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CURRICULUM REFORM AND TEACHERS: DIFFICULTIES AND DILEMMAS IN PRIMARY MATHEMATICS EDUCATION
Fatih OzmantarErhan BingolbaliHatice AkkoUniversity of Gaziantep, TurkeyUniversity of Marmara, Turkey
In this paper we attend to the question: What are the key problems and challenges faced in primary school education in your country? As the writers of this paper are from Turkey, we deal with Turkish primary education, which lasts for 8 years. Initial five years, students are usually taught by one classroom teacher who is expected to teach a wide range of subjects including Turkish Reading and Writing, Science and Mathematics. During the last three years, students are taught by different teachers who are specialists in their subject areas.
Turkish primary education has been the subject of a curriculum reform since 2004. The new curriculum defines and determines new roles for teachers and students, which are quite different from what they are used to. The new curriculum aims to create classroom environments where students actively participate in learning, ask questions, are held responsible for their learning, solve and construct mathematical problems collaboratively. The teachers are expected to adopt an open approach to teaching which values different solutions, encourages students to make connections amongst mathematical ideas and concepts, and design activities according to students (mathematical) needs and potentials.
Recently we have been involved in a research (still ongoing) to see how well equipped the classroom teachers are to cope with the demands of the new curriculum (Bingolbali, Ozmantar & Akkoc, submitted). The reason that we focus on teachers rather than students is because primary teachers in Turkey show a great variety in terms of their subject matter knowledge in mathematics and pedagogical content knowledge. This is due to the fact that many inservice primary teachers until 1990s were qualified to teach at primary level only after graduating from a twoyear educational institution programme which then became a fouryear faculty programme to which it was possible, and was in fact the case that, students with no basic mathematical understanding were admitted. Furthermore, during the period between 1992 and 2000, graduates of various university departments including engineering, business, mathematics, physics, chemistry, sociology, psychology were entitled and assigned to be primary teachers only after a very short period of teacher training. This, in and of itself, poses serious challenges to teaching mathematics at primary level. With all these in mind, it was hence important to see the situation of primary teachers subject matter knowledge and pedagogical content knowledge (Shulman, 1987) both of which are particularly important to achieve the vision of the new curriculum.
To this end, we designed a study which broadly speaking consists in two stages. The first was composed of data collection through questionnaires and the second of following teachers in their actual classroom settings during their teaching of mathematics. We aimed for the first stage to reach a large number of primary teachers and hence developed two questionnaires, which consisted of openended questions on basic mathematical concepts (e.g., probability, decimals, word problems and multiplication), all of which were part of the primary curriculum. Our intention was to gain insights into teachers knowledge on these concepts, into their views on different solution strategies, into their evaluation of erroneous student answers and into their awareness to common student misconceptions in mathematics. Openended questions in questionnaires present a mathematical problem with different (hypothetical) student solutions, some of which were erroneous and teachers were asked to evaluate these solutions. In this paper, within the allowed space, we analyse two particular items, selected from each questionnaire (please note that each questionnaire was applied to different populations of teachers). The first item was related to alternative solutions to a multiplication and the second involved calculating the dimensions of a rectangle. We next present each item respectively, and summarise and discuss our analyses of teacher responses.
Item 1:
Below students three different responses to this multiplication are presented. All three students have reached the same result. Please evaluate each response and explain which one or ones you would accept as an answer and why? (adopted from Ball & Bass, 2003).
This item was part of the first questionnaire and responded by 216 primary teachers, from 104 different schools. We analysed the frequency of teacher preferences for any of A, B and C or any combination of these, see Table1 for the summary.
A A&BA&B&CNo answer Total Number
Percentage 145
%67 33
%15 36
%17 2
%1 216
%100Table1: Teachers responses (frequencies and percentages) to Item1
As seen, majority of the teachers preferred only A as an acceptable solution (%67). A and B were accepted by %15 and all A, B and C by %17. This analysis effectively indicates the teachers difficulty in adopting an open approach to different solutions of the same mathematical problem. Majority of the teachers privileged solution A while rejecting B&C despite the fact that B&C were correct but just different from the standard solution. This becomes especially evident when we examined the reasoning of those teachers who accepted A as the only solution. This examination yielded three main reasons for teacher preference; these are rule (e.g., Solution A follows the rule), practicality (e.g., Solution A is easy, practical and takes less time) and difficulty of B&C (e.g., B&C are difficult to understand and/or teach). Frequency of the three main reasons is presented in Table2.
Those teachers who chose only solution A (145)
Number
Percentage RulePracticalDifficulty of B and C79
%5439
%2717
%12Table2: Main reasons for accepting only solution A
The dominant belief, as our analysis suggests, was that mathematical solutions should be practical, follow routines and take little time. For example, one teacher accepts only A as a correct solution because there is only one way to the truth (right conclusion) and the other noted that if students try to do the multiplication like in B and C, he would interfere at the very beginning not to do so. This analysis suggests that the reason for the rejection of B&C is linked to teachers customary mathematical solutions and to their already formed personnel views on (teaching and learning of) mathematics.
Item2
Fourth and fifth grade students are presented with following problem:
What can be the dimensions of a rectangle with exactly half the area of this rectangle? Please explain your answer.
The responses of two students to this problem are presented below. How would you grade these students responses over a range from 0 to 10? Please explain why (adopted from Hansen et al., 2005).
Students Answer and Explanation ScoreReasonStudent
KTo find out half area of the rectangle, I do this:EMBED Equation.3. Then each dimension can be 5 cm.Student
LI would have the half of each dimension: 6EMBED Equation.32 = 3 and 4EMBED Equation.32 = 2. Then I would come up with a rectangle with a one side being 3 cm and the other 2cm. And draws the following figure:
This item was part of the second questionnaire and responded by 148 primary teachers from 10 different schools. As mentioned before, use of openended questions for both formative and summative assessments is particularly emphasised within the new curriculum scripts. We wished to see how primary teachers evaluate students solutions to openended problems, especially if they are inaccurate. We also desired to see on what basis primary teachers make decisions while grading student solutions. Analysis of the teachers responses in terms of frequency of the scores given to the solution of student K&L is presented in Table3.
Scores012345678910Student K76
%5113
%97
%57
%55
%313
%94
%33
%24
%31
%115
%10Student L35
%244
%36
%46
%46
%418
%123
%23
%21
%10
65
%44Table3: Teachers responses (frequencies andpercentages) to Item2
Table3 displays a great variation in teachers scoring of the erroneous solutions. For example, while %51 of teachers awarded StudentK with an exact 0, noting that solution was wrong; others still being aware of the inaccuracy of the solution graded the same solution ranging from 1 to 9 (similar observation is also true for the scoring of studentL). Such a great disparity is certainly striking; but on what basis the teachers decided on these scores? Those grading the solution K with 0 seem to think resultcentred and give such reasons as because the solution is wrong or because student didnt understand the calculation of area. Others seem to put the student efforts in the centre and give such reasons as at least student tried or student didnt remain indifferent to the problem; however even then their grading ranged from 1 to 9. Surely subjective judgements are and hence certain variation in scores is understandable; yet such a huge gap points out teachers lack of assessment criteria for student responses, solutions and strategies. However assessment criteria are particularly important because teachers grades provoke certain thoughts for students regarding what is important in mathematics, e.g. making effort or finding the accurate result.
Our analysis of teacher scores also indicates primary teachers difficulties in mathematics. This is especially evident considering the full grades given to inaccurate solutions. For instance %44 of the teachers rewarded the solution of studentL with an exact 10 (this is %10 for the solution of studentK). We are not tempted to think that, on the basis of our data, these figures hold for the whole population of primary teachers in Turkey. Yet this is an important proportion and needs to be taken seriously. As a matter of fact, teacher difficulties in mathematics especially at primary level are reported in studies from different countries (see for example Manouchehri, 1998; Ma, 1999; Ball & Bass, 2003). Given that primary teachers need to specialise in many subject areas, these difficulties might be understandable; nevertheless, teacher difficulties in mathematics raise serious concerns with teaching mathematics for conceptual understanding. A deep understanding of key facts, principles and the rules of evidence and proof (Manouchehri, 1998) are all essential to a sound knowledge of mathematics at any level, but how much mathematics content primary teachers need to acquire is not an easy question to answer. We believe that this issue warrants further considerations and mathematics education community need to think seriously to find ways to enrich, and perhaps strengthen, particularly inservice, yet without disregarding preservice, primary teachers content knowledge. Designing training programs for inservice teachers might be a step towards this direction but this is not an easy task either due to the fact that inservice teachers often become resentful when they are asked to attend a program to learn mathematics. During our data collection we at times received serious reactions from those who tended to think that we were measuring their mathematical understanding. Hence we believe that for any training program to be a success we, as teacher educators, must think carefully while deciding upon the method of training teachers, without making them reactive, the depth of mathematical content and the breadth of the programs.
In conclusion, this paper reported on the findings from the initial analysis of data collected through questionnaires with openended questions. Our initial analysis suggests that teachers found it difficult to adopt an open approach which values different solutions of the same mathematical problem. They privileged and preferred certain solutions while rejecting others despite the fact that the solutions were correct but just different from the standard solution. The reason for the rejection teachers cited is often linked to their customary mathematical solutions. Our analysis also suggests that teachers experienced difficulties while evaluating the student responses to the openended mathematical problems. The difficulty was related to the fact that teachers do not have a wellestablished assessment criteria for student responses. Lack of assessment criteria is another challenge to be noted here: A great variation was evident in teacher evaluation; for example, erroneous answers received grades ranging from 0 to 10. Finally teachers themselves experienced mathematical difficulties in some basic concepts, especially when met openended or nonstandard mathematical problems. These findings reveal certain difficulties and obstacles in implementing the new curriculum as well as in primary level mathematics education.
REFERENCES
Ball, D.L. & Bass, H. (2003). Toward a practicebased theory of mathematical knowledge for teaching. In E. Simmt & B. Davis (eds.), Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group CMESG, Edmonton, AB, pp. 314.
Bingolbali, E., Ozmantar, F. & Akkoc, H. (submitted). Curriculum Reform in Primary Mathematics Education: Teacher Difficulties and Dilemmas. The paper is submitted to the Joint Meeting of the 32nd Conference of the International Group for the Psychology of Mathematics Education, and the XX North American Chapter. Morelia, Mxico.
Hansen, A., Drews, D., Dudgeon, J., Lawton, F.& Surtees, L. (2005). Children's errors in mathematics : understanding common misconceptions in primary schools, Exeter : Learning Matters, 2005.
Ma, L. (1999). Knowing and teaching elementary mathematics : teachers' understanding of fundamental mathematics in China and the United States. Mahwah, N.J.: Lawrence Erlbaum Associates.
Manouchehri, A. (1998). Mathematics curriculum reform and teachers: what are the dilemmas? Journal of Teacher Education, vol. 49, no. 4, pp276286.
Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57, 122.
This paper has some commonalities with a paper (Bingolbali, Ozmantar, & Akkoc; submitted) we have submitted to PME 32 but has a different focus.
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