Do you know what values you teach when you teach mathematics? Maybe you don’t even think you are teaching any values? What values are your students learning from you? Does everyone teach to the same values as you do when they teach the same mathematics? These questions, and others like them, have motivated us to develop a research project on this topic and in this paper we shall outline the main ideas involved.

At present there is little knowledge about what values teachers are teaching in mathematics classes, about how aware teachers are of their own value positions, about how these affect their teaching, and about how their teaching thereby develops certain values in their students. Values are rarely considered in any discussions about mathematics teaching, and a casual question to teachers about the values they are teaching in mathematics lessons often produces an answer to the effect that they don’t believe they are teaching any values. It is a widespread belief that mathematics is the most value-free of all school subjects, not just among teachers but also among parents, university mathematicians and employers.

Current developmental policies in state and national programs are focused on improving achievement outcomes of students, and although their statements of intent often mention the encouragement of ‘desirable’ values, the curriculum prescriptions which follow have little to say about their development (see for example, Australian Education Council, 1991). Values teaching in mathematics certainly seems to happen more implicitly than explicitly and at least one study suggests that there is not necessarily a one-to-one correspondence between what is intended and what actually occurs (Sosniak, Ethington and Vareless, 1991). We can certainly assume, therefore, that values teaching inevitably affects the achievement of other curricular outcomes, and therefore those with responsibility for developing state and national programs in mathematics should also be concerned about the poor state of knowledge generally about values teaching in mathematics.

Furthermore there are important ideas which have been developed in the last few years which could have widespread benefits for mathematics learners around the world. In the areas of technology (see Noss and Hoyles, 1996), ethnomathematics (see Gerdes, 1995) and critical mathematics education (see Skovsmose, 1994) the role of mathematics teachers in values teaching is being critically examined. Also at this year’s Psychology of Mathematics Education conference in South Africa there was considerable discussion of implicit values in various sessions under the overall theme of the conference which was: Diversity and Change in Mathematics Education. What is of particular interest about all these developments is that there is a strong concern both to question, and also to change, the values currently being taught.

This clearly requires even more detailed knowledge at the classroom level, since not only do we have little knowledge about what values teachers are teaching when they teach mathematics, we have even less idea of how potentially controllable such values teaching is. Therefore an additional issue concerns whether teachers have, or can gain, control over their values teaching, which would theoretically enable them to teach other values besides those which they currently teach.

So what are the values we are talking about here? Values in mathematics teaching are conceptualised as the deep affective qualities which teachers promote and foster through the school subject of mathematics. Our initial analyses reveal that there are three kinds of values which teachers convey: the general educational, the mathematical, and the specifically mathematics educational. For example, when a teacher admonishes a student for cheating in an examination, the values of ‘honesty’ and ‘good behavior’ derive from the general educational and socialising demands of society.

Then when a teacher proposes and discusses a task such as the following: "Describe and compare three different proofs of the Pythagorean theorem" the mathematical values of ‘rationalism’ and ‘openness’ are being conveyed (see Bishop, 1988, and below). However there are other values being transmitted which are specifically associated with the norms of the institutions within which mathematics education is formally conducted. For example, the values implied by the following instructions from the teacher: " Make sure you show all your working in your answers", "Don’t just rely on your calculator when doing calculations, try estimating, and then checking your answers", are about ‘examination-wiseness’ and ‘efficient mathematical behavior’. If these different values are considered important in good practice teaching then improving teachers’ knowledge of their values teaching will surely improve their mathematics teaching.

At present there are three principal literatures which we have used to frame our research. These are the literatures on the affective domain and values education generally (e.g. Krathwohl, Bloom and Masia, 1964; Raths, Harmin and Simon, 1987; Tomlinson and Quinton, 1986), on affective aspects of mathematics education (Buxton, 1981; Fasheh, 1982; McLeod, 1992; Thompson, 1992; Sosniak et al, 1991), and on social and cultural aspects of mathematics education (Bishop, 1988; Davis and Hersh, 1981 and 1986; Joseph, 1991; Wilson, 1986).

With regard to the first literature, Krathwohl’s (1964) analysis of the Affective domain of Bloom’s well-known Taxonomy first introduced the ideas of ‘values’ and ‘valuing’ as important educational objectives. Their analysis suggests five levels of response to a phenomenon in increasing degrees of commitment. Of particular interest here are levels 3 and 4 that are summarised as follows:

Going on from there, Raths, Harmin and Simon (1987), summarising their oft-quoted book, offer seven criteria for calling something a value. They say (p199): "Unless something satisfies all seven of the criteria noted below, we do not call it a value, but rather a ‘belief’ or ‘attitude’ or something other than a value." They summarise their criteria in the following terms:

Both the Taxonomy and the criteria emphasise certain aspects of valuing which seem important to focus on, namely;

In relation to values education more generally, the work of Tomlinson and Quinton (1986) was particularly important since it moved the discussion from earlier reliance on the work of Kohlberg (1984) and his followers into the mainstream subject curriculum. They argued strongly that when considering this area due attention should be paid to three elements (p3): aims or intended outcomes; means or teaching/learning processes; and effects or actual outcomes.

Regarding the second literature, there have been several relevant studies, such as Buxton (1981) and Fasheh (1982), but McLeod (1992), in one of the most up-to-date and comprehensive research summaries of the area separates the field into studies of beliefs, of attitudes, and of emotions. He, like others who have surveyed the area, cites no research on values, although the tone of his discussion makes it clear that ideas about both beliefs and attitudes towards mathematics do relate to values held by both teachers and students.

In another chapter in the same book Thompson (1992) also discusses the research on teacher beliefs, particularly in relation to teachers’ actions in the classroom. She points to a repeated finding that teachers’ actions frequently bore no relation to their professed beliefs about mathematics and mathematics teaching. The research by Sosniak et al (1991) also found striking inconsistencies between different and sometimes conflicting belief statements given by the same teachers. We would contend that this discrepancy is precisely why it is necessary to study values rather than beliefs, in order to determine the deeper affective qualities that underpin teachers’ preferred decisions and actions. You can believe different things to be true or important, but when you teach you have to make decisions and choices on-the-spot, and this literature suggests that it is your deeply held values which determine the choices you make.

The third literature, although limited, has been helpful in clarifying the mathematical focus of the values we are interested in. Although being interested in the first of the three sets of values, the mainly educational, we are predominantly concerned with those values which relate to either mathematics or mathematics education. If we are trying to improve mathematics teaching we will need to keep the focus on the mathematical aspect of teachers’ values.

Sources such as Davis and Hersh (1981, 1986) and Joseph (1991) have proved helpful, although they don’t address values directly. Wilson’s (1986) review, whilst pointing out the paucity of writing and research on values in mathematics teaching did discuss two values, a respect for truth, and the authority of mathematics. Later analyses by Bishop (1988 and 1991) sought to build more broadly on the wide literature on mathematical history and culture. Using White’s (1959) three component analysis and terminology, he proposed that, in ‘Western’ mathematics development, the predominant ideological values concern the ideas of ‘rationalism’ and ‘objectism’, the sentimental values (which is White’s term for individuals’ feelings about their relationship to knowledge) are those of ‘control’ and ‘progress’, while the sociological values refer to societal relationships regarding mathematical knowledge, such as ‘openness’ and ‘mystery’. Wilson’s first value is an ideological one, while the second fits comfortably within White’s ‘sentimental’ component.

There is a definite need in our view to develop professional development programs. Our review of the literatures, and our broad experience of practices in initial-training and in-service courses in mathematics education suggest that there is a particular need for activities which focus on the behavioral aspects of values in mathematics education, such as the choosing, the preferring, the consistency of behavior, etc. A strand of previous research which can be built on is that of ‘teacher decision-making’. Bishop’s (1976) original research work in this area laid the foundations for other research studies which have demonstrated that a simple documentation of teacher behavior is not adequate for assisting with development. One also needs also to know why teachers choose to do what they do, what they think they then do, what they appear to do, and what alternatives they can generate. These ideas are now well known in professional development work and in the context of this topic they will give rise to various values-clarification activities of relevance to teacher development at all levels.

An important component of these activities is the critical incidents which happen in the classroom, and on which they are based. Critical incident analyses always raise issues about just what makes any incident critical to the sequencing and flow of teaching, about the choices facing teachers at such moments and about the criteria for choosing the appropriate option at that moment. In our experience of using such material, teachers find it stimulating to engage with and explore in a neutral and supportive environment the complex issues involved. Hopefully in this context, the teachers’ values will become overt. This is important because previous instruments focus largely on the teacher’s behavior separate from that of the learners. We are aware however, from one of our current projects (see Brew, Pearn, Bishop and Leder, 1995) that peers influence students’ affective responses every bit as much as does the specific behavior of the teacher, and therefore the students and the teacher together socially construct the affective and value-laden environment in the mathematics classroom.

For teachers then, recognizing moments in the flow of a lesson in which they will make a decision can be called a critical moment. Such a moment and what results from it can be termed a critical incident. It is at this point in the lesson that the values of teachers may become clearly exposed to an observer, and to the teacher themselves if they are aware of what is happening. In other words, in such incidents as these, if teachers know that there are alternative actions from which they can choose which will influence the flow of the lesson, and they make a purposeful choice which they act on and do so in a consistent pattern over a period of time, then purposeful reflection on such critical incidents may expose the values they hold. Hence their implicit values may become explicit to them.

Take the following moment in a lesson. The students are working on routine examples from their text books. A student asks for help. What should the teacher do? Some teachers may, because of a habit that has been with them for many years, follow a set routine and think little of what they do. They may go to the student and start suggesting a particular set of steps that will lead to the solution found in the back of the text. What value does this action by the teacher imply? Is the teacher aware of the implied value? May be the teacher is aware of the value that is implied at this moment, and is quite content with this state. We would contend however that if the teacher was not aware of the value implied by the action, then for them to become aware of this ‘hidden’ aspect of the situation, would further empower their teaching.

Clearly at the moment in which the teacher responds to the student’s request for help there are a number of possible courses which could be taken. Some such are that the teacher may simply ignore the request, may direct the student to ‘have another go at the problem themselves’, may suggest that the student seek help from another named student in the class, may request that the student ask a peer of their own choice for help, and so on. Each of these actions in themselves imply different or overlapping values. We are interested in first knowing whether the teacher is aware of the possibilities available at that moment. If so, we are also interested in what choices the teacher believes are available to select from. Finally we are particularly interested to know whether the teacher recognises any values implied by the act of and implementation of the subsequent action.

We wish to emphasize that we are not interested here in making an ethical judgment on the value(s) that the action of the teacher implies. We are interested in whether the teacher knows whether, and if so what, values are implied by their teaching.

We are also very aware that at a moment of choice, clearly the teacher’s values will be not the only factor influencing the choice. But the values are, we believe, an important and often hidden aspect of the situation.

There are various values mentioned in recent mathematical curriculum documents such as the CSF Mathematics (Board of Studies, 1995), The national statement on mathematics for Australian schools (Australian Education Council, 1991), the document from the USA Assessment standards for school mathematics (National Council of Teachers of Mathematics, 1995), as well as more general material on mathematics teaching and learning (see Clarke, 1988). Some of the values that are suggested in these documents are: ‘clarity’, emphasized in the scope statements of The national statement, but not prominent in the content oriented sections; or ‘flexibility’ and ‘rationalism’ implied in the non content section of the CSF by highlighting the need for students to test different mathematical procedures; ‘openness’ and ‘consistency’ are found in the glossary of the NTCM document; and Clarke (1988, p.4) notes the need to assess "persistence, systematic working, efficient and effective organization, accuracy, conjecturing, ... creativity", although sadly in the rest of this valuable monograph there is little emphasis on these values. There is no doubt that this aspect of mathematics teaching is recognised by the profession, but it is relegated to the periphery, perhaps because no one is really sure of how to structure teaching for these values.

To explore these notions further, the reader may wish to develop a ‘mind game’. Take one of these values and work out what your actions could be that would affirm the value.

• First, what situation could arise ‘spontaneously’ in the classroom which would give you an opportunity to signal to your students that this value was important to you, and in the study of mathematics?
• Would your action involve;
• verbal comment, a question, or a particular way of answering a question; and/or
• a repositioning of yourself, a raised eyebrow, or other body language; and/or
• some delayed action for the student, the small group of students, or perhaps the whole class; and/ or
• something else?
• It seems to us that such an exercise as this may be one way to help teachers in classrooms become more aware of the values that they will inevitably be teaching.

It is one thing for a teacher to become more aware of the values that are built into the fabric of teaching mathematics. However it appears to us that if values teaching is to be taken seriously, it may mean another set of scenarios need to be devised to help students take notice of these values and examine them for what they are. There has been a long history in education of assuming that the teachers’ actions, often incidental in nature, will influence students’ behaviors, beliefs and values. This is undoubtedly true. However the extent of this influence is open for debate. For students to really change, we suspect much more overt action is required.

So the reader may wish to enter into another mind game. What value would fit into the section of work that you are currently teaching in your classroom? Clearly there may be more than one value, but for this exercise just concentrate on one and make the beginning of this type of thinking a little easier. For example take ‘clarity’ as a value to begin with.

• What do you mean by clarity?
• Will everyone use this meaning, or are the range of acceptable meanings?
• Is there anything special about this value when it is used in the peculiar situation of teaching school mathematics?
• Why is clarity important for school mathematics?
• Who are we trying to make things clear for? In other words, for what audience(s) is the clarity required; the individual learner, other peers, parents, the teacher?
• Does the change of audience make a difference?

We would suggest that starting off the lesson explaining to your group of students why ‘clarity’ is important in mathematics will be some what of a waste of time. Alternatively, how can you plan your lesson so there hopefully will be experiences for the students that will actively and inevitably involve notions of ‘clarity’ during the natural course of the lesson? What are the possible responses of the students? What are your options at these points which will affirm that you think ‘clarity’ is important, but also get your students thinking more deeply about this value and its place in the doing of mathematics? What are follow up ideas, starting from and drawing on the situations in this first lesson, that could be ‘set up’ in subsequent lessons? How can you circle back to this notion next month when you are working on another topic area? Is there any assessment issues that should be included in your planning?

The primary question, and the one which initiated this research was: what values are mathematics teachers teaching? Our concern with this question is that although values teaching and learning go on inevitably in all mathematics classrooms, most of it appears from our preliminary studies to be done implicitly, and therefore there is only a limited understanding of what values are being taught, and of how much mathematics teachers are aware of what values they are encouraging (Abreu, Bishop and Pompeu, 1992; Clarkson, 1998). The intention here is to move teachers’ knowledge from implicit to explicit.