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&,9Teachers beliefs about MATHEMATIcs and its teaching A qualitative study in germany
Katja Maass, HYPERLINK "mailto:katja.maass@ph-freiburg.de" katja.maass@ph-freiburg.de
University of Education, Freiburg, Germany
Teachers beliefs about effective mathematics teaching have a major influence on the question of why changes in day-to-day teaching practice are rare. In order to gain a deeper insight into teachers beliefs about effective mathematics teaching, a qualitative study has been carried out. The relevant research questions of the study were: What beliefs about effective mathematics teaching do teachers have? What differences exist between teachers from different types of schools in Germany? Can a relationship between the various types of beliefs be reconstructed? Can different types of teachers be identified? The paper gives the theoretical and methodological framework and the results. The results of the study indicate that there are mainly two types of teachers. One type of teacher regards effective mathematics lessons as being those in which the teacher explains mathematical results which are then used by students in stereotype exercises. The second type focuses on learning processes, problem-solving processes and on students working independently.
THEORETICAL FRAMEWORK
Teachers beliefs about mathematics and mathematics education are thought to have a major impact on the implementation of innovative ways of teaching into day-to-day mathematical lessons (Bishop, Seah & Chin 2003, p. 718, Chapman 2002, p. 177, Llinares 2002, p. 195, Ponte et al. 1994, p. 356, Gellert 1998, p. 84). Lloyd (2002) found out that teachers beliefs about mathematical education have a strong connection to the mathematical lessons they experienced as children.
However, different definitions of beliefs and different concepts are used within the discussion about mathematics education (Pehkonen & Trner 1996, p. 101, Opt Eyde, de Corte & Verschaffel 2002, p. 13). Furinghetti & Pehkonen (2002, p. 51) compare various definitions and explore the question of whether beliefs are incontestable personal knowledge. They also look at the relationship between beliefs and knowledge. From an epistemological point of view, they suggest differentiating between objective and subjective knowledge. Beliefs should be regarded as subjective knowledge, and affective components should also be taken into consideration. They also differentiate between deeply rooted beliefs and surface beliefs, the first of which are regarded as harder to be changed. Following Trner, the term beliefs will be used here. According to Trner and Pehkonen (1996, p.6) beliefs will be defined as follows: Beliefs are composed of a relatively lasting subjective knowledge of certain objects as well as the attitudes linked to that knowledge. Beliefs can be conscious or unconscious; the latter are often distinguished by an affective character.
In general, it is believed that beliefs are hard to change (Trner 2002, p.117), but some studies indicate that this can be possible under certain circumstances (Kaasila, Hannula, Laine & Pehkonen 2006, Maass 2004). Certain types of beliefs about mathematics can be distinguished in Trner (2002): beliefs about mathematics as a science; beliefs about mathematics as a subject at school; beliefs about the learning of mathematics; beliefs about the role of mathematics teachers and beliefs about the role of students. As part of an empirical study (Grigutsch, Raatz & Trner 1998), the following categories of teachers mathematical beliefs could be reconstructed: the aspect of scheme (mathematics is a fixed set of rules); the aspect of process (in mathematics problems are solved); the aspect of formalism (mathematics is a logical and deductive science); and the aspect of application (mathematics is important for our lives and for society). Similar categories can be found within the international discussion: Ernest (1991) and Dionne (1984) differentiate between a traditional perspective, a formalist perspective and a constructivist perspective, which seem to correspond to the aspects of scheme, formalism and process. These beliefs, however, mainly refer to mathematics as a science and not to mathematics education. In addition to these, further categories for teachers beliefs can be reconstructed. In a study about the teaching of probability, categories referring to effective mathematics teaching and the usefulness of mathematics lessons for students (Eichler 2006, pp. 154), and categories referring to students and the general framework of school, have been identified (Eichler 2002, p. 26).
However, concerning the distribution of the various aspects in teachers beliefs, there are different results. Grigutsch, Raatz & Trner (1998, p. 36) found out that the majority of teachers taking part in a teacher training course seem to have many application-oriented and process-oriented beliefs. By contrast, teachers who had been investigated by Kaiser (2006) had mainly formalist and scheme-oriented beliefs. Based on these categories, Kaiser (2006) found out that innovations required by the curriculum are interpreted by the teacher in such a way that they fit into their belief system. The same holds true for the mathematics tasks chosen for the lessons. The widespread scheme of belief-categories described above was used as a basis for data evaluation in this study. Additionally, within this system of Grigutsch et al, a difference was made between beliefs referring to mathematics education and those referring to mathematics as a science (Trner 2002).
THE GERMAN SCHOOL SYSTEM
In Germany there are three kinds of secondary schools. Students are separated at the age of 10 11 according to their ability. Those with low abilities go to the Hauptschule for another 5 years, those with average abilities go to the middle school called Realschule for another 6 years, and those with high abilities go to the Gymnasium for another 9 years. All three types of schools aim to provide students with a general education but at different levels.
Teacher training for the three types of schools is very different. Teachers for the Hauptschule are educated together with Primary School teachers and to some extent with the Realschule teachers. The mathematical education they get is especially in comparison with the teachers at the Gymnasium low level. In addition, a lot of teachers teaching mathematics at the Hauptschule are not trained to be mathematics teachers. However, they do get a lot of pedagogical instruction. By contrast, those who want to teach mathematics at a Gymnasium (up to higher secondary level) get a scientific education in mathematics, whereas their pedagogical education is often not regarded as important.
Methodological Background
The study described here is a qualitative study. An elementary goal of qualitative research is to explain complex relations within a day-to-day context instead of explaining singular relations by isolation (Flick, Kardorff & Steinke 2002). It is rather to discover new things than to prove things that have already been discovered. In this study, teachers beliefs about mathematics and its teaching are to be explored in detail. What are teachers beliefs about mathematics and its teaching? What teaching do teachers regard as effective? Can different types of teachers be identified?
An essential characteristic for the selection of the sample survey and evaluation methods was the principle of openness: Since there is not much known about teachers beliefs about mathematics teaching, hypotheses should not be brought to the study but rather developed whilst dealing with the data and then formulated as results.
The sample group studied included 20 Gymnasium teachers and 20 Hauptschule teachers. Teachers of both school types were chosen in order to have as wide a variety of German teachers as possible. Additionally, teachers were selected according to their experience (from not very experienced to very experienced), their gender and the subjects studied at university (mathematics as a subject or not). To get information about teachers principles of different schools have been asked with the help of a little questionnaire. Semi-structured interviews were used in order to collect data about teachers beliefs. The interview questions were designed to leave enough space for a wide range of teachers answers at the same time as providing as much information as possible about the teachers mathematical beliefs (Flick 2000). Data analysis was carried out according to the analysis of content of Mayring 2002 (p. 468). Within this analysis, you proceed in three steps: 1. Summarizing analysis of content: The original data is summarized with the help of paraphrasing central aspects of the data. 2. Explicit analysis of content: Parts of the data which seem to be unclear are analysed with the help of additional information about the person interviewed (such as observations during the interview, information about his context) 3. Structuring analysis of content: After an analysis of the whole data, typologies can be created.
The categories described above within the theoretical framework have been used as preliminary and rough categories. All categories were filled with codes derived from theory as well as with in-vivo-codes. These are codes which emerged directly out of the data as teachers formulations. Quotations from the interviews were used to explain the codes. Data analysis was carried out in a team. Members of the team analysed interviews individually, then cases were discussed. With the help of the codes and the categories, case-comparing and case-contrasting analyses were conducted and lead to the reconstruction of different groups of teachers. To highlight the results, typologies were created. Procedures of creating types, comparing cases and contrasting cases play an important role in qualitative research because in this way the complex reality is reduced and made concrete (Kelle & Kluge 1999). A typology is the result of a grouping process, in which one or more characteristics are used to allocate objects to groups in such a way that the objects within the groups are as similar as possible and the actual groups are as different as possible. The newly formed groups are called types. At the same time, the groups can have partial overlap (Kelle und Kluge 1999, S. 75 ff.). This means that on a continuum, types can also represent different characteristics on a gradualistic scale.
RESULTS OF THE STUDY
The data analysis showed that within the given cohort of 40 teachers differences between teachers from the Hauptschule and from the Gymnasium concerning the beliefs about mathematics teaching were relatively small, whilst among both groups the same two types of teachers in relation to their beliefs about effective teaching could be identified. For this reason, differences between Hauptschule and Gymnasium will be named first. Then we will look at the different types of teachers who could be reconstructed in both types of school.
Differences between teachers of the Gymnasium and the Hauptschule
Beliefs about mathematics as a science: The mathematical beliefs of many teachers at the Gymnasium seem to consist of formalism-orientated beliefs. In addition to that, the teachers seemed to have a strong feeling about the usefulness of mathematics for the development of our society and also for day-to-day life. Concrete examples, however, were seldom given. Only few process-orientated beliefs could be reconstructed and almost no scheme-orientated beliefs. The teachers of the Hauptschule seemed to have less pronounced beliefs about mathematics as a science than the others. Most of the answers they gave referred to mathematical education. They called mathematics logical and structured, but did not become more explicit. Although stressing the usefulness of mathematics in daily life and professions, they apparently only saw the usefulness of elementary mathematics.
Interest in mathematics: All teachers of the Gymnasium showed a high interest in mathematics. Hauptschule teachers did not point out any interest in mathematics but mostly an interest in teaching mathematics.
Beliefs about the aims of mathematics education: Most teachers of the Gymnasium seem to regard the preparation of the students for university as an important aim of their lessons, regardless of the fact that only a part of the students will go to university. By contrast, teachers of the Hauptschule often named calculating and preparation for life.
Beliefs about motivation: The motivation of students did not seem to be very important for teachers of the Gymnasium. Partly students were expected to be motivated simply because they are at a Gymnasium. In contrast, motivation was regarded as very important to make teaching effective by teachers of the Hauptschule. As ways of motivating pupils they named showing students that mathematics can be fun, that fearing mathematics is not necessary and giving them opportunities to be successful in finding results of tasks.
These differences between the two types of teachers may be seen in connection with their education at university and their teaching experience at school. However, neither the different forms of teacher education, nor the different beliefs about mathematics as a science that teachers of both school types have (pronounced vs. no pronounced beliefs), seem to have a significant impact on what teachers regard as effective mathematics teaching. Reasons for this could be further influencing factors, for example their own experience mathematics lessons (see theoretical framework), or, despite all differences, similar features in their teacher education.
Two main groups of teachers concerning their beliefs about effective mathematics teaching could be found. Major characteristics of these two groups will be illustrated by two case studies first. Both of them have been chosen from the teachers from the Gymnasium in order to not mix up differences related to the two groups and those related to the type of school. The evaluation of the case studies is based on the whole interview. Due to limited space, we will only describe the results of this evaluation and give some quotations as an illustration. For the same reason we will focus on beliefs about effective mathematics teaching.
Case example Claus (male)
Claus is a teacher at a German Gymnasium. The subjects he studied at University are mathematics and physics. He has been teaching for decades. Claus regards the training of students logical thinking as a main goal of mathematics education.
Mathematics education is first of all a basic logical education, exercises in logical thinking.
He names further aims such as taking mathematics as a basis for other sciences or calculating in daily life. These aspects however do not seem to be very important for him because he keeps coming back to logical thinking. He also explains how he exercises logical thinking effectively.
[When choosing tasks] it is very important to proceed in tiny little steps, from simpler tasks to more difficult tasks, that you are not too hasty []
In his lessons he clearly distinguishes between explanation and exercise.
The lesson before clearly determines what I do in the following lesson: Whether I have to do exercises or whether I can proceed with content.
In this system of exercises and proceeding with content he regards his own explanations as highly effective.
The explanations of mathematics are the most important things; the teacher must be able to do this. [] this comes first!
He seems to regard problem solving as highly ineffective, if not as impossible.
The teacher cannot do mathematics lessons as some optimistic pedagogical experts may wish. [] Mathematics has been developed over hundreds of years. How can students invent this by themselves every year? This [mathematics education] is only possible if the teacher directs the lessons and [] explains the things.
In his opinion, open tasks and mathematics do not belong together, and for this reason he does not choose them for his lessons. Additionally, context-related tasks do not seem appropriate to him for mathematics lessons.
[Context-related tasks] are not really important in my lessons, because the most important aim I see is the logical education.
Other parts of the interview (which are not given here) clearly show that he regards context-related tasks as too complicated for his students. The teaching methods he prefers relate to his way of proceeding.
Students have to be able to write down something at the place where they sit [] and they need a direct view to the blackboard, because there everything is explained. So they cannot sit around a big table as in conferences. []The work at the blackboard, with chalk is the most efficient way to explore things to a big number of students.
Additionally, he regards group work as highly inefficient.
Normally [] there are groups where there is one good and two or three weak students,[] Then nothing happens, because the good student loses interest, because the weak students do not want to engage themselves.
Other parts of the interview show that he does not seem to differentiate between the different levels of his students ability in his lessons. Furthermore, motivation does not appear to be relevant for him and he only briefly refers to motivation given by the subject itself and the grades students get. Altogether, for Claus effective teaching seems to consist of explanations given by the teacher and of exercises. In his view, students are not able to solve given problems; he regards context-related mathematics as not necessary to teach students logical thinking.
Case-example Peter (male)
Peter studied Physics and Mathematics to become a teacher at a Gymnasium. He has 13 years teaching experience and is also involved in teacher training. Peters beliefs about effective mathematics teaching differ significantly from those of Claus. First of all, he sees totally different aims for mathematics education.
You have to make students see that mathematics is fun. And this is possible also for low achieving students. [] And it is one big aim for me that my students say [] Okay, this is a problem, I will try to solve it. [] That is all I want.
He obviously wants to enable students to solve problems and this aim seems to be significant for his beliefs about effective mathematics teaching.
Just imagine you would not allow a child to make any mistakes [when trying to learn to walk] or you would say, You have to do it exactly as I do. You can be assured that no child in the world would be able to walk [] and this is how we do mathematics education.
In his opinion, students need to experience things themselves when trying to learn mathematics. Additionally, Peter regards making mistakes as an important part of learning mathematics effectively. In order to give the students the opportunity to make mistakes he often uses group work.
I divide students into groups [] and then they have to work on a problem I give them. Afterwards the groups present the solutions. Some of them may be correct, some brilliant, others will be a catastrophe and others will have interesting mistakes.
References to reality are also important for Peter.
I start off from the situation [] and develop mathematics out of it. And it is a real application which is relevantI to life [] and very often it is an application which is not only relevant to life but also of interest for the students.
In his view, applications do not discriminate low achieving students.
If you give the students a context-related task, then some students of course have difficulties. If I give them other tasks [], other students have difficulties []. You cannot say that especially low achieving students have problems.
The whole interview shows clearly that it is very important for Peter that students work together without the teacher and that he only has to help in cases of urgency. In order to support these aims he names a wide range of methods and he emphasises the necessity to select a method which fits to the chosen task. So, in his opinion, group work, for example, is appropriate when the students need to cooperate to discover new things. Additionally, he uses several media, such as the calculator, the computer, the overhead projector and, of course, the blackboard. If students are to present their group work they are supposed to use posters.
Altogether, Peters beliefs system about effective mathematics teaching focuses on giving the students the opportunity to work independently and to have fun discovering things. So, Peters and Claus beliefs about effective mathematics teaching contrast significantly. They are typical examples of two of the groups of teachers which could be identified. These two groups will now be described in a more general level with the help of a typology.
Types of teachers - their beliefs about effective mathematics teaching
Type I - Transmission teacher: According to the beliefs of this teacher, logical thinking, calculation, spatial thinking and knowledge in mathematics are seen as the most important aims in mathematics education. With the intention of reaching these goals, teacher explanations as well as stereotype exercises are considered to be highly effective. Mathematics lessons which are highly pre-structured by the teacher and leave little leeway for the student to work independently are regarded as highly successful. In contrast, open tasks and problem solving tasks are seen as ineffective. Different levels of student performance seemed to be met by a different amount of tasks while exercising. The teacher seems to prefer the blackboard and the textbook to other media. The main teaching methods chosen are teacher-student talk in plenary and individual work. As reasons for not integrating open problem solving tasks or modelling tasks he referred to the students or the general framework of school or mathematics as a subject. The teacher is of the opinion that students are not able to solve these tasks, do not like solving these tasks or do not understand long texts. He also names a lack of time in lessons, a lack of applications of mathematics, a lack of tasks, too many students per class and an overlap with other subjects. He sees the ability to apply mathematics in daily life as an aim and seems to expect students to do the transfer themselves. This teacher regards the lessons he experienced as a child as good. If he teaches at a Gymnasium, he has formalism-orientated beliefs about mathematics as a science.
Two subtypes could be identified: I.a: Transmission teacher with focus on context-free mathematics: The teacher focuses on context-free mathematics. Context-related mathematics is regarded as not effective according to his aims (see Claus). I.b: Transmission teacher with focus on context-related mathematics: The teacher wants the students to learn how to apply mathematics later in life. To reach these aims, tasks which are related to daily life are regarded as important. However, they have to be directly connected to certain mathematics content; emphasis is put on finding an exact solution. Open context-related tasks and modelling tasks are not considered to be important.
Type II Learning process teacher: This teacher names as the aim of mathematical education the ability to solve problems, to apply mathematics to real life, to see the various characteristics of mathematics and to see mathematics as a cultural heritage. To reach these goals, open tasks and problem solving tasks are regarded as important, and methods where students work independently such as group work are chosen. Explanations by the teacher are not regarded as effective. The teacher tries to meet the students various levels of performance by letting students work on open tasks independently and on their own level. Several media are used. He does not like the mathematics lessons he experienced as a student and became a teacher to improve mathematics education. If he teaches at a Gymnasium, he has process and application-orientated beliefs about mathematics as a science.
Two subtypes could be identified: II.a: Learning process teacher with focus on context-free mathematics: The teacher mainly focuses on context-free mathematics. He seems to use context-related tasks only occasionally to make mathematics more meaningful. II.b: Learning process teacher with focus on context-related mathematics: One aim of the teacher is to show how mathematics can be applied in life. To fulfil this aim, open context-related tasks and modelling tasks are chosen as often as possible. Emphasis is put on tasks which start with the experience of the students (see Peter).
Although, of course, not every teacher matches exactly one of the types, the two groups of teachers can be identified very clearly. In the chosen sample, mostly type I could be reconstructed, both in the Hauptschule and in the Gymnasium. In both types of schools, teachers of type II could rarely be reconstructed. However, despite these similarities, some little differences could be reconstructed between the teachers from the different types of schools. Within type I, teachers of the Hauptschule seemed to put even more emphasis on transmission teaching. Other than the teachers for high achieving students, they pointed out the importance of repeated explanations, of exercising a lot, of having a special exercise book only filled with rules and the importance of explaining only one way of solving a task in order to give the weak students a clear structure. Whereas in the Gymnasium type I.a could be reconstructed quite often, it was type I.b in the Hauptschule.
CONSEQUENCES
The data analyses lead to the construction of two main types in relation to beliefs about effective mathematics teaching. The results indicate an interrelation between the beliefs about teaching and the aims of mathematical education seen by the teacher. In cases where the beliefs about the aims of mathematics education do not seem to meet the beliefs about effective mathematics teaching this may be due to the fact that the teacher does not reflect on how to reach these aims. Furthermore, teachers of Type I simply do not seem to know enough reality-related or open tasks. Often they regarded students as not being able to work independently in groups. Altogether, this could be not only a question of beliefs but also a question of the educational and mathematical competency of the teacher. Differences between teachers beliefs about effective mathematics teaching from different types of school were small. This could be due to the mathematics education the teachers experienced as students (see theoretical framework). The data show that mainly those who liked their mathematics lessons felt encouraged to become teachers, whilst only a few of those who did not like their mathematics lessons became teachers in order to change the situation. This leads to an assumption about why transmission teaching still has a huge relevance. Additionally, for Gymnasium teachers an interrelation between beliefs about effective mathematical teaching and the beliefs about mathematics as a science could be reconstructed, which seems to be in accordance with the results of Kaiser 2006.
Concerning the aim to integrate innovation into day-to-day teaching practice, this refers to important aspects of teacher training: The aims of mathematical education should be discussed with prospective teachers (as well as with in-service teachers in a teacher training course), and there should be some reflection on how to reach these goals. Methods to find and to develop reality-related tasks and open tasks should be a further issue. Links between the content of mathematical lessons and methods should be highlighted (e.g. Peter sees these links). Additionally, the positive aspects of letting students work independently on open tasks should be highlighted. It has to be made clear that strong guidance of weak students leads to dependence, as for example Peter explains. Finally, teacher training should include reflections about the nature of mathematics and its usefulness on a meta-level, together with reflections on the mathematics education they experienced as students, in order to create an awareness of themselves as teachers which allows them to change their beliefs about mathematics teaching developed during school time. Teacher training for teachers at the Gymnasium should include pedagogical aspects, such as motivation. These aspects seem to be relevant for pre-service and in-service training. As beliefs need a long period to be changed, in-service teacher training has to mean long-term training. Pre-service training has to start with these aspects from the very beginning. Some of these demands do not seem to be new. Firstly, however, the results of the study validate them empirically. Secondly, the details of the beliefs reconstructed in the study and the described interrelations show the importance of paying attention to details, such as the aims of mathematical education or reflection about mathematics, and not only to postulate the integration of problem solving into teacher education.
The results of the study also have consequences in terms of research in mathematics education: It is important to evaluate how teachers beliefs about effective mathematics teaching can be changed by long-term intervention. Additionally, it is also important to assess under what conditions a change of teachers beliefs about effective teaching leads to a change of classroom practice. Studies comparing the training of prospective teachers for the Gymnasium and the Hauptschule could help to identify the best components of both systems and lead to an optimization of the training.
References
Bishop, A., Seah, W. & Chin, C. (2003). Values in mathematics teaching the hidden persuaders? In A. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, F. Leung (Eds.), Second international handbook of mathematics education (pp. 717-765). Dordrecht: Kluwer Academic Publishers.
Chapman, O. (2002). Belief structure and inservice high school mathematics teacher growth. In G. Leder, E. Pehkonen & G. Trner (Eds.), Beliefs:, A hidden variable in mathematics education? (pp.177 193). Dordrecht: Kluwer Academic Publishers.
Dionne, J. (1984). The perception of mathematics among elementary school teachers. In J. Moser (Eds.), Proceedings of 6th Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA) (pp. 223228). Madison: University of Wisconsin.
Eichler, A. (2006). Individuelle Stochastikcurricula von Lehrerinnen und Lehrern. Journal fr Mathematikdidaktik, 27 (2) 140-162.
Eichler, A. (2002). Vorstellungen von Lehrerinnen und Lehrern zum Stochastikunterricht. Der Mathematikunterricht, 45, 26-44.
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Appendix: Questions for the interview
Which subjects did you study?
How many years of teaching practice do you have?
Try to explain what mathematics is in your eyes!
In which areas of life can mathematics be used?
Why did you study mathematics?
What are the main aims of mathematics education?
What should a mathematics lesson look like?
Which criteria are relevant for choosing tasks?
How important are context-related tasks for your lessons? Are they more complicated or easier for students?
How important are open tasks for your lessons? How important is problem solving in your lessons?
Which media do you use?
How important is it for you to motivate students?
Which ways of working do you use?
How do you deal with the different levels of performance that students show?
Describe the mathematics lessons you experienced as a child. How did they influence your choice of profession?
(continued)
notes
Katja Maass
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