The assessment culture of mathematics in Finland: a student perspective

ABSTRACT

This study outlines the assessment culture of mathematics in Finnish basic education (grades 6 and 9) based on a nationally-sampled dataset (N = 673). The study draws on the theoretical framework of assessment cultures as defined through cultures of learning, compliance and fear. We approach the assessment culture of mathematics through the student perspective. Students responded to a survey of how often summative and formative assessment practises were used in mathematics, and how useful they perceived them to be. A cluster analysis was conducted to understand the various student sub-groups on the basis of their perceptions of assessment. Overall, the results highlight the dominating role of exams as the backbone of mathematics assessment. Based on these findings, we conclude that in Finland, where assessment in basic education is mainly low-stakes, and where there are no national tests at this level, the assessment culture of mathematics largely represents a culture of compliance.

KEYWORDS:

• Assessment of mathematics
• assessment culture
• student perspective

Introduction

There have been numerous calls for mathematics educators to better align their assessment practises with the ideals of Assessment for Learning (AfL) and formative assessment (Goos et al., 2020; Suurtamm & Koch, 2014; Suurtamm et al., 2016). The assessment culture of mathematics education has been noted to build on Assessment of Learning (AoL) - namely, testing - rather than on AfL. Teacher surveys indicate that tests and quizzes are the most frequently used assessment practises around the world (Suurtamm, Koch, & Arden, 2010). This has been reported at various levels of education in North and South America (Camargo & Ruthven, 2014; McMillan, Myran, & Workman, 2002) and in Africa (Sunzuma, Ndemo, Zinyeka, & Zezekwa, 2012). Similar results have also been published from Asia (Zhao, Van den Heuvel-Panhuizen, & Veldhuis, 2018) and Europe (Atjonen et al., 2019; Iannone & Simpson, 2011, 2021; Veldhuis, Van den Heuvel-Panhuizen, Vermeulen, & Eggen, 2013). Regardless of a strong preference for AfL in modern educational policies, major shifts in the exam-driven assessment cultures of mathematics are yet to be seen.

In our study, we examine the assessment culture of mathematics in Finland by paying attention to students, whose perspective on mathematics assessment has hitherto remained marginal, both in Finland and internationally. Finland offers a particularly interesting context, as neither national testing nor strict grading policies restrict the classroom assessment conducted by the teachers. The Finnish National Core Curriculum strongly links classroom assessment of mathematics with the active role of students through formative assessment and dialogic assessment practises, strongly advocating for AfL. However, summative assessments and written exams still have a strong position among all Finnish teachers, not only among mathematics teachers, as reported by Atjonen et al. (2019) and Mäkipää and Ouakrim-Soivio (2020).

We conduct a person-oriented analysis based on a nationally-sampled dataset collected in order to understand the perceptions of student sub-groups in terms of these assessment perceptions. We supplement earlier research on mathematics assessment in two ways. First, we approach the phenomena through the students’ eyes, which has thus far been much less studied than the perspective of teachers (Kyaruzi, Strijbos, Ufer, & Brown, 2019). Second, we build on earlier research on student perceptions of mathematics assessment by shifting the attention from secondary and post-secondary education to both lower (students aged 7–12 years in grades 1-6) and higher (years 12–15 in grades 7-9) levels of basic education.

Cultures of assessment from the students’ viewpoint

Conceptualising assessment cultures

Assessment cultures are subsets of educational cultures defined by the values, beliefs and assumptions concerning assessment held by teachers, principals and students (Banta, 2002). Assessment cultures influence how assessment is conducted and how schools respond as pedagogical communities to change or to perceived threats regarding assessment (Maki, 2012). Sometimes cultures may be tacit or taken for granted in ways that are difficult to recognise, consciously or critically. This is what Shulman (2005) calls a “signature pedagogy”, the habits of mind, heart and hand in certain professions or disciplines. Different school subjects have their own signature pedagogies that determine how students are taught and assessed, with some signature pedagogies perhaps stronger than others. Recently, Pitt and Quinlan (2021) conceptualised “signature assessments” based on Shulman’s (2005) work, noting that in a certain disciplinary culture, assessment reflects, for example, conceptual (content knowledge), epistemological (what counts as valid knowledge in a discipline), and social (social structures of disciplinary practice) dimensions. Of these dimensions, the epistemological aspect of mathematics assessment was recently discussed through the perspective of university students by Nieminen and Lahdenperä (2021). Nieminen and Lahdenperä analysed how assessment both reflected and constructed the epistemic nature of mathematics. As mathematical knowledge was considered as stable and objective, and solutions as being either right or wrong, exams were seen as an appropriate method for demonstrating mathematical knowledge. At the same time, test-driven assessment was analysed to further strengthen such an epistemology, as valid mathematical knowledge was understood only in terms of what timed tests can measure.

Fuller and colleagues (2016) differentiated between three cultures of assessment (see also Skidmore, Hsu, & Fuller, 2018). While these cultures overlap, one of them often dominates in a certain national and disciplinary context. The first, culture of learning, envisions quality as improved learning. In such a culture, assessment focuses on improving learning through diverse assessment practises (Dochy, Segers, Gijbels, & Struyven, 2007). Learning is conceptualised as an active construction of knowledge that asks students to practise self-assessment and reflection, and to collaborate with the teacher and schoolmates regarding feedback. Furthermore, students are provided with genuine opportunities to participate in all phases of assessment and to rehearse versatile assessment skills in practice (Falchikov, 2004).

The benefits of the assessment culture of learning have been reported widely in the discipline of mathematics at various levels of education. Student-centred assessment practises (e.g. formative assessment, AfL) support, for example, the quality of learning (Nieminen, Asikainen, & Rämö, 2021), achievement (Palm, Andersson, Boström, & Vingsle, 2017), self-efficacy and interest (Rakoczy et al., 2019) and motivation (Faber, Luyten, & Visscher, 2017). Outside mathematics, it has been showed that collaboration (e.g. peer-assessment), higher-order thinking skills (e.g. challenging assignments), and self-assessment may deepen the students’ engagement in assessment, which improves their results in formative and summative assessments (e.g. Bae & Kokka, 2016; see also Falchikov, 2004).

In addition to the culture of learning, the other two cultures as defined by Fuller, Skidmore, Bustamante, and Holzweiss (2016) and Skidmore et al. (2018) are the cultures of compliance and fear. These focus on rules and regulations over students’ learning, rather than on learning itself (Skidmore et al., 2018). A culture of fear punishes teachers’ innovative ideas for assessment, which is reflected in unilateral and teacher-driven assessment practises such as exams. Within such a culture, students may see testing, grading and teacher-centred assessment as the most important practises. One might find a culture of fear in high-stakes assessment cultures where grades have a great impact on the students’ later life (Fuller et al., 2016). This may cause stress and anxiety in assessment situations, which hinders students in expressing their best learning and promotes impractical study habits (Löfgren, Löfgren, & Lindberg, 2019; OECD, 2015). Because assessment is important to students (Atjonen et al., 2019; Lord & Jones, 2006; Löfgren et al., 2019), the pressure to succeed in assessment situations may become high.

Finally, in a culture of compliance, teacher-driven assessment practises are over-emphasized, although not based on fear but on compliance. Whereas cultures of fear and compliance might both cause issues from the viewpoint of learning, compliance, rather than drawing on force (culture of fear), is related to systemic hindrance of students’ critical reflexivity. As such, over-reliance on compliance reflects earlier literature on restricted student agency and epistemic injustice in mathematics (Tanswell & Rittberg, 2020) and in mathematics assessment (Nieminen, 2020; Nieminen & Lahdenperä, 2021). Ethical issues arise if students are not taught to act agentically and autonomously in assessment. Indeed, Skidmore and colleagues (2018) note that while it is sometimes necessary to comply, such an assessment culture hinders the development of critical thinking skills of both students and teachers. Robinson and Fielding (2007) reiterated that the culture of compliance might teach students to be too dependent on the teacher’s acceptance of whether they have done all that was required and whether their work is “right” or “wrong” (see also Black, 2015). A culture of compliance might reduce assessment interaction and on-going observation of students’ activities, as overemphasis is given to summative assessment rather than to formative assessment with diverse feedback practises, which are of crucial importance for enhancing students’ responsibility for their own learning in mathematics (Kyaruzi et al., 2019).

Student perspectives on classroom assessment

Students’ perceptions of assessment play a significant role in assessment behaviour. For example, students who perceive assessment as important or who want to improve their performance tend to obtain higher achievement scores (e.g. Brown & Hirschfield, 2007; Eklöf & Nyroos, 2013). Perhaps even more importantly, assessment cultures that emphasise testing (namely, assessment cultures of fear and compliance) may promote superficial learning for grades (Dochy et al., 2007; Nieminen et al., 2021) and students’ short-sighted behaviour by last-minute preparation for exams (Eklöf & Nyroos, 2013). If students have sufficient relevant knowledge of the main functions of assessment, there are multiple benefits for their deeper learning (Smith, Worsfold, Davies, Fisher, & McPhail, 2013).

Regarding perceptions of assessment methods, Rieg (2007) compared teachers’ and students’ views of the perceived usefulness of various methods for learning. Rieg found that students wanted, for example, opportunities to take tests in pairs or small groups, and to have oral or take-home exams, but teachers did not approve these practises. In a review by Lord and Jones (2006), students reported that coursework, practical examinations and computer-based formats fit their needs, and they appreciated a range of formal and informal, both teacher- and student-led assessments. On the other hand, some students might prefer clear-cut exams, numerical grades and ordinal ranking of their work, and even condemn qualitative written and oral comments as containing insufficient information regarding what is “good or bad” (Lord & Jones, 2006).

The notion of student assessment literacy - students’ skills and capacities to utilise assessment for the purposes of their learning - is also closely associated with students’ perceptions of assessment. According to Smith et al. (2013), assessment literate students understand the purposes of assessment, are aware of the processes of assessment, and are able to judge their own work and develop strategies to improve it. They are able to regulate their own beliefs about assessment based on their perceptions (Charteris & Thomas, 2017).

Student perspectives to mathematics assessment

Students are not simply passive reflectors of the assessment cultures of mathematics. Rather, they take an active part in constructing these cultures (Nieminen, 2020). It is thus crucial to understand students’ perceptions of mathematics assessment when analysing mathematical assessment cultures. Many studies have indicated the dominance and students’ preference for traditional summative assessment in the context of post-secondary education (Iannone & Simpson, 2015). The student perspective has been studied through many concepts. For example, Iannone and Simpson (2019) studied students’ preferences and beliefs, finding no correlation between university students’ summative assessment preferences and their beliefs about mathematics. It has been noted that mathematics students are often unfamiliar with formative assessment processes and have little experience of self- and peer-assessment (e.g. Brown & Lally, 2018).

It appears that research on students’ perceptions (or other related concepts such as conceptions or beliefs) of mathematics assessment have been mostly conducted in the context of post-secondary education (e.g. Iannone & Simpson, 2015, 2019; Nieminen, 2020; Nieminen & Lahdenperä, 2021). We were able to identify only a few mathematics studies that investigated the students’ perspective on assessment at lower levels of education. In one study by Martínez-Sierra and colleagues (2016), several social representations of assessment in mathematics were identified among Mexican high-school students. Students in the study felt that assessment in mathematics must be different from that employed in other subjects because mathematics is different, and that assessment indicates what else is needed to “advance” or “acquire”. Kyaruzi and colleagues (2019) surveyed Tanzanian secondary-school students and highlighted that students appreciated formative feedback practises as long as they were perceived as warm and supportive and as monitoring rather than scaffolding. The researchers concluded that “student perceptions of their teachers’ formative assessment practises have a large impact on the effectiveness of instructional processes” (p. 294).

Research objective

Based on the theoretical framework of assessment cultures of learning, fear and compliance (Fuller et al., 2016; Skidmore et al., 2018), our study sheds light on the assessment culture of mathematics assessment in Finnish basic education. Finland offers an interesting context for a study on mathematical assessment cultures due to its overall low-stakes assessment culture. As there are no national exams in basic education, and as assessment at this level of education is largely low-stakes, one might hypothesise that Finland offers a fertile ground for an AfL-oriented culture for mathematics assessment. Our characterisation of the assessment culture of mathematics contributes to earlier literature on national surveys of mathematics assessment (e.g. Camargo & Ruthven, 2014; Zhao et al., 2018) as we analyse students’ perceptions regarding mathematics assessment. Overall, student perceptions of assessment are an understudied field in mathematics education, particularly outside post-secondary education. Furthermore, assessment research has often conceptualised “students” as one homogeneous mass and under-emphasised the multiple potential sub-groups of students with various perceptions of assessment (Nieminen et al., 2021). To address these research gaps, we draw on a student-centred perspective of mathematics assessment in Finnish basic education, clustering students based on their assessment perceptions. Our research questions are as follows:

RQ1) According to the students, how often were various assessment and feedback practises used in mathematics? Were there differences between lower and higher levels (6th and 9th grades; students ages 12 and 15) of basic education?

RQ2) What kinds of student sub-groups could be identified in terms of students’ assessment perceptions?

RQ3) Were there differences between students’ perceptions of mathematics assessment in different sub-groups?

Thus, RQ1 characterises the assessment culture of mathematics as seen from the perspective of students, and RQ2 and RQ3 map out students’ perceptions of such an assessment culture. Overall, our approach is explorative. We emphasise that our analysis focuses on students’ perceptions rather than on “the reality of assessment”. Our findings only represent the assessment culture through the students’ perspective; although this view might be “skewed”, it represents how students see and understand such an assessment culture, and is thus worth studying. Similarly, our analysis of the assessment culture itself is only mediated through students’ perceptions.

Methodology

Context of the study

In Finland, basic education lasts for 9 years (from ages 7-15), and comprises grades 1-9. Finland relies on a decentralised educational system in which the local educational authorities have considerable leeway for action. In close collaboration with teachers, local educational providers must modify various parts of the FNAE’s (2014) national curricular guidelines (including parts of the contents and objectives of school subjects) and design the local curriculum.

Overall, assessment in Finnish basic education is low-stakes: “Evaluation is not exercised to control or sanction, but, rather, to develop education at all levels of the system, creating the best learning opportunities for every learner.” (Kumpulainen & Lankinen, 2016, p. 72) Finnish teachers must use the subject-specific objectives and assessment criteria set in the national curricular guidelines but are given considerable professional freedom to decide on their assessment methods and practises. No school inspections are used to check whether the national or local curriculum guides are being followed in terms of assessment. Instead, education providers must self-evaluate the quality of the education they provide (see Kumpulainen & Lankinen, 2016, for further details).

Finland does not apply national high-stakes exams in basic education. Instead, for developmental evaluation purposes, Finland utilises sample-based assessments of learning outcomes in selected school subjects at the end of basic education. As low-stakes tests, these do not have any impact on students’ further choices of schooling. The principles of the FINEEC, the Finnish Education Evaluation Centre, which implements the assessments of learning outcomes, rely on developmental evaluation and information guidance rather than accountability and control. Overall, the Finnish school culture does not include a strong tendency towards test anxiety. For example, the anxiety index in PISA measurements has been markedly low (OECD, 2015). However, while providing a low-stakes assessment culture, recent studies have noted that summative assessment and closed-book exams are still over-emphasised in Finnish basic education, according to reports by both students and teachers (Atjonen et al., 2019; Mäkipää & Ouakrim-Soivio, 2020). One might characterise the context as promising for further development of AfL, even though such approaches are not currently used extensively.

Data collection and participants

The data were collected in a broad national evaluation project that mapped the frequency and perceptions of assessment practises used in Finland by principals, teachers, students and parents (Atjonen et al., 2019). The report by Atjonen and colleagues was the first national-level evaluation of educational assessment and feedback practises in Finland. The original student questionnaire was designed and led by a team of researchers nominated by the Finnish Education Evaluation Centre (FINEEC). The team of researchers, nominated by the FINEEC, designed the student questionnaire by combining parts of several existing research-designed instruments and practice-based, tailored items of assessment. The student instrument was tested with 96 students (three age groups), on the basis of which some item clarifications and removals were made before the final data was gathered in January–February 2018. Particular attention was paid to item development in a way that would enable students to recognise formative assessment practises, as students might not always be aware of all such methods (Atjonen et al., 2019). This was achieved through language that was accessible for students.

The data were collected by first sampling randomly from all Finnish- and Swedish-speaking (the two official languages of Finland) schools of basic education (grades 1–6, lower level; and grades 7-9, higher level). Second, the teachers were sampled so that they evenly represented six groups of school subjects. In the third phase, the students were sampled based on the disciplines which the teachers taught. Overall, the student data consisted of 4882 replies covering a broad range of school subjects. From this data, in order to answer our research questions, we focused on students in basic education (grades 6 and 9) who had responded to the electronic questionnaire for the subject of mathematics. Moreover, from all the sections of the student questionnaire, we focused on those that concerned the frequency and perceptions of assessment, guided by our RQs.

All the survey items utilised a Likert scale from 1 to 5, in which the names of the scales varied depending on the item (e.g. “I completely agree” - “I completely disagree”; “always” - “never”). To answer RQ1, we focused on two sets of variables: students rated the frequency (how often they were used) of i) ten assessment practises and ii) nine feedback practises. The following ten summative and formative assessment practises were offered as a list for students in terms of the first two sets of items: closed-book exam, exam with materials, portfolio, essay, product (e.g. a computer program), performance (e.g. drama, music), self-assessment, peer-assessment, feedback from the teacher to the whole group, and assessment discussions between the teacher, the parents and the students. Students’ assessment perceptions (RQ2) were measured in terms of these specific assessment practises. Furthermore, the nine feedback practises listed were as follows: points or grades, instructions for learning, teacher indicates the errors, information about the goal of a task, oral feedback, model answers, descriptive comments, teacher provides a grade but does not point out the errors. Students’ perceptions of assessment were measured based on the above-listed ten assessment practises. Students rated the extent to which they thought each assessment practice supported their learning, again using a 5-point Likert scale (from “Does not help at all” to “Helps very much”). To further determine students’ perceptions of the assessment culture of mathematics (RQ3), we drew on eight items on perceptions of teachers’ formative assessment actions, and five items on pressures related to assessment, respectively (see Tables 6 and 7).

Students’ grade (6th or 9th grade) and gender (girl, boy, I don’t want to answer) were collected as background variables. We sought for gender differences because earlier research in Finland has identified profound gender differences in students’ socio-affective trajectories in mathematics education (e.g. Metsämuuronen & Tuohilampi, 2014). In addition to grade and gender, the following five single-item background variables (using a scale of 1–5) were also used: self-reported study success, general enthusiasm towards learning, interest in mathematics, importance of mathematics, and perceived personal importance of assessment.

Altogether, 828 students completed the survey regarding mathematics. We conducted a missing value analysis for the clustering variables and, due to the exploratory nature of the quantitative analysis, removed all the cases with missing data (155 cases). The final N was 673, consisting of 320 girls and 311 boys, and 395 sixth and 278 ninth graders. A missing value analysis was conducted. No differences were identified between those who had missing values and those who did not in terms of gender (χ²(2, N = 822) = .22, p = .89). Sixth graders were slightly over-represented amongst those with missing values (χ²(2, N = 828) = 7.15, p < .01, Cramer’s V = .09). In terms of the ordinal background variables (see above), no statistically significant differences were identified.

Analysis methods

For RQ1, we first reported students’ responses to how often various assessment and feedback practises had been used in mathematics education and compared grades 6 and 9 (the ends of the lower and upper levels of basic education). In the T-tests (and in the further post-hoc tests for ANOVA in terms of RQ3), effect sizes are reported using Cohen’s d (Cohen, 1988). According to Cohen, values up to 0.2 refer to small, 0.5 to moderate and 0.8 to large effect sizes.

For RQ2, we drew on a person-oriented approach, using cluster analysis to examine student sub-groups especially in terms of their assessment perceptions. We used an analysis procedure similar to that used by Iannone and Simpson (2015, 2019) in their studies, in which they have used the Assessment Preferences Inventory (API) instrument in post-secondary mathematics education. However, we also included formative assessment methods in the questionnaire items.

We conducted a principal component analysis (PCA) with a varimax rotation to investigate the structure of the students’ responses (RQ2). We anticipated that the items would reflect a similar component structure to that in the Iannone and Simpson study in post-secondary education, even though our study was conducted in the context of basic education. The number of components was fixed at four, after which the components explained 71.4% of the total variance of the data. The Kaiser-Meyer-Olkin Measure of Sampling Adequacy (KMO) for the dataset was .88 and Bartlett’s test of sphericity indicated p < 0.001, both supporting the performance of the component analysis. The next step was to investigate the internal consistency of each of the factors using Cronbach’s Alpha. Compared to Iannone and Simpson (2019), we also identified a component called “dialogue”, consisting of the items of formative assessment. In addition, “closed-book exam” formed its own separate component. In Table 1, we present the component structure and the Alphas.

Table 1. Loadings on components after the varimax rotation.

Item	Exam	Access to materials	Individual product	Dialogue
Closed-book exam	.94
Exam with materials		.89
Portfolio		.63
Essay			.77
Product			.79
Performance			.71
Self-assessment				.81
Peer-assessment				.71
Feedback from the teacher to the whole group				.75
Assessment discussions*				.70
% of variance explained	7.3	8.5	12.5	43.2
α	–	.55	.78	.81

*discussions between the teacher, the parents and the students

After the PCA, we used cluster analysis to examine the student sub-groups in the data, based on the four components. Hierarchical cluster analysis (Ward’s method, squared Euclidean distance) was used to determine the number of clusters, and K-means cluster analysis to identify the cluster membership. The clusters were characterised in terms of the background variables by using Pearson’s chi-squared test. The differences between the clusters in terms of their perceptions of assessment were identified using one-way ANOVA testing and, further, Tukey’s post-hoc tests.

Findings

RQ1) Outlining the assessment culture of mathematics

Closed-book exams were overwhelmingly the most used assessment practice compared to all other forms or methods (Table 2). Half of the students reported that closed-book exams were used “always” (also both mode median). Feedback and discussions were used quite often, but process-based assessments (e.g. portfolios) were less frequent. It is worth noting that the standard deviations varied considerably.

Table 2. The frequency of the assessment practices in the mathematics classroom (1 = never; 5 = always), according to the students, and their statistically significant (p < .001) differences between sixth and ninth graders.

							6th grade		9th grade		Independent samples test
Assessment practice	Mean	SD	Mode	Median	% of “never”	% of “always”	Mean	SD	Mean	SD	t	Cohen’s d
Closed-book exam	4.07	1.16	5	5.00	5.7	50.1	4.22	1.10	3.87	1.21	t (667) = 3.49	.30
Self-assessment	2.72	1.18	3	3.00	20.6	6.9	2.76	1.10	2.67	1.29	–	–
Feedback from the teacher to the whole group	2.59	1.23	3	3.00	26.8	6.6	2.59	1.12	2.58	1.26	–	–
Assessment discussions	2.23	1.27	1	2.00	40.0	7.9	2.44	1.26	1.93	1.22	t(660) = 5.23	.43
Peer-assessment	1.83	1.00	1	1.00	51.2	1.1
Portfolio	1.63	1.15	1	1.00	72.1	4.9	1.74	1.24	1.46	.99	t(665) = 3.08	.25
Product	1.57	.96	1	1.00	67.7	1.5
Performance	1.46	.94	1	1.00	76.5	1.5	1.60	1.03	1.26	.75	t(662) = 4.68	.38
Essay	1.43	.93	1	1.00	77.9	2.1
Exam with materials	1.39	.86	1	1.00	76.9	2.1	1.30	.73	1.53	1.00	t(659) = 3.34	.26

Regarding statistically significant differences between the sixth and ninth grades (Table 2), younger students reported the use of almost all assessment practises more than their older schoolmates, except for “exams with materials”. The broadest variation was in terms of assessment discussions. All effect sizes of the comparisons were moderate (.25 < d < .43).

Regarding feedback practises in mathematics (Table 3), points and grades were used most often, according to the students, with more practises aligning with the ideals of formative assessment (e.g. instructions for learning). Oral feedback and verbal comments were also noted quite often.

Table 3. The frequencies of different feedback practises, according to the students (1 = never; 5 = always).

Feedback practice	Mean	SD	Mode	Median	% of “never”	% of “always”
Points or grades	3.93	1.08	5	4.00	4.0	37.6
Instructions for learning	3.46	1.11	4	4.00	6.8	17.3
Teacher marks the errors and says what the error was	3.39	1.20	3	3.00	8.6	20.2
Information about the goal of the task	3.14	1.15	3	3.00	9.6	12.8
Oral feedback	2.94	1.01	3	3.00	9.8	5.6
Model answers	2.85	1.16	3	3.00	16.3	7.3
Verbal, descriptive comments	2.55	1.08	3	3.00	20.1	3.8
Teacher provides a grade but does not point out the errors	2.31	1.21	1	2.00	32.5	5.7
Errors have not been marked	1.80	1.09	1	1.00	54.8	3.5

The only statistically significant (p < .05) differences were that sixth graders reported a higher frequency of points and grades (mean = 4.01, SD = 1.03) than ninth graders (mean = 3.82, SD = 1.15), and also a higher frequency of verbal comments by the teacher (6th grade: mean = 2.64, SD = 1.02; 9th grade: mean = 2.42, SD = 1.16). The effect sizes were small in each case (d = .17 for points and grades, d = .20 for verbal comments).

RQ2) Examining student sub-groups

The students were clustered according to their responses to ten items concerning how useful each assessment practice was for learning. We conducted a hierarchical cluster analysis by using four components as grouping variables. Based on the dendrogram, a three-cluster solution was chosen as the most appropriate because it differentiated the extremes more successfully compared to the tested four-cluster solution. A K-means analysis with three clusters was conducted (Table 4). A further discriminant function analysis showed that 100% of the participants were correctly classified in their clusters.

Table 4. Comparisons of the three student clusters, based on the four components of students’ assessment preferences.

	Cluster 1^a		Cluster 2^b		Cluster 3^c		Total		ANOVA		Post-hoc testing
	M	SD	M	SD	M	SD	M	SD	F(2, 672)	η^2	Tukey HSD
Closed-book exam	3.75	.73	3.69	.83	1.49	.50	3.27	1.16	500.1*	.60	1, 2 > 3*
Access to materials	1.99	.78	3.36	.79	2.18	.94	2.58	1.04	208.4*	.38	2, 3 > 1*2 > 3*
Individual project	1.67	.64	3.24	.76	1.73	.76	2.32	1.04	381.4*	.53	2 > 1, 3*
Dialogue	2.20	.82	3.27	.77	1.85	.77	2.56	.99	192.6*	.37	2 > 1 > 3*

a = Exam-oriented pupils, b = Pupils who benefit from multiple forms of assessment in their learning, c = Pupils who do not feel they benefit from assessment in their learning. * p < .001

We named and characterised the clusters as follows:

• Cluster 1: Students (N = 263) who reported benefiting from exams in their learning (in short: exam-oriented), meaning that they reported benefiting mostly from exams in their learning.

• Cluster 2: Students (N = 274) who reported benefiting from multiple forms of assessment in their learning (multi-benefiters), meaning that they reported high means for all four components of assessment practises.

• Cluster 3: Students (N = 136) who do not feel that they benefit from assessment in their learning (by-passers), meaning that they reported low levels for all four components of assessment practises.

The clusters were compared and characterised in terms of the background variables. Based on a Chi square test (χ²(3, N = 668) = 17.72, p < .001), there were more girls among multi-benefiters than boys; however, Cramer’s V = .16 indicated that the difference was not great. In terms of age, sixth graders were over-represented among the exam-oriented students (χ²(2, N = 673) = 9.77, p < .01, Cramer’s V = .12).

Exam-oriented students and multi-benefiters did not differ remarkably in terms of the categorical background variables (Table 5).

Table 5. Comparing the three student clusters in terms of the ordinal background variables.

Background variable	Cluster 1		Cluster 2		Cluster 3		Total		ANOVA		Post-hoc testing
	M	SD	M	SD	M	SD	M	SD	F(2, 672)	η^2	Tukey HSD
Study success^a	3.89	.90	3.90	.95	3.17	1.04	3.75	.99	31.64*	.09	1, 2 > 3*
Importance of assessment	3.61	1.13	3.87	1.01	3.11	1.26	3.61	1.14	20.82*	.06	1 > 2**2 > 3*
Excitement about learning	3.52	.89	3.67	.91	2.64	1.12	3.41	1.02	56.76*	.15	1, 2 > 3*
Interest in mathematics	3.59	.95	3.67	1.00	2.73	1.28	3.45	1.10	40.21*	.11	1, 2 > 3*
Importance of mathematics	4.38	.74	4.35	.74	3.73	1.14	4.24	.88	30.63*	.06	1, 2 > 3*

a = self-reported on a scale of 1–5. *p < .001; **p < .05

The only statistically significant difference (p < .05) was that multi-benefiters reported a higher level for importance of assessment than exam-oriented students. By-passers formed a “risk group” in terms of lower reported study success, enthusiasm about learning, and interest and importance of mathematics (p < .01), as they could not recognise the relevance of assessment for their learning.

RQ3) Perceptions of assessment and differences between the clusters

We compared the assessment perceptions of students between the three student clusters. As shown in Table 6, the multi-benefiters (Cluster 2) perceived their teacher’s actions rather more positively than exam-oriented students when it came to the teacher informing them about their strengths in assessment. By-passers (Cluster 3) reported more negative perceptions about their teacher’s assessment actions. However, the effect sizes were small across the ANOVA comparisons.

Table 6. Comparing the perceived formative assessment actions of the teacher between the student clusters.

The teacher has …	Cluster 1		Cluster 2		Cluster 3		Total		ANOVA		Post-hoc testing
	M	SD	M	SD	M	SD	M	SD	F(2, 672)	η^2	Tukey HSD
helped me to revise my mistakes and learn from them	3.57	1.10	3.85	.93	2.95	1.27	3.56	1.12	31.83*	.09	2 > 1 > 3*
encouraged me to try harder	3.37	1.18	3.63	1.10	2.64	1.29	3.33	1.22	32.03*	.09	1, 2 > 3* 2 > 1**
told me how to progress in my studies	3.36	1.14	3.58	1.05	2.90	1.14	3.35	1.13	17.30*	.05	1, 2 > 3*
helped me to set my own goals	3.31	1.08	3.58	.96	2.71	1.16	3.30	1.10	30.76*	.09	2 > 1 > 3*
helped me to develop my ways of learning	3.26	1.11	3.62	.95	2.78	1.19	3.31	1.11	28.47*	.08	2 > 1 > 3*
informed me about my strengths	3.14	1.25	3.42	1.14	2.42	1.26	3.11	1.26	31.17*	.09	1, 2 > 3* 2 > 1**
guided me to assess if I have reached my goals	3.01	1.11	3.34	1.00	2.61	1.15	3.06	1.11	21.56*	.06	2 > 1 > 3*
told me how to prepare for assessment	2.87	1.10	3.25	1.13	2.46	1.20	2.94	1.17	23.03*	.07	2 > 1 > 3*

* p < .001; ** p < .05

We examined students’ perceptions of assessment-related issues: the findings are presented in Table 7. No statistically significant differences were found between exam-oriented students and multi-benefiters. By-passers stood out from the other two clusters, as the students in that cluster reported more pressures related to assessment. Yet again, effect sizes for the ANOVA comparisons were close to 0.

Table 7. The perceived critical pressures of mathematics assessment in each of the student clusters.

	Cluster 1		Cluster 2		Cluster 3		Total		ANOVA		Post-hoc testing
	M	SD	M	SD	M	SD	M	SD	F(2, 672)	η^2	Tukey HSD
My teacher is too demanding or strict in assessment	2.35	1.07	2.37	1.15	2.79	1.28	2.45	1.16	7.04*	.02	1, 2 > 3*
Assessment situations frighten and distress me beforehand	2.10	1.20	2.30	1.22	2.75	1.41	2.31	1.27	12.10*	.04	1, 2 > 3*
There are too many exams and assessed tasks	2.56	1.20	2.59	1.17	3.17	1.28	2.70	1.23	13.07*	.04	1, 2 > 3*
The teacher gives me too much negative feedback	1.79	1.04	1.93	1.06	2.56	1.34	2.00	1.15	22.47*	.06	1, 2 > 3*

* p < .001; ** p < .05

In summary, the students generally reported relatively low levels of critical assessment pressure. For example, students in all clusters reported that mathematics assessment did not cause them distress beforehand. The number of exams and assessed tasks was the item closest to raising issues, according to the students, but even in Cluster 3 (by-passers), the mean was not alarming (3.17).

Discussion

The exam-dominated culture of mathematics assessment

We have outlined the assessment culture of mathematics in the context of Finnish basic education (RQ1). This study adds to our understanding of the signature pedagogies (Pitt & Quinlan, 2021; Shulman, 2005) of mathematics assessment through a student perspective. Our findings indicate that, according to the students, assessment in mathematics was mainly conducted by means of closed-book exams. The role of assessment discussions, peer-assessment, or portfolio, for example, was quite minor in the students’ responses. Younger students (sixth graders) reported higher frequency of closed-book exams than their older peers (ninth graders), but the overall repertoire of assessment practises was more widely reported by the sixth graders. According to students, “points or grades” were the most frequent form of feedback, rather than formative practises (e.g. “instructions for learning” or “oral feedback”). While comparing the use of assessment and feedback practises between the lower and higher levels of basic education, it was found that younger students reported that closed-book exams were used more frequently. However, as Suurtamm and Koch (2014) remind us, assessment practises can be diverse even when exams dominate a certain culture. International teacher surveys have highlighted the dominance of exams in mathematics education even when formative assessment practises have been implemented (e.g. Camargo & Ruthven, 2014; Zhao et al., 2018). In our study, a wider set of practises was indeed used more often at the lower level of basic education, according to the students.

These findings imply that students perceive the assessment culture of mathematics in Finland as exam-driven: a closed-book exam appears to be the “default practice”. This finding reflects the overall assessment culture in Finland. As Mäkipää and Ouakrim-Soivio (2020) showed, students’ perceptions of assessment largely reflected Assessment of Learning rather than Assessment for Learning (see also Atjonen et al., 2019). Such over-representation of closed-book exams in students’ perceptions is in clear contradiction with the national guidelines, in which students’ active role and participation in mathematics assessment are emphasised through versatile assessment and feedback practises.

Our finding might reflect the disciplinary assessment culture of mathematics. Perhaps exams are deemed suitable for assessing mathematical knowledge. At the same time, our findings represent the complexity of implementing formative assessment and AfL practises even in a low-stakes context that allows teachers a considerable amount of autonomy over their assessment practises: it might not be easy to change students’ perceptions of assessment. The perceived broader method spectrum of sixth graders may be explained by the broader educational and pedagogical studies of their teachers during initial teacher education. Furthermore, teachers of ninth graders may be more tied to the traditional assessment culture of mathematics.

Digging deeper: the three student clusters

Based on a cluster analysis of students according to their assessment perceptions, three student clusters were identified: exam-oriented students (Cluster 1; N = 263); multi-benefiters (Cluster 2; N = 274); and by-passers (Cluster 3; N = 136). Students in Clusters 1 and 2 (79.8% of our population) highly appreciated exams as a practice that supports their learning, but the multi-benefiters also understood the value of formative assessment practises. Therefore, the strong exam-orientation discussed in the previous section is also seen in the cluster formation. The third cluster, by-passers, constitutes a “risk cluster”, as these students perceived assessment the most negatively. However, although these students reported more negative perceptions concerning formative assessment practises, they also reported a notably low level of issues in mathematics assessment.

Several pedagogical implications follow. First, the sub-group of multi-benefiters should ideally be as large as possible, as numerous benefits of assessment literacy are well known (Smith et al., 2013). Training students to actively take advantage of and to engage with versatile assessment practises has been shown to be beneficial for learning (e.g. Bae & Kokka, 2016; Lord & Jones, 2006; Palm et al., 2017), and the idea indeed lies at the heart of AfL. This cluster stood out, as the students reported more positive perceptions of assessment in terms of whether the teacher helped them to correct their mistakes and to try harder, and whether the teacher informed them about their strengths. These aspects refer to the potential, substantial benefits of the AfL approach in mathematics assessment (Goos et al., 2020; Suurtamm & Koch, 2014).

Second, the by-passers might need special pedagogical attention in order to encourage them to engage with assessment (Bae & Kokka, 2016). The by-passers reported lower study success, lower enthusiasm about learning, and lower interest in and importance of mathematics, which implies that perhaps their educational trajectories were overall negative. How to promote positive perceptions and, further, assessment literacies of such “by-passers” constitutes an important goal for future research on mathematics assessment. Several previous studies (e.g. Lord & Jones, 2006; Löfgren et al., 2019) have indicated that assessment as such is important for students, yet the by-passers did not perceive assessment as beneficial for their mathematics learning. A similar notion could be made concerning the exam-oriented students, who might benefit from an appreciation of more diverse assessment practises. Interestingly, exam-oriented students only differed from the multi-benefiters marginally in terms of assessment perceptions. This finding emphasises that the exam-oriented students did not perceive significant issues in the general assessment culture of mathematics. As there is a convincing body of research highlighting the evidence of the benefits of AfL (Black, 2015; Black & Wiliam, 2018), it remains an interesting question for future research to determine how exam-oriented students could be supported through student-centred AfL practises.

Mathematics assessment and the culture of compliance

The general overview of used assessment practises (RQ1), according to the students, paints a picture of an exam-based culture, which is further reflected in students’ assessment perceptions (RQ2, RQ3). However, as Table 7 shows, mathematics assessment did not generally frighten or distress students in any cluster, nor was negative feedback a significant issue (although Cluster 3 still formed a “risk cluster”). These findings reflect the low-stakes, non-stressful culture of both school and assessment culture in Finland (OECD, 2015). Furthermore, the findings reflect earlier research in post-secondary education that has emphasised how mathematics students themselves might wish for closed-book exams (Iannone & Simpson, 2015, 2019).

Our results imply that the assessment culture of mathematics in Finland builds on compliance (Fuller et al., 2016; Skidmore et al., 2018). Issues emerge as the culture of compliance might teach students to rely on the teacher’s actions (Black, 2015; Robinson & Fielding, 2007) rather than aiming to develop their own skills by taking responsibility for their own actions, which is a basic principle of AfL (Black & Wiliam, 2018). It should be asked whether mathematics assessment enables critical reflexivity, as students’ perceptions reflected such a culture of compliance; this is most evident with the exam-oriented student cluster. Over-reliance on compliance might provoke the backwash effect of assessment that leads students to study for exams, even in low-stakes cultures. We emphasise that students are active co-creators of mathematical assessment cultures. Under the culture of compliance, teachers’ efforts to develop their assessment practises may even face resistance from students (Nieminen, 2020).

We have studied the culture of mathematics assessment in the intersection of two contexts: mathematics education and Finnish basic education. Although assessment literature has started to conceptualise disciplinary assessment cultures (Pitt & Quinlan, 2021), it is necessary to consider the socio-historical and -political contexts of assessment as well. For example, few other school disciplines are tested as frequently as mathematics in our current societies. The framework of assessment cultures has offered us a useful tool to understand students’ perceptions. The culture of compliance reflects the overall low-stakes assessment culture of Finland. As it is mathematics in particular that is often tested in national examinations, it might be that in high-stakes assessment cultures, the mathematical assessment culture might reflect that of fear. This aspect is crucial to consider while comparing apparently similar findings about exam-driven assessment practises in mathematics, as introduced in the very beginning of our study. Although Finland appears to provide a fertile ground for AfL implementation as outlined in the National Core Curriculum, AoL still largely dominates assessment. This tension seems contradictory; we interpret it to represent the complexity and slow pace of national AfL implementations. AoL and exams have strong socio-cultural and -historical roots in Finnish education, even without high-stakes testing systems. This historical weight should not be underestimated.

We also emphasise the disciplinary assessment culture of mathematics. When looking at our results in parallel with the international findings, we cannot help but ask: is there something universal about how mathematics is assessed? Does mathematics have a particularly strong signature pedagogy (Pitt & Quinlan, 2021; Shulman, 2005)? Although Tanswell and Rittberg (2020), amongst others, have discussed ethical issues of positivist and “objective” epistemologies in teaching mathematics, perhaps not enough attention has been paid to the role of assessment in the construction of such epistemologies. Nieminen and Lahdenperä (2021) discuss the epistemological dimension of assessment cultures, noting that even assessment practises that appear “objective” (namely, exams) are not neutral but actively contribute to students’ perceptions of the epistemic nature of mathematics. The unilateral assessment culture may be risky for various objectives in mathematics education and its main aim of promoting mathematical thinking and reasoning (Goos et al., 2020) and transversal skills. Surely the culture of compliance is not only an issue of mathematics assessment; indeed, in the Finnish report by Atjonen and colleagues (2019), other STEM disciplines drew on heavily similar assessment practises. Future research could elaborate on the intersections and boundaries between various assessment cultures in order to determine what makes the assessment culture of mathematics mathematical.

A crucial task for future research is to explore how AfL could be promoted under mathematics assessment cultures of compliance. In this quest, listening to pupils’ perspectives – and understanding their views both as a product and as a building block of assessment cultures – is crucial for teachers to enhance their understanding of the benefits of AfL. Perhaps the intersection of literatures on assessment literacies (Smith et al., 2013) and cultures (Skidmore et al., 2018) offers affordances for such future quests. Furthermore, it should be noted that previous studies in post-secondary education have highlighted the need for radical assessment changes to enable students’ critical reflexivity (Nieminen, 2020; Nieminen & Lahdenperä, 2021): change does not always have to occur one step at a time, especially if assessment draws on unethical premises.

Limitations and implications for future research

Several limitations should be highlighted. First, we emphasise that our findings might not report the “reality” of how mathematics assessment is presented in surveys of professional teachers. However, students’ voices are, of course, as “true” as those of adults: they reflect mathematics assessment as students perceive it. It is important to note that students were only taking part in this study from within the culture of compliance, and this notion means that, by definition, the findings are “biassed”.

Our exploratory study drew on an instrument developed by a team nominated by FINEEC (Atjonen et al., 2019). Due to the lack of validated research instruments, we cannot directly compare our findings with those of studies using instruments such as the Students’ Conceptions of Assessment instrument (e.g. Brown & Hirschfield, 2007) or the Assessment Preferences Inventory (Iannone & Simpson, 2015). However, the component structure as identified in our study reflects that of Iannone and Simpson (2019), and the new component “Dialogue” fits our theoretical underpinnings of formative assessment. Our strength is the inclusion of formative assessment practises, not only summative ones. We have only constructed the “student perspective” through our limited approach that draws on perceptions. Due to the exploratory nature of the study, we provide a “big picture” of the assessment culture of mathematics in Finland; our exploratory study is an opening point for future research rather than the final word.

When it comes to the research instrument, various limitations should be noted. The FINEEC comparative evaluation survey was conducted using the same items for both sixth and ninth graders, which means that typical respondents were 12 and 15 years old. It was challenging to design items that are consistently interpreted in both age-groups regarding concepts such as “assessment”, “useful”, and “learning”. Even though the issues of language were carefully considered while constructing the questionnaire, and even though the questionnaire was rigorously piloted, our findings only represent what students understood as “assessment”. Our findings only represent the student perspective, for better or for worse: the students’ perceptions of assessment were mediated through the language of the questionnaire and through the students’ sense-making (see e.g. Iannone & Simpson, 2019).

Future studies could develop our view by utilising the Assessment Preferences Instrument supplemented with formative mathematics assessment practises. The prospective studies could theorise how students’ assessment perceptions are constituted within the prevalent assessment cultures. Both qualitative and quantitative approaches are needed to further conceptualise the role of students in co-creating the assessment culture of mathematics. Deeper data (e.g. interview data) would enable a further analysis of more nuanced concepts such as beliefs, conceptions and experiences.

Conclusion

Finally, we do not only call for further developments toward AfL in mathematics, but also for greater consideration of the students’ perspective within such developments. As Martínez-Sierra and colleagues (2016) put it, “it is possible to change the learning concepts of students by modifying the assessment methods” (p. 257). The student perspective should thus not be neglected during reforms of mathematics assessment. Any national AfL reform is likely to fail if students prefer traditional assessment practises, being left without support to develop their critical thinking and assessment literacies. At the same time, mathematics students might even strive for change rather than trying to resist it (Nieminen & Lahdenperä, 2021). We supplement the view by Martínez-Sierra and colleagues by noting that likewise, it is possible to change assessment practises and cultures based on students’ values, beliefs and perceptions. Perhaps the student cluster of multi-benefiters, as identified in our study, might include agents for change who would push teachers towards student-centred assessment cultures.

Disclosure statement

No potential conflict of interest was reported by the author(s).