the International Group for the Psychology of Mathematics Education, Vol. ... to Cobb (1986) there is a relation between beliefs and learning of mathematics. In.
STUDENTS’ BELIEFS AND ATTITUDES ABOUT STUDYING AND LEARNING MATHEMATICS Kapetanas Eleftherios and Zachariades Theodosios University of Athens, Greece In this paper we focus on students’ beliefs and attitudes which concern studying and learning mathematics. The sample of this study was 1645 students of 10th, 11th and 12th grade. From our data two factors of beliefs and three factors of attitudes were traced. We investigate whether these factors correlate, whether there are any differences of students’ beliefs and attitudes according to their social status and gender and whether they influence students’ performance and ability to understand mathematical proofs. INTRODUCTION There are many studies concerning students’ beliefs and attitudes about mathematics. In Shoenfeld (1989), Mc Leod (1992) and Broun et al. (1988), it is verified that there is a link between students’ attitudes and their performance in mathematics. According to Cobb (1986) there is a relation between beliefs and learning of mathematics. In Schoenfeld (1989) it is demonstrated that students’ beliefs about Euclidean Geometry is a consequence of the teaching of mathematics. Some researchers agree that students’ attitudes can be changed into more positive ones. Regna and Dalla (1992) assert that when teachers are enthusiastic in their teaching and plan activities which are accessible to students, then students’ attitudes can be improved. In Kifer & Robitaille (1989) and in Philipou & Christou (2000) it is verified that students’ beliefs are influenced by their social surrounding. According to Dematte et al. (1999) it seems that students’ beliefs about mathematics are influenced by the educational system of their country. In Pehkonen (1995) students’ beliefs from eight countries are investigated. In Christou C. & Philipou G. (1999) factorial structure of 13 years old students’ beliefs among four countries (Cyprus, Finland, U.S.A., and Russia) are investigated. In this paper we investigate 10th, 11th, 12th grade students’ beliefs and attitudes about studying and learning mathematics and we examine their correlation. We also investigate whether they influence students’ performance and ability to understand mathematical proofs. THEORETICAL BACKGROUND As it comes from the literature, there are various opinions concerning the notion of “beliefs”. According to Goldin (1999), a belief may be “the multiply encoded cognitive configuration to which the holder attributes a high value, including associated warrants”. Cooney (1999), asserts that a belief is “a cluster of dispositions to do various things under various circumstances”, which leads to the acceptance that “different circumstances may evoke different clusters of beliefs” (Presmeg 1988). It is widely accepted that beliefs are the individual’s personal cognitions, theories and 2007. In Woo, J. H., Lew, H. C., Park, K. S. & Seo, D. Y. (Eds.). Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Vol. 3, pp. 97-104. Seoul: PME.
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Kapetanas & Zachariades conceptions that one forms for subjective reasons. Their nature is partly logical and partly emotional. According to Mc Leod (1992) “beliefs are largely cognitive in nature and are developed over a long period of time”. We will use the term “beliefs” in the meaning of personal judgments and views, which constitute one’s subjective knowledge, which does not need formal justification. As it happens with the notion “beliefs”, there is also luck of consensus about the notion of “attitudes”. Many researchers use attitudes as a term which includes beliefs about mathematics and about self. Mc Leod (1992) accepts that attitudes “refer to affective responses that involve positive or negative feelings of moderate intensity and reasonable stability”; they may appear as a result of the automation “of a repeated emotional reaction to mathematics” or of “the assignment of an already existing attitude to a new but related task”. According to Hannula (2002) “attitude is not seen as a unitary psychological construct but as a category of behavior that is produced by different evaluative processes. Students may express liking or disliking of mathematics because of emotions, expectations or values”. Hannula declared that attitudes can change under appropriate circumstances. In this study we investigate 10th, 11th, 12th grade students’ beliefs and attitudes, which mainly concern studying and learning mathematics and we explore their factorial structure; we investigate whether there are any differences in student’s beliefs and attitudes, concerning their social status and gender; we examine whether these factors correlate and influence students’ performance at school and their ability to understand mathematical proofs. THE STUDY Methodology Data reported in this paper was collected by a questionnaire administered to 1645 students of 10th, 11th and 12th grade. These students were from 25 high schools in the district of Athens in Greece, which were selected by the stratified - two stages cluster sampling method. This study is a part of a broader one, the aim of which is to investigate students’ beliefs and attitudes concerning mathematics, how they are evoked and affect students’ understanding, performance and ability in mathematics. We constructed the questionnaire taking into account analogous questionnaires from the literature, as in Schoenfeld, (1989). The questionnaire consists of 28 questions (statements), 10 of which concern beliefs and 14 concern attitudes about mathematics. The 25th question concerns students’ performance in mathematics at school in the previous year. There are three more tasks, the 26th, 27th and 28th, called mathtest in this paper, which measure students’ ability to understand mathematical proofs. These last three tasks were differentiated according to the students’ grade. Below we present one task of this type for each grade, because of lack of space. Students were asked to choose one of the numbers 1, 2, 3,…, 9 that best describes what they feel or think about each one of the first 24 statements, using number 1 to declare “I don’t agree at all” and number 9 to declare “I absolutely agree”. We used a scale range from 1 to 9, 3-98
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Kapetanas & Zachariades because we believed that with this scaling, students would express their views precisely. Twenty one of the questions-statements of our questionnaire are presented in table 1. These are the ones which constitute the five factors (see table 1 below).Three of the statements of the questionnaire are omitted, because of their low loadings in the factors, while statements 25, 26, 27 and 28 are presented below: Q25. Your overall grade average in mathematics last year was : …………. Q26. For α, b>0, if α > b, then α+4>b+4 (1). So,
(a + 4)a b+4 a > b + 4(2) . Thus < (3). b a+4 b
Explain why relations (1), (2) and (3) hold. (This task was for 10th grade students). Q27. Let a, b , c be real numbers such that a − b ≤ 5 and b − c ≤ 5 . Then the following hold: b − 5 ≤ a ≤ b + 5 (1), −b − 5 ≤ −c ≤ −b + 5 (2). So, we obtain −10 ≤ a − c ≤ 10 (3). Therefore a − c ≤ 10 (4). Explain why relations (1), (2), (3) and (4) hold. (This task was for 11th grade students). Q28. Let f be a real function, defined by f ( x) = x3 + 1, x ∈ R . We observe that f (−1) = 0. We suppose that there is p ∈ R , with p ≠ −1 , such that f ( p ) = 0 .Then, if p < −1 it holds that f ( p ) < f (−1) (1) and if p > −1 , it holds that f ( p ) > f (−1) (2). In any case there is a contradiction. Explain why the relations (1) and (2) hold and what the contradiction is. (This task was for 12th grade students). Data analysis Exploratory factor analysis which was applied, led us to five factors, with sufficient internal consistency and reliability. Factors F1 and F2 concern beliefs and factors F3, F4 and F5 concern attitudes. In order to investigate whether there are differences in students’ beliefs and attitudes concerning their social status and gender, we applied multivariate analysis of variance (manova).We also calculated Pearson correlations for these factors and variables 25 and the mathtest, in order to investigate which of them correlate and whether they correlate positively or negatively. RESULTS Table 1 shows the five factors, the related items, means, standard deviations, factor loadings and Cronbachs’ alpha. Factors
Cronbach’s Mean St.D Loada ings F1 “Utility of proofs and mathematics’” 0.604 6.584 1.58 0.665 Q24 “You study the proof of a theorem, because you believe that the understanding of proofs can give you ideas, which will help you in problem solving” PME31―2007
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Kapetanas & Zachariades Q3 “Mathematics which I learn at school contributes to improving my thinking” Q23 “You study the proof of a theorem, because you believe that the understanding of the proof will help you to understand the theorem” Q4 “Mathematics which I learn at school is useful only for those who will study mathematics, sciences and engineering in the university”(reversed) F2 “Mathematical understanding through procedures” Q20 “If you are able to write down the proof of a theorem, then you have understood it” Q21 “If you are able to express a definition, then you have understood it” Q19 “Studying mathematics means you learn to apply formulas and procedures” F3 “Love of mathematics” Q6 “You loved mathematics in junior high school” Q5 “You loved mathematics in elementary school” Q7 “You love mathematics nowadays in senior high school” F4 “First level of studying mathematicsstudying mathematics with understanding” Q10 “Whenever you study mathematics you try to understand the proofs of theorems”
0.634 0.631
-0.573
0.639
5.812 1.35 0.751 0.717 0.575
0.735
5.642 2.23 0.869 0.812 0.665
0.783
7.110 1.52 0.726
Q9 “Whenever you study mathematics you try to understand what the theorems say”
0.690
Q8 “Whenever you study mathematics you try to understand definitions”
0.650
Q12 “Whenever you study the proof of a theorem you try to understand the successive steps of the proof”
0.648
Q11 “Whenever you study mathematics you try to prove the theorems by yourself”
0.599
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Kapetanas & Zachariades Q13 “Whenever you study the proof of a theorem you try to understand the reason for which we follow this procedure towards the proof” F5 “Second level of studying mathematicsstudying mathematics with reflection”
0.510
0.703
5.09
1.59
Q15 “When you have done an exercise you examine whether it could be done in a different way”
0.794
Q18 “When you have done an exercise you examine whether you could extend it by adding some new questions”
0.736
Q16 “When you have done an exercise you think again about the steps you have taken, reflecting on them”
0.645
Q14 “When you have studied a proof of a theorem you think again about the whole proof, reflecting on it”
0.514
Q17 “When you have done an exercise you examine whether the result you have found is logical”
0.468
Table 1: The five factors Table 2 shows the results of manova analysis with factors F1 – F5 as dependent variables and “gender” and “social status” as independent variables. As it is shown in this table there is a significant statistical difference between female and male students concerning factors F2 (p=.03
