Education Research International

Volume 2015 (2015), Article ID 213429, 8 pages

http://dx.doi.org/10.1155/2015/213429

## A Survey on the Permanence of Finnish Students’ Arithmetical Skills and the Role of Motivation

^{1}School of Applied Educational Science and Teacher Education, University of Eastern Finland, P.O. Box 86, 57101 Savonlinna, Finland^{2}School of Information Sciences, University of Tampere, 33014 Tampere, Finland

Received 24 November 2014; Accepted 30 December 2014

Academic Editor: Gwo-Jen Hwang

Copyright © 2015 Timo Tossavainen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This study concerns the permanence of the basic arithmetical skills of Finnish students by investigating how a group () of the eighth and eleventh year students and the university students of humanities perform in problems that are slightly modified versions of certain PISA 2003 mathematics test items. The investigation also aimed at finding out what the impact of motivation-related constructs, for example, students’ achievement goal orientations, is and what their perceived competence beliefs and task value on their performance in mathematics are. According to our findings, the younger students’ arithmetical skills have declined through the course of ten years but the older students’ skills have become generic to a greater extent. Further, three motivational clusters could be identified accounting for 7.5 per cent of students’ performance in the given assignments. These results are compatible with the outcomes of the recent assessments of the Finnish students’ mathematical skills and support the previous research on the benefits of learning orientation combined with the high expectation of success and the valuing of mathematics learning.

#### 1. Introduction

In this millennium, the success in PISA (Programme for International Student Assessment) surveys has induced much positive interest in Finnish school and mathematics education. However, critical voices have also been heard. Already in 2005, based on the experience of their regular work, 206 Finnish mathematics professors and other mathematics educators from all Finnish universities and polytechnics published a letter [1] in a leading Finnish newspaper where they expressed their deep concern over Finnish students’ real mathematical skills, claiming that PISA only tells a half truth about students’ mathematical knowledge. The publication of the latest PISA results in December 2013 has only increased this criticism; the scores for Finland in the PISA 2012 mathematics test fell significantly [2].

Further, The Finnish National Board of Education, which is responsible for drawing up the national core curricula for primary and secondary education in Finland, regularly surveys primary and secondary students’ mathematical achievement with quite extensive samples. In the most recent reports [3–5], the general view is that the Finnish students’ mathematical skills have darkened in all areas of school mathematics. The most significant changes are related to arithmetic although basic arithmetic operations are still managed rather well [4, 5]. Geometry and the operations with the concept of per cent seem to be especially problematic issues for Finnish students.

The principal purpose of the present study is to provide a complementary view of the quality of Finnish mathematics education by examining how permanent or generic the results of teaching primary and lower secondary mathematics in Finland are. Since Finnish students have traditionally been quite good in the basic arithmetic operations and statistical reasoning, we decided to focus on such test items that merely require applying these skills. In order to make the recent development comparable with the situation a decade ago, at least, at an approximate level, some of the test items on our questionnaire are slightly modified versions (i.e., translated and recontextualized in the Finnish setting) of certain questions in the PISA 2003 mathematics test; for further details, see Sections 3 and 4.

The participants of the study consist of a group of the eighth grade students (), eleventh grade students of upper secondary school () studying the advanced () or basic () syllabus in mathematics, and university students () majoring in other than a mathematical subject. Consequently, our data do not represent a genuine longitudinal study and hence we cannot measure the permanence of the students’ arithmetical skills at an individual level. However, the comparison of the participating groups with one another and with the Finnish students in PISA 2003 is sensible because the contemporary national core curriculum for primary and lower secondary school has been implemented since August 2006 and the differences between that and the previous version from 1994 are very modest. In practice, all participants both in our study and in PISA 2003 have essentially studied the same mathematical contents in comprehensive school.

Another goal of the present paper is to draw an overview of the participants’ motivation in mathematics in order to better understand their performance in the test problems. Consequently, our research questions are as follows.(1)*How does students’ performance in the given mathematical assignments (of which three correspond to the test items M413, M467, and M468 in PISA 2003) vary across the educational level?*(2)*How does the students’ motivation in mathematics relate to their performance in the given mathematical assignments?*

#### 2. Theoretical Framework

Although there is much empirical evidence indicating the important role of motivation in the study of achievement in mathematics, the relations between motivation-related constructs, such as students’ achievement goals and motivational self-perceptions and success in mathematics, call for further research (cf. [6]). A contemporary view in research on motivation emphasizes the role of achievement goals. A large body of research on achievement motivation during the past few decades has demonstrated that goals, values, and competence beliefs are primary effects on achievement motivation [7]. Still, according to Conley [8], little is known about how these components of motivation function as a coherent set within individual students. In her recent study, Conley [8] suggested a model of combining achievement goals and expectancy-value perspectives. This is also a focus in the present study.

Goal orientation theory has emerged as an important theoretical perspective regarding students’ motivation in school [9]. This theory provides a framework for extensive research on motivational orientations that contributes to students’ academic achievement, persistence, and performance. Researchers have differed about the number and the definitions of the orientations that people may adopt in achievement situations [10]. However, despite these differences, most researchers have mostly focused on two main orientations, a mastery orientation and a performance orientation, which were found to differently relate to adaptive and maladaptive engagements. In the literature, there is a diverse terminology used in labelling them. Often, the words “approach” and “avoidance” have been used as affixes to the performance orientation.

A mastery goal orientation (also called learning orientation and task orientation) has been defined as a student’s incentive to learn and develop competence. In the engagement in academic tasks, the focus is on learning per se or on task mastery instead of, for example, a competition with other students. In performance-approach orientation, the focus is on demonstrating one’s competence relative to others or in competition for good grades and on outperforming others. In performance-approach orientation, labelled as “self-enhancing ego orientation” by Skaalvik [11], the student’s goal is directed toward attaining favourable judgments of competence, whereas in performance-avoidance orientation, labelled as “self-defeating ego orientation” in [11], the aim is at avoiding unfavourable judgments of competence. Skaalvik [11] has also identified a fourth goal orientation, particularly in mathematics learning, which has been named as work avoidance orientation, or avoidance orientation, because a student’s ultimate goal is to invest a minimum amount of effort, gain an easy success, and only reach a passing grade [9, 11, 12].

As for contemporary research on goals, it has focused on a “multiple goals perspective.” This approach on goals has demonstrated that individuals can and do operate according to multiple salient goals and that both mastery and performance-approach orientation result in higher achievement (e.g., [13]), whereas performance-avoidance and work avoidance orientation consistently lead to maladaptive outcomes such as low perceived competence and poor performance. However, findings for performance-approach goals have been much less consistent [9, 11, 12].

In achievement motivation research, student performance is assumed to be mediated also by their motivational beliefs such as efficacy expectations and task values [7, 14]. Theoretically, the expectancy of success is closely related to other conceptions of self and ability beliefs, such as self-efficacy [15]. Eccles and her colleagues’ model of achievement is known as the expectancy-value model of achievement and is comprised of two related components: expectancy for success and subjective task value. Accordingly, expectancy for success is defined as individuals’ beliefs about how well they will do in an upcoming task. It also relates to their perception of being able to carry out their academic tasks successfully. Subjective task value refers to the qualities of tasks that increase or decrease the probability of their selecting the task or putting effort on learning it [14]. Subjective task value has four subcomponents: intrinsic or interest value, attainment value, utility value, and cost.

*Intrinsic value* refers to the student’s enjoyment of performing the task or the subjective interest they have in the subject.* Attainment value* is the importance of doing well in the task. Similarly,* utility *(*instrumentality*)* value* refers to how useful and important a school subject is for the students’ future goals, such as career plans.* Cost value* refers to negative (undesirable) aspects in engaging in a task, such as performance anxiety, the fear of failure. Cost can also be conceptualized in terms of the loss of time and energy for other activities [7].

Evidence is quite strong that the expectancy of success, interest, intrinsic motivation, and intrinsic value predict greater academic engagement and success in learning (see [7]). For example, in their recent study, Trautwein et al. [16] showed that expectancy and value components predict significantly German upper secondary school students’ () performance in mathematics based on TIMSS (Trends in International Mathematics and Science Study) standardized test and that the prediction power of expectancy of success is larger than that of value components.

#### 3. Materials and Methods

The data of this survey were collected from four different lower secondary schools, four upper secondary schools from the different parts of Finland, and one university. The size of the sample () in this survey is not as large as in the surveys mentioned in the introductory section, yet it can be seen as sufficient to provide a quite realistic overview of Finnish students’ arithmetic skills. The participating university students represent the humanities: the students from three different courses at one typical multifaculty Finnish university were invited to participate. Those who did, did so on a voluntary basis without any reward.

We have organized the participants into four groups and, in the following sections, use the following abbreviations of them: PK, the eighth graders (“peruskoululaiset”); MAA, the eleventh graders studying the advanced syllabus in mathematics (“pitkä matematiikka”); MAB, the eleventh graders studying the basic syllabus in mathematics (“lyhyt matematiikka”); UNIV, the university students. The difference between the MAA and MAB syllabi in upper secondary school is that the MAA students have ten compulsory courses but they often take three or four specialisation courses, too, and the MAB students study six compulsory courses and typically 0–2 specialisation courses. Moreover, the MAB courses are more focused on basic arithmetic, problem solving, and the real life phenomena.

In the PISA, the Finnish participants are 15-year-olds from the eighth and ninth grades. In order to minimize the organizational troubles at the participating schools, we focused only on the eighth graders. Since most children start school in Finland at the age of seven, the PK students are fourteen or fifteen years old. Most of the MAA and MAB students are seventeen years old. The age of the UNIV students was not asked but mostly they are second or third year university students.

The participants were given 60-minute time to answer the form. The use of a calculator was allowed. The form contained, first, a section surveying the participants’ educational history and other typical background information. In the second section, the students’ motivation-related beliefs were measured using a modified questionnaire based on a well-established achievement goal instrument by Skaalvik [11] and Lukin [6] and an instrument targeted to measure students’ competence and task value beliefs which was adapted from Wigfield’s and Eccles’s [14] expectancy-value model of motivation.

The students’ goal orientations were measured using a 27-item Likert-type instrument with a five-point scale ranging from 1 which is “Strongly disagree” to 5 which is “Strongly agree.” The students’ perceived competence (the expectancy of success) and the value of learning mathematics were measured in the same range using a 19-item scale. Both measures were adapted to the Finnish educational context. The items were written in a domain-specific form (see [10]), for example, “*In my mathematics class, I like to solve problems by working hard.*” Other exemplar assertions will be given in Section 4.2.

The data of this section were analysed in sequential steps consisting of an exploratory factor analysis, -means cluster analysis, and two-way analysis of variance (two-way ANOVA). First, an exploratory factor analysis was applied to develop two sets of scales of the students’ motivation-related beliefs and to serve in identifying two sets of latent constructs underlying the batteries of measured variables. Second, a centroid-based clustering, that is, -means clustering, was used to identify the groups of students based on the six scales derived from the factor structures of the motivation instruments. The third phase of the data analysis was to test mean differences in students’ mathematics performance using two-way ANOVA with a motivation group (three clusters) of a student and his/her educational level as independent variables.

The final section of the form consisted of altogether seven mathematical assignments. The items were given to the students in Finnish but we now give their English translations.(1)* (Corresponding to M413Q01 in PISA 2003) Anne and her two friends decided to spend their holidays in Sweden.*(a)* How many Swedish Kronas did they get for 2000 Euros as the exchange rate was 1 EUR = 9.3 SEK?*(b)* A month later the rate was 1 EUR = 9.5 SEK. How many Kronas more or less would they have gotten with this rate?*(2)* (The first part corresponding to M467Q01 in PISA 2003) There are differently coloured balls of the same size in a box. The number of balls of each colour is given in the following*. *Red*:* 6** **Yellow*:* 5** **Orange*:* 3** **Green*:* 3** **Blue*:* 2** **White*:* 4** **Purple*:* 2** **Brown*:* 5* *Bertil takes one ball out of the box without looking at the colour.*(a)* What is the probability that the ball is red?*(b)* How many red balls should you insert into the box in order to guarantee that the probability is at least 25 per cent?*(3)* (The first part corresponding to M468Q01 in PISA 2003) Camilla takes part in a quiz, which consisted of five rounds, each one giving 0–100 points. After four rounds, the average of her points is 60. Her scores for the final round are 80 points.*(a)* What is the average of Camilla’s points in the whole quiz?*(b)* What could have she been able to deduce on that average already before the final round?*(4)* On a summer day, you are at the market square with Diana and Edward. Edward proposes: “If I buy us 1/2 kg of strawberries, which we all share one by one, how about you two together pay me the price of the portion multiplied by the remainder of the division?” Should you suggest to Diana that you accept this offer?*

The fundamental idea in designing the test items was as follows. First we selected three mathematics questions from the PISA 2003 test (M413Q01, M467Q01, and M468Q01). They were only translated and recontextualized in the Finnish setting. Then we generated three follow-up assignments to measure the same content knowledge. The purpose of this manoeuvre was to measure to what degree success in the given task depends on that how it is given. The last test question was intentionally designed to be more challenging than the others in order to see how many students are motivated enough to take on the challenge.

The first six test items were scored on the scale 0–3, where 0 is a wrong conclusion with no explanation or with several serious defects in the explanation or a missing answer, 1 is a correct conclusion with no explanation or an incorrect conclusion with a partly reasonable explanation, 2 is a correct conclusion with a few minor defects in the explanation, and 3 is a correct conclusion with a reasonable explanation. In the corresponding PISA 2003 test items, the evaluation of the responses was dichotomous (1 is full credit; 0 is no credit); that is, no explanations were required. Therefore, in the comparison of our data with that of PISA 2003, we recoded and . The scale on the last and a more challenging assignment was 0–6.

#### 4. Results

##### 4.1. Analysis of Students’ Mathematical Performance

We begin with answering the first research question. The distributions of the means and the percentage of the full credit responses of each group (the figures in the parenthesis) are given in Table 1.