TEACHERS’ BELIEFS AND KNOWLEDGE ABOUT THE PLACE VALUE SYSTEM

Janne Fauskanger and Reidar Mosvold

University of Stavanger

This paper describes the foundations and considerations of a research project that has just been started. The focus is on the beliefs and knowledge (CKT-M) of 1st and 2nd grade teachers, but the long term goal is to improve in-service teacher education.

INTRODUCTION

Teaching is a complex affair, and the factors that influence the teaching of mathematics are many. Teachers’ beliefs and knowledge about the subject are two important factors of influence. In this proposed study we focus on teachers of 1st and 2nd grade (primary school).

As teacher educators we are working in both pre-service and in-service teacher education. As a new curriculum reform - Knowledge Promotion (UFD, 2005) - is now being put into action, teachers, schools and local governments are asking for in-service education. Our aim is to investigate and try to identify features of the best in-service education. A sensible starting point in such a project is to learn more about the teachers’ knowledge and beliefs (Ball, Hill, & Bass, 2005; Jacobs & Morita, 2002; Pehkonen, 2003).

Our plans are to divide the proposed project into two parts:

1. Research into teachers’ knowledge and beliefs

2. A working model for in-service education

This paper describes the introductory phase of the project, and we focus on the plans, research questions, foundations and considerations of our research project.

Richardson (1996) presumes that the knowledge that student teachers acquired before they enter a course influences what they learn and how they learn it. There is no reason to believe that this is not also true for teachers as well. This emphasises the importance of learning to know teachers before they enter in-service education. Our preliminary research questions are:

1. What are the 1st and 2nd grade teachers’ expressed CKT-M about the place value system?

2. What are the 1st and 2nd grade teachers’ professed beliefs about effective mathematics teaching?

THEORETICAL LENS

Teachers’ beliefs

Many aspects influence teaching, and researchers agree that the teachers’ beliefs have an impact (Leder et al., 2002). Still, there is a lack of agreement on how to define the concept. The suggested definitions are many, and researchers often end up describing belief systems rather than beliefs (Furinghetti & Pehkonen, 2002; McLeod & McLeod, 2002). It appears that each individual has a system of beliefs. This system of beliefs is complex and non-trivial for outsiders to grasp. Each individual continuously tries to maintain the equilibrium of their belief systems (Andrews & Hatch, 2000). Pajares (1992) suggests that beliefs are filters through which our experiences are interpreted, and this relates to Scheffler’s (1965) idea about beliefs as dispositions to act in certain ways. Based on these ideas, the suggestion about possible inconsistencies between beliefs and practice has arisen. A teacher might claim to have a problem solving view on mathematics whereas his classroom practice displays a strong emphasis on procedural knowledge. The researcher might then claim that this represents an inconsistency between the teachers’ beliefs and practices (cf. Cooney, 1985; Raymond, 1997). In response to this, Leatham (2006) presented an alternative framework for viewing teachers’ beliefs as sensible systems.

Within a sociocultural framework, knowledge is believed to be a social construct. Beliefs, on the other hand, appear to be an individual construct (Op’t Eynde et al., 1999). We might therefore say that beliefs are subjective knowledge and include affective factors. Others have tried to exclude the emotional aspect from beliefs, and suggest that a single belief might be associated with an emotion (Hannula, 2004). Various factors like these make it hard to study beliefs. Teachers might not always be conscious of their beliefs, and they might also try to hide their beliefs in cases where these do not appear to fit the researchers’ expectations. In Mosvold (2006), for instance, a teacher replied in a questionnaire that he often emphasised real-life connections in his mathematics teaching. In the following interview, though, he expressed a completely opposite view.

Complete agreement about a single definition of beliefs is probably neither possible nor desirable, but it is important for a researcher to be aware of the various aspects of the concept (Mcleod & McLeod, 2002).

Teachers’ Knowledge

The Norwegian Ministry of Education and Research (KD, 2006) draws attention to the fact that Norwegian teachers’ education in mathematics and mathematics education is below the international average, and that the mathematics teachers to a strikingly small extent participate in relevant in-service education (UFD, 2005; Grønmo et al., 2004). Research from the last 15 years shows that teachers (from the U.S.) do not know enough mathematics, and the students in consequence do not learn enough (Ball, Hill, & Bass, 2005; Ma, 1999).

When analysing 700 1st and 3rd grade teachers (and almost 3000 students), researchers found that teachers’ knowledge have an effect on students’ knowledge (Hill, Rowan, & Ball, 2005). Stigler and Hiebert (1999, p. 10) decrease the impact of this statement by saying: “Although variability in competence is certainly visible in the videos we collected, such differences are dwarfed by the differences in teaching methods that we see across cultures”. But even though research indicates that teachers’ knowledge has a positive influence on students’ learning, it is not obvious what the content of this knowledge is. There are also no clear guidelines about what to focus on in in-service education.

Formerly one thought that if the teachers knew enough mathematics, their teaching would be good and their students would learn mathematics. The content of in-service education then became purely mathematical (Cooney, 1999). On the other extreme, there appeared to be consensus in Norway that it was possible to become a good mathematics teacher without knowing any mathematics at all.

Begle (1968) and Eisenberg (1977) have called teacher educators’ attention to the notion that effective teaching is about more than the teachers’ mathematical competence. Shulman (1986) addresses four questions, one of which is “what are the sources of the knowledge base for teaching?” He tries to put teacher knowledge into category headings. Content knowledge, pedagogical content knowledge, and curricular knowledge, pointed to the fact that mathematical knowledge alone does not automatically transfer into better teaching. Teaching methods makes a difference (Stigler & Hiebert, 1999).

Researchers such as Ball (Hill, Rowan, & Ball, 2005) have based their work on Shulman’s, and tried to identify and specify the mathematical knowledge they think a teacher needs. They call this mathematical content knowledge for teaching (CKT-M), and it is defined as:

(…) the mathematical knowledge used to carry out the work of teaching mathematics. Examples of this “work of teaching” includes explaining terms and concepts to students, interpreting students’ statements and solutions, judging and correcting textbook treatments of particular topics, using representations accurately in the classroom, and providing students with examples of mathematical concepts, algorithms, or proofs (Hill, Rowan, & Ball, 2005, p. 373).

The research connected with CKT-M suggests that teachers in the lowest third of the distribution of knowledge may benefit most from in-service education. Efforts to recruit teachers into in-service education might focus most heavily on those with weak CKT-M. We need ways to differentiate and select teachers, and we need to know more about teachers’ CKT-M in order to use the in-service time more efficiently (Ball & Bass, 2002; Da Ponte & Chapman, 2006).

DISCUSSION OF METHODS

In this study, data will be gathered in two phases. First, we will conduct a survey in order to identify some of the teachers’ beliefs and knowledge. The part of the questionnaire which relates to beliefs has been inspired by a Finnish study of beliefs (Hannula et al., 2005), whereas the part concerning knowledge will be developed during the fall of 2007 in cooperation with a representative of Deborah Ball’s team from the University of Michigan. This phase will not be discussed further in this paper. The second phase of the data collection has been inspired by the study of Jacobs and Morita (2002), and we are going to investigate the teachers’ beliefs and knowledge through their own analysis of videotaped lessons. The teachers will be shown two different videotaped lessons and they will be asked to compare and contrast the lessons. Unlike Jacobs and Morita, our intention is not to compare teachers’ ideas, but rather to learn more about the teachers’ CKT-M and beliefs from what they are saying when commenting on the videos.

The major goal for our project will be to examine teachers’ evaluations of videotaped lessons at a level of detail that facilitates inferences about their underlying “scripts” regarding effective mathematics instruction and their CKT-M.

Methods of data-gathering

The premise upon which this method is based, is that teachers’ opinions can be activated through the process of analysing and criticising a video of an actual lesson.

Several researchers argue that it is best to use indirect tasks that then enable the researcher to make inferences from the data generated (Leinhardt, 1990; Pajares, 1992; Prawat, 1992). Access to teachers’ beliefs is probably best achieved through indirect, rather than direct, questions followed by interpretation, or through evaluation of videotaped mathematics lessons. Wagner (1997) claims that practitioner resistance may be relatively low in low-involvement arrangements. Our plans are to work with a few groups of 1st and 2nd grade teachers in two schools. The videotaped lessons might be from schools using different methods.

Analysing videos from the work of colleagues is a sensitive issue. In order to do this in a productive way, the group must be attuned to each other and have complete confidence. Lesson-study is a good example of a community working together analysing each other’s lessons. The teachers need to be used to doing this regularly, as a practice for professional development.

Research reveals the complexity of mathematical knowledge for teaching, and indicates a shift from regarding mathematical knowledge independent of context to regarding teachers’ knowledge situated in the practice of teaching (Llinares & Krainer, 2006). If teachers’ knowledge is situated in the context of teaching, stimulating teachers to communicate their own “scripts” will be appropriate.

Beforehand, the teachers will be asked to pause the video whenever they want to make a comment about instruction or about the content knowledge presented. They

will be asked to specify whether the comment is going to relate to a “strength”, a “weakness”, and then offer their critical comment. The teacher’s comments will be given while the researcher is present in the room, so that the teacher(s) could ask questions or invite the researcher to a form of conversation. This makes it an unstructured interview according to Bryman (2004). The teachers will be given the opportunity to make summarised comments in the end.

Critical reflection of practice is an essential part of teachers’ learning (cf. Llinares & Krainer, 2006). We are going to let the teachers comment on the videos in groups, to see if this elicits more interesting data than individual comments alone (cf. Jacobs & Morita, 2002).

Researchers are faced with challenges regardless of which methods they choose. Our research is no exception. An important aspect will then be to conduct a pilot study (Bryman, 2004), and that may change some of the methodological assumptions.

Methods of data coding and analysis

Our plan is to approach the data according to “Grounded theory”, just like Jacobs and Morita (2002) did. Grounded theory is defined as theory that was derived from data, systematically gathered and analysed throughout the research process. Following this approach, there is a strong relationship between data collection, analysis, and theory (Strauss & Corbin, 1998).

The teachers’ comments will be recorded, transcribed and classified into idea units in two different columns: beliefs and CKT-M. The idea units will be sorted into a hierarchy of categories derived from the data. Since we are two researchers working together, we can both classify the data and see if inter-coder reliability in relation to the idea units and the categories will be represented (Bryman, 2004).

We will carefully study teachers’ idea units in an open coding process until we are able to produce broad and meaningful categories. Jacobs and Morita (2002) yielded eight main categories that were not mutually exclusive, into which any idea unit was coded. The final stage employed was axial coding. In axial coding one puts the data back together in new ways after the open coding, by making connections between categories and is done in order to produce a grounded theory about the teachers’ evaluations (Strauss & Corbin, in Jacobs & Morita, 2002). We will immerse into the data and let some features and patterns emerge and be crystallised as following loosely a grounded theory approach to the analysis of data (Bryman, 2004; Kvale, 1996).

Grounded theory is not without its limitations (Bryman, 2004), and those limitations may also restrict our analysis, and thereby our results. We hope that two researchers working together will be useful in this connection. We will also invite the teachers to read our interpretations and comment on them as a process of validation.

Contributions with respect to previous research

There is a lack of research on in-service education in Norway, and there has been little or no research with a focus on the knowledge and beliefs of 1st and 2nd grade teachers. It is in this area that our research will hopefully make some important contributions.

REFERENCES

Alseth, B., Brekke, G., & Breiteig, T. (2003). Endring og utvikling ved R97 som bakgrunn for videre planlegging og justering: matematikkfaget som kasus. Notodden: Telemarksforsking-Notodden.

Andrews, P., & Hatch, G. (2000). A comparison of Hungarian and English teachers' conceptions of mathematics and its teaching. Educational Studies in Mathematics, 43(1), 31-64.

Ball, D. L., & Bass, H. (2002). Toward a Practice-Based Theory of Mathematical Knowledge for Teaching. Paper presented at the 2002 Annual Meeting of the Canadian Mathematics Education Study Group, Edmonton.

Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing Mathematics for Teaching. Who Knows Mathematics Well Enough To Teach third Grade, and How Can We Decide? American Educator (Fall 2005), 14-17+20-22+43-46.

Begle, E. (1968). Curriculum research in mathematics. In H. Klausmeier & G. O’Hern (Eds.), Research and Development toward the Improvement of Education (pp. 44-48). Madisom, WI: Dembar Educational Research Services.

Bryman, A. (2004). Social Research Methods (2. ed.). Oxford: University Press.

Cestari, M. L., Santagata, R., & Hood, G. (2004). Teachers Learning from Videos. In B. Clarke, D. M. Clarke, G. Emanuelsson, B. Johansson, D. V. Lambdin, F. K. Lester, A. Wallby & K. Wallby (Eds.), International Perspectives og Learning and Teaching Mathematics (pp. 489-502). Göteborg: National Center for Mathematics Education.

Cooney, T. J. (1985). A beginning teacher's view of problem solving. Journal for Research in Mathematics Education, 16, 324-336.

Cooney, T. J. (1999). Conceptualizing teachers' ways of knowing. Educational Studies in Mathematics(38), 163-187.

Da Ponte, P. J., & Chapman, O. (2006). Mathematics teachers' knowledge and practices. In A. Gutiérrez & P. Boero (Eds.), Handbook of Research on Psycology of Mathematics Education (pp. 461-493). Rotterdam: Sense Publishers.

Eisenberg, T. (1977). Begle revisited: Teacher knowledge and student achievement in algebra. Journal of Research in Mathematics Education(8), 216-222.

Furinghetti, F., & Pehkonen, E. (2002). Rethinking characterizations of beliefs. In E. Pehkonen, G. Törner & G. C. Leder (Eds.), Beliefs: A Hidden Variable in Mathematics Education? (pp. 39-57). Dordrecht: Kluwer Academic Publishers.

Grønmo, L. S., Bergem, O. K., Kjærnsli, M., Lie, S., & Turmo, A. (2004). Hva i all verden har skjedd i realfagene: norske elevers prestasjoner i matematikk og naturfag i TIMSS 2003. Oslo: Institutt for lærerutdanning og skoleutvikling, Universitetet i Oslo.

Hannula, M. S. (2004). Affect in mathematical thinking and learning. University of Turku, Åbo.

Hannula, M.S., Kaasila, R., Laine A., & Pehkonen, E. (2005). Structure and typical profiles of elementary teacher students’ view of mathematics. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education Vol 3 (pp. 41-48). Melbourne: PME.

Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers' Mathematical Knowledge for Teaching on Student Achievement. American Educational Research Journal, 42(2), 371-406.

Jacobs, J. K., & Morita, E. (2002). Japanese and American Teachers' Evaluations of Videotaped Mathematics Lessons. Journal for Research in Mathematics Education, 33(3), 154-175.

KD. (2006). Et felles løft for realfagene. Strategi for styrking av realfagene 2006-2009. Kunnskapsdepartementet.

Kvale, S. (1996). InterViews. An Introduction to Qualitative Research Interviewing. London: SAGE Publications.

Leatham, K. R. (2006). Viewing Mathematics Teachers' Beliefs as Sensible Systems. Journal of Mathematics Teacher Education, 9(1), 91-102.

Leder, G.; Pehkonen, E., & Törner, G. (Eds.) (2002). Beliefs: a hidden variable in mathematics education? Dordrecht: Kluwer Academic Publishers.

Leinhardt, G. (1990). Capturing craft knowledge in teaching. Educational Researcher(19), 18-25.

Llinares, S., & Krainer, K. (2006). MATHEMATICS (STUDENT) TEACHERS AND TEACHER EDUCATORS AS LEARNERS. In A. Gutiérres & P. Boero (Eds.), Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future (pp. 429-459). Rotterdam: Sense Publishers.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, New Jersey: Lawrence Erlbaum Associates.

McLeod, D. B., & McLeod, S. H. (2002). Synthesis - Beliefs and mathematics education: implications for learning, teaching, and research. In E. Pehkonen, G.

Törner & G. C. Leder (Eds.), Beliefs: A Hidden Variable in Mathematics Education? (pp. 115-123). Dordrecht: Kluwer Academic Publishers.

Mosvold, R. (2006). Mathematics in everyday life. A study of beliefs and actions. Unpublished The degree doctor philosophiae, University of Bergen, Bergen.

Op't Eynde, P., De Corte, E., & Verschaffel, L. (1999). Balancing between Cognition and Affect: Students' Mathematics-Related Beliefs and Their Emotions during Problem-Solving. Paper presented at the Conference on Mathematics Beliefs and Their Impact on Teaching and Learning of Mathematics, Mathematical Research Institute, Oberwolfach, Germany.

Pajares, M. F. (1992). Teachers' Beliefs and Educational Research: Cleaning up a Messy Construct. Review of Educational Research, 62(3), 307-332.

Pehkonen, E. (2003). Lærere og elevers oppfatninger som en skjult faktor i matematikkundervisningen. In B. Grevholm (Ed.), Matematikk for skolen (pp. 154-184). Bergen: Fagbokforlaget.

Prawat, R. S. (1992). Teachers' beliefs about teaching and learning. American Journal of Education(100), 354-395.

Raymond, A. M. (1997). Inconsistency between a Beginning Elementary School Teacher's Mathematics Beliefs and Teaching Practice. Journal for Research in Mathematics Education, 28(5), 550-576.

Richardson, V. (1996). The role of attitudes and beliefs in learning to teach. In J. Sikula, T. Buttery & E. Guyton (Eds.), Handbook of research on teacher education (pp. 102-119). New York, USA: Macmillan.

Scheffler, I. (1965). Conditions of knowledge. Chicago: Scott, Foresman.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher(15), 4-14.

Stiegler, J. W., & Hiebert, J. (1999). The Teaching Gap. Best ideas from the world's teachers for improwing education in the classroom. New York: The Free Press.

Strauss, A., & Corbin, J. M. (1998). Basics of Qualitative Research: Techniques and Procedures for Developing Grounded Theory (2. ed.). Thousand Oaks, California: Sage.

UFD. (2005). Kunnskapsløftet. Utdannings- og forskningsdepartementet.

Wagner, J. (1997). The Unavoidable Intervention of Educational Research: A Framework for Reconsidering Researcher-Practitioner Cooperation. Educational Researcher, 26(7), 13-32.