One way adaptive expertise in school mathematics, specifically arithmetic, can manifest is in students’ Adaptive Number Knowledge (ANK), defined as the well-combined knowledge of numerical properties and arithmetic relations (McMullen, Brezovszky, et al., 2016). Beyond arithmetic with natural numbers, substantial individual differences in adaptive number knowledge have been found in the context of rational numbers, referred to as Adaptive rational Number Knowledge (ArNK; McMullen et al., 2020). Individual differences in ArNK have been found even among those with a higher level of performance on more routine tasks (McMullen et al., 2020). Indeed, routine arithmetic and numerical knowledge and skills are necessary for supporting A(r)NK, but they alone do not constitute these manifestations of adaptive expertise in mathematics (McMullen, Brezovszky, et al., 2016).

Without understanding procedures and their underlying principles, it is difficult to apply the skills to situations where there are new types of problems and different task-solving styles are required (Hatano, 1988). In fact, high levels of rational number knowledge or achievements in school mathematics in general do not necessarily lead to adaptive expertise in mathematics (McMullen et al., 2020). While rational numbers are an important part of mathematical development (Siegler et al., 2012), general upper-secondary school students, who have been studying rational numbers since the last years of primary school, still struggle to understand them (Van Hoof et al., 2015). However, little is known about A(r)NK in general upper secondary school students as most studies are made with students in lower grades (e.g., Halme et al., 2022; McMullen, Brezovszky, et al., 2016; McMullen et al., 2020).

Resnick (1991) introduced the abstract entities hypothesis in which the problem is that there is a discontinuity between school mathematics and everyday mathematics. This may explain why some students feel that formal mathematics does not prepare them for real-world mathematics. One issue is that in everyday situations, mathematics is not the only aspect you can focus on. In many situations, mathematics is an important factor and can help see or solve problems, but explicit attention must be focused on the mathematics for it to be used. As such, higher tendencies to spontaneously focus on mathematical features appear to be an important factor in mathematical development (e.g., Hannula et al., 2010; Hannula & Lehtinen, 2005; McMullen, 2014).

Research on spontaneous mathematical focusing tendencies emphasises the idea that individuals differ in their tendency to spontaneously focus on mathematical features in situations that are not explicitly mathematical (Hannula & Lehtinen, 2005; McMullen et al., 2019; Verschaffel et al., 2020). Spontaneous Focusing on Numerosity (SFON) is defined as “a process of spontaneously (i.e., in a self-initiated way not prompted by others) focusing attention on the aspect of the exact number of a set of items or incidents. (…) SFON indicates the amount of a child’s spontaneous practice in using exact enumeration in her or his natural surroundings” (Hannula et al., 2010, p. 395). For instance, a child could spontaneously, without prompting, report that there are two bananas and four pears in the bowl. In many everyday situations, there are multiple possible and relevant aspects to focus on (e.g., colour, shape, size, and taste), unlike in most mathematics textbook exercises.

As mathematical concepts become more difficult and students develop their mathematical skills and knowledge, the importance of more complex spontaneous mathematical focusing tendencies may also increase (McMullen, 2014). Spontaneous Focusing on Quantitative Relations (i.e., SFOR) describes the tendency to recognise and use quantitative relations without external guidance in situations that are not explicitly mathematical (McMullen, 2014; Van Hoof et al., 2016). Whereas SFON involves paying attention to exact numbers, SFOR requires paying attention to the relations between different quantities. For instance, in the above example noticing that there are twice as many pears as bananas, or that one-third of all the fruits are bananas.

SFOR and SFON both uniquely predict mathematical development (McMullen et al., 2019). SFON is a predictor of counting skills (Hannula & Lehtinen, 2005) and studies have found relations between SFON tendency in young children and later arithmetic skills (Hannula et al., 2010) and rational number conceptual knowledge (McMullen et al., 2015). Earlier research has discovered that adults have higher SFON than children in the same tasks (Chan et al., 2020; Chan & Mazzocco, 2017; Mazzocco et al., 2020) and SFON tendency in adults is related to higher mathematical academic achievement (Ben-Shachar et al., 2020). On the other hand, SFOR is shown to be related to students’ rational number knowledge development (Halme et al., 2022; McMullen, Hannula-Sormunen, et al., 2016) and it is a unique predictor of ArNK in primary and lower secondary school (McMullen et al., 2022). Whether the relation between SFOR and ArNK persists still in upper secondary school or is only relevant within the period of rational number learning, has not been researched.

The relations between spontaneous mathematical focusing tendencies and mathematical skills appear to be reciprocal. For example, SFON tendency predicts higher early numerical skills, which leads to a stronger SFON tendency (Hannula & Lehtinen, 2005). Likewise, SFOR tendency predicts the development of rational number knowledge, while rational number knowledge predicts the development of SFOR (McMullen, Hannula-Sormunen, et al., 2016). However, strong SFON and SFOR tendencies may not naturally develop in all students through typical instruction. It was shown that activities specifically designed to support SFOR had a stronger effect on SFOR tendency than promoting rational number knowledge through a number-line estimation game (Määttä et al., 2025). This implies that SFOR and SFON tendencies may not only be relevant within mathematical development but also desired outcomes in themselves that can be supported by instruction. Thus, it is important to investigate individual differences even among students in upper secondary education.

We propose that SFOR, SFON and ArNK all represent abilities to use mathematical knowledge outside of well-rehearsed routine problems. For example, SFOR and SFON are expected to support mathematical development through self-initiated practice (Hannula & Lehtinen, 2005; McMullen, 2014) meaning students with higher SFON or SFOR tendencies are expected to be more likely to recognise opportunities to use their existing mathematical knowledge in everyday situations. In addition to supporting mathematical development, self-initiated practice is also a desired outcome of mathematics education. Being able to see mathematics in everyday life and understanding the relevance of mathematics in different situations and problems is outlined as a goal of mathematics instruction (National Council of Teachers of Mathematics, 2014; Opetushallitus, 2014). Similarly, ArNK represents the use of mathematics in new situations and it may be supported by self-initiated practice with non-routine mathematics questions also outside the mathematics classroom (McMullen et al., 2022). Thus, we consider the possibility that SFOR, SFON and ArNK could all be seen as different aspects of adaptive expertise in mathematics.

One reason why SFOR was argued to be a supporter of the development of ArNK was that the self-initiated practice that could arise from a higher SFOR tendency may also support more richly connected knowledge about rational numbers (McMullen et al., 2022). SFON is connected to the development of rational numbers conceptual knowledge (McMullen et al., 2015) and conceptual knowledge plays an important part in forming adaptive expertise (Hatano, 1988). In addition, students who also experience reasoning and using rational numbers outside their mathematics instruction may gain a richer understanding of the concepts about the size of and operations with rational numbers (Määttä et al., 2025; McMullen, Hannula-Sormunen, et al., 2016). Thus, these students may be more likely to develop stronger ArNK. However, the exact relations between SFOR, SFON, and ArNK are unclear at the beginning of general upper secondary school.

In Finland, students choose either a vocational institution or a general upper secondary school after comprehensive school. The purpose of general upper secondary school is to strengthen students’ general knowledge and ability, deepen their interest in the world of sciences and arts, and develop skills for life management and working life (Opetushallitus, 2019). In general upper secondary school students can choose either the advanced (includes ten mandatory courses; in Finnish: pitkä matematiikka) or the basic (includes six mandatory courses; in Finnish: lyhyt matematiikka) mathematics syllabus track. This selection can be seen as a sign of more mathematical motivation and advanced knowledge (Metsämuuronen & Tuohilampi, 2017). Therefore, studying the differences between these two groups regarding SFOR, SFON, and ArNK will also shed more light on the development of adaptive expertise in school mathematics.

Adaptive expertise in mathematics is a field that needs more research since one societal goal of mathematics education is that students can solve novel problems flexibly and effectively in and outside the classroom. This study aims to expand our knowledge of adaptive expertise in SFOR, SFON and ArNK in general upper secondary school students. In this study, adaptive number knowledge (ANK) is focused on Adaptive rational Number Knowledge (ArNK) and spontaneous mathematical focusing tendencies measures include Spontaneous Focusing on Quantitative Relations (SFOR) and Spontaneous Focusing on Numerosity (SFON). 

The research questions:

As mentioned earlier, SFOR and SFON have reciprocal relations with mathematical knowledge and skills (e.g., Hannula & Lehtinen, 2005; McMullen, 2014), these relations may become less important as mathematical topics become more advanced. Nonetheless, there remain individual differences in SFON in adults (Chan et al., 2020; Mazzocco et al., 2020). Therefore, we aim to examine if SFOR and SFON are also such general outcomes of mathematics education that there would no longer be individual differences in them after nine years of comprehensive school.

Among adults and students even with a strong routine and conceptual knowledge of mathematics there have been individual differences in A(r)NK (e.g., Halme et al., 2022; McMullen, Brezovszky, et al., 2016; McMullen et al., 2020). While general upper secondary school students should have adequate basic mathematical skills, it is not a guarantee that their rational number knowledge and skills will be highly adaptive. Thus, we expect that there will still be individual differences in students’ ArNK in general upper secondary school.

General upper secondary students may freely choose between the advanced and basic mathematics tracks; thus, it could be a sign of motivation towards mathematics. In addition, the choice appears to reflect existing differences in mathematics skills (Metsämuuronen & Tuohilampi, 2017; Niemi et al., 2021). While it has not been previously examined, we expect that there are differences between the student groups in their adaptive expertise in school mathematics.

Earlier studies show that SFOR (McMullen et al., 2022; McMullen, Hannula-Sormunen, et al., 2016; McMullen & Siegler, 2020; Van Hoof et al., 2016) and SFON (McMullen et al., 2015) have correlations to rational number knowledge and skills. Prior research has also found significant connections between ArNK and skills and spontaneous mathematical focusing tendencies in other age groups, such as primary school and secondary school (e.g., Halme et al., 2022; McMullen et al., 2015; McMullen, Hannula-Sormunen, et al., 2016; McMullen & Siegler, 2020). However, less is known about these connections with students who have more advanced levels of mathematical knowledge. As these measures are proposed to capture manifestations of adaptive expertise, we expect SFOR and SFON to be related to ArNK in general upper secondary school students.

The mathematical proficiency of Finnish students has been consistently demonstrated to be strong in international assessments. They have performed above the OECD average in the Programme for International Student Assessment (i.e., PISA) since Finland first joined the study, with particularly high rankings in the early 2000s (OECD, 2023a, 2023b). In the Trends in International Mathematics and Science Study (i.e., TIMSS), which evaluates more curriculum-based mathematical knowledge, Finnish students have scored slightly above average (Von Davier et al., 2024). Although results in both PISA and TIMSS have declined in recent years, Finnish students continue to perform better in PISA, which focuses a bit more on applying mathematical skills in real-life contexts, compared to TIMSS, which emphasises more the knowledge acquired through formal education (OECD, 2023a, 2023b; Von Davier et al., 2024).

Participants (N = 106, 48% male; 46% female; 6% other or no response) were first-year students in general upper secondary school (approximately 15-17-year-olds) from three different schools in Finland. The participating schools are in smaller municipalities, each of which has only one upper secondary school. As a result, students who choose to continue in upper secondary education most often proceed to these local schools almost by default. Consequently, the schools are not particularly selective in terms of admission requirements and can be regarded as fairly typical local institutions. Of the participants, 57 were studying or were going to study the advanced mathematics syllabus track (includes ten mandatory mathematics courses; in Finnish: pitkä matematiikka) and 40 chose the basic track (includes six mandatory mathematics courses; in Finnish: lyhyt matematiikka) and nine of them did not answer.

Due to the necessity to accommodate the teachers’ schedules, 36 of the students were tested at the end of their first school year in the spring of 2021 and 70 were tested at the beginning of their first school year in the autumn of 2021. Since the percentages of the students in the advanced (59%) and basic (41%) syllabus tracks were the same at both testing times and there were no statistically significant differences in any variables between the testing times, we combined them into a single sample within the analyses.

We received permission to conduct the study from the teachers during whose lessons the testing was held. The students gave their consent for the research at the beginning of the study. Since the students were over 15 years of age, consent was not required from their guardians. At the start of the test, students were informed that taking part in the research was voluntary, that they could stop at any point if they wanted to, and that participation in the study would not affect their grades or other schoolwork. The study followed the ethical guidelines of the Finnish National Board on Research Integrity (Tutkimuseettinen neuvottelukunta; TENK, 2019).

Due to COVID-19 restrictions for school visitors, the testing was done on a digital platform instructed by the teacher. The teachers obtained the instructions in a written information package, and they could contact the researcher at any time via phone if needed. The teachers were allowed to help the students with practical issues, such as technical problems, but not with issues concerning how to solve the task. Students completed the test individually on computers and the test took approximately 25 minutes. Since one goal was to measure spontaneous mathematical focusing, students were not informed about the mathematical nature of the study before the testing, and the testing was held during a Finnish language lesson to avoid any association with mathematics. The teachers were aware of the nature of the study, but they were informed not to mention it to the students. Students were told that the study investigated students’ perception skills and at the end of the test the mathematical nature was revealed. The measures were in the following order: the Photo description task (six items), the Teleportation task (two items), and the Arithmetic sentence production task (four items). Before each task, there were instructions on what to do in the task and all tasks and items were presented to students in the same order.

Before the analyses, all answers were imported to Microsoft Excel and were coded according to coding rules used in previous studies for SFOR, SFON, and ArNK (Määttä et al., 2022; McMullen et al., 2020; McMullen, Hannula-Sormunen, et al., 2016). Analyses were made with IBM SPSS Statistics 27 (IBM Corporation, 2020) and figures with JASP (JASP Team, 2021). We used descriptive statistics and t-tests to address individual differences in SFON, SFOR and ArNK (Research Questions 1 and 2). We conducted independent samples t-tests for examining group differences between mathematics tracks (Research Question 3) and Pearson’s correlation coefficients for relations between dependent variables (Research Question 4). There were some missing values because not everyone had answered all the questions. No missing value replacement was used.

Based on descriptive statistics, SFOR, SFON and ArNK variables were normally distributed (skewness [-.852, 1.59] and kurtosis [-1.06, 2.17]) hence the correlation analysis was conducted using the Pearsons’s correlation coefficient. The results were also checked with non-parametric tests, and they did not differ from the parametric results.

Tables 2 and 3 show the descriptive statistics for the Photo Description (Items 1 to 6) and Teleportation (Items 1 and 2) task for SFOR and SFON, respectively. The distribution of SFOR and SFON mean scores in the Photo description and the Teleportation task can be seen in Figure 3. The average values are lower for both SFOR and SFON in the Photo description task than in the Teleportation task (SFOR t(102) = -9.285; p < .001, d = 1.1219 and SFON t(103) = -5.360; p < .001, d = 2.055). In addition, there were more SFON descriptions in both tasks than SFOR descriptions (Photo description task t(104) = 17.317; p < .001, d = 0.995 and Teleportation task t(103) = 16.849; p < .001, d = 1.788).

A closer analysis revealed that the Teleportation task included only SFOR descriptions related to multiplicative relations (e.g., “items have tripled”) with no descriptions of fractions or part-whole relations. In contrast, the Photo description task Items 3 to 5 included only a few descriptions of multiplicative relations and the rest were descriptions of part-whole relations (e.g., “three turtles from which two are bigger and one smaller”). In the whole sample, there was only one fraction description, which was in Item 1 of the Photo description task.

When the descriptive results were examined on an individual basis, it was noticed that 86.7% of the sample had at least one SFOR description and 99.1% had at least one SFON description. In the Photo description task 1.9% and in the Teleportation task 12.3% of the students gave no SFOR or SFON descriptions.

The descriptive statistics for ArNK measures are in Table 4. There was high individual variance in students’ performance on all the measures of ArNK: number of correct answers, equivalent solutions, and multiple representations (Figure 4). Only 3.8% of the sample did not produce mathematically correct answers, 18% did not have even one equivalent solution and 41% did not have even one multiple representation. Overall, there were still substantial individual differences in ArNK among the general upper-secondary students.

Descriptive statistics for SFOR, SFON, and ArNK between the advanced and basic mathematics syllabus tracks can be seen in Table 5 and the distribution of mean scores in Figure 5. The students in the advanced and basic mathematics syllabus tracks did not statistically differ in SFOR. However, the students in the advanced mathematics track had higher SFON, but only on the Teleportation task. In addition, they had a higher ArNK than those in the basic syllabus track.

The correlations between ArNK and the two measures of spontaneous mathematical focusing, SFOR and SFON, can be seen in Table 6. There were weak to moderate positive correlations between ArNK and both SFOR indices and ArNK and SFON on the Teleportation task. ArNK and SFON on the photo description task were not statistically significantly related.

Earlier studies have discovered that the salience of the features in the stimuli affects how much students focus on mathematical features, for example, numerosity (e.g., Chan & Mazzocco, 2017; Viarouge et al., 2019). The diversity of photographs across these items may stimulate different types and amounts of SFOR and SFON based on the pictures' characteristics, which may have affected the measures' validity. We had to analyse the spontaneous mathematical focusing tasks separately since the reliability was not sufficiently high to combine them. This may be due to the differences in the structure of the tasks (e.g., the Teleportation task was more structured). Nevertheless, the reliability for SFOR in the Photo description task was rather low and could not be improved by removing items. This could be partially explained by the floor effect on the number of SFOR descriptions. Using more items, or more systematically varying pictures, could increase the reliability of the measures. However, reliable indicators of students' self-initiated practice in recognition and use of exact numbers and multiplicative relations require using different tasks (Hannula & Lehtinen, 2005; McMullen, 2014). Notably, the Photo description task is based on earlier measures of guided focusing on quantitative relations and has been used in intervention studies with younger students (Määttä et al., 2022, 2025). When Määttä et al. (2025) used the Photo description task in their study on pre-tests and post-tests, Cronbach’s alpha was very low on the pre-test (.36) because of low SFOR values. After the intervention, there were more SFOR values and the alpha was at a good level (.78) (Määttä et al., 2025). However, in other studies of SFOR tendencies with tasks that do not solely rely on providing written descriptions of multiplicative relations to describe everyday situations internal consistency appears higher, even before explicit training (e.g., McMullen et al., 2014; McMullen, Hannula-Sormunen, et al., 2016). This indicates that it might be that without intervention explicitly linking school math to everyday situations SFOR responses in Photo description task might be fairly fragmented (Edelsbrunner et al., 2025). Thus, one possible future research direction could be to further examine the relation of SFOR responses across a variety of tasks. SFOR and SFON were examined with the same task, which might have created more overlapping connections between them than if they were measured with separate tasks. For more insightful research on the specific relations between SFOR and SFON, and other related mathematical skills, it would be meaningful to also study them by creating tasks that specifically measure only SFOR or SFON. On the other hand, using the same tasks and pictures to measure SFOR and SFON prevents stimulus characteristics from shadowing individual differences and allows for examining the individual differences of multiple spontaneous mathematical focusing tendencies simultaneously.

Adaptive expertise in mathematics is one important and desired outcome of mathematics education (e.g., National Council of Teachers of Mathematics, 2014; Opetushallitus, 2014, 2019) and the results of this study combined with earlier research suggest that SFOR, SFON and ArNK are parts of adaptive expertise in mathematics. Thus, we would hope that students already in general upper secondary school have some SFOR and SFON as an outcome of their mathematics learning. As well, that students would have strong ArNK after many years of experience with school mathematics. Nonetheless, students still have individual differences in how mathematical they see the world around them and the level of well-connected knowledge of rational number characteristics and arithmetic relations. These results suggest that we should still consider how to promote adaptive expertise in school mathematics even towards the end of comprehensive school.