MATHEMATICS AND MATHEMATICS TEACHING:
PERCEPTIONS OF BEGINNING SCOTTISH PRIMARY SCHOOL TEACHERS
Professor Donald S Macnab and Mrs. Frances Payne
Northern College
Paper presented at the European Conference on Educational Research, Edinburgh, 20-23 September 2000
NOTE:
An expanded version of this paper is being prepared for journal publication. The material contained in the paper is Copyright and may not be quoted or reproduced without the written permission of the authors.
ABSTRACT This paper presents the results of the first stage of an investigation into the attitudes towards and feelings about mathematics and mathematics teaching of Scottish first and final year Primary student teachers following a 4 year Bachelor of Education (BEd) degree course and Primary students completing a one-year Postgraduate Certificate of Education (PGCE) course. The research, which is part of a wider investigation into factors and processes underlying and affecting curriculum change with particular reference to mathematics education, seeks to locate the views of beginning primary schools teachers within a broad range of attitudes to mathematics and to styles and approaches to the teaching of mathematics in Primary schools, and to relate these to attitudes of beginning teachers in other countries and to issues of curriculum development and implementation of policy. Data collection is by questionnaire survey of all beginning teachers in Scotland, with subsequent selective interviewing.
INTRODUCTION
The process of curriculum change and development involves curriculum in a number of levels or aspects. There is the intended curriculum as formulated by policy-makers, the curriculum as implemented by teachers, the curriculum as experienced by pupils, and the attained curriculum - the learning that pupils achieve, (Robitaille, 1993).
Connections between these levels are mediated by teachers who, alongside their professional and cognitive skills and understanding, bring to the task a variety of beliefs and attitudes which interact with these skills and understanding to provide an interpretative framework within which implementation takes place, and which has a substantial influence on the pupil experience. There is an extensive literature dealing with these interactions; some exemplars, relating both to student teachers and teachers in practice, are listed below.
Brown,(1999), for example, argues from what is essentially a socio-cultural position that there is a dichotomy between perceptual and structural aspects of mathematics in the minds of student teachers which requires resolution, while Schuck, (1996), refers to chains of belief which enmesh students' thinking. Fleener et al., (1997), investigated students' views of actual and ideal classroom environments. Kyriakides, (1996), emphasises that the transformation of curriculum reform into practice depends, at least in part, on teachers' perceptions of the purposes of mathematics; Stables, (1999), quotes Stones (1983), to the same effect.
Foss and Kleinhasser, (1996), refer to the complexities of the relationship between teachers' knowledge, beliefs, and professional practice. Tann, in Calderhead (Ed.), (1993), uses the term 'personal theory' to encompass the beliefs, values, understandings and assumptions which teachers (and student teachers) possess and which to a greater or lesser extent determine their views of teaching and learning in the classroom. She argues that such personal theory usually exists at an implicit level and hence may be difficult to articulate and identify and hence difficult to unearth and examine.
A general overview in a European context is contained in Hannalu (Ed) 1998. The Association for Teacher Education in Europe (ATEE) is also investigating the area, (Jong and Brinkman, 1999).
The present study seeks to:
• document general characteristics of beliefs and attitudes (personal theory) of students completing Primary school teacher education courses in Scotland, both to mathematics itself, and to styles of and approaches to mathematics teaching and learning;
• compare and contrast beliefs and attitudes between BEd and PGCE students;
• compare and contrast beliefs and attitudes between first year and final year BEd students;
• relate these beliefs and attitudes to
those of beginning teachers in some other countries,
issues of curriculum development and implementation of policy;
• make recommendations for
courses of teacher education for Primary schools,
processes of curriculum change in schools.
It is part of a wide-ranging investigation into factors and processes which influence curriculum change with particular reference to mathematics.
The paper presented here outlines some initial findings from the study, concentrating on the first two of the above aims.
METHODOLOGY
Organisation
In Scotland, there are two entry routes to Primary school teaching - the four-year BEd. degree course and, for those already possessing an appropriate degree, the one-year Post-graduate Certificate of Education (PGCE) course. Both types of course are provided in five institutions. Four of these are Universities - Edinburgh, Glasgow, Paisley, and Strathclyde. The fifth - Northern College - has twin campuses in Aberdeen and Dundee, and is currently in the process of demerging and amalgamating with the Universities of Aberdeen and Dundee respectively.
The principal official document relating to mathematics in Primary schools in Scotland is Scottish Office Education Department (1991) - Curriculum and Assessment in Scotland National Guidelines: Mathematics 5-14, and it is this document which underlies the mathematical component of teacher education courses. More recently, HM Inspectorate of Schools (Scotland) issued a further document - Improving Mathematics Education 5-14 (Scottish Office Education and Industry Department, 1997), at least partly as a consequence of the relatively poor performance of Scottish children both in national tests of mathematics attainment (the Assessment of Achievement (AAP) Programme), and in the Third International Mathematics and Science Study (TIMSS). Further background and detail on these and other matters relating to curriculum development in primary school mathematics education in Scotland can be found in Macnab (1999).
The co-operation of staff in all five of the institutions was sought and obtained to collect data by questionnaire survey of their BEd Year 4 and PGCE (Primary) students, together with, for comparison purposes, a smaller scale survey of students in the 1st year of their BEd. degree course. Some subsequent selective interviewing was also carried out. The BEd Year 1 survey took place in December 1999, with BEd Year 4 and PGCE (Primary) in the spring of 2000.
To ensure reasonable independence of response and to maximise the percentage return, institutions were asked to administer the questionnaires to a complete student year group at the same time and to collect completed questionnaires at that time. The authors are very grateful for the willing co-operation they received from their colleagues in the carrying out of these arrangements.
Overall, the percentage returns with institution spreads were as follows:
• BEd Year 1: 79% Spread 53% - 91%
• BEd Year 4: 72% Spread 28% - 97%
Note: Omitting the institution with a return of 28%, the overall return was 78% with a spread of 66% to 97%
PGCE: 62% Spread 56% - 86%
Note: One institution with a return of 10% was omitted from the PGCE figures.
The subsequent interviewing was carried partly face-to-face, and partly by telephone. Only a few self-selected students were involved, but it was felt valuable to have the opportunity to explore in more depth some of the questionnaire issues.
Questionnaire construction
The questionnaire was designed to give respondents the opportunity to express their views on a range of matters connected both with their own experience of mathematics and with mathematics teaching in the Primary school.
By its nature, a questionnaire cannot achieve the degree of in-depth insight which is potentially possible in an extended discussion or interview. On the other it can access the views of a much larger number, and, if effectively constructed, can lay claim to provide an comprehensive picture of the views of the groups under survey.
Three question styles - choosing words from lists, four-point value scales, and rank ordering - were used in the questionnaire, both to encourage engagement with the import of the questions, and in an attempt to elicit underlying and possibly not well articulated aspects of the students' views and feelings about mathematics and mathematics teaching. As far as possible the questions were framed to be neutral with regard to approaches to and methodologies of mathematics teaching; they did not emphasise to any one approach or methodology over others. Information was also sought on age, gender and school level of attainment in mathematics.
Critical comment on early drafts was provided by colleagues; later drafts were piloted on BEd Year 3 students.
FINDINGS
Background data
Overwhelmingly (90%+) students in Primary teacher education courses in Scotland are female, and 50% possess mathematics qualifications in excess of the minimum required for entry. While BEd course students are drawn mainly from schools with 80% in the 17-23 age-group, the pattern in PGCE course is more diverse - approximately 20% in the 17-23 age-group, 40% aged 24-30, and 40% aged 30+, indicating that few are admitted to the course end-on to the conclusion of their higher education studies.
Feelings About Mathematics
Respondents were asked to select two words from a list to characterise their feelings about mathematics when they were in Primary school, when they were in Secondary school, and now. The words provided were:
Fun, Useful, Incomprehensible, Interesting, Boring, Routine, Exciting, Difficult.
For each student group, the four most frequently chosen words were as follows (figures in percentages):
BEd 1 BEd 4 PGCE
In Primary school: Fun 45 38 30
Useful 37 39 41
Interesting 35 39 41
Routine 42 45 40
In Secondary school: Useful 32 27 26
Boring 32 26 27
Routine 30 38 36
Difficult 53 54 49
Now: Useful 67 70 71
Interesting 52 43 49
Fun 16 23 31
Routine 16 24 15
The least chosen words for all groups were exciting and incomprehensible. In general the choice patterns for Primary school and Now were broadly similar with positive words predominating, while for the Secondary school by far the most frequent choice was difficult. The overall impression gained is the students' view of mathematics was positive in Primary school, turned negative in Secondary school, and became positive again in their teacher education course.
Students were also asked about their feelings now when working on mathematics tasks; the format again a word choice from a list. The first of these questions asked generally about feelings when working on a mathematics task, providing the list:
Stimulated, Worried, Unenthusiastic, Confident, Bored, Determined.
The top three choices were as follows, in order of choice:
BEd1: Determined 63% Worried 46%, Stimulated 36%,
BEd4: Determined 54% Confident 40%, Stimulated 38%,
PGCE: Determined 61% Stimulated 51%, Confident 41%,
Very few students chose bored (below 10%).
The second of the questions asked more particularly about students' feelings when they encountered difficulty in a mathematics task, the word list provided being:
Discouraged, Indifferent, Frustrated, Ashamed, Motivated, Annoyed.
Here the top three responses for each group were:
BEd 1: Frustrated 78%, Annoyed 43%, Discouraged 32%
BEd 4: Frustrated 72%, Annoyed 34%, Motivated 32%
PGCE: Frustrated 71%, Motivated 40%, Annoyed 31%
Students' Own Understanding of Mathematics
From a wide range of thirteen topics drawn from the Primary mathematics curriculum, over 70% of BEd 4 and PGCE students rated their understanding as very good/good. This ranged from 99% (PGCE) for whole number/arithmetic to 70% (BEd 4) for algebra. The only topic where this level of understanding fell below 70% was ratio (58%). However for BEd 1 students, in a number of topics: fractions, decimals, positive and negative numbers, and information handling/statistics, their rating of very good/good was below 70%, (range of 53% to 67%).
Except for the later section “Teaching and Learning in Mathematics” the remaining sections deal only with findings from BEd 4 and PGCE students because questions were not asked of BEd 1 students, due to their insufficient experience of the teaching of mathematics.
Attitudes Towards Teaching Mathematics
With regard to the four principal components of the Primary school curriculum - Environmental Education, Expressive Arts, Language, and Mathematics, respondents were asked to choose which words from a list provided, best described their feelings about teaching each of these components. The list consisted of:
Easy, Difficult, Dull, Exciting, Worrying, Enjoyable, Boring
For both BEd 4 and PGCE students, the responses of enjoyable (75%+) predominated for Environmental Education, Expressive Arts, and Language, followed by exciting (60%+) for the first two of these, and to a lesser extent for Language. For mathematics, while enjoyable predominated (~70%), exciting did not (~30%), and was ranked on a par with difficult and worrying. Dull and boring were chosen by very few students (7% and less) in all of the components.
Teaching Methodology and Classroom Organisation
Both BEd 4 and PGCE students had positive feelings about and were confident in the use of a variety of methodological approaches ranging from the highly teacher structured (exposition 90%+) to individual work programmes (50%+), although they felt more confident with the more structured approaches.
Both groups of students felt more confident with teaching the more structured aspects of mathematics such as facts and procedures rather than those requiring for example, improvisation and investigation, although they felt somewhat more confident with teaching mental methods of calculation (BEd 4, 84%, PGCE 76%) than with mathematical procedures (BEd 4, 60%, PGCE 68%).
Teaching and Learning in Mathematics
Students were asked to rank a variety of aspects of teaching and learning mathematics in terms firstly of what they as prospective teachers thought important, and secondly in terms if what they considered Primary schools thought important. The first of these, but not the second, was also in the BEd 1 questionnaire.
For BEd 4 students the learning of mathematical facts (65%), mental methods (62%), and procedures (54%) predominated in the top three positions, compared with methods of problem-solving (26%), and applying mathematics in unfamiliar contexts (25%). Other aspects such as improvisation and investigations appeared in the top three positions with a frequency, 10% or less. For PGCE students the picture was broadly similar.
The BEd 1 data, however, were quite different. Here the most frequent choices in the top three positions were methods of problem-solving (66%), applying mathematics (44%), and learning mathematical procedures (31%). Mental methods of calculation appeared with a frequency 13%, and learning mathematical facts 27%.