On Teachers’ Beliefs and Changing Mathematics Teaching

177

Erkki K. Pehkonen

On Teachers' Beliefs and Changing Mathematics Teaching

Summary:

The paper deals with the problem of changing mathematics teaching within the frame

work of teachers' belieft. Firstly, the concept of belief is discussed briefly, e.g. distinc

tions between knowledge and belieft are explained Also, the central role of belieft for

teaching mathematics is considered Secondly, the problems involved helping teachers

to change their teaching are discussed, and some models describing teacher change are

dealt with. In both parts of the paper, a variaty of research papers are presented In

addition, some open questions concerning belieft are discussed

Introduction

The conception of good teaching changes with the view of a human being. Since the

ideal view of a human being is changing with time, the only sure and constant matter

seems to be the change. But the direction in which the change is moving - let us say

within the next twenty years - and how it will happen, cannot be estimated exacdy.

Thus, there cannot be any universally valid "recipe" for good teaching. In practice,

teaching situations should be confronted as new ones, i.e. it is not possible to predict

exacdy how the lesson will function. Therefore, the solutions used may be changed and focussed until the teaching in the classroom really happens, and so the purpose of

teaching "pupils' learning" should guide teachers' decisions. Actually, teaching may be

compared with creative problem-solving (Schultz 1991), where the knowledge one has

should be applied (usually in a new way) creatively and purposefully. Teaching is some

thing one cannot plan exact1y beforehand.

It is impossible to prepare teaching material which may be successfully used by every

one. Even the most creative and varying textbook can be used in such a way that pupils

will consider mathematics boring and dul!. On the other hand, a competent teacher with insight and courage can use any material with good results, and his pupils will be fascinated by mathematics. Of course, it is worthwhile to develop good materials, and to

check their applicability through research (cf. Zech & Wellenreuther 1992). In this type

of research, the methodology of action research seems to be appropriate, but it is not

realistic to anticipate that deve10ped materials will be omnipotent.

Therefore, a competent teacher who is engaged in using different instructional methods

and has the ability for innovative teaching seems to be the only factor which may cause

an instructional change. The crux of the problem is how we can he1p our teachers and

prospective teachers to become reflective and innovative. Here, beliefs seem to be suited

to deal with problems the teacher encounters, many of which are ill-defined and deeply

entangled (Nespor 1987).

In the following, we will consider the problem of changing teaching, especially in the

(JMD 15 (94) 3/4, S. 177-209)

178 Erkki K. Pehkonen

framework of teachers' beliefs. In addition, we will restriet ourselves here to the case of mathematics teachers; teachers' beliefs has been dealt with more generally e.g. by

Nespor (1987) and Pajares (1992). Insofar as we consult literature we restrict ourselves

mainly on American research reports.

1. On the basic concepts Research has revealed that knowing the right facts, i.e. algorithms and procedures, does

not necessarily guarantee success in solving mathematical problems. There are other

factors - such as decisions made by the solver and the strategies he uses, as weIl as his

emotional state at the time he is solving mathematical tasks - which have a major effect

on the performance of a solver (e.g. Schoenfeld 1985, Garofalo 1989). "Purely cogni

tive" behavior is rare. Belief systems shape cognition, even though some people may not

be consciously aware of their beliefs (Schoenfeld 1985). Furthermore, Lerman (1983)

pointed out that the teachers' philosophy (or the view) of mathematics has influences

and shapes their teaching practice.

During this century, beliefs and belief systems were to some extend examined in the

beginning of the century, mainly in social psychology (Thompson 1992). But shortly

after that behaviorism spread to the research in the psychological domains. Then the

focus was on the observational parts of human behavior, and beliefs were nearly forgot

ten. New interest in beliefs and belief systems emerged mainly in the 1970's, through the

developments in cognitive science (Abelson 1979). Some papers on mathematics educa

tion described, with "horror stories", how teachers or pupils were not able to find the

right ans wer or method (e.g. Schoenfeld 1983, Graeber & al. 1986), or how they might

have fantasy ideas about mathematical problem solving (e.g. Erlwanger 1975).

All accept that a teacher's knowledge and skills - both mathematical and pedagogical -

are the most important determinants affecting the quality ofhis teaching. But when Fen

nema & Franke (1992) discussed the meaning of teachers' knowledge, they pointed out

that one could not separate the impact of beliefs from knowledge. Although beliefs are

popular as a topic of study, the theoretical concept of "belief' has not yet been dealt

with thoroughly. The main difficulty has been the inability to distinguish beliefs from

knowledge, and the question is still unclarified (e.g. Abelson 1979, Thompson 1992). In

order to show the problems, we will discuss here briefly both these basic concepts.

1.1. What are beliefs? An individual continuously receives perceptions from the world around hirn. According

to his experiences and perceptions, he makes conclusions about different phenomena

and their nature. The individual's personal knowledge, i.e. his beliefs, is compound of

these conclusions. Furthermore, he compares these beliefs with his new experiences and

with the beliefs of other individuals, and thus his beliefs are under continuous evalua

tion and change. When he adopts a new belief, this will automatically form a part of the

Teachers' beliefs 179

larger structure of his personal knowledge, of his belief system, since beliefs never appear fully independently. Thus, the individual's belief system is a compound of his conscious or unconscious beliefs, hypotheses or expectations and their combinations which he has earlier adopted. (Green 1971)

Different conceptions of beliefs

With beliefs, one means different matters depending on the discipline and researchers

who will deal with them. Beliefs are considered equal e.g. to concepts, meanings, propositions, rules, preferences, and mental images (Thompson 1992). In everyday langua

ge, the word "belief' may be used in many different ways. It could express on the one .

hand faith and conviction, and on the other hand opinion and doubt or hesitation (Jones

1990).

Beliefs are in near connection with schemata, which refer to the construction of know

ledge in human memory. They represent a mental model with which an individual or

ganizes information (McDonald 1989). With the aid of schemata, an individual can re

cognize environmental aspects, and operate with them. Thus, schemata dominate an in

dividual's interaction with his environment (Marshall 1989). These schemata may con

tain information from the past which is often unconscious. In connection with them,

there are attitudes, beliefs and emotions which influence an individual's expectations in future situations (McDonald 1989). The meaning of schemata will be emphasized in

problem solving situations, where reactions are expected from the individual. In these

situations, the schemata act as tools with which the individual constructs these reactions

(Marshall 1989).

There are different contents for the concepts "belief' and "belief system" used in studies

in the field of mathematics education. As a consequence of the vague definition of the concept, many researchers have formulated their own definition for "belief'. For exarn

pIe, Schoenfeld states, in order to give a first rough impression, that "belief systems are

one's mathematical world view" (Schoenfeld 1985). He later modifies his definition, interpreting beliefs as an individual's understandings and feelings that shape the way that

the individual conceptualizes and engages in mathematical behavior (Schoenfeld 1992).

Hart (1989) uses the word belief to represent a certain type of assessment pertaining to a

group of conceptions. Lester & al. (1989) explain that "beliefs constitute the individual's

subjective knowledge about self, mathematics, problem solving, and the topics dealt

with in problem statements". Whereas Thompson (1992) understands beliefs as a

subclass of conceptions. Yet another different explanation is given by Bassarear (1989)

who sees attitudes and beliefs on the opposite poles of a bipolar dimension.


A working definition for belief Here, we understand beliefs1 as one's stable subjective knowledge of a certain object or

concern to which tenable ground may not always be found in objective considerations.

The reasons why a belief is adopted are defined by the individual self - usually uncon

sciously. The adoption of a belief may be based on some generally known facts (and

beliefs) and on logical conclusions made from them. But each time, the individual makes his own choice of the facts (and beliefs) to be used as reasons and his own eva

luation on the acceptability of the belief in question. Thus, a belief, in addition to knowledge, also always contains an affective dimension. This dimension influences the role and meaning of each belief in the individual's belief structure.

As earlier stated, an individual's beliefs are usually in connection with each other. Some

beliefs depend on the other, for the individual, more important beliefs. Thus, they form

different belief systems which might be in connection with other belief systems. The notion of belief system is a metaphor used for describing how one's beliefs are organi

zed (Green 1971; see also Rokeach 1968).

Beliefs and belief systems are affected by the way people understand themselves and their environments. In his study, Saari (1983) tried to structure the central concepts of the affective domain. He grouped them into three categories: feelings, belief systems,

and optional behavior. Belief systems can be seen to be developed from simple percep

tual beliefs or authority beliefs - via new beliefs, expectations, conceptions, opinions

and convictions - to a general conception of life. In his structure of concepts in the af

fective domain, Saari understands e.g. attitude as a component-structured concept which has a component on each of the three dimensions: feelings, belief systems and optional behavior. In that sense, we understand that beliefs form a component of attitudes. In

accordance to Saari (1983), we may explain conceptions as conscious beliefs, i.e. we

understand conceptions as a sub set of beliefs. Conceptions are higher order beliefs which are based on such reasoning processes for which the premises are conscious.

Therefore, there seems to be a basis for conceptions, at least they are justified and

accepted by the person himself.

1.2. What is knowledge?

The concept of knowledge will be discussed very briefly here stressing its near con

nections with beliefs. Those interested in a broader discussion may refer to the litera

ture, e.g. Goldin (1990), Steffe (1990), Fennema & Franke (1992).

According to the c1assical definition "knowledge is a well reasoned true belief', i.e. all

knowledge (also scientific) is based on beliefs. Apremise is that all the beliefs which

I For the concept "belief', one may find several different translations into German. For example, the following translations were found in the International Review ofMathematical Education (ZDM-journal): Einschätzung, Einstellung, Meinung, Sichtweise, Überzeugung, Vorstellung (in alphabetical order). Tbe concept "belief' has been discussed by Bernd Zimmermann in his habilitation work (1991), especially from the view point ofGerman speaking countries.


fonn the basis are logically true and justified, in the sense that the other facts in the phenomenon world speak for them. Thus, beliefs are individuals' subjective knowledge which expressed as sentences might be (or might not be) logically true. Knowledge has,

instead ofthat, always this property (Lester & a1. 1989). One reason for the difficulty to distinguish beliefs and knowledge is the relativity of

knowledge, in the sense that knowledge is historically changing. It means that our con

ception ofknowledge is changing all the time. For example in the 1700's, one generally

accepted knowledge among the mathematicians was that all infmite series with the limit

zero of the general tenn are convergent. This belief was rejected as a knowledge when

the well-known counterexample L(lIn) was found at the end of that century, and in

consequence, the theory of infinite series was developed. Another example is the very

general belief nowadays that boys are on the average more talented in mathematics than girls. Though research, however, has not given any support to this belief, the belief is

nevertheless still very strong.

Teachers' knowledge When one discusses the issue of a mathematics teacher's knowledge, mathematics itself

is automatically present. Among mathematics educators, one may find strong be1iefs

ab out the importance of mathematical knowledge to teachers. The mathematicians usually share this belief. The lack of teachers' mathematical knowledge is often used as an explanation for their pupils' low achievement in mathematics.

Knowledge of

mathcmatics

Knowledge of learners'

cognitions in mathematics

Pedagogical knowledge

Fig. 1.1. A model on teachers' knowledge (Fennema & Franke 1992).


Teachers' knowledge of mathematics teaching indudes their knowledge of mathematics and pedagogy, as weIl an understanding of pupils' cognitions. Always underlying

teachers' knowledges are their beliefs. The triangle in the center indicates teachers'

knowledges and beliefs in context (or as situated).

1.3. Comparing beliefs and knowledge

When discussing teachers' beliefs in the light of research results, Thompson (1992) also

states that distinctions between knowledge and beliefs are fuzzy (Fig. 1.2), because of

their dose connections. Skemp (1979) tried to dear up the problem of distinguishing

between knowledge and beliefs, as follows: "Knowledge is the name we give to concep

tual structures built from and tested against our own experiences of actuality. Beliefs are

what we have accepted as facts for other reasons. These are frequently used in combi

nation as the basis for the functioning of a director system."

BELIEFS

Fig. 1.2. Distinctions between knowledge and beliefs are fuzzy.

Properties of belief systems

In order to solve the problem of distinguishing knowledge and beliefs, some structural

differences between belief systems and knowledge systems have been noticed. For example, Rokeach (1968) organized beliefs along a dimension of centrality to the indivi

dual. The beliefs that are most central are those on which there is a complete consensus;

beliefs about which there is some disagreement would be less central. Whereas Green

(1971) discusses three dimensions of belief systems: quasi-logicalness, psychological

centrality, and cluster structure, which will be considered here more closely.

Quasi-logicalness. Knowledge systems are formed logically from premises and from

conclusions deduced from them. Whereas, the relationships between beliefs cannot be

said to be logical, since beliefs are arranged according to how the believer hirns elf sees

the connections between them. In other words, each person has in his belief system a

structure which can be called quasi-logical, with some primary beliefs and some deriva

tive beliefs. This quasi-logical order is unique for each person.

Also Abelson (1979) pointed to this lack of logic in belief systems: Within a belief sy

stem, beliefs are not necessarily held in consensus with other beliefs. Therefore, one

could have beliefs which contradict other beliefs held by the same person at the same


time. Furthennore, the believer is usually aware that others may have different beliefs. Whereas, one important feature of knowledge systems is that it cannot contain contradictions.

Psychological centrality. The dimension of psychological centrality is lacking in knowledge systems. One cannot say that somebody knows a topic strongly. But beliefs have

their own psychological strength, i.e. the degree of conviction with which beliefs are

held. Beliefs can be held with varying degrees of conviction. The most central beliefs

are held most strongly, whereas the peripheral ones may be changed more easily

(compare with Rokeach' s concept of centrality) ..

Cluster structure. Beliefs are held in clusters which are not necessarily connected with

each other. "Nobody holds a belief in total independence of all other beliefs. Beliefs al

ways occur in sets or groups." (Green, 1971) This cluster structure enables the individual even to hold conflicting beliefs within his own belief system (cf. quasi-logicalness).

The clustering property may help to explain some inconsistencies found in an indivi

dual's belief system.

In addition to the cluster structure, Abelson (1979) pointed out that belief systems e.g. rely heavily on evaluative and affective components. A belief system typically has ex

tensive categories of concepts which are grouped into "good" and "bad". As a typical

example, those who support so-called "green values", also usually believe that nuclear

power is bad, materialism and waste are bad, natural alternative energy sources are good, re-cycling is good etc. Knowledge systems are lacking such evaluations.

Finally, it should be noticed that not all researchers are taking the problem described so

seriously. Some have argued that it is not important to distinguish between knowledge

and beliefs, but rather to find out how belieflknowledge systems influence teachers' behavior in mathematics classes (Thompson 1992).

2. Teachers' mathematical beliefs

Concerning teachers' mathematical beliefs, one should note that it is only a label for a

great variety of beliefs. Usually, one classifies teachers' mathematical beliefs in four

groups (e.g. Underhill 1988): beliefs about the nature of mathematics, beliefs ab out

teaching mathematics, beliefs about learning mathematics, and beliefs about oneself in a

social context.

However, there are many other beliefs of importance which are connected with those

mentioned. For example, there are the teacher's beliefs ab out teaching and what it is

appropriate to do in the classroom, about schooling and its purpose in society, about

learners, their abilities, and their expectations of the teacher, about hirnself as a teacher,

the amount of authority he has, and the roles he must play, and about a host of personal

and professional factors that may influence his teaching.

As an example of teachers' mathematical beliefs, we will consider the research results of


Alba Thompson. She condensed teachers' beliefs about problem solving, as fo11ows (Thompson 1989):

"1. It is the answer that counts in mathematics, once one has an answer, the problem

is done.

2. One must get an answer in the right way.

3. An answer to a mathematical question is usually a number.

4. Every context (problem statement) is associated with a unique procedure for "getting" answers.

5. The key to being successful in solving problems is knowing and remembering

what to do."

Besides it seems that traditional mathematics teaching supports e.g. the development of

such beliefs both in pupils and in teachers.

The mathematical beliefs given by Thompson (1989) are very generaiones and pos

sessed by many teachers. If we want to understand a teacher's behavior and classroom

practices we should investigate his beliefs in depth. Such an investigation is done e.g. in the dissertation of Iones (1990).

2.1. A teacher's mathematical belief system In his dissertation, Iones (1990) investigated the belief systems oftwo female mathema

tics teachers (Darla and lodi) in amiddie school. He focused particularly on their beliefs

conceming mathematics, themselves as teachers, and teaching mathematics, and he

wanted to sketch a model of a teacher's belief system within the framework of Green's

theory (1971). As research methods, he used observations in schools, about two weeks at a time, interviews conceming the teachers' mathematics teaching, and interviews

which were not explicitely connected with their school practice. In the following, we

will discuss his model of Darla's belief system about teaching mathematics as an exampIe of the earlier theoretical considerations.

Iones (1990) found four major themes in Darla's belief system about mathematics

teaching: Interrelatedness ("If you leave out one thing, you're gonna come across some

thing else that's related and you' re gonna have to end up teaching that little bit anyway",

# 1), Different Perspectives ("It's important that they have interactions with their peers in

hopes that they' 11 get a different perspective of the concept", #2), Organization ("You

have to have an organized way ofpresenting ideas or you lose people", #3), and Think

for Yourself ("If a student has their own thoughts on math and how it's related, chances

are they have a good understanding of it", #4). These themes were not isolated from

each other.

Quasi-Iogical relationships

Besides these four primary themes, there were a number of derivative themes in Darla's

belief system (Fig. 2.1); for example: Applications to Everyday Life (#5) derives from


theme 1 ("I' m always trying to relate things to math - you know, put it in a mathematical sense"), and Closure (#7) derives from a combination of themes 2, 3, and 4 ("I like

to come to a solution", "You have to be very precise in how you set up a problem 'cause the least little thing can throw our answer off', and "Math is open to different ways of

thinking, but those ways ofthinking have to bring you to a certain place").

2 4

INTERREL~TEDNESS J DIFFERENT 3 THINKFOR PERSPECTIVES ORGAMZATION YOURSELF

I I I I "- /1 , '- " 5 10 Applications The Nature of

to Everyday Life Mathematics

7 • • 6 I CI~ure I 8 9 Teacher Must Nature of Dedication

Play Many Roles Teaching to the Job Mathematics ,

11 The Structure

of Middle School Mathematics

~ ~ 12 13 14 15

Experience Relationships Help Students Students' Is Not Passive Between Teacher Develop Socially Sense

and Students Making

Fig. 2.1. Quasi-Iogical relationships in a teacher's belief system (Jones 1990)

As another example, the theme that Experience Is Not Passive (#12) is derived from

theme 4 ("In math, you gotta do it to leam it") and theme 2 ("That common sense [in

mathematics] isn' t there because it has to be there; it's there because someone he1ped

them put it there").

Psycbological relatioßships

Next, we will look at what kind of psychological organization Darla's be1iefs have, as

described in Jones's dissertation (1990). In Fig. 2.2, there is Darla's belief system con

cemißg mathematics, viewed from the point ofpsychological centrality.

186

DIFFERENT PERSPECI1VES

The Nature

Erkki K. Pehkonen

of Mathematics INTERRELATEDNESS

GRGANIZATIOY .. Apphcations

The Structure of Middle School Mathematics

Closure

to Everyday Life

Fig. 2.2. Psychological relationships in a teacher's belief system concerning mathematics (Jones 1990).

When considering the psychological relationships within one's beliefs, we may take dif

ferent viewpoints. For example conceming mathematics, the most important belief for

Dada seemed to be Organization. Jones (1990) had also discussed Darla's belief system

from the viewpoint of Self-As-Teacher and Teaching Mathematics. In the former he explored that her most important belief is The Nature of Being a Teacher, and in the latter Different Perspectives.

2.2. The role of heliefs in teaching When considering the theory of constructivism (e.g. Davis & al. 1990) as a basis for the understanding of teaching and learning mathematics, it follows that teachers' and pupils'

mathematical beliefs take on a key role when trying to understand their mathematical

behavior. This is also valid in research: In order to understand the mathematical behavior

in classrooms, we also have to investigate teachers' and pupils' mathematical belief sy

stems (Noddings 1990). When mathematics educators are explaining school instruction

using the constructivist framework, teachers' and pupils' beliefs are necessarily involved

(see e.g. Leder 1992).

The conceptions L~e teacher has about mathematics and its teaching strongly influence

his c1assroom management (Goldin 1990), as weB as the effectivity of his teaching

(Lester 1989). The National Council of Teachers of Mathematics (NCTM 1989) asserts

that teachers play an important part in the formation of pupils' be1iefs ab out mathematics.

Therefore, teachers' be1iefs form a powerful factor in pupils' leaming. In her dissertation,


Martha Frank (1985) introduced a scheme for some factors affecting pupils' problemsolving behavior. Here, her scheme has been reorganized and adapted to the case of a teacher's actions during mathematics lessons (Fig. 2.3).

Beliefs have a central role as a background factor for a teacher's thinking and acting. A teacher's mathematical beliefs act as a filter which deals with almost all his thoughts and actions conceming mathematics. A teacher's prior experiences in mathematics teaching

and leaming, wbich strongly guide his teaching behavior (e.g. through models), fully act

on the level of beliefs - usually unconsciously. When he is using his mathematical and

pedagogical knowledge, bebefs are strongly involved. On the other hand, the teacher's

motivation and needs as a mathematics teacher are not only connected with his mathematical beliefs. For example, a teacher's need to receive a salary for the work done

may affect bis actions in classroom, and is not necessarily connected with his

mathematical beliefs. In addition, there are different factors in the societal environment

affecting the teaching situation, which will set limits for a teacher's actions: Besides

different administrative orders, such as the mathematics syllabus, the number of lessons,

and the lesson-break-cycle, which determine the situation, there are societal

mathematical expectations and myths, e.g. mathematics is calculations (for more myths

A teachetf$ aetions during math lessons

experiences in math

Mathematical and pedagogical

knowled e

A teacher's mathematical heUefs

Administrative orders

as a math teacher

Fig. 2.3. Factors affecting a teacher's mathematical behavior.


The net of factors affecting via beliefs a teacher's mathematical behavior, described in Fig. 2.3, will reveal only a part of the truth. In fact, the situation is more complicated: The teacher functions in a complex net of influences, Underhill (1990) talks ab out a

web of beliefs - there are colleagues, the school principal, the school administration,

math supervisors, teacher educators, and parents who all have their own beliefs about

the nature of mathematics and the nature of learning and teaching mathematics.

2.3. Empirical research on teachers' beliefs

In the literature of research on teaching and teacher education within last fifteen

years,one may see some development or a shift in emphases: "Research has moved from

on analysis of what teachers were to what teachers did to what teachers decide to the

more contemporary emphasis on what teachers believe" (Cooney 1993) The research

has clearly moved away from deterministic methodologies toward more descriptive

ones.

Several researchers have investigated teachers' mathematical and pedagogical know

ledge. In their synthesis of research on teachers' knowledge, Fennema & Franke (1992)

expressed that it is impossible to separate the impact of beliefs from that of knowledge.

In the field of teachers' beliefs, reseachers have introduced a variety of closely related

theoretical notions including teachers' subjective theories (e.g. Tietze 1990) and teachers' beliefs (e.g. Thompson 1984, lones 1990).

During the last decade, several studies on teachers' beliefs were undertaken. In the Uni

ted States, there are tens of dissertations on the topic (see the references e.g. in lones

1990). Furthermore, there are many studies concerning the change of teachers' beliefs

which will be discussed separately later on. Thompson (1992), and earlier Underhill

(1988), have both compiled a review of results done (mainly in the U.S.) on teachers'

beliefs.

In the empirical research done on teachers' beliefs, the methodology has consisted of

interviews and observations, but also of questionnaires - sometirnes combinations of

these. The number of test subjects varies from a few teachers (N smalI) to some tens of

teachers (N large). A great many of the studies are realized in schools, but there are also

studies conducted during teacher pre-service training. The purpose of research works also varies from a general view of beliefs to beliefs about some specific questions. For

example, a wide interest in conceptions about problem solving exists. In the following,

some exarnples of the different types of studies are discussed briefly. Table 2.4 gives an

overview of the studies under discussion, regarding their methodology and sarnple size.

Teachers' beliefs

interviews &

obsetvations

questionnaire

combination

ofboth

N sma11 « 10)

Thompson 1984

Civil1989 Najee-u11ah & a1. 1989

Kaplan 1991

N large (2: 10)

Grouws & a1. 1990

Bottino & al. 1991

Zimmermann 1991

Pehkonen 1993b

Emest 1988

Even 1988

Table 2.4. A classification, concerning the methodology and sampie size of some research work done on teachers' beliefs.

Studies in schools and in-service training

189

General features. In these studies, we may focus on research on a general view of beliefs

(Thompson 1984, Zimmermann 1991). In addition, Zimmermann (1991) tried to connect

pupils' beliefs with their teachers' beliefs.

Thompson (1984) explored mathematical beliefs of three female mathematics teachers

in the middle schoo!. She followed their everyday teaching practice in mathematics for

four weeks and interviewed them regularly. As she investigated connections between the

teachers' explicated beliefs and their everyday teaching practice, she found that the

teachers' beliefs, attitudes and preferences regarding mathematics and its teaching signi

ficantly dominate the form of their teaching behavior.

Using a questionnaire, Zimmermann (1991) gathered the data from 2658 pupils in gra

des 6 - 9 and from their 85 mathematics teachers in a11 different types of school

(Gymnasium, Realschule, Hauptschule, Gesamtschule) in Hamburg. His purpose was to

compare the pupils' beliefs related to mathematics teaching with the responses of their

teachers to the corresponding statements. With the use of cluster analysis, he could ex

tract six groups of pupils and five groups of teachers, and find a certain correspondence

between two of them: problem-oriented and schema-oriented.

Specific questions. Most of the reseach done focussed on some specific questions, e.g.

Kaplan (1991) exarnined the consistency between teachers' beliefs and practices. Fur

thermore, a wide interest in conceptions about problem solving exists (Najee-ullah & al.

1989, Grouws & a1. 1990, Pehkonen 1993b). Whereas Bottino & a1. (1991) wanted to

reveal teachers' conceptions in relation to certain mathematical topics.

The study of Kaplan (1991) exarnined the consistency between the beliefs and practices


of two elementmy mathematics teachers through an analysis of interviews and classroom behaviors. Beliefs and practices were described on two levels, and each level was

coded as empirieist, maturationist, or constructivist. Findings suggest that when defined

in these terms, beliefs are generally consistent with practice, but that surface beliefs tend

to be more consistent with superficial practices while deep beliefs tend to be more consistent with pervasive behaviors.

The study of Najee-ullah & al. (1989) reports evidence of beliefs held by two high

school basic skills mathematics teachers observed while solving mathematical problems. In particular, beliefs ab out attributions of success and failure are related to a variety of

achievement and performance outcomes. The objective of this study was to document

evidence of attributions of success and failure and their relationship to the problem

solving behavior exhibited by these two teachers. Observations revealed that 1)

attributions were made as explanations of their performance and 2) attributions of suc

cess were classified differently with respect to the locus of control dimension of cau

sability while attributions of failure were classified similarly for the stability and controll ability dimensions of causality.

Grouws & al. (1990) interviewed 25 mathematics teachers in junior high school con

cerning their beliefs and teaching practice, with special attention on problem solving.

The teachers could be grouped according to their definition on problem solving into four

classes: Problem solving means (1) a word problem, (2) to find a solution to problems,

(3) to solve practical problems, (4) to solve thinking problems.

The purpose of Pehkonen (1993b) was to clarifY what Finnish teacher educators (N =

42) in mathematics think about problem solving. The gathering of teacher educators'

views was carried out in two stages with the aid of questionnaires. The most important result was as follows: Problem solving is important, since it helps the fostering of pupils'

cognitive readiness, and helps pupils to use the mathematics they have leamed.

Bottino & al. (1991) gathered data during a teacher in-service course (N = 79) in Italy.

With a questionnaire, they aimed at determining teachers' conceptions of 14 - 16 year

old pupils' possibilities to leam certain topics (manipulating with algebraic expressions

and basic geometry) as weIl as of the level to teach them during the first two years of

upper secondmy school (age 14 - 16). The results obtained suggest that teachers' choices

seem to be more affected by pressures from their coIleagues in successive school years

than by educational considerations.

Studies with prospective teachers

There are a few research works on prospective teachers' mathematical beliefs (Emest 1988, Even 1988, CiviI1989).

Emest (1988) explored, with a questionnaire and observations, a group of primary

school student teachers (N == 30) with regard to: their knowledge of mathematics, atti

tude to mathematics, displayed confidence and liking of mathematics teaching, and ap-


proach to mathematics teaching. The mathematics specialists tended to have positive

attitudes to mathematics, and to its teaching, but varied in their approach to teaching

mathematics: Only 40 % adopt a creative, problem solving approach. Students with low

levels of knowledge of mathematics are more varied in their responses. It seems that attitudes to mathematics are less significant for these students than attitudes to teaching

mathematics, which correlate with the teaching approach.

Even (1988) investigated the knowledge and understanding of the relationships between

functions and equations as perceived by prospective secondary mathematics teachers

(N = 152) in the last phase of their professional education. As a method, she used an

open-ended questionnaire, and interviewed about 10 % of the subjects. They were asked

to write adefinition of a function, to indicate how functions and equations are related to

each other, and to find the number of solutions to a quadratic equation, given a positive

value and a negative value of the quadratic expression. The findings suggest that these

students hold a limited view of functions as equations only; they do not have a way of

making sense of the modem definition and lack the ability to relate solutions of equa

tions to values of a corresponding function.

The paper of Civil (1989) deals with pre-service elementary teachers (N = 8). Sources of

data incIuded: tape recordings of the students' in-cIass group work and of task-based in

terviews; written homework problems, essays, their diaries, and in-class observations.

Though the teachers in this study generally knew how to solve the mathematical pro

blems given to them, they had little to say about the validity of different methods. To

understand their pupils' methods of solution should be important to them, but this may

not be so, given their views about teaching mathematics.

3. Teacher change

For many enterprises to reform instruction in school, the typical way of thinking has

been that one can control teaching and learning in schools with regulations outside the

school (i.a. with the curriculum). For example, the change in the school system in Fin

land in the early 1970s, from a parallel school system to a comprehensive school, was

realized as an administrative renewal in which individual teachers would have no pos

sibility to influence anything. As another example, one may mention the enterprise to

influence the realization of the curriculum (teaching) in the 1980s through textbooks.

They were so ready-made that teachers lost the opportunity to control their own work

and apply their teaching skills. (cf. also Apple & Jungck 1990) Today, such develop

mental solutions are being imposed where the initiatives for change, at least some ex

tent, are in the hands of practicing teachers - the so-called bottom-up method (e.g.

Schultz 1991, Shaw & aJ. 1991, Silver 1991).

One of the most up-to-date fields of emphasis in the research of mathematics education

has been the exploration of teachers' beliefs and of possibilities to change them


(Thompson 1992). Within the PME (Psychology of Mathematics Education) conferen

ces, "1987 marked the first time interest in teacher beliefs in relation to curriculum in

novation was specifically noted, when teachers were viewed as potential obstacIes to in

novation, as - something - to take into account and to be changed" (Hoyles 1992). And

through 100king at the PME proceedings, one may realize that, from that year, there

have been an extensive number of research papers about teacher change in the PME

conferences.

However, a suspicious question has also been put forward: "Is it at all possible to chan

ge teachers' beliefs?" (Thompson 1991). That question has remained yet unanswered.

Actually, we may ask as Mason has (1991): "Is it ethically right to change people?"

Therefore, it is more appropriate to speak only about teacher change which will be un

derstood as teachers' voluntariness to change. Thus, we as teacher educators are only

"offering teachers opportunities for change" instead of changing teachers.

3.1. Surface beliefs vs. deep beliefs Teachers' conceptions of "good mathematics teaching" have been so deep rooted that

surface changes - as changing outer conditions (i.a. curriculum, teaching materials) -

cannot influence them. If a teacher is compelled to undergo a change, he will adapt to

the new curriculum, e.g. by interpreting his teaching in a new way2, and absorb some of

the ideas of the new teaching material into his old style of teaching. In fact, there seems

to be a gap between teachers' expressed beliefs and their teaching practices (lones & al.

1986). For example, a teacher may express the belief that exploring mathematical situa

tions is more important than rote practice, but nevertheless assign about 50 exercises for

work during cIass (Shaw 1989). Another teacher may believe that he is allowing pupils'

ideas to guide cIassroom discourse, but in reality he will only recognize those ideas

which fit into his prepared plan.

When aiming a change also in a teacher's practices, one should get into the deep level of

teachers' beliefs (cf. Kaplan 1991) about mathematics and its teaching and learning, and

not to be satisfied with surface changes in beliefs. Or talking with the language of Green

(1971): We hope that teachers are not satisfied in changing only their peripheral beliefs

about mathematics te ac hing, but they are also ready to change some central beliefs, if

necessary.

U sually in developing teaching, teachers' deep beliefs about mathematics te ac hing have

not been taken into account, and these deep beliefs are factually guiding teachers' ac

tions (e.g. Kaplan 1991). However, many methods of developing teaching try to change

the teachers immediately. It has been thought that if teachers could see, understand, and

intemalize the need for change and if methods to realize the change were provided, the

curriculum and teaching materials would not form any obstacIes. But only lecturing

2 E.g. in Victoria IAustralia, when the teaching ofproblem-solving was made obligatory in the curriculum, teachers interpreted most of their earlier routine tasks to be problems (Stacey 1991).


about the need for change and demonstrating the methods to realize it are not enough,

since we are then still on the surface level ofbeliefs.

3.2. On the conditions for teacher change

In order to reach a change, one should keep an eye on several factors. For example, the

experiences showed that the support given to teachers in their schools was essential for

teacher change. It was important to visit other c1assrooms and discuss observations

made with colleagues, in order to achieve a change of beliefs on the practical level. It

also seemed to be important to organize such situations in which teachers could reflect

on their thinking and actions (Schultz 1991). Furthermore, research has shown that the

change in schools should be dealt with more as a process than as a product. School

seems to be a more proper unit of change than a school district or an individual teacher

(Silver 1991).

Based on research experiences, the following conditions of change could be explicated

(Shaw & al. 1991): In order to affect a successful and positive change, (1) teachers need

to be pelturbed in their thinking and actions, and (2) they need to comrnit to do some

thing about the perturbance. In addition, (3) they should have avision of what they

would like to see in their c1assrooms, and (4) develop a plan to realize their vision (Fig.

3.1).

Cultural Environment

Relle",on /

Reflection

Fig. 3.1. Framework for teacher change (Shaw & al. 1991).

In the framework (Fig. 3.1), there are four central concepts. The cultural environment is

different for each teacher. For each teacher in the project, some central cultural elements

were noticed which have their impact on the process of change. Such elements are e.g.

the support given by others, time, money, other resources, taboos, customs, and com

mon beliefs. The change cannot happen without a perturbance in the teacher's thinking

and actions. For example, pupils, colleagues, parents, administrators, teacher-educators,


books, articles, and self-reflection may act as sources of perturbance. Comrnitment is a

personal decision to realize the change as a result of one or more perturbances. For

teachers in the process of change, they need to form a personal vision of what mathe

matics teaching and leaming should look like in their classes. Thus, if we want to have a

change in teaching, teachers should already be actively involved in the planning stage of

the innovation.

As an example of the practical situation, the teacher change described by Cobb & al.

(1990) fits very weIl the explained theory. The teacher was willing to cooperate and

planned new ways to te ach with the researchers. But the researchers noticed that, in

classroom situations, she was using her old conceptions of mathematics teaching to

make her decisions. The change occured when her thinking became perturbed through

that she saw that the teaching strategies she used were not powerful.

3.3. A model for the development levels in the change Another interesting attempt for a theoretical model of teacher change was exposed by

Thompson (1991). It deals with the levels in the development of teachers' conceptions of

mathematics teaching. She proposed a framework for the development, which was based

on reflections from work carried out with twelve pre-service and in-service teachers

over the past five years.

"The proposed framework consists of three levels. Each level is characterized by

conceptions of:

1. What mathematics iso

2. What it means to learn mathematics.

3. What one teaches when teaching mathematics.

4. What the roles ofthe teacher and the students should be.

5. What constitutes evidence of student knowledge and criteria for judging cor-

rectness, accuracy, Of acceptability ofmathematical results and conclusions."

Her model is partly similar to the earlier published considerations of Schram & Wilcox

(1988) and Schram & al. (1989).

In her paper, Thompson (1991) gave a brief verbal characterization of each level. Based

on these, a table of the characterizations (Table 3.2) has been elaborated, in order to

give a better overview. Thompson's five points mentioned above are reduced to four by

combining the second and third points (leaming and teaching mathematics). In addition,

the development of conceptions of problem solving is added as a fifth column.

Teachers' beliefs

0

-' ~ ~ -'

.... -' ~ ~ -'

....

.....

'" ~ ...

Wh.t i. Wh.t i. Wh.t .... Whata ... Wh.I i. mathem .. tical I •• mingll •• ehing Ihe rol .. ot Ihe erileri. tor Ihe problem oolvina?

mathematicI7 leocher .nd .ludenla? judaing cernodn_?

• common uses of • mcmorizltion of • the tacher ia • • lhe leIeher is an • gelhng .nswerB .uilhmetic skills in nlllcctions o( (acts, demonstrator of IUthority tor correcl- to "story problems" daily situations rules, formulils and well-eslablished ness • helpin~studenls

procedures procedures • accurate anawers 1.5 to idenli lhe righl • mathemaucal • teaching se9uences • students imitate the goolot m.them.- procedur. f'rules knowledge means of lopics .nd skills lies instruction ot lhumb") rote, procedural specified in • book profiacncy

• rules continue to • an ernergmg • much u in Level 0 • authority for • viewed.5 a govern aU work in awareness of the • the tucher attend. correctness still separate curricuJar mathematics use oE instructional to the "rea50n& lies with experls strilnd

representation6 behind the ruleo" • tau11tt separ.tely • appreciation for • U~ of manipula- • _ludenla indude • pro lern_ unrel.ted

understanding the tives fn instructlon same understanding to malhem.tical Iopics being studied concepts .lnd prin- • promote the view

ciple behind rules thil "math is fun" • teaehing ".bout" problem solving

• understanJing • teach for under- • Ihe teaeher sleenl • lhe process 01 • problem sol ving malhematics as a slinding sludenls' thinking doing m.lhem.lies is used .s • leIehing complex sYbtem of • understanding in rn.thematically i. the goolot melhod

• te.ching "via" different intercon- grows out of engage- proouctive w.Y. teaching nected conccpls. ment in Ihe process • he li.len81o • students thern- problem solving pronodures .nd of domg rn.lh. ltudon'" ide .. selv .. check their rcprl~l.!nl"tIOiIS malles • Ituden".pr_ anewers for correct-

the.r id •• n ...

Table 3.2. The levels in the development of teachers' conceptions

of mathematics teaching.

195

In the first column, the conception of mathematics is developed from rote calculations to

the complex system of interconnected mathematical entities. In the second colurnn, the

conception of leaming/teaching mathematics is changed from mere memorization to

understanding via doing mathematics. In the third colurnn, the role of the teacher deve

lops from ademonstrator to a facilitator of learning, and the role of pupils from imita

tors to active leamers. In the fourth colurnn, criteria for judging correctness is changed

from the teacher's authority to independent action. In the fifth column, the conception of

problem solving is developed from solving separate "story problems" to a teaching

method.

According to Thompson (1991), there are three levels in Table 3.2. It is worthwhile no

ting that actually the researcher's beliefs give here the framework, within which the de

velopment of teachers' conceptions on mathematics teaching are considered. One may

imagine that, in the future, some teachers will develop themselves further, e.g. into the

fourth or fifth level - level 3 and 4.

3.4. Empirical research on teacher change

For some time now, research concerning teacher change has been very vivid. Especially

in the PME-NA conferences (North American Chapter of the Psychology of Mathema

tics Education), teacher change has been a central theme during recent years. In the fol

lowing, we will brietly decribe some studies.


There exists little knowledge on the process of change in teachers' beliefs (Thompson

1992). As a first attempt, Thompson (1991) has sketched three levels through which

teacher change seems to happen (cf. section 3.3). Thus, she offers an answer, at least

partially, to the question "How do teachers' conceptions change?" Furthermore, Shaw &

al. (1991) have collected some conditions for teacher change: perturbance, willingness,

vision (cf. section 3.2). Therefore, we are provided with some answers to the question

"Under which conditions does teacher change occur?"

In the empirical research done on teachers change, the researchers have used interviews

and observations, but also questionnaires - sometimes combinations of these. The num

ber oftest subjects varies from a few teachers, N small « 10), to some tens ofteachers,

N large (- 10). A great many of the studies are realized in schools, but there are also

studies conducted during teacher pre-service and in-service training. In many cases, the

researchers have realized special pro grams aiming for change in teachers' (or pro

spective teachers' ) beliefs.

In the following, some examples of the different types of studies are discussed briefly.

Table 3.3 gives an overview of the studies to be dealt with, conceming their methodo

logy and sampie size.

interviews &

observations

questionnaire

combination

ofboth

N small « 10)

Cooney 1985

DeGuire 1991

Hart 1991

Schram & al. 1991

Wood & al. 1991

Hart & Najee-ullah 1992

Rice 1992

N large (~ 10)

Albelt & al. 1988

Russell & Corwin 1991

Dougherty 1992

Dionne 1984

Crawford 1992

lakubowski & Chappell 1989

Schram & al. 1989

Bishop & Pompeu 1991

Ben-Chaim & Fresko 1992

Table 3.3. A classification concerning the methodology and sam pie size of some research work done on teacher change.

The emphases on teacher change continued in the PME-NA conference 1993. In the

proceedings (Becker & Pence 1993), one may find the newest research work on teacher

change.


Studies in schools

Change in general. The programs for change in schools with experienced teachers were aimed mainly to develop a conceptually based approach to mathematics teaching (e.g.

Wood & al. 1991). Such a developmental program is e.g. the Atlanta Math Project; its preliminary results are described in the following papers: Hart (1991), Hart & Najee

ullah (1992).

The purpose of the case study of Wood & al. (1991) was to examine a teacher's learning

in the setting of the classroom. In an ongoing mathematics research project based on

constructivist views of learning and set in a second-grade classroom, the teacher chan

ged her beliefs about learning and teaching. These alternations occured as she resolved

conflicts and dilemmas that arose between her previous form of practice and the

emphasis ofthe project on children's construction ofmathematical meaning.

Hart (1991) presents the theoretical and conceptual frameworks for assessing teacher

change within the Atlanta Math Project. As a very preliminary result, she mentioned a

teacher who commented that for her one of the most valuable aspects of change has

been in the communication established between teachers in her department.

The paper of Hart & Najee-ullah (1992) explored aspects of change in the learning en

vironment and in teacher knowledge for one teacher (Margaret) who was in her second

year in the Atlanta Math Project. The data was gathered from two videotapes of Marga

ret teaching her grade 6 class and from her responses to a project instrument. The most

important finding was that Margaret's surface beliefs and deep beliefs became with time

more consistent with each other.

Change via a specific feature. In programs used, there was usually one feature of the

new rationale which was emphasized: For example, developing metacognitions (e.g.

DeGuire 1991), promoting communication (e.g. Russell & Corwin 1991), working to

gether in study groups of teachers and researchers (e.g. Schram & al. 1991, Rice 1992),

co-teaching in the classroom (e.g. Ben-Chaim & Fresko 1992).

The paper of DeGuire (1991) aimed for teacher change through developing their me

tacognitive processes. She presents two case studies which offer a glimpse into the me

tacognitive processes oftwo subjects (experienced teachers) with some previous experi

ence in problem solving. The techniques used were as folIows: journal entries, written

problem solutions with explicit "metacognitive reveries", videotapes of talking aloud

while solving problems, and general observation of the subjects. The data were gathered

throughout a semester-Iong course on the teaching of problem solving. The other case

study gives clear evidence of the automatization of metacognitive processes.

Russell & Corwin (1991) tried to promote communication skills in the c1assroom, in or

der to achieve teacher change. They present a study of teachers attempting to develop

better mathematical discourse in their c1assroom. The project began with interviews and

observations of all participating teachers. The subjects were a group of 12 elementary teachers (grades K-7) who were investigating ways to develop mathematical discourse


in their classrooms. The method is a compound of two phases: The period of "going

slow" is characterized by gradual changes in the amount of time devoted to mathema

tics, types of questions, and the nature of mathematical problems presented. A more

complex and difficult phase of change, "Ietting go", involved giving up planned goals or

topics to pursue ideas arising from students' mathematical work. In their analysis of re

sults, the researchers reported seeing the wide range of individuals, teaching practices,

changes in classroom practices, and changes in beliefs.

The studies of Schram & al. (1991) and Rice (1992) used the method of collaborative

study: In the paper of Schram & al. (1991), they use a theoretical framework developed

from the literature on teacher empowerment to describe changes in the discourse process

in which six elementalY teachers and four researchers pa11icipating in a mathematics

study group were engaged. The data sources include tape recorded Math Study Group

sessions and additional relevant documents. The purpose of this collaborative effort is to

try to move toward a more conceptuaJly-based approach to mathematics instruction,

curriculum development, and student assessment.

Rice (1992) reports on the effectiveness of the so-caJled Key Group model. Key Group

is a school-based professional development model where each group consists of three

teachers from one school, an outside consultant and, in many cases, a parent. The data

was gathered with individual interviews from five teachers of grade K-l. The study in

dicates that Key Group is effective because it provided the oPPOltunity for teachers to

construct new understandings about teaching and learning, the roles they ass urne and the

nature of change.

The method of co-teaching in the classroom was used in the study of Ben-Chaim &

Fresko (1992). A form of co-teaching was utilized as one mode of intervention in a

project to improve mathematics instruction in Israeli secondary schools. The project has

been on-going in six comprehensive secondary schools. The data was coJlected through

questionnaires to pupils, consultants and teachers, as weil as through interviews with

teachers, consultants and school principals. The initial reactions of pupils, teachers,

school principals, and co-teaching consultants suggest that, on the whole, this is a viable

in-service approach for demonstrating instructional strategies to teachers and for in

creasing their involvement in reflection and planned instruction.

Consistency of change. Cooney's study (1985) deals with the problem of the consistency

of teacher change. He explored the change of one young mathematics teacher's (Fred)

beliefs conceming problem solving. The study lasted one year during which Fred fi

nished his pre-service studies and began independently to teach in school. The analysis

of results revealed contradictions between Fred's idealism and his classroom practice:

Pupils were not always ready to receive his teaching in problem-solving.


Programs for teacher in-service training The objective of the pro grams for change was mainly to develop a conceptually-based approach to mathematics teaching (e.g. Albert & a1. 1988, Dougherty 1992). Different methods were used, usually in the form of a mathematics education in-service course with an innovative approach, in order to achieve the goal of the research (e.g. Dionne

1984, Bishop & Pompeu 1991). The project of Albert & a1. (1988) was based on in-service teacher education in lower

secondary classes. The project staff defined some criteria for the effective teaching of

mathematics - e.g. variety of teaching style, cognitive level of questioning, etc. The

main intervention relied on observation-based counselling. The analysis of project re

cords led to the creation of teacher profiles (ineffective, effective, desirable) which were

used to evaluate the effect of the project on each teacher (N=20). As a result of two

years of project activities, the researchers report progress both in teacher change and

student achievement.

The study of Dougherty (1992) investigates teacher change in intermediate and secondary classrooms (N =13). Using methodology consisting ofinterviews and observa

tions, movement to a process teaching model is documented. Data have revealed that

teachers can make behavioral changes but the richness of those changes is related to the

match between teacher philosophical structures and the teaching approach. Additionally,

materials supporting both the philosophy and specific pedagogical actions is an im

portant contributing factor in the change process.

Dionne (1984) evaluated the perception of mathematics which the teachers have with a

new tool called "Questionnaire sur la perception des mathematiques". He tested this tool

with a group of teachers enrolled in an in-service course in mathematics education

(N = 18). These teachers were taking a 45-hour mathematics education course in which a special approach was used. The "questionnaire" was used immediately before and after the course. The same tool was used in similar conditions with a control group

(N = 16) taking a course in a field other than mathematics. The results obtained in the

post-test showed that teachers in the experimental group had a more constructivist per

ception of mathematics than teachers in a control group.

Bishop & Pompeu (1991) described arecent study carried out in Brazil conceming the

influences of an ethnomathematical teaching approach on teacher attitudes to mathe

matics education. There were 19 teachers who planned and developed together with the

researcher six different "ethnomathematical" teaching projects. The teachers applied

these te ac hing projects in their classes (pupils' ages varied from 6 - 15 years), and each

project was. used for about 3 - 5 weeks, i.e. about 15 - 25 hours of mathematics class

time. The changes in the teachers' attitudes were checked with questionnaires and in

terviews. The researchers noticed that after the research study, these teachers became

much more concemed with many aspects emphazised by the ethno-mathematical approach.


Programs with prospective teachers

Methods similar to those used with in-service teachers were tried in teacher pre-service

education. Usually the intervention happens in the form of an "innovative teaching"

course offered to prospective teachers (e.g. lakubowski & Chappell 1989, Schram & al.

1989, Crawford 1992). The basic goal of such a course is "demonstrating the feasibility

of creating in new teachers a more conceptual level of knowledge about mathematics,

mathematics learning and mathematics teaching" (Schram & al. 1989,296).

The paper of lakubowski & Chappell (1989) deals with prospective elementary teachers'

beliefs about mathematics and about mathematics learning. Attitudinal surveys were

used to identifY a diverse sampie of 22 students from 186 students enrolled in a "How

Children Learn Mathematics" course. The informants were interviewed and asked to re

spond to a questionnaire which helped to identifY beliefs about mathematics and

teaching mathematics. Changes in beliefs were evidenced over the semester.

The study of Schram & ai. (1989) examined an intervention in an elementary te ach er

education program. The intervention - a sequence of mathematics courses and an inte

grated methods course - emphasizes the conceptual foundations of mathematics. The

data consisted of classroom observations from a cohort of 24 pre-service elementary

teachers. Additional data exists from an intensive sampIe of four students that include

tape-recorded interviews, writing assignments, observations of their student teaching,

and tape-recorded conferences with mentor teachers. The paper investigates some of the

changes in teacher candidates' beliefs about how mathematics is learned, what it means

to know mathematics, and the role of the te ach er in creating effective mathematical ex

periences for children.

Crawford (1992) deals with final year student teachers (N = 45) for the elementary

school. Earlier school experiences (often traditional) forms the basis of student-teachers'

ideas about teaching and mathematics, and has a powerful influence on their initial

classroom behavior. The results of an educational intervention aimed at providing

experiences as a basis for an alternative rationale are reported. Initial results, which are

based on a survey, suggest that many pre-service student teachers are able to develop a

rationale for teaching practice based on their knowledge of how learning occurs, and

apply their developing rationale in practice.

4. Discussion

About thirty years ago, there was a sharply delineated distinction between the cognitive

and affective domain. Through the research work done in the 1980s, our view of the si

tuation got clearer, and the boundaries between these two domains became increasingly

blurred. Today, we have a fairly extensive literature on pupils' beliefs, and a moderate

but growing literature on teachers' beliefs (Schoenfeld 1992).

We will conclude the paper presenting some open questions conceming beliefs, and a

Teachers' beliets 201

summary of implications from research for teacher in-service training which aims for

teacher change on the level ofteaching practice.

4.1. So me open questions concerning beliefs Here, we will discuss briefly three groups of underrepresented questions in the field of

mathematical beliefs: the birth of beliefs, the international comparison of beliefs, and

beliefs of non-mathematicians.

Birth of beliefs

A few research results exist on teacher change (cf. section 3.4). But the central question

still seems to be open: "What does give birth to beliefs?", "How will an individual's be

liefs take their shape?" We do not know very much about how beliefs come into being.

The social context of the school and other processes and constraints have been sugge

sted as likely sources of beliefs, but just how they operate on beliefs is far from c1ear

(Nespor 1987). In the case ofpupils' beliefs, D'Andrade (1981) suggests that beliefs de

velop gradually through a process much Iike "guided discovery" where children respond

to the situations in which they find themselves by developing beliefs that are consistent

with their experiences. Schoenfeld (1992) states that beliefs are abstracted from one's

experiences and from the culture in which one is embedded. But an open question is:

"What kind of experiences are crucial for change?"

One can read in the literature or hear from a colleague about excellent teachers who

have used e.g. new teaching practices. But how these teachers have developed themsel

ves in their teaching - or is it an inherent propelty? - is an unc1ear question. No answers

can be found in the literature to the question: "Which factors have influenced teachers'

beliefs about mathematics teaching and resulted in change?" Researchers speak much

about change and changing (cf. Thompson 1992), but the interconnection of those

conditions and factors through which the change has actual1y happened is stil1 fuzzy.

Are these factors perhaps connected with the structure of one's general beliefs system

(cf. Green 1971) which is unique for each individual?

International comparison of heliefs

An international comparison of teachers' beliefs (and resp. pupils' beliefs) has been an

almost unexplored field. The main question here is: "Are there essential differences in

conceptions ofmathematics teaching in different countries?" We know that mathematics

can be understood as a universal discipline. So, the question arises whether teachers'

(and pupils' ) conceptions on mathematics and on mathematics te ac hing and learning are

also universal or culture-bound.

The Second International Mathematics Study (Robitaille & Garden 1989) indicates that

there are large differences between countries on measures of mathematicaI beliefs and

attitudes. In the international comparison of pupiIs' conceptions about mathematics


teaching (Pehkonen & al. 1993), preliminmy findings suggest that the differences in

pupils' conceptions are essentially larger between countries than within a country

(Pehkonen 1993a, Graumann & Pehkonen 1993). There is no reason to anticipate that

the situation would be different in the case of teachers' conceptions. Some findings in

this direction are found by Pehkonen & Lepmann (1993) who compared teachers' ma

thematical conceptions in Finland and Estonia.

Beliefs of non-mathematicians

In our society, attitudes towards mathematics are ambivalent. On the one hand, our

highly developed and complex society needs and relies very much on mathematics and

techniques, but on the other hand, most individuals have a frustrated view of leaming

mathematics as an endless collection of rules and procedures to memorize. To inquire

into these attitudes seems to be important for various reasons.

Usually the studies carried out on mathematical beliefs dealt with pupils or mathematics

teachers. The question of what non-mathematicians - e.g. the man on the street - think

about mathematics is almost uninvestigated. A few pioneer studies exist in this field

(Möller 1989, Jungwirth 1993).

4.2. Concluding remarks

The model of beliefs has implications for our understanding of teaching and teacher

education. If we are interested in why teachers organize and run classrooms as they do,

we must pay much more attention to their mathematical beliefs, e.g. their subjective

interpretations of classroom processes. It seems that classroom structures have sources

in teachers' beliefs. (Ne spor 1987) In order to und erstand teaching from the teachers'

perspective, we have to understand the beliefs through which they define their work.

A summary of implications for teacher in-service training

One problem in teaching lies in the teachers' desire to teach actively. Rogers (1992)

states that - a teacher's self-concept usually requires hirn or her to be actively engaged in

teaching". The question is how we can help teachers "to keep their mouths shut" and to

let their pupils work actively. Rogers (I992) suggests that an important objective of any

curriculum project should be to help teachers make public and exarnine their own belief

systems. When they are doing so they may develop their own ideas about mathematics,

its teaching and leaming.

Theoretical background. For teacher change, there are some theoretical considerations

found in the literature which seem to have more connections than others with change on

a deep level. Self-reflection seems to be a powerful method for change on the deep level

(e.g. Hart & al. 1992). And the teacher should leam to be aware of his actions to the

extent that he reflects on his doing and undoing. According to the model of experiential

leaming (e.g. Kolb 1984), the stages of doing and of reflecting will follow each other.


The learning just happens when an individual reflects on his actions. From such self-re

flections, the awareness of one' s own opportunities may arise (Mason 1991). As a mat

ter of fact, the central question is how to develop teachers' metacognitions (Schultz

1991 ).

In the literature, a few existing research results about teacher in-service change point out

that it is difficult to accomplish change and to maintain it (Richardson 1990). But there

are experiments which have used the methods of interference described above and have

been successful. For example, Cobb, Wood & Yackel (1990) reported from their pro

ject, where thirty second-grade teachers were using the following problem-centered in

structional activities in their c1assrooms:

"We first conducted a one-week summer institute with the teachers and then visited

their c1assrooms at least once every two weeks during the first year in which they

participated in the project. The teachers also met once a week in small groups to

discuss their c1assroom experiences. In addition, the teachers participated in four af

ter-school working sessions during the school year."

About five years later, Yackel (1993) reported that of these thirty teachers only two ha

ve retumed to the textbook-centered teaching.

In summarizing the research results of teacher change, we may state that aprerequisite

for a teacher to change is perturbance in his teaching practice. In addition, he should be

ready to change and have avision of a new practice. This could happen through self-re

flection where experiental leaming (in his own c1ass and in those of his colleagues)

plays a central role. Teacher change is a long-lasting process where one may see chan

ges first in surface beliefs, and far later in deep beliefs.

End note

If we want to help teachers to develop their deep beliefs (i.e. their teaching practice), we

will need to provide them during in-service training besides theoretical knowledge also

experiential knowledge. This could be realized e.g. by participating in the everyday

work ofteachers by going to their schools and offering them help in the problem situati

ons of their teaching practice. It can mean e.g. to ren der a model lesson in the c1ass, to

provide hints for the use of certain materials and ideas, and/or to plan together the next

lesson. Thus, solutions to practical situations will be shown to the teacher in order to

realize in c1assrooms many-sided teaching situations which emphasize interactive

teaching.

The change in teachers' conceptions and beliefs seems to happen until he is compelled

to solve those contradictions and problems which are formed between his teaching

practice and the needs of the pupils' own constructing of their knowledge. The study of

Wood & al. (1991) showed a good example: A teacher in their research project was

easily ready to change her surface beliefs in discussions with researchers, hut not her

teaching practice, since she thought "I know the school reality and I am a good teacher".


On the deep level of her beliefs (teaching practice), she was ready to change, until she

saw the contradiction between her teaching goals and the results of her pupils.

Acknowledgements are grateJully extended to Prof Dr. Uwe Tietze (University oJ Göt

tingen) Jor giving many valuable comments about the structure oJthe paper.

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On Teachers’ Beliefs and Changing Mathematics Teaching