(A paper presented to the 28th AARE 1998 Conference, Adelaide, 30 November-3 December.)

Judy Anderson

Australian Catholic University

This study aimed to explore primary school teachers' beliefs about the role of problem solving in learning mathematics and the extent to which they claim to incorporate problem-solving approaches in their planning and teaching of mathematics. Survey research methods were used to gather data from teachers in metropolitan and country schools in New South Wales (NSW). The questionnaire was an appropriate instrument to use as it revealed the diversity of beliefs held by teachers as well as the use of a range of problem-solving tasks and teaching strategies.

Introduction

There has been substantial advice to teachers to teach problem-solving skills and to use problems as a focus of learning in mathematics (Cobb, Wood & Yackel, 1990; Schoenfeld, 1992). This advice has been provided in papers in research and professional journals (e. g. Bickmore-Brand, 1998; Sullivan & Mousley, 1994), in national curriculum statements (Australian Education Council, 1991) as well as in curriculum documents in NSW (Board of Studies NSW, 1998; NSW Department of Education, 1989). Such advice has been accompanied by considerable efforts through preservice and inservice programs to change teaching practices from more traditional approaches to contemporary or reformed methods (Crawford & Deer, 1993; Van Zoest, Jones, & Thornton, 1994). It has been argued that the use of non-routine problems and problem-centred activities form the basis of classroom activity in a reformed or inquiry-based classroom (Clarke, 1993 & 1997; Cobb, Wood & Yackel, 1991).

The level of implementation of mathematical problem solving in primary classrooms in NSW was the focus of a larger study. This paper reports on part of the first stage of the investigation that explored teachers' beliefs about the role of problem solving in learning mathematics as well as their classroom practices. Survey research methods were used to collect data about teachers' knowledge and beliefs regarding the discipline of mathematics, teaching mathematics, and learning mathematics since all of these factors impact on what occurs in teachers' classrooms (Ernest, 1989; Thompson, 1992). It was appropriate to use a questionnaire to try to seek the breadth of teachers' problem-solving beliefs and practices as very little is known about teachers' responses to the advice they have received, particularly within the context of Australian teachers.

This paper provides an overview of the larger study and describes the model that was used to guide the research. Two perspectives about mathematics teaching and learning are described; these provided a framework for the development of part of the questionnaire. A rationale for the use of a questionnaire to seek data about teachers' beliefs and practices is provided. The limitations of using a questionnaire are described and associated issues with administration are outlined. Questionnaire items that related to beliefs and practices, types of problem-solving tasks, and teaching approaches are presented, followed by a description of some preliminary results. Finally, the implications of using a questionnaire to seek information about teachers' beliefs and practices are discussed.

The Overall Study

The research reported in this paper is part of a larger investigation that aimed to explore the range of beliefs about problem solving that primary school teachers hold as well as to describe their classroom practices. In the first stage of the overall research, data were collected from 174 classroom teachers in NSW using a questionnaire designed by the researcher. The second stage of the research involved interviews and classroom observations in an effort to gain richer information about the constraints and opportunities teachers' experience as they attempt to implement innovative practices.

From the questionnaire responses, nine teachers holding a variety of beliefs that were representative of the spread were selected for participation in the second stage of the study. These teachers were interviewed about their questionnaire responses in order to confirm interpretation of items and to explore in more detail some of the issues that had been raised. From the initial interviews, two teachers were selected to participate in classroom observations and further interviews. These teachers had highlighted a variety of important issues related to possible constraints and opportunities that impacted on their efforts to implement problem-solving approaches. The combination of information collected from questionnaires, interviews and observations provided a detailed picture of the factors that have influenced beliefs and practices for these two teachers. The variety of data collection methods also provided an indication of the level of coherence of their beliefs and practices.

Essentially this research is about what teachers believe and how they view their own practices. Many factors influence teachers' problem-solving beliefs and practices. A proposed model of the relationship between these factors is presented in Figure 1.

Figure 1. A model of the factors that impact on reported beliefs and practices.

Teachers' reported beliefs are influenced by their actual beliefs (Thompson, 1992), by their knowledge and interpretation of advice about teaching problem solving (Fennema, Carpenter & Peterson, 1989), by their use and understanding of curriculum documents (Morine-Dershimer & Corrigan, 1996), and by their own experiences as learners of mathematics as well as by their experiences in classrooms (Ball, 1988). Reported classroom practices are influenced by reported beliefs, by actual practices in classrooms as well as by the constraints and opportunities that occur within the school context (Tobin & Imwold, 1993). Another purpose of the overall study was to explore the validity of this model.

It is acknowledged that models may represent an oversimplification of relationships between factors (Keeves, 1997). In this study, the model delineates components of the research and provides a framework based on information from previous studies (eg. Ernest, 1989; Fennema et al, 1989; Flexer, Cumbo, Borko, Mayfield, & Marion, 1994; Koehler & Grouws, 1992; Raymond, 1997). Each of the factors in the model acts as a filter that impacts on teachers' decision making and cannot be easily separated as is suggested in the diagrammatic representation. In fact there are complex interrelationships between factors that may change depending on the context that teachers are operating within (Hoyles, 1992). In this study, the model was initially used to assist in the planning of data collection about teachers' problem-solving beliefs and practices.

To facilitate a discussion about problem-solving beliefs and practices, an artificial dichotomy about teaching and learning, described here as perspectives about the role of problem solving, was used. One end of this dichotomy is the belief that mathematics is a fixed body of facts to be delivered by teachers and internalised by students. This belief is often associated with classroom practices involving individual student work with rehearsal of routine questions and reliance on textbooks or worksheets. This view may be accompanied by a belief that problem solving is an end (Wright, 1992) and that problems should be presented to students after they have mastered basic facts and skills. This perspective was described by Ernest (1989) and is referred to in this paper as a traditional teaching approach.

Another perspective, referred to as a contemporary teaching approach, has been described as representing a reformed classroom (Clarke, 1997). Such teachers believe that mathematics is a dynamic subject to be explored and investigated. Classroom practices associated with this perspective usually involve more group work and the use of non-routine questions that promote mathematical thinking and the development of problem-solving skills. This view may be accompanied by a belief that problem solving is a means (Wright, 1992) and that problems can be the focus of learning in mathematics lessons. These two perspectives represent end-points of beliefs about mathematical problem solving with many teachers holding beliefs that are situated somewhere between. These perspectives were used as the basis of the construction of the initial questions in the questionnaire.

Questionnaire Design

In mathematics education research, questionnaires have been used to collect data about teachers' beliefs and practices. For example, The Beliefs Survey used by Peterson, Fennema and Carpenter (1987) was designed with four subscales including the role of the learner; the relationship between skills and understanding; the sequencing of topics; and the role of the teacher. Another example is the Beliefs About Teaching Mathematics questionnaire designed by Van Zoest et al (1994) which incorporated a range of views adapted from Kuhs and Ball (1986). More recently, questionnaires have been used by Perry, Howard and Conroy (1996), Howard, Perry and Lindsay (1997), and Raymond (1997).

For this study, the questionnaire was used to gather a wide variety of data from the target population of primary school teachers in NSW. It was used to both describe teachers' reported beliefs and practices and to examine the relationship between particular variables that represented some of the factors in the model (Rosier, 1997). The questionnaire was considered to be a less intrusive method of data collection that could be administered in a relatively short period of time in a selection of country and city schools. As all teachers would be responding to the same set of questions, it was anticipated that the responses might provide a reliable set of data for that group of teachers.

Using a questionnaire to ask teachers what they believe can be fraught with difficulty for several reasons. It is possible that some teachers have never really thought about what beliefs they hold and whether there may be different ways of looking at the same thing. These teachers may teach according to the way they were taught mathematics or they may just do the same thing each year without reflecting on their practices. Also, beliefs are not static, but fluid, and are subject to change based on the experiences and influences that may be taking place (Thompson, 1992), or, they may be context specific (Hoyles, 1992). Therefore, the completion of questionnaires will only represent a snapshot of beliefs at a particular point in time. These concerns need to be considered when designing the instrument and analysing the results.

When using a questionnaire to gather data, particular assumptions are made. Wolf (1997) describes the assumptions as:

(a) the respondent can read and understand the questions or items;

(b) the respondent possesses the information to answer the questions or items;

(c) the respondent is willing to answer the questions or items honestly (p. 422).

Each of these assumptions needed to be addressed in the design of the questionnaire and in the planning and administration of this stage of the study.

The final questionnaire, entitled Problem Solving in Mathematics Teaching: Teachers' Views and Teaching Practices, was compiled with reference to earlier studies (Perry et al, 1996; Peterson et al, 1987; Van Zoest et al, 1994), outcomes from an exploratory investigation by the author (Anderson, 1996) and trialing of the instrument. It included five questions that used a Likert scale and five open-ended questions. The Likert scale questions included those relating to beliefs, teaching strategies, preferred question types and sources of problems. Open-ended questions requested that teachers describe a recently used problem, explain why they prefer particular types of questions, and describe the professional development needs of the staff at their school in relation to the implementation of problem-solving approaches. Near the end of the questionnaire, teachers were provided with the opportunity to reject problem-solving approaches to teaching mathematics. The question stated:

The following statement was made recently at a teacher inservice course:-

"People who push problem solving in mathematics obviously don't work in classrooms. It is a waste of time."

How do you react to this statement?

Responses to this question enabled the researcher to check for consistency with responses to earlier belief statements.

To encourage teachers to respond as honestly as possible, questions about beliefs were presented as representing the views of two imaginary teachers, Naomi and Gwendolin. Naomi's statements represented a belief system that suggested that she uses problem solving as an end in learning mathematics. This perspective is based on a traditional approach to teaching and learning mathematics (Thompson, 1992). Gwendolin's statements suggested that she uses problem solving as a focus for learning mathematics. This perspective is based on a more contemporary view (Clarke, 1993).

So that teachers could more readily relate to each of these perspectives, the statements were presented in the context of teaching two digit addition to a Year Three class. An example of a statement attributed to Naomi was "students should learn basic number facts before they do application and unfamiliar problems" and an example of a statement attributed to Gwendolin was "mathematics lessons should focus on problems rather than on practice of algorithms". Respondents were required to record their level of agreement with each of the statements. There were four levels to select including "strongly agree", "agree", "disagree", and "strongly disagree". A middle position of "unsure" was not used in an effort to encourage respondents to make a definite decision of either agreement or disagreement.

An important consideration in the design of the instrument was to ensure that each item was interpreted in the same way by all of the respondents and that the meaning of terms and expressions was clear. An earlier investigation highlighted the need to address teacher interpretation of the term problem (Anderson, 1996). To overcome the diverse meanings attributed to this term, a set of question types and an example of each was presented at the beginning of the questionnaire as background information. The question types were chosen on the basis of typical questions or problems that are discussed in the literature (Charles & Lester, 1982; Clarke & McDonough, 1989) as well as in resource materials and textbooks in NSW. It was indicated that the term "exercise" was to be used for algorithms and was not being referred to as a problem in this questionnaire. The problem types were "application problem", "open-ended problem" and "unfamiliar problem"; the latter was described as "a problem type the students haven't seen before". The example for each problem type related to two digit addition since this was the chosen topic of reference in the statements about beliefs.

Several questions on the questionnaire required teachers to indicate frequency of use of particular question types. The question types were those listed in the background information (Table 1) and teachers were required to select from the frequency categories of "often", "sometimes" and "never". This item was intended to gauge the types of questions teachers prefer to use and a subsequent question requested reasons for preferred choices. Most teachers probably use each of these types of questions at some stage in their teaching and so it is the frequency of use of these types which provides a greater indication of the importance a teacher places on the role of problem solving in learning mathematics.

Table 1

Another question listed twenty items that related to teaching strategies. This question aimed to explore teachers' practices when teaching mathematics, with most items involving specific reference to problem-solving approaches. The items were chosen on the basis of strategies mentioned in the literature (Clarke, 1997; Koehler & Grouws, 1992; Van Zoest et al, 1994) as well as in curriculum documentation (NSW Department of Education, 1989). Strategies included grouping of students, types of problem-solving tasks, teacher explanations about problem-solving, and use of concrete materials and calculators. Respondents were requested to rate the frequency of use of each of the strategies using the categories of "hardly ever", "about once a month", "about once a week", and "almost always". It was anticipated that the frequency with which teachers report that they use these strategies would provide an indication of their perceived importance and usefulness.

Administration

Initially, principals from departmental schools in metropolitan and country NSW were invited to participate in the study through contact with a selection of district mathematics consultants. Of the twenty-nine principals contacted, seven did not agree to participate for reasons of other priorities or participation in earlier studies. Twelve principals invited the researcher to address the staff about the study and two of them provided time for the whole staff to complete the questionnaire. The remaining ten principals discussed the study by telephone and then agreed to distribute questionnaires to staff for completion. This initial distribution yielded a total of 132 completed questionnaires from teachers in 21 schools. To increase the size of the data set, further groups were invited to participate. These included a group of teachers who were studying a mathematics education course while completing a Bachelor of Education degree on a part-time basis, a group of beginning teachers who had recently been appointed to schools in NSW, and a group of teachers participating in a teacher inservice course.

Many teachers feel uncomfortable and possibly even threatened by the notion of describing their teaching practices, particularly with regard to mathematics. In an effort to overcome the potentially threatening nature of completing a questionnaire about mathematics and problem solving, the researcher spent considerable time talking to teachers about the value of providing honest information. It was made clear to potential respondents that all views would be valued and that participation was voluntary. Respondents were encouraged to contact the researcher if further information was required and in addition, the researcher volunteered to speak to participating teachers about the results when they were analysed. Although few contacts were made, it seems that most teachers who responded had a particular interest in teaching mathematics or in the issues surrounding the use of problem-solving approaches.

The questionnaires provided a profile of the respondents. Of the 174 responses received from teachers, 28 (16%) were male and 146 (84%) were female. The teachers were from 40 schools; these included Catholic and departmental schools that were located in country and urban settings in NSW. Three categories of teaching experience were presented in the questionnaire. Results indicated that 32 (18%) teachers had been teaching from zero to four years, 30 (17 %) from five to nine years, and 111 (64%) for more than ten years. Table 2 provides background information about current roles as reported by respondents. Approximately one third were teaching kindergarten to Year 2, about one quarter were teaching either Years 3 or 4, and slightly less than one quarter of respondents were teaching in Years 5 or 6.

Table 2.

It should be noted that the final group of respondents is not a representative sample of primary school teachers in NSW. The schools were not randomly sampled and many teachers did not respond to requests to complete the questionnaire. Some teachers offered reasons for lack of participation in the study. These included: a need to complete other more urgent tasks; a belief that the children in their current class or mathematics group are not capable of doing problems; and the belief that as kindergarten teachers, it is not appropriate to engage in problem-solving activities with the children. The final set of questionnaires still yielded a spread of teachers' beliefs about problem solving, a substantial variation in preferred problem-solving tasks, and considerable diversity in frequency of use of a variety of teaching strategies.

Analysis and Preliminary Results

Table 3

The data suggest that for the majority of surveyed teachers, number facts and some basic mathematics are needed before students can tackle problems. Also, many teachers supported the notion that problems can motivate and challenge students. Most teachers rejected the view that to solve problems, students need to know the four mathematical operations and that problems should be left to the end of the topic. Encouraging students to invent their own methods was also rejected. The responses suggest that traditional approaches gained more support from teachers than contemporary views. Given the overwhelming support for the item about language difficulties, it is clear that most teachers believe that problems can be made more difficult by the language that is often used. This is not surprising as placing mathematical problems in real-life contexts usually requires the use of much more language than is often required in mathematical exercises.

Several items yielded a more balanced set of responses as is indicated in Table 4. These are the items that had 60% or less for the level of agreement. From this data it can be seen that there is some support for the more traditional practices of: learning algorithms before doing problems; relating problems to the specific content of lessons; and focusing on practising skills. Equally, there is some support for the more contemporary practices of: beginning topics with unfamiliar problems; focusing on problems rather than exercises; working out and exploring individual methods.

Table 4

The only item that did not spread the respondents was Naomi's statement involving the language involved in problems. All other items provided a spread of responses suggesting that the questionnaire was a useful instrument to gain information about the range of teachers' problem-solving beliefs.

The items that explored frequency of use of different types of questions also revealed a range of responses from teachers. Table 5 lists the four question types and the frequencies expressed as a percentage. Overall, open-ended problems and unfamiliar problems are less frequently presented to students than application problems and exercises. These results suggest that while the majority of teachers use more traditional questions in mathematics lessons, some teachers appear to use open-ended and unfamiliar problems on a regular basis. It is possible that for these teachers, problem solving is viewed as a means rather than as an end in the learning of mathematics.

Table 5.

Frequency of use of each of the question types (expressed as a percentage).

The strategies that lecturers rated as being appropriate for regular use are recommended in advice given to teachers about problem solving in the reformed classroom and are consistent with advice provided in curriculum documentation. These include: whole-class discussion for sharing solutions and strategies; small, cooperative group discussion; provision of concrete materials and calculators; recognition of the need to encourage individual student recording of methods and procedures; and encouragement of the use of problems that have been posed by students as well as the use of problems that relate to students' interests.

The strategies that teachers report that they frequently use showed a reliance on more traditional teaching strategies. From the questionnaire responses, most teachers reported regular use of skills practice, whole-class discussion, teacher modelling of problems, discussion of problem-solving strategies, and reliance on concrete materials. The strategies that were infrequently used by teachers included allowing students to use calculators, encouraging students to work alone, presenting problems to students with little guidance of appropriate procedures, allowing students to choose problems to solve, spending considerable time on one problem, and encouraging students to pose their own problems.

There are clearly differences between strategies recommended by mathematics education lecturers and those preferred by teachers. It seems that teachers are not yet convinced that calculators can be an integral part of the primary mathematics classroom. Also, there is less frequent use of individual student methods of recording and student created problems. This may suggest the persistence of a reliance by teachers on standard algorithmic procedures since teachers may believe that students need to learn the traditional or textbook method. It is also possible that teachers might feel threatened by unfamiliar methods and believe that it is important to maintain control of what children learn and how they represent their mathematics.

Conclusion and Recommendations

A small number of teachers seem to have responded to advice and do report using many of the teaching strategies that the literature claims promotes effective learning in mathematics. This does not of course guarantee that this is what is happening in their classrooms. Part of the larger investigation is to observe teachers as they implement problem-solving strategies and to compare reported practices with observed practices. If teachers are in fact implementing these strategies then it is of interest to explore why other teachers have not responded to the same advice. Or, is it the case that teachers' beliefs about teaching and learning mathematics filter the information resulting in quite different interpretations of the advice? If teachers hold similar beliefs, it may be possible that constraints in some school cultures are much stronger than in others. Alternatively, particular school cultures may actively support a problem-solving approach and encourage teachers to implement the strategies explored in this questionnaire.

The questionnaire responses have provided valuable information about reported teaching strategies in relation to the use of problem solving in learning mathematics. Considerable analysis has still to be completed on both Likert items and open-ended questions. At this stage, frequency tables indicate that the items relating to teachers' beliefs have yielded a spread of beliefs and that teachers use a variety of question types in their classrooms. The open-ended questions have also provided a rich source of data although several respondents did not complete these questions. Interviews and observations will be used to confirm and enrich the questionnaire analysis.

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