. N. Carraher, A. D. Schliemann, British Journal of Developmental Psychology 3, 21-29 (1985). 24. F. Brawner, et al., TIMSS-R Philippine Report Volume 2: Mathematics. (Department of Education, Culture, & Sports (DECS), Department of Science and Technology – Science Education Institute (DOST-SEI), & University of the Philippines National Institute of Science and Mathematics Education (UP-NISMED), Manila, 2000). 25. C. P. Vistro, doctoral dissertation, University of Georgia (1991); Dissertation Abstracts International, 52, 6, 2059A, DA9133542 (1991). 26. M. D. Ibe, “The First Licensure Examination for Teachers (LET): Implications for Teacher Education Initiatives”, in Proceedings of the MATHTED ’97 Conference, (De La Salle University, Manila, 1997), pp. 41 – 57. 27. M. D. Ibe, “The Scenario in Mathematics Teacher Education. In the Proceedings of the Conference”, in Strengthening Collaboration in Mathematics Teacher Education, (Ateneo de Manila University, Quezon City, 1995), pp. 11 – 29. 28. M. D. Ibe, “An Analysis of Examinee Performance in the 1991 PBET”, (University of the Philippines, Diliman, 1991). 29. T. F. Green, The Activities of Teaching, (McGraw-Hill, NY, 1971). Acknowledgements This paper was taken from a research funded by the Celestino M. Dizon Endowed Professorial Chair and the Senator Gil J. Puyat Endowed Professorial Chair, Ateneo de Manila University, 2000 – 2001. The author also wishes to acknowledge Ms. Josephine Chua of the Ateneo Mathematics Department for her invaluable assistance and insights.