Following what has been presented with respect to theory of mathematics-related self-beliefs, this study presents the evaluation and the comparison of Chilean and Finnish 9-year old students’ self-efficacy feelings, effort, mastery goal orientations, and enjoyment of mathematics. Further, we will examine the connections between the four factors of interest, and compare whether the connections are similarly structured and correspondingly strong in both countries.
Methodology
It is clear that we cannot be sure whether we have been able to catch all the aspects of affect and beliefs at present. Accordingly, Leder (2006) reminds that the limitations of instruments, designed with the help of previous information, may influence on how the topic is measured, recognized and discussed further on.
Beliefs in quantitative studies are often measured by a questionnaire. This is an economic, fairly simple method that is familiar to many students. However, asking students to evaluate their beliefs through a questionnaire does not (necessarily) give information about the context at the moment. Further, it is possible that students’ beliefs are affected by the questionnaire: for example, emotions may become stronger when there are provocative claims nourishing them. In this study, we acknowledge, that students’ beliefs cannot undoubtedly be measured directly, but need to be inferred through students’ self-reports or behavior (Leder, 2006). Consequently, we accept that the interpretation may be false, when the concern is on a beliefs per se, but if the focus is on what can be said about the reported beliefs’ connections with mathematics learning or other beliefs, we rely
on previous findings: it is empirically and theoretically acknowledged, that data of students-reported beliefs gathered by empirical studies are real, existent and have implications to learning. For example, Leder (2006) marks that PISA results show that motivation, self-related beliefs and emotional factors have linked to students’ learning strategies and thus to lifelong learning: the interpretation has been done using the gathered data, given by students.
In our study, we are interested on Finnish and Chilean 9-year old pupils’
mathematics-related beliefs about self, and in particular self-related beliefs about self-efficacy, mastery goal orientations, effort, and enjoyment regarding mathematics and mathematics learning.
By examining beliefs about self-efficacy, we will find out the state of students feelings of their ability. We have measured students’ perception about their self-efficacy using items concerning self-confidence (e.g. “I am sure that I can learn math”) and self-competence (e.g. “I have made it well in mathematics”).
To find out the state regarding students’ mastery goal orientations, we have items made explicitly to measure students’ intentions to deeply orientate in mathematics learning (e.g. “On every lesson, I try to learn as much as possible”. This will tell us about students’ values with respect to mathematics learning.
By the items designed to measure effort, we will interpret what a student can expect to learn in mathematics, as effort is a trait of motivation. Enjoyment will tell us about the emotional circumstances of a student. This can be seen as an attitude, a trait aspect of emotions. Revealing the state of emotions we can infer what students’ relation to mathematics is. Items to measure effort and enjoyment of mathematics have been designed to that particular purpose only (e.g. ”I always prepare myself carefully for exams”; “I have enjoyed pondering mathematical exercises”).
The data used in this study was gathered within an ongoing research aiming to develop mathematics learning in Finland and Chile. The data was collected during the academic year 2010-2011: September-October 2010 in Finland (Regions near to Helsinki) and February-March 2011 in Chile (Santiago). In Finland, the number of participants was 466, and in Chile 459, this makes the altogether number of participants 925. The project is funded by the Academy of Finland (project #1135556) and Chilean CONICYT. The overall aim is to develop a model for improving the level of pupils’ mathematical understanding by using open problem tasks in mathematics teaching.
All the items used are part of a questionnaire developed by Hannula & Laakso (2011). The measurement was done using 3-point Likert scale: this is a common approach to the measurement of affect (Leder, 2006).
To analyze the sample, we first calculated the sum variables, and then checked the normality of the distributions. All the sum variables were skew to the left: this means, that the answers were mainly positive, and almost none of the students chose the first category (”I don’t enjoy mathematics”, ”I can’t do mathematics”
etc.). The reliabilities of the sum variables were between .512 - .663 for Chilean pupils and between .606 - .832 for Finnish pupils (Cronbach’s alphas). In sum variables, the amount of missing cases varied between 30-46 (6%-10%) with regard to Finnish pupils, and between 58-94 (13%-20%) with regard to Chilean pupils.
We continued calculating the propositions of the answers in all of the categories.
To make the comparison between the countries, a t-test was made. We chose the t-value according to the similarity or the non-similarity of the variances: Levene test was made to find out the case. As the distributions were skew to the same directions, it was allowed to make parametric comparisons. Still, the results were confirmed using non-parametric tests.
Finally, the connections between the variables were examined. First, we checked the type of the possible connections from scatter plots: the connections were not clear, but if there was any, it was rather linear than something else. This suggested using Pearson correlation (parametric), though the confirmation was again done with Spearman correlation (parametric, based on order). When a non-parametric comparison was made, in all cases the results got confirmed. Because of that, all the results presented are based to parametric calculations.
Results
The variables examined showed a very positive picture. With all the variables, the deviations were small, and the answers were almost thoroughly in the highest category. In the following, the explicit propositions, as well as the results of the t-tests between the countries according to each of the sum variables are presented.
Self-efficacy
Finnish 9-year old pupils have high feelings about their self-efficacy. Most of the pupils (65 %) rate their self-efficacy feeling to be in the highest category. Only
few pupils (2 %) experience the opposite, i.e. rate their self-efficacy feeling to be in the lowest category.
Chilean 9-year old pupils have a bit more moderate self-efficacy feelings. About a third of the pupils (37 %) rate their self-efficacy feelings into the highest category.
Most pupils (62 %) place their self-efficacy feeling into the middle category, while only some of the pupils (1 %) have the lowest self-efficacy feelings. See the table 1 for exact percentages.
A statistically significant difference between the two countries was found: Finnish students tend to have slightly stronger confidence on their self-efficacy (mean [F]
= 2,63; mean [C] = 2,36; p < 0,001).
Table 1. Self-efficacy.
Self-efficacy (%) low middle high
Finland 2 33 65
Chile 1 62 37
effort
With respect to effort, the difference is quite the same. Most Finnish pupils (58
%) place the amount of their effort to be in the highest category. Few (2 %) state their effort to the lowest one.
In Chile, about a third (35 %) set their effort into highest category, and most pupils (64 %) into the middle category. Only one percent rates the amount of effort to the lowest category. To see the percentages, see table 2.
Again, a statistically significant difference between the two countries was found:
Finnish pupils make stronger effort (mean [F] = 2,56; mean [C] = 2,34; p < 0,001).
Table 2. Effort.
Effort (%) low middle high
Finland 2 40 58
Chile 1 64 35
Mastery goal orientation
Speaking of the mastery goal orientation, the picture is remarkably positive.
Nearly all Finnish pupils (90 %) rate their orientation to be in the highest category. Almost none of the Finnish pupils (0,5 %) rate their orientation to the opposite category.
Also in Chile, almost every pupil (87 %) experiences the highest orientation.
Only few pupils (1 %) feel the opposite; percentages are presented in table 3.
According to t-test, no statistically significant difference was found. Students have high and equal orientations in both countries (mean [F] = 2,9; mean [C] = 2,9; p=0,5).
Table 3. Mastery goal orientation.
Mastery goal orientation (%) low middle high
Finland 0,5 9,5 90
Chile 1 12 87
enjoyment of mathematics
Finnish pupils’ emotions towards mathematics are mainly positive. Most pupils (66 %) enjoy mathematics, while quite a few pupils (6 %) do not. The picture is pretty same with Chilean pupils: Most pupils (61 %) enjoy mathematics, while only a handful (1 %) does not. See table 4 for the percentages.
An interesting detail is, that what comes to enjoyment, there are more pupils not enjoying in Finland, in spite of the fact that in Finland there are also more pupils enjoying. The situation is more polarized in Finland. However, a t-test result argues that there is not a statistically significant difference between the countries:
on average, students enjoy equally in both countries (mean [F] = 2,6; mean [C]
= 2,6;. p=0,5).
Table 4. Enjoyment of mathematics.
Enjoyment of mathematics (%) low middle high
Finland 6 28 66
Chile 1 38 61
Connections of the variables
Six type of connections were examined: MGO (=Mastery Goal Orientation) – effort; MGO – EoM (=Enjoyment of Mathematics); MGO – S-E (=Self-Efficacy);
effort – EoM; effort – S-E; EoM – S-E.
According to scatter plots, the connections were either linear or there seem to be no connection at all. In all cases, strong connections were not visible in scatter plots. Still, all the correlations were statistically significant (**). See table 5 for correlations.
In Finland, almost all the connections have quite similar correlations (r ≈ 0,5).
This is the case in all connections except MGO – S-E: both goal orientations and self-efficacy go better in line with effort and enjoyment than they go in line with each other. This is not the case in Chile: MGO – S-E is the third strongest connection, though all in all, the connections are weaker in Chile than in Finland. What is noteworthy about the connection of MGO – S-E is that if there is a discrepancy between the two (goals are either unrealistic or a student do not really feel that he/she is able to reach them), it affects students enjoyment and achievement (Tuohilampi, 2011). A student needs to feel that the goals are achievable to not fall to helplessness.
Altogether, the connections are not remarkably high. The coefficient of determination (r2) range from 0,09 to 0,29, so none of the variables can clearly predict another one. One reason for that is that almost all the answers were in the highest category in all variables, and almost none of them were placed to the lowest one: for such a few cases existing in the categories outside the highest one it is hard to find a connection even there was one.
Discussion
The self-related beliefs concerning mathematics seem to be delightfully positive with regard to Finnish and Chilean 9-year old students. Especially mastery goal orientations rated into the highest category almost exclusively in both countries:
young pupils are eager to learn mathematics and they want to understand it deeply. Still, it is shown by Tuohilampi (2011) that it is important that the self-efficacy feelings follow the aspirations, otherwise students may lose their satisfaction to do mathematics, and even achievement may get worse.
Some differences between the countries were found: In Finland, pupils had little higher beliefs about their self-efficacy and effort. The differences were not remarkable, but in Chile, the pupils were a bit more heterogeneous, and the differences favored categorically Finnish pupils. As the belief structure may be less organized with regard to primary pupils than with respect to adolescents (Hannula & Laakso, 2011), the heterogeneity within the self-beliefs, as well as the weaker connections between them, seem to suggest that pupils in Finland may have developed their belief system more at the age of 9.
Students believe “what is first told”. This changes only when new information conflicts with previous perceptions (Op ’t Eynde & al, 2002; quotation marks by authors). As young pupils in this study had positive self-beliefs relating mathematics, they easily seem to accept that mathematics is nice, it is worthwhile to work with it, and they are able to do it. Secondary school students’ attitudes towards mathematics are poor compared to primary level students (Tuohilampi, 2011). A feeling of being able gets colored with uncertainty, the feeling of amusement moves towards hate or desperation, and many students wish to perform well, not necessarily learn well. Obviously, there are factors that impact the positive self-belief structure during the school years, making it more negative.
Yet, to have a positive structure at the beginning is the proper starting point.
Primary school pupils are enthusiastic to learn mathematics, so we need to be very careful to provide accessible mathematics in the following school years to make sure their capability feelings accompany.
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