McLeod (1992) divides the affective responses to mathematics education into beliefs, attitudes and emotions. These three terms are all connected, but vary in stability, intensity, cognitive appraisal and the time they take to develop. Tough there is a general agreement to divide the affective domain into these three constructs, the definition of them is not undisputed and many times the words
“beliefs” and “attitudes” are used synonymously (Di Martino & Zan, 2011). In this paper we do not try to formulate a comprehensive definition of the concepts, but merely point out some aspects that are of relevance to our study.
In mathematics education two various definitions of attitude are particularly popular; the multidimensional and the simple definition. In this study we assume the simple definition where an attitude is seen as “a general emotional disposition toward a certain subject” (Di Martino and Zan, 2001, p.18). Hannula (2002, p.29) points out that in the case of a questionnaire:
… the first reaction is usually emotional and based on associations. These automatic associations are a product of the student’s previous experiences with mathematics. This second process falls under the simple definition of attitude as an emotional disposition.
According to McLeod (1992), one’s attitude towards mathematics can involve aspects such as liking geometry, enjoy problem-solving and being bored by algebra. We explore students’ attitudes towards their mathematics education and which ways of working they find enjoyable.
In the field of mathematics education there are many variations of the concept belief and it is difficult to give an explicit and shared definition (Furinghetti &
Pehkonen, 2003). In many studies the lack of an explicit definition can make it hard to understand what is being investigated (Di Martino & Zan, 2011). As advised by Furinghetti and Pehkonen (2003) we consider beliefs as belonging to subjective knowledge. The beliefs studied are seen as affective and possible to change under the right circumstances. Otherwise, the didactic activities would seem useless.
The concept beliefs concern different fields of mathematics education and McLeod (1992) divides students’ beliefs into four categories: beliefs about mathematics, beliefs about the contexts in which mathematics education occurs, beliefs about self and beliefs about mathematics teaching. We focus on the latter two. Beliefs about self include concepts like self-concept and confidence, for example believing that you are good at mathematics. Beliefs about mathematics teaching involve, among others, beliefs about instructions and we investigate which mathematics instruction the students find instructive. However, students’
ability to evaluate their own achievements and learning can be a complicated issue. Boekarts & Corno (2005) stress that the learning goals teachers have in mind are not always adopted by their students. And if they are adopted, students sometimes find it difficult to work effectively towards these goals since they value things like entertainment and well-being higher. This may cause students to choose less effective strategies to achieve the learning goals. Furthermore, students’ beliefs about themselves and mathematics teaching can affect their self-regulation. For example, if a student believes learning mathematics should be easy and that effort is unimportant, this could lead to a choice not to engage in effective learning strategies (Butler & Winne, 1995). Hence, what students think are good strategies are not necessarily always the most effective for achieving the learning goals.
Methodology
The students participating in the study came from two Swedish lower secondary schools situated in small cities (see manufacturing municipalities in Johansson, 2011). Compared to the national average, one of the schools had a higher proportion and the other a lower proportion of students passing the subject of mathematics in the ninth grade. Both schools had been granted money from the National Agency for Education and participated in the national mathematics initiative. The questionnaire used in the study was developed during a final degree project (see Neuman, 2011). The teachers at the participating schools were allowed to propose changes to the questionnaire, and it was piloted. Almost the
same questionnaire, with a few changes, was used in a similar study at another school (Cervin, 2011).
The questionnaire consisted of a number of statements about students’ relation to mathematics and mathematics teaching. The students were asked to rate their agreement with these statements on a four-step scale. We also posed questions about commonly used working methods in mathematics education. On a five-step scale, the students were to classify the working methods as enjoyable or boring and instructive or not instructive. A total of 188 students aged 12-16 (grades 6-9) took part in the study. Three questionnaires were only partially filled in, so the total number of analyzed questionnaires was 185. Not all the results obtained from the studies (Neuman, 2011; Cervin, 2011) are presented in this paper. The gathered data allowed us also to conduct further statistical investigations.
In our data analysis we first used descriptive statistics to summarize and organize the data. Some initial answers to our research questions could be given by this analysis. We are now in the beginning of a deeper data analysis and present some of the results obtained so far. A hierarchical cluster analysis using Ward´s method was run on 7 cases responding to items on the students’ attitudes toward their mathematics education and beliefs about them self as mathematic learners.
Only 172 students could be involved in the cluster analysis because it requires an answer for every variable. Independent sample t-tests were conducted to compare the mean values of the cases for the clusters and to see if there were any significant difference between the clusters attitudes and beliefs about different working methods. The correlation presented in this paper was tested using a two-tailed Spearman’s rho test. No significant differences have been found thus far between the two schools concerning the results of the data analysis we have conducted. Hence, we present the results for the two schools together.
Results
Table 1 shows that mathematics is not especially popular among the students at the two focus schools. 63% disagree or strongly disagree with the statement Mathematics is one of the topics I like best in school. Less than half of the students look forward to their mathematics lessons, and 59% agree or strongly agree with the statement Most of the tasks in mathematics are too easy for me. However, at the same time, a strong majority are satisfied with their mathematics education in general as 88% agree or strongly agree with the statements I am satisfied with the mathematics teaching I receive today and In school I get enough help in mathematics.
Just like in the national evaluation from 2003, the students consider mathematics
useful for them as 92% agree or strongly agree with the statement I will have use for the mathematics I learn in school. The students’ self-conception is good:
three out of four students believe they are good at mathematics. There is also a significant positive correlation (0.40 at the 0.01 significance level) between the statements I am good at mathematics and Most of the tasks in mathematics are too easy for me. This shows that students who believe they are good at mathematics, compared to those who believe the opposite, more often feel the mathematics tasks are too easy for them.
Table 1. Students’ responses to statements about attitudes towards mathematics education and beliefs about themselves as mathematics learners. The maximum value for the mean is 4 and the minimum value is 1.
Statement Strongly agree
or agree
Strongly disagree or
disagree Mean Standard
deviation Mathematics is one of the
topics I like best in school 37% 63% 2,17 0,98
I look forward to
math-ematics lessons 41% 59% 2,31 0,77
I will have use for the
math-ematics I learn in school 92% 8% 3,43 0,70
Most of the tasks in
math-ematics are too easy for me 59% 41% 2,66 0,78
I am satisfied with the mathematics teaching I receive today
88% 12% 3,20 0,76
In school I get enough help
in mathematics 88% 12% 3,28 0,73
I am good at mathematics 75% 25% 2,83 0,91
A cluster analysis run on the seven statements in Table 1 produced two clusters, between which the variables were significantly different in the mean (Appendix 1). The first cluster was predominant (n=129) and characterized by more positive attitudes towards their mathematics education and more self-confidence in mathematics than cluster two (n=43).
Regarding students’ attitudes towards different working methods in mathematics, use of other material than textbooks (computer, games, etc.) was seen as the most enjoyable with a mean-value of 4,16. Concerning instructiveness, less than half the students assigned the working method a 4 or a 5 which gave it a mean-value of 3,46. This means that three other working methods were considered more instructive. Compared to the other working methods, group work and working in pairs were also regarded as quite enjoyable. One of the two working methods most of the students found instructive was joint review with the teacher. Yet, with a mean-value of 2,89 less than a third of the students experienced it as enjoyable.
The least enjoyable working method, according to the students, was homework.
However, working with textbooks and individual work were also considered boring, as a significantly larger proportion of students gave these working methods a 1 or 2 instead of 4 or 5. But if we instead choose to look at how instructive they found the working forms individual work and working in textbooks, they regarded these as the most and third most instructive methods (Table 2).
Table 2. How enjoyable and instructive the students found different working methods. A rating of 5 reflects the most enjoyable or instructive alternative. SD stands for standard deviation.
Enjoyable Instructive
Working method Have worked with Mean SD Mean SD
Individual work 100% 2,72 1,21 3,78 0,89
Work in pairs 81% 3,52 1,01 3,38 0,91
Group work 78% 3,48 1,05 3,32 0,94
Work with textbooks 98% 2,64 1,22 3,72 0,95
Other material than textbooks 75% 4,16 0,96 3,46 0,98 Joint review with the teacher 99% 2,89 1,11 3,78 1,07
Problem-solving 81% 2,97 1,34 3,27 1,09
Laboratory work 54% 3,14 1,11 3,16 0,92
Interdisciplinary thematic work 37% 3,00 1,22 2,97 0,86
Homework 95% 1,95 1,05 2,99 1,01
Independent sample t-tests were run to determinate if there were any significant differences between the two clusters regarding their beliefs and attitudes towards the different working methods. There were significant differences between the clusters for individual work, work in textbooks, joint review with the teacher, problem-solving, laboratory work, interdisciplinary thematic work and homework.
The students in Cluster 1 regarded these working methods as both more enjoyable and instructive than the students in Cluster 2 (Appendix 2).