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CHAPTER 4

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CHAPTER 4

    PRIMARY SCHOOL TEACHERS’ PERCEPTIONS ABOUT THEIR NEEDS

    CONCERNING MATHEMATICS TEACHER EDUCATION

    Solange Amorim Amato

    Universidade de Brasília, Brasília, Brazil

In this paper I describe some of the results of interviews held with seven primary school teachers. I

    was interested in uncovering any difficulties they had experienced at the beginning of their careers

    and how an in-service course taught by me had influenced their understanding of mathematics. The

    intention was to use this information to help me develop a teaching programme for a similar course

    in pre-service teacher education. I also describe how the interview results and the literature were

    combined to provide support for a re-structuring of the way pre-service primary school trainees

    studied the teaching of mathematics. Key findings were: (i) most teachers thought that in pre-

    service teacher education trainees should acquire a more practical knowledge base about how to

    teach mathematics; (ii) some teachers revealed that they avoided teaching mathematics whenever

    possible at the beginning of their careers and (iii) most teachers wished they had had more time in

    the in-service course to discuss the teaching of certain content in greater detail. These findings led

    me to decide that the programme should focus on: (a) helping primary school trainees to acquire a

    strong relational understanding and some pedagogical content knowledge of most content in the

    primary school curriculum and not just a sample of it and (b) providing them with direct experience

    of appropriate ways to teach particular topics.

Issues For Debate Raised By This Paper

    (1) What do primary school trainees need to learn about subject matter knowledge and pedagogical

    content knowledge in teacher education in order to start teaching in a simple but relational way so

    that their first students do not face many problems when trying to make sense of mathematics and of

    further related content?

    (2) Some teacher educators believe that working towards developing teachers who are autonomous

    and who seeks study groups and other means of learning and growth is incompatible with the idea of

    learning about subject matter knowledge through formal instruction. In which ways does trainees‟

    acquisition of subject matter knowledge in pre-service teacher education hinders or helps their future

    autonomy as teachers?

    (3) What are the social-political implications of teachers‟ instrumental understanding of mathematics?

    (4) Bennett (1993) recommends that primary school teachers should have the necessary subject

    knowledge for teaching the mathematical content to the highest level expected of students doing that

    stage of schooling. What other aspects does a strong relational understand for the purpose of

    teaching mathematics entail?

Keywords That Describe The Focus Of This Paper

    (1) Primary pre-service teacher education, (2) Subject matter knowledge, (3) Pedagogic content

    knowledge, (4) Mathematical representations and (5) Relational understanding of mathematics

Background And Purpose

    Wilson et al. (1987) identified seven knowledge components which teachers may use in order to make decisions for the purpose of teaching and to help them promote understanding on the part of

    their students. One of these components is subject matter knowledge (SMK) which includes both the

    substantive and syntactic structures of the discipline. The focus of this paper will be on teachers and

    trainees‟ acquisition of substantive understanding of the mathematics they will teach. However,

    pedagogical content knowledge (PCK) which “includes an understanding of what it means to teach a

    particular topic as well as knowledge of the principles and techniques required to do so” and general

    pedagogic knowledge or “knowledge of pedagogical principles and techniques that is not bound by

    topic or subject matter” (p. 113) are also mentioned.

    Twenty one years ago I started teaching mathematics to 11-17+ year olds with a very positive attitude towards the subject. Good performance in very demanding mathematics course components

    at university level made me think that I had acquired enough content knowledge for teaching the

subject at school level. It did not take long after I started teaching to notice that I did not have the

    kind of knowledge necessary for teaching even the most basic content to my first class of 11 year

    olds. I could present my students with correct procedures, but could not answer most of their

    questions concerning the reasons for using certain steps in the procedures. I remember well the first

    student‟s question which made me aware of such fact. After just 3 months I started teaching, an 11

    year old was puzzled by the result of 1/2 x 1/3 = 1/6 and asked “1/6 is smaller than 1/2 and 1/3. Why

    do we get a smaller result number when multiplying fractions?” Neither the school and university

    mathematics, nor the more general pedagogic knowledge, I learned in the departments of psychology

    and education helped me answer this question.

    The several mathematics components I did in the mathematics department were not directly related to school content. Studying more advanced mathematics, like calculus and analysis, often

    does not provide trainees with enough opportunities to revise and deepen the content they will have

    to teach (Brown et al., 1997). I had one course component which focused on methods for teaching

    mathematics at schools, but this was neither enough to discuss methods for teaching all the

    mathematics contents taught to 11-17+ year olds, nor to address the mathematical deficiencies I

    presented with respect to the mathematics contents I would have to teach in the future. The problem

    is not much different in primary school teacher education. While mathematics teacher education

    focuses on mathematics components, primary school teacher education focuses on general pedagogic

    knowledge. There is only one course component which focus on methods for teaching mathematics

    to 7-10 year olds.

    My initial experiences as a school teacher and much later my experiences as a teacher educator led me to think that both mathematics and primary school trainees may not have an appropriate

    understanding of the mathematics content they have to teach. The problem also seems to exist in

    developed countries and have been discussed in the literature about teacher education. It is widely

    recognised, that primary school teachers and trainees‟ knowledge of mathematics is, on the average,

    insufficient (e.g., Philippou and Christou, 1994; Ma, 1999 and Goulding et al., 2002).

    For Skemp (1976) relational understanding involves knowing both what to do and why it works, while instrumental understanding involves knowing only what to do, the rule, but not the

    reason why the rule works. Research on teachers and trainees‟ understanding of mathematics tends

    to show that they often only have an instrumental understanding of the content they have to teach

    (e.g., Ma, 1999 and Goulding et al., 2002). Research has also demonstrated a relationship between

    teachers‟ SMK and teaching effectiveness (e.g., Putnam et al., 1992 and Rowland et al., 1999).

    Wilson et al. (1987), Fennema and Franke (1992) and Bennett (1993) review some studies which

    started to investigate teaching practices as a mediator between teachers‟ knowledge and student

    learning. The results suggest that there is a positive relationship between teachers‟ SMK and their

    instructional practices. One of the most important factors in such a relationship seems to be teachers‟

    organisation of substantive knowledge. The collective message gathered from those studies is that

    teachers who possessed more organised and interrelated knowledge tended to be more relational in

    their teaching, while those with a poor relational understanding tend to base their teaching mainly on

    instrumental teaching. Students‟ learning from school may, therefore, be influenced by the

    “inadequacy of their teachers‟ knowledge of mathematics” (Fennema & Franke, 1992, p. 147).

    McDiarmid and Wilson (1991) maintain that SMK is not sufficient to teach mathematics but it is a necessary condition to relational teaching. They argue that only a few students can discover

    things by themselves. The majority, however, need the help of someone who has the mathematical

    understanding to translate concepts and operations into representations that are mathematically

    appropriate and that can be helpful to a variety of learners. Much SMK is needed to select

    appropriate mathematical representations and make an effective use of them (Leindhart et al., 1991).

    Ball and Feiman-Nemser (1988) found that trainees‟ instrumental understanding produced incorrect

    representations of mathematics (“one-fourth of 100 equals 25” was represented as 1/4 ? 100 = 25

    instead of 1/4 x 100 = 25 or 100 ? 4 = 25). Putnam et al. (1992) conclude that teachers who have

    limitations in their SMK may not realise that they or their students have made incorrect calculations

    turning the mathematical content as secondary to the classroom activities they engage their students

    in. Thus teachers' instrumental understanding may not only affect students‟ opportunities to learn

    mathematics relationally, but also contribute to the development of misconceptions in their students'

    minds.

    My experiences of teaching in pre- and in-service teacher education led me to undertake a research project with the main aim of investigating ways of helping primary school trainees to

    improve their understanding of mathematics in pre-service teacher education. Ball and McDiarmid

    (1990) argue that continued documentation about teachers and trainees‟ lack of mathematical

    knowledge will not contribute much to ameliorate the problems encountered in teacher education

    and teaching. The implication is that research-based methods of tackling the problem are required.

Methodology

    I carried out an action research (Amato, 2001) with the aims of improving primary school trainees‟ understanding of, and attitudes to, mathematics. In a first phase of the reconnaissance stage,

    trainees‟ liking for and understanding of mathematics were investigated empirically through two

    small questionnaires that were administered to two different samples of trainees (n1 = 224 and n2 =

    184 respectively). In a second phase of the reconnaissance stage seven primary school teachers were

    interviewed in the hope that their answers and suggestions would provide more information in order

    to clarify the two research problems and to improve the plan for the first action step. As this action

    research was concerned with my teaching practice as a teacher educator, they were teachers whom I

    had taught in in-service courses with similar content to the course component I would be teaching in

    the action steps of the research. Those teachers had also done a similar course component at pre-

    service teacher education and had had experience in teaching mathematics to children, so they were

    thought to be more able to provide information about novice teachers‟ initial difficulties in the

    teaching mathematics than trainees or teacher educators. They could also reveal their opinions about

    how their pre-service teacher education could have been different and prevent them from having

    some of those difficulties. So they could provide me with useful suggestions for the action steps of

    this research.

    There were two main action steps and each had the duration of one semester thus each action step took place with a different cohort of trainees. A teaching programme was designed in an

    attempt to: (a) improve trainees‟ relational understanding of the content they would be expected to

    teach in the future and (b) improve their liking for mathematics. Much information was produced by

    the data collected in the reconnaissance stage and in the two action steps but, because of the

    limitations of space, this paper only presents some of the results from the interviews with the seven

    teachers (second phase of the reconnaissance stage) and indicate how the analysis of the data

    affected the action steps of the research. An interview schedule was designed to be used as an aide-

    memoir. The questions were separated into three main sections which asked the teachers about: (1)

    their first teaching experiences and difficulties, (2) teaching for relational and instrumental

    understanding and (3) the in-service course and their suggestions about how to develop a new

    programme for teaching trainees in pre-service teacher education.

Some Results

    The teachers who took part in the reconnaissance interviews had all taken in-service courses taught by me which varied in content. Teachers 1, 2 and 3 had participated in a 120-hour course

    totally dedicated to mathematics teaching in grades 1 to 4. The rest of the teachers had participated

    in a course about the teaching of numeracy and literacy in grades 1 and 2. Only half of that course

    (60-hours) was dedicated to the teaching of mathematics. With the exception of Teacher 7, who was

    a novice teacher, the teachers had experienced many years of teaching primary school classes. All

    teachers had done both a vocational course at school level and a college pre-service teacher

    education course. Several categories emerged from the teachers‟ answers to the reconnaissance

    interviews. However, the categories presented in this paper are only those which had a more direct

    effect on the teaching programme developed for the action steps of the research. The three

    categories selected to be reported in this paper were:

    ? (i) The need for an initial knowledge base

    ? (ii) Avoiding the teaching of mathematics

    ? (iii) Not having discussed all mathematical content they had to teach

(i) The need for an initial knowledge base

    Only Teacher 7 said she did not have any difficulties in starting to teach mathematics. She was

starting her career as a primary school teacher and her responses differed in some senses from the

    more experienced teachers. The other six teachers said that they had experienced many difficulties

    and blamed mainly their training in teacher education. They complained that their college teacher

    education courses were too theoretical and that they did not learn much about how to teach

    mathematics in their pre-service courses. Teacher 1 said she did not know how to work with the

    content: “As I did not know another option to work, I would teach in the way I was taught”.

    Most of the time during the interviews the teachers emphasised the positive aspects of the in-service course. It seems that they were so much in need of practical suggestions for teaching

    mathematics that they did not pay much attention to the negative aspects of the course. When asked

    what could be improved in the programme of the course Teacher 3 said: “The course came in good

    time to help us change our teaching in the classroom. Then we could not find any flaws in the

    course”. It seems that, to learn from teaching, teachers need first an initial knowledge base to rely on.

    Teacher 1 argued:

    To teach from your own experiences you are going to use what you have. I used what I had.

    I do not know about other teachers but I needed some guidance. Today perhaps I do things

    that would allow me to learn by myself but when you start it is very difficult. What are you

    going to use? You are going to use what you have.

    Some teachers like Teacher 1 seemed to both need and wish for some guidance. Ignoring such needs seems to be ignoring teachers and trainees‟ prior knowledge (Ausubel, 2000). Apart from

    SMK and other knowledge components, teachers also need PCK which is knowledge about how to

    teach specific subject matter. The need for guidance, mentioned by Teacher 1, seems to be related to

    the need for acquiring some initial PCK in the case of a novice teacher. Because PCK is directly

    related to teaching practices, Ernest (1989) has called it practical knowledge. Dunne (1993) found

    that the trainees with less confidence in their mathematical knowledge, those who had less

    experience with children or less experience with teaching, were the ones who most often required

    “more „ideas on a plate‟ from tutors, more clear cut ways of operating” (p. 103). The discussion of

    teaching strategies and materials for teaching particular mathematical content are sometimes

    interpreted by some academics in a negative way and as providing trainees with recipes or

    procedures. I tend to interpret teachers‟ wishes for directions and suggestions in a very positive way,

    as a demonstration of interest, desire to improve education and professionalism.

    Decisions made: Trainees‟ acquisition of an initial form of PCK in pre-service education

    involves acquiring a repertoire of representations and activities that can be transformed by the

    teacher for classroom use. Novice teachers‟ reflections on what happened in the classroom as a

    result of their decisions and actions seem to be more important than where the teaching strategies

    and materials they use comes from. Slowly they can start combining the ideas gathered from

    textbooks, teacher educators and more experienced teachers with their own ideas. I thought that the

    trainees in the action steps of the research would need to experience and reflect about some practical

    activities that they had probably not experienced as school students and, in this tacit way, to acquire

    an initial repertoire of mathematical representations and activities that have the potential to develop

    relational understanding of the primary school curriculum.

(ii) Avoiding the teaching of mathematics

    I did not ask the teachers about their attitudes to mathematics. In another phase of the reconnaissance stage attitudes to mathematics was investigated with a sample of primary school

    trainees. The focus of the interviews with the teachers was on their initial experiences in teaching

    mathematics and on the in-service courses. However, Teacher 2 revealed how her negative attitudes

    to mathematics had affected her initial teaching. Teacher 3 commented that her weak PCK also had

    affected her teaching in a similar way. Both teachers said that, at the beginning of their careers, they

    avoided teaching mathematics when possible and spent more time on the teaching of literacy:

    Teacher 2: I never enjoyed mathematics much. When we entered the classroom, at the

    beginning [of our career], we started spending more time on teaching the subject we liked

    more. ... Primary school teachers only wish to teach the child how to read and write. The

    teaching of mathematics is frequently forgotten. Either we only deal with it in a superficial

    way or we deal only with the most basic topics, those that I feel myself obliged to teach.

    This is the result of the failure we ourselves experienced in mathematics, failure between

    inverted commas. ... Failure as a student. We end up spending more time on teaching the

    subject we feel better about. Often we end up making the student dislike mathematics

    because we ourselves do not identify with the subject.

    Teacher 3: Generally, when we have some problem, we try to run away from it. It is what

    happens with the teacher that starts teaching. In my case it was 1979. I ran away from

    mathematics. It was necessary for the co-ordinator to ask us what we were teaching in

    mathematics to remind us to teach it. The greatest worry was in teaching the students how

    to read and write. However, unconsciously it was an escape because we did not know how

    to work with mathematics, how to teach it. It was as if we were running away from failure.

    Teacher 4 said she tended to postpone the teaching of certain content because she did not master them:

    We never finished teaching all the content of a grade. We kept teaching only the most basic

    content, the four operations. We did not teach geometry which is very interesting. The

    content became very restricted. Perhaps because of the lack of materials or even the lack of

    knowledge. Many times we postponed the teaching of topics we did not master.

    Teacher 6 said that liking mathematics was the most important aspect of teaching and learning about mathematics teaching. She expressed her perceptions about her colleagues‟ feelings towards

    mathematics and how this appears to affect their teaching in terms of time allocation: “I hear my own

    school colleagues saying that they emphasise something else because they do not like mathematics.

    Then they do not teach much mathematics to their students”. Teachers‟ attitudes to mathematics are

    said to affect the way they will teach in the future (Ball, 1988) and the classroom ethos (Ernest,

    1989). According to Bromme and Brophy (1986), teachers model their attitudes and beliefs during

    their teaching. In most cases messages are conveyed without teachers‟ awareness. Yet the most

    direct influence of primary school teachers‟ negative attitudes to mathematics on their students‟ learning appears to be time allocation. Bromme and Brophy point out that “such teachers have been

    found to allocate more instruction time to subject-matter areas that they enjoy, and less to areas that

    they dislike” (p. 122). Low time allocation was found to restrict students‟ opportunities to learn (e.g.,

    Fisher, 1995). Therefore, teachers need to improve their liking for mathematics and to be aware of

    the benefits of high time allocation especially for activities which have the potential to develop

    relational understanding.

    On the other hand, Widmer and Chavez (in Jordan, 1987) maintain that it can not be assumed that all teachers who did not like mathematics as students, will dislike teaching the subject. “Many, in

    fact, seem eager to break the cycle of poor attitudes engendering poor attitudes and are determined

    to provide their pupils with positive experiences in learning mathematics” (p. 239). So they may be

    more interested in learning different teaching methods from the ones they experienced at school. One

    of the trainees interviewed by French, (1988) remarked: “Perhaps the fact that I‟ve had problems

    with maths but feel better about it now will help me to be more understanding with the pupils‟

    different abilities” (p. 80). Yet to feel better about mathematics involves experiencing new ways of

    learning mathematics for understanding in pre-service teacher education. All the teachers interviewed

    said that they enjoyed the more practical aspect of the in-service course. They said that learning how

    to use the concrete materials improved their own understanding of mathematics and their teaching

    practice: "Teacher 1: After each meeting we could easily apply the ideas in the classroom. I did it and

    I saw the results with my students”. Teacher 2 also mentioned the role of concrete materials in

    improving her attitude to mathematics: “I started liking mathematics together with my students. I

    perceived that, when you lean on something concrete, when you saw the „whys‟ in mathematics, it

    was not a beast with seven heads”.

    Decisions made: Most of the time in the in-service courses the teachers only saw the

    manipulation of concrete materials by other colleague teachers who volunteered to perform the actions to the whole group on a large place value board placed near the chalkboard. However, it seems that just seeing those representations helped them in gaining some confidence in teaching mathematics. In order to help future trainees become more confident, I decided to make an effort to let them manipulate concrete materials to represent concepts and operations much more often than it had been done in previous courses. It would be also important to discuss the presentation of some mathematical concepts to pre-school children in a spiral way (Bruner, 1960) in order to increase students‟ opportunities to learn. In this way trainees could also notice how the construction of concepts which were considered complicated could be started much earlier at school if done in a very informal way through practical activities with concrete materials, simple spoken language and games.

(iii) Not having discussed all mathematical content they had to teach

    With the exception of Teacher 3, all the teachers I interviewed said that everything in the in-

    service course had been very helpful to their work, but they complained that the in-service course had been too short to discuss all they needed for teaching mathematics to initial grades. The in-service courses they had attended varied in the amount of teaching time and so in the amount of mathematical content and representations discussed. Teacher 3 was more interested in learning about the teaching of mathematics to the first and second grades and so mainly about the teaching of place value and operations with natural numbers. According to her, the content of the in-service course had been enough to help her teach those grades and learn more about mathematics teaching from teaching such content. Most of the teachers, however, wished they had had more time to discuss the teaching of certain content in greater detail.

    Teacher 1 did not seem to be able to transfer the ideas discussed in detail for natural numbers,

    fractions and measurement of length to the teaching of decimals, area and volume. She said she enjoyed the practical activities and applied most of them in her classroom, but the ideas that were only superficially discussed in the course were not put into practice and she still had difficulties in teaching them: “There are things that I continue to do in the same way as before. The course was not

    enough. Time was scarce and we did not discuss those things in detail”. She also complained about not having had a school co-ordinator to discuss her doubts while attempting changes in her practice.

    Teachers 4, 6 and 7 also expressed their worry about not having discussed all the content in the

    whole primary school curriculum. Teachers 5 and 7 thought that the course should have included more about fractions. The main mathematical content of grades 1 and 2 is place value and the four operations with natural numbers. Only the concept of “half” is included in the curriculum of grade 2. Although the relationship between fractions and division was discussed in detail, the teachers were afraid of having to face a third or fourth grade (9 and 10 year olds respectively) in the following years and not having learned how to work with fractions: “Teacher 5: I have noticed that teachers have greater difficulty in teaching fractions ... Novice teachers are not used to working with fractions in a more concrete way”.

    Decisions made: It became apparent that rational numbers, geometry and measurement of

    length, area and volume were important issues for teachers. So I decided to place a great emphasis in the teaching of these topics in the action steps of this research. Most of the teachers seemed to have the common desire to learn much more than they knew about the teaching particular mathematical content. Their main suggestion was to increase the duration of the in-service courses. In the past I had opted for teaching less content, to be able to do it in a more detailed way. After the interviews, it became clear that in pre-service teacher education trainees needed to: (i) acquire a strong relational understanding to be able to teach the curriculum as a coherent and organic whole by emphasising the relationships among concepts and operations, (ii) acquire a repertoire of representations about most contents in the primary school curriculum and not just a sample of it and (iii) experience a range of children‟ activities that could be appropriate to teach some concepts and operations from very informal pre-school stages to final stages in formalisation.

    Chinn and Ashcroft (1993) write about the dynamic interactions of the parts and the whole in

    the learning of mathematics: “It is a subject where one learns the parts; the parts build on each other

    to make a whole; knowing the whole enables one to reflect with more understanding on the parts, which in turn strengthens the whole” (p. 3). So a strong relational understanding of mathematics (the whole) involves knowing much of its content (the parts) and how the content has been put together (the connections). As many trainees would be teaching different grades in a very near future, it was decided to make a greater effort in improving their relational understanding of a good range of mathematical content in the primary school curriculum. In the case of primary school teachers a strong relational understanding for teaching mathematics also involves having the necessary subject knowledge for teaching the mathematical content to the highest level expected of students doing that stage of schooling (Bennett, 1993) so the programme also had to include children‟s activities for the

    formalisation of concepts and procedures in the primary school curriculum.

    Teaching time was anticipated to be the greatest problem in the action steps of this research

    because all teaching strategies I know to help students acquire relational understanding require a good amount of time to be put into practice. In order to have more time for the re-teaching of more complex topics in the primary school curriculum, I reduced the discussions about theories of

    teaching and learning mathematics. I decided to present only some theoretical ideas about the use of representations in the teaching of mathematics. This small theoretical part of the teaching programme would be presented at the beginning of the semester and revised after some related activities. I also decided not to include any discussions about the teaching of topics that were considered easy for teachers like small numbers, money, measurement of capacity, mass and time.

    The idea of in depth discussions about mathematical learning and teaching theory was not thought to be appropriate, especially if the data collected in the action steps of the research revealed that the trainees needed to improve their understanding of the content they would have to teach in the future. A more practical approach was thought to help them learn some SMK and PCK in a more experiential (Brown, 1992) or tacit way (Sotto, 1994). So I opted for focusing the teaching on helping trainees acquire relational understanding of the mathematical content they had to teach in the future and not on theories of teaching and learning mathematics. Providing examples of the theories in practice was thought to be more profitable than asking the trainees to read books and articles about them. A more theoretical approach was thought to be appropriate only for future course

    components about mathematics teaching or for future in-service courses after the trainees had: (i) more relational understanding, (ii) acquired a repertoire of mathematical representations and activities and (iii) had experienced some of those theories in a more tacit way through the activities in the programme. In other words, I decided to turn around the previous ways of learning: theoretical in the pre-service phase, practical in the in-service phase.

    Final decisions made about the teaching programme: In developing a new programme for a

    pre-service course I thought that trainees needed more than anything to feel confident about their capacity to learn and teach the mathematics in the primary school curriculum. Therefore, I had to decide what were the most basic aspects of PCK in order to help novice teachers start teaching in a more relational way. I also had to consider the short time available in Brazilian teacher education for course components about mathematics teaching and trainees‟ weak relational understanding to deal

    with certain teaching strategies. Another problem was the large number of trainees enrolled in each class (about 44 trainees).

    In the action steps of the research the re-teaching of mathematical content was integrated with

    the teaching of pedagogy by asking the trainees to perform children‟s activities which have the potential to develop relational understanding of the subject. So the main strategic action was an attempt to improve trainees' relational understanding through methods similar to those that could be used with their future students. Four other more specific teaching strategies were selected for the programme and they had the following aims in mind: (i) promote trainees‟ familiarity with several

    mathematical representations for each concept (real world contexts, concrete materials, pictures and diagrams, spoken languages and written symbols); (ii) expose trainees to several ways of

    representing and performing operations (with the aid of concrete materials, mentally and with written symbols); (iii) help trainees to construct relationships among concepts and operations and (iv) facilitate trainees‟ transition from concrete to symbolic mathematics. These teaching strategies were thought to provide greater short-term benefits for the trainees‟ relational understanding and for their

    future practice than other strategic actions that could have been adopted. The results of all the

decisions presented in this paper will be published elsewhere.

Discussion

    My belief that instrumental teaching was going on in schools in Brasília and in teacher

    education was confirmed by what the teachers said in interviews. What I was not expecting was the

    allocation by teachers of less teaching time to mathematics than to other subjects. More research is

    certainly needed in order to investigate the extent of this “avoidance” problem and how it can be

    prevented. In this study, similarly to the results found by Good and Brophy (in Bromme and Brophy,

    1986), low time allocation to the teaching of mathematics seemed to be related to teachers' negative

    attitudes, instrumental understanding and lack of PCK. Avoiding the teaching of certain content

    could be even more damaging to children‟s learning than only teaching them instrumentally. It is worrying to have found this result in Brasília and makes me think that the problem can be even more

    acute in less developed regions of Brazil where some primary school teachers have not done any

    teacher education course and others have not even finished school. Taking into consideration that the

    teachers who participated in this research all had done a teacher education course at tertiary level

    and that the avoidance problem has already been mentioned in the literature in English language (e.g.,

    Bromme and Brophy, 1986), it seems appropriate to suggest that teacher educators and researchers

    from developed countries should also investigate its existence.

    Learning some SMK and PCK from my own teaching experiences and from other teachers

    proved to be a very slow process. It took me a long time and a great effort to acquire some relational

    understanding and PCK while teaching several large classes simultaneously. Learning mathematics

    from teaching also seems to be a slow process for primary school teachers, as they have to teach

    several subjects simultaneously. When teachers find the time to work together in study groups they

    should be discussing complex problems related to their practice and not dedicate their precious time

    trying to acquire SMK of the mathematics they teach which I consider a basic part of their

    professional knowledge and so the responsibility of pre-service teacher education. An initial

    knowledge base, which I think it is a combination of a strong relational understanding of

    mathematics (SMK) and knowledge of a repertoire of representations (PCK), must be available to

    trainees in pre-service teacher education when they are supposed to have the time to dedicate

    themselves to such learning (Ma, 1999). Otherwise their first students may well be led to think that

    mathematics is a complicated and unreachable form of knowledge because their teachers have not yet

    learned ways of communicating the subject.

References

    Amato, S. A. (2001). Brazilian Primary School Student Teachers’ Understanding of, and Attitudes

    to, Mathematics, Thesis Submitted for the Degree of Doctor of Philosophy, Linacre College,

    Oxford University.

    Ausubel, D. P. (2000). The Acquisition and Retention of Knowledge: A Cognitive View, Dordrecht,

    The Netherlands: Kluwer Academic Publishers.

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