Combined conceptualisations of metacognitive knowledge to understand students’ mathematical problem-solving

Abstract

This study examined the metacognitive knowledge of Finnish comprehensive school students and explored whether grade-based differences in metacognitive knowledge exist among 6th, 7th and 9th graders. Integrating qualitative (declarative, procedural, conditional) and contextual (person, task, strategy) frameworks, the research aims for a comprehensive understanding. We employed mixed-methods approach; qualitative data from 225 student interviews underwent qualitative theory-driven analysis, followed by quantitative methods to explore variations and relationships. Results showed prevalent procedural and strategic metacognitive knowledge. Utterances on problem-solving often linked to task-specific knowledge, emphasising the need for a varied metacognitive knowledge set. 9th graders excelled in explaining strategy use, while 7th graders demonstrated proficiency in understanding when and why to employ specific strategies. Although a qualitative level approach can aid in understanding the development of metacognitive knowledge, combining qualitative and contextual conceptualisations provides a better overview of metacognitive knowledge. This study suggests that metacognitive knowledge-supporting elements are needed in math learning materials and teachers’ pedagogical actions.

Keywords:

• Metacognitive knowledge
• math education
• comprehensive school
• student
• problem-solving

REVIEWING EDITOR:

• Ana Paula Lopes, Polytechnic Institute of Porto: Instituto Politecnico do Porto, Portugal

SUBJECTS:

• Mathematics Education
• Educational Psychology
• Child Development

Introduction

Metacognition, widely recognised as a critical component of learning and academic success, empowers students to develop a profound understanding of their own cognitive processes (Händel et al., 2013). By fostering this understanding, students gain insights into their learning strengths and weaknesses, enabling them to adapt and refine their approaches to learning (Veenman et al., 2006). Notably, metacognition holds particular significance in mathematics, as emphasised by Muncer et al. (2022) meta-analysis. Within the domain of mathematical problem-solving, metacognition becomes an even more crucial factor (Anif et al., 2021). To illustrate, Radmerhr and Drake (2017) explained that metacognition is a process of personal thinking about their way of thought in building a strategy to solve a problem.

Within the framework of metacognition, researchers have identified two primary components of metacognition: (i) knowledge of cognition and (ii) control and regulation of cognition (Flavell, 1979; Veenman et al., 2006). Several math researchers (Chytrý et al., 2020; Desoete et al., 2001; Mihalca et al., 2017) perceive both knowledge and regulation to be compatible with the demands of mathematical expertise. Regulation involves the ability to control actions and processes based on knowledge of cognition. Whereas knowledge of cognition, known as metacognitive knowledge, refers to awareness of cognitive actions and processes. In this paper, we focus on metacognitive knowledge since it is identified as a crucial factor in predicting students’ academic achievement, especially in mathematics (Chytrý et al., 2020; Muncer et al., 2022; Tang et al., 2021; Tay et al., 2024). Metacognitive knowledge relevant to school-related domains can be effectively trained in a comprehensive school (Desoete et al., 2001). However, metacognitive knowledge has received relatively little attention in mathematical learning research (Neuenhaus et al., 2011). Concurrently, Veenman et al. (2006) called for a more detailed examination of metacognitive knowledge. Specifically, research is necessary to analyse how metacognitive knowledge is activated through mathematics.

Within the scope of mathematics, our previous research (Toikka et al., 2022) focused on primary school students’ self-assessment capabilities. Our findings revealed that the students were primarily able to engage in self-assessment through emotional reflections related to mathematics and the experiences produced by math education. Building on this foundation, we sought to research whether older students exhibit cognitive complexities beyond that of their younger counterparts. In this study, our primary aim is to recognise students’ metacognitive knowledge in mathematics and compare the observed differences in metacognitive knowledge between students at the sixth, seventh and ninth grades in Finnish comprehensive school. We combine two theoretical conceptualisations of metacognitive knowledge to scrutinise our empirical data: while Flavell (1979) focuses on describing the context of knowledge, Schraw and Moshman (1995) emphasise the qualitative aspect of knowledge in their work. We expect the combination result in more profound illustration of metacognitive knowledge in mathematics.

Conceptualisations and empirical challenges of metacognitive knowledge

The conceptualisation of declarative-procedural-conditional knowledge made by Schraw and Moshman (1995) is considered hierarchical because each level builds on the previous level, with conditional knowledge being the highest level of metacognitive knowledge. Thus, it can be described as a qualitative level approach to metacognitive knowledge.

First, (i) declarative refers to an individual’s knowledge and awareness of cognitive processes in general (Schraw & Moshman, 1995). Declarative knowledge includes general knowledge, for example, about oneself as a learner and the task or topic being learned. Research has indicated that declarative knowledge is an important predictor of mathematical achievement and is particularly related to mathematical reasoning (Özsoy, 2011).

Second, (ii) procedural knowledge refers to an individual’s understanding of how to use different cognitive strategies and skills to solve problems and learn new knowledge. Procedural knowledge includes knowledge of how to use specific strategies, for example, to solve a mathematical task. Wijns et al. (2019) have found that mathematically skilled students have a larger repertoire of procedural knowledge than less skilled individuals.

Third, (iii) conditional knowledge refers to an individual’s understanding of when and why to use, for instance, specific cognitive strategies. Conditional knowledge includes knowledge of the situational and contextual factors that influence learning, such as the difficulty of the task or the available resources. Smith and Mancy (2018) reported that skilled mathematicians have a better understanding of when and how to use strategies and procedures than less skilled individuals.

The second theoretical tool of our research is designed by Flavell (1979). He emphasised the importance of considering the interaction between three factors, a learner, a learning task and a learning strategy, in developing metacognitive knowledge. Therefore, he recognised it as important to focus on contextualising knowledge by examining the specific circumstances and conditions in which knowledge arises, thereby enabling a deep understanding of the meaning and significance of knowledge. The conceptualisation of person, task and strategy knowledge is considered categorical because each category is distinct and does not necessarily build on the others.

Based on Flavell (1979), firstly (I) knowledge of a person deals with how individuals learn, as well as with an individual’s learning processes. Person-focused knowledge can take the form of comparing one’s own abilities to those of others, or knowledge can be generalisations about learning and learners. Individuals may be aware of whether they will succeed or fail in a specific cognitive process. This is related to individuals’ understanding of their own weaknesses and strengths.

Secondly, (II) task knowledge refers to the learning content, difficulty or requirements of a task that is to be completed or is currently being completed (Flavell, 1979). Understanding task-related knowledge helps to identify more effective strategies for solving a task (Desoete et al., 2001), but such understanding can have an impact on performance from an emotional perspective. Efklides and Vlachopoulos (2012) displayed that individuals’ beliefs about a task’s worth, emotional reactions to it and attributions based on previous success or failure in similar tasks influence performance. For instance, a student who perceives a task to be important and valuable is likely to be more motivated to complete it and may therefore perform better than a student who does not attach such importance to the task. Similarly, an individual who has experienced success in similar tasks in the past may be more confident and perform better than someone who has had negative experiences with similar tasks.

Thirdly, (III) strategy knowledge includes a great deal of knowledge on deciding which learning strategy is the most efficient (Flavell, 1979). A person with advanced strategy knowledge will be able to determine why one strategy is more appropriate in a particular process (Whitebread et al., 2009). Strategy knowledge also refers to an individual’s understanding of specific strategies learned through experience or education, as well as how each strategy can be applied (Flavell, 1979). These strategies encourage learners to think critically about their own attitudes towards learning and how they approach tasks, instead of relying solely on teacher-led instruction or explanation (Carr, 2010).

In the majority of studies where the context of knowledge has been delineated, the predominant emphasis has been on the studying of strategy knowledge, as demonstrated in studies by Aydın and Dinçer (2022), Karlen et al. (2014) and Nusantari et al. (2021). Studies delving into strategy knowledge typically incorporate a qualitative framework, aligning with Artelt and Schneider (2015) assertion that all three qualitative levels of knowledge are essential for the effective application of strategies.

Although the emphasis has been on strategy knowledge, a thorough examination of previous studies shows that indications to the presence of person and task knowledge also exist behind strategy knowledge. For instance, Artelt and Schneider (2015) suggested that individuals need to regulate their thoughts regarding used strategy and adjust it to the situation to be applied. Building on the findings, the selection and application of strategies depend not only on strategy knowledge but also on individual goals and task demands. Aydın and Dinçer (2022) confirmed that in terms of mathematical skills, individuals with advanced knowledge of tasks will be aware of their beliefs about mathematics and opinions about mathematical tasks; hence, they are able to determine strategies with less effort.

Moreover, Pintrich (2002) illustrated that learners systematically assess their individual characteristics, encompassing strengths, weaknesses, and motivation, and compare them with the demands of the given task. If they discern alignment between their existing knowledge base and their interest in the subject matter, they may adapt their task execution strategy. Additionally, learners access procedural or conditional knowledge that is context-specific, such as recognising that multiple-choice assessments prioritise recognition over recall. This finding implies that contextual knowledge related to person, task, and strategy cannot be fully separated from one another but is always intertwined in some way. In the context of previous research (for instance Aydın & Dinçer, 2022; Pintrich, 2002), we may infer that there is an indication of interplay between qualitative and contextual frameworks.

The fundamental complexity of metacognition, which cannot be directly observed, requires innovative approaches to its measurement. Scholars often rely on identifying elements that provide insight into or are indicative of metacognitive processes based on a solid theoretical foundation (Craig et al., 2020).

The examination of the complexities surrounding metacognitive knowledge frequently centres on the selection between employing quantitative or qualitative research methods. Research conducted by Efklides and Vlachopoulos (2012) and Ozturk (2017) indicated that qualitative methods may excel in capturing the multifaceted nature of metacognition and its real-world applications, whereas quantitative methods may be better suited for discerning trends in metacognitive knowledge across larger sample sizes. The former methodology reaches into an individual’s capability to discern suitable strategies or methods in specific task scenarios, while the latter method places emphasis on the frequency and practical scope of those strategies.

To illustrate the previously mentioned methodological choices and their impacts on studying metacognitive knowledge, Schraw et al. (2006) conducted a comparative study involving think-aloud protocols (qualitative method) and questionnaires (quantitative method) to assess metacognitive knowledge within a college student cohort. Their findings revealed that think-aloud protocols were more adept at revealing the profound depth and complexity of students’ metacognitive knowledge, while questionnaires provided a more efficient means of assessing metacognitive knowledge across a broader spectrum of students.

The evolution of metacognitive research over the past few decades is evident in the development of advanced approaches, including the formulation of metacognitive knowledge tests (Händel et al., 2013; Karlen et al., 2014; Neuenhaus et al., 2011). This trajectory signifies a dynamic shift in researchers’ strategies to better capture individuals’ conscious perceptions of their learning and cognitive processes.

Amidst this methodological diversity, the disclosure of thought processes through the thinking aloud method remains a central approach (Efklides & Vlachopoulos, 2012). Participants verbalise their cognitive processes as they engage in specific tasks, offering researchers insights into their strategic selection and performance assessment. Based on the benefits of qualitative approach, this study is founded on the premise that expressing one’s thoughts verbally represents a viable method for deep understanding of metacognitive knowledge.

The role of metacognitive knowledge in enhancing mathematical problem-solving

Metacognitive knowledge has proven to be a vital part, especially in mathematical expertise (Baten et al., 2017; Muncer et al., 2022; Schneider & Artelt, 2010; Siagian et al., 2019). The development of metacognitive knowledge is linked to the progression of major mathematical competence. Researchers are mainly agreeing that metacognitive knowledge develops over time; young children often demonstrate limited awareness of their cognitive processes, while older children and adolescents showing greater consciousness of their learning (Björklund & Causey, 2017; Schneider & Löffler, 2016). In particular, conditional knowledge tends to develop later in childhood and adolescence, as individuals become more adept at monitoring their own learning and adapting to new situations (Lai, 2011).

Schneider (2008) indicated a steady improvement in declarative metacognitive knowledge from childhood to adolescence. For example, decades ago, Kreutzer et al. (1975) demonstrated that kindergarten and the first-grade children exhibited a relatively advanced understanding of declarative knowledge, although the third and the fifth graders had better comprehension than younger children. Kreutzer et al. (1975) observed also that 9- and 11-year-olds were better able to understand that different individuals have varying ways of remembering and that these ways can change depending on the situation. Furthermore, Veenman and Spaans (2005) discovered that metacognitive knowledge and strategic behaviour in mathematical problem-solving improved significantly between grades 5 and 7.

Researchers have stated that metacognitive knowledge may not develop linearly or in a predictable way and the development can be influenced by several factors, such as individual differences in cognitive and socioemotional development and in cultural and educational contexts (Efklides & Vlachopoulos, 2012; Schneider, 2008; Veenman et al., 2006). Although many factors influence the development of metacognitive knowledge, some developmental pathways have been observed, such as its development from domain-specific to domain-general. Neuenhaus et al. (2011) implied that metacognitive knowledge seems to improve from domain-specific knowledge to domain-general knowledge as the number of learning experiences increases. Students’ metacognitive knowledge becomes more domain-general, allowing them to apply their knowledge and strategies more effectively in different contexts. Both Vo et al. (2014) and Schneider (2008) confirmed this by suggesting that metacognitive abilities develop concurrently with domain-specific changes in young children’s mathematical knowledge.

Stillman and Mevarech (2010) claimed that metacognitive knowledge is a good predictor of mathematical expertise since pre-school. The results of Stillman and Mevarech (2010) study align with Schneider’s (2008) findings, that metacognitive knowledge not only affects young, school-aged children but also determines their performance in mathematics at the secondary school level, regardless of individual students’ intellectual abilities.

In line with the conclusions of Stillman and Mevarech (2010), Özsoy (2011) discovered a strong relationship between metacognitive knowledge and mathematical competence while studying fifth grade students. Building upon the findings of the previous study, the ability to predict, monitor and evaluate one’s work, as well as employ procedural metacognitive knowledge, are the most powerful aspects of metacognition that influence mathematical competence. Xue et al. (2021) recently confirmed that metacognitive knowledge explains a significant amount of individual test score differences and accurately predicts school success.

Both Hargrove and Nietfeld (2015) and Jagals and van der Walt (2016) have demonstrated the importance of developing a comprehensive understanding of declarative, procedural and conditional knowledge to effectively solve mathematical problems. In terms of mathematical problem-solving ability, declarative knowledge refers to an individual’s knowledge of mathematical concepts, principles and formulas (Little & McDaniel, 2015). Without a clear understanding of these fundamentals, individuals may struggle to solve mathematical problems effectively. On the other hand, procedural knowledge enables a learner to apply problem-solving strategies and techniques to solve mathematical problems (Hargrove & Nietfeld, 2015). Lastly, conditional knowledge involves understanding when and why to use different problem-solving strategies and techniques in problem-solving (Hargrove & Nietfeld, 2015). This requires individuals to evaluate the complexity of the problem, the resources available and the time constraints to determine the most appropriate approach.

Many studies have highlighted the importance of metacognitive knowledge in mathematics (Chytrý et al., 2020; Tay et al., 2024), particularly in problem-solving performance (Desoete et al., 2001; Kuzle, 2018; Mihalca et al., 2017). Metacognitive knowledge enables students to be aware of cognitive processes. This awareness allows them to select appropriate problem-solving strategies and adjust them as needed (Desoete et al., 2001). By fostering metacognitive knowledge, cognitive regulation improves, thereby enhancing problem-solving proficiency (Kuzle, 2018). Therefore, we conclude that cognitive regulation and metacognitive knowledge both correlate with success in mathematical problem-solving from primary school to high school.

The present study

Research questions

Based on the theoretical framework of metacognitive knowledge and its age differences concerning mathematics learning in particular, the following research questions were formulated: (1) What kind of metacognitive knowledge can be recognised from the students’ utterances in interviews about problem-solving, math and its learning? (2) What kinds of grade-based differences in metacognitive knowledge can be observed in the 6th, 7th and 9th graders?

Participants

The discretionary group-based sample consisted of 225 students (43% girls and 57% boys) from a Finnish school, comprising 71 sixth graders (approximately age 12), 81 seventh graders (approximately age 13) and 73 ninth graders (approximately age 15). To study the students in these specific grade levels stems from the perspective of Finnish school system, as sixth graders are at the end of primary school. Furthermore, seventh graders are at the beginning of lower secondary school, while ninth graders are in the final stage of lower secondary school.

The entire age cohort from a single school was included in the study, representing a diverse and natural range of socio-economic backgrounds, where students have been selected to the school based on their residential area. The school involved in the study was a teacher training school, operating as a publicly funded and offering tuition-free education.

Students participated voluntarily based on informed consent, and guardians supported their children’s participation by providing their consent. The interviewer and students did not meet before the interviews, and they did not know each other in advance from private or school-based life. By adopting this approach, we were able to mitigate the influence of potential expectations (Powell et al., 2012).

The anonymity of the students was protected by assigning pseudonymised codes consisting of three or four digits to each student (for instance, 9110), rendering them unidentifiable, but there was access to the original data. The first digit indicated the grade of the student, while the other digits were running numbers starting from a randomly chosen student to be the number one.

Procedure

In this study, all 225 students took part in two-phase data collection process. The data collection for this study was part of a school development project funded by the Finnish National Agency for Education, where school practices were being improved. This enabled the participation of all sixth, seventh, and ninth-grade students from this specific school. The data collection procedure was conducted over a period of almost 3 months during mathematics class hours in various locations within the school premises. Because of the data collection focused on mathematics, the teachers agreed the data collection during math classes. The study was conducted as part of schoolwork as a self-assessment task for students.

The data were produced through face-to-face meetings in which students individually solved a mathematical problem (phase 1), followed by individual interviews with a researcher (phase 2). The duration of the recorded data collection sessions varies from 5 to 23 minutes. The transcribed data generated a total of 173 pages.

In the first phase of data collection, the problem-solving task was chosen to stimulate metacognitive knowledge since studies have shown a connection between metacognitive knowledge and problem-solving (Desoete et al., 2001; Kuzle, 2018; Mihalca et al., 2017). The students verbally solved a given mathematical problem-solving task. Also, they were encouraged to verbalise their own thinking process aloud.

The problem-solving task involved percentage calculations but was adapted for the grade level. Seventh graders were given the following task: ‘Mathew is selling his bike for 100 euros. There are many potential buyers and thus, Mathew decides to raise the initial price by 10%. However, the next day, Mathew decides to decrease the price he raised by 10%. Mathew thinks the price for the bike is now 100 euros. Does Mathew reason correctly? Justify your answer’. The task was written on a paper in Finnish, and the researcher verbally repeated it.

The problem-solving task presented to the 6th and 9th graders was modified. The task given to ninth graders was made more difficult by changing the percentages from 10% to 20%. For younger students, the task was made easier by giving intermediate steps in percentage reduction:

‘Mathew justified the price request as follows:

1. If I take 10% of 100 euros, it is 10 euros.

2. If I discount 10% first, the price will be: $100\mathrm{ }\mathrm{\text{euros}}-10\mathrm{ }\mathrm{\text{euros}}=90\mathrm{ }\mathrm{\text{euros}}.$

3. If after this I add 10% to the price, it is:$\mathrm{ }90\mathrm{ }\mathrm{\text{euros}}+10\mathrm{ }\mathrm{\text{euros}}=100\mathrm{ }\mathrm{\text{euros}}$’.

After a student said their solution to the problem-solving task out loud, the second phase of data collection was conducted via a semi-structured interview. During the interviews, the students had an opportunity to use a self-assessment tool called Reflection Landscape (see www.arviointimaa.fi/arviointimaa-reflection-landscape/) to help externalise their thoughts. Although the utilisation of the tool was discretionary, approximately 90% of the participants opted to employ it, conditional on their familiarity with the tool in their previous school activities.

As a result of the cognitive stimulation caused by the problem-solving task, the students were directed to reflect their experiences and learning in mathematics. The interview questions were centred on three parts: the problem-solving task, mathematics and its learning. The questions covered, for instance, the students’ view of whether they correctly solved the mathematical problem, how they usually progress in problem-solving tasks, what are their self-assessed strengths and weaknesses in mathematics and what kinds of goals in math they have.

The semi-structured interviews were conducted by the first author of this article. The advice given by Harter (2012) and Jacobse and Harskamp (2012) was followed as they recommend interviews as an effective way to gain insight into students’ metacognitive knowledge. Interviews encourage students to verbalise their thinking in a way that is recognisable to the researcher. Jacobse and Harskamp (2012) suggested that students must be asked to verbalise their thoughts while working on a task or after working on a task.

On the other hand, the systematic review of Craig et al. (2020) indicated questionnaires to be a common way to study metacognition. We abandoned questionnaires, because they are seen as problematic for a variety of reasons, such as validity issues. Veenman et al. (2006) stated that scores on these questionnaires often do not align closely with actual behavioral measures during or after task performance.

Multiple analyses of data

We employed a mixed-methods approach, combining quantitative procedures with qualitative data obtained from problem-solving tasks and interviews with students. The qualitative data were analysed using a qualitative theory-driven content analysis (Krippendorff, 2018). The analysis unit was an utterance with a single meaning, typically consisting of a sentence or two. In practice, utterances referring to metacognitive knowledge were sought from the interview data.

We first examined the data from two perspectives: qualitative (Schraw & Moshman, 1995) and contextual (Flavell, 1979). Once these perspectives were identified, the utterance was further examined in terms of three dimensions: qualitative perspective was viewed through the declarative, procedural, and conditional levels, and meanwhile contextual perspective was explored through the task, person, and strategy dimensions. Since the utterances of metacognitive knowledge appeared to include overlapping from both qualitative and contextual frameworks, a coding protocol (see Table 1) was developed during the analysis, integrating both Schraw and Moshman (1995) and Flavell’s (1979) frameworks of metacognitive knowledge.

Table 1. Students’ metacognitive knowledge (f = 1,463 interview utterances) by its qualitative level (see Schraw & Moshman, 1995) and context (see Flavell, 1979).

Qualitative		Context
level	Person	Task	Strategy	Total
Declarative	General description of a) individual’s own competence in math or b) importance of math (f = 246)	General description of ways to solve tasks in all-purpose (f = 141)	General description of learning strategy (f = 131)	518
Procedural	Describing one’s view of mathematics through practical knowledge (f = 180)	Practical reframing of ways to solve a task (f = 80)	Practical description how to use specific learning strategy (f = 357)	617
Conditional	Outlining a) different learners and ways to support them or b) detailed description of view of mathematics (f = 63)	Accurate explanation of how to solve a specific task (f = 142)	Selecting appropriate learning strategy in specific situation and justifying the decision (f = 123)	328
Total	489	363	611

With the help of the following citation from 7th grader (721), we illustrate how the data was analysed from two perspectives and from three dimensions: ‘How do I learn? Well… I guess either by solving exercises or by coming up with some sort of personal mnemonic for certain things’. First, from the citation, we identified whether there is any indication about the qualitative of metacognitive knowledge and its context. Since we observed both perspectives (qualitative and contextual) from the citation, we examined in more detailed level what kind of qualitative level (declarative, procedural or conditional) and contextual dimensions (person, task or strategy) are evident in the student’s utterance. In the citation, from qualitative perspective the student demonstrated procedural knowledge, as the student understood how to utilise knowledge in this case in their learning process. From a contextual perspective, the student exhibited knowledge of strategy by explaining their approach to learning.

The first author conducted the primary analysis, which was then refined by discussions between the first and second authors. To validate the analysis, the third and fourth authors were invited to discuss the selected set of utterances to ensure that the coding principles were consistent and reliable.

To investigate whether there were any significant differences in metacognitive knowledge across grade levels, we conducted a quantitative analysis. First, we employed cross-tabulation to examine the association between grade level and metacognitive knowledge, creating a contingency table that displayed the frequency of utterances by grade level. To further investigate statistically significance potential differences via chi-squared test in metacognitive knowledge across grade levels, we divided our sample into three groups based on grade level. This allowed us to compare the utterances across groups and determine whether there were any significant variations in metacognitive knowledge based on grade level (Sharpe, 2015).

Notions of limitations, reliability and ethics of the methodology

We see theory-driven analysis as a strength, but it may have excluded some interesting perspectives from the analysis. Despite the strong theoretical basis, the approach was challenging because students’ expressions required careful interpretation in relation to the categories of metacognitive knowledge. To improve the validity, the analysis performed by the first author has been reviewed and confirmed by the other authors. In addition, this study was cross-sectional; therefore, we can only deduce the outcome and not the development of metacognitive knowledge.

Also, the study’s reliability is influenced by its domain-specific focus on examining students’ metacognitive knowledge within the context of mathematics. It is widely agreed upon that the growth of metacognitive knowledge starts in a specific field and then, with increasing experience, becomes more adaptable and extends beyond that initial domain (Aydın & Dinçer, 2022). While a domain-general approach might yield broader results, it could lack depth and fail to capture the nuanced influence of metacognitive knowledge on learning and performance within subjects.

This study has been conducted following the responsible conduct of research guidelines of Finnish National Board on Research Integrity TENK (2023), ensuring scientific criteria and ethical sustainability, obtaining the necessary research permit and informing all parties involved. The students were asked for verbal informed consent to participate in the study in accordance with the guidelines set out by Finnish National Board on Research Integrity TENK (2023). Students were informed about the content of the research, the pseudonymised and anonymised processing of personal data and the practical implementation of the study. Based on this information, students made a voluntary decision to participate and were allowed to refuse at any time. Guardians were also approached for consent.

Findings

Observed metacognitive knowledge in students’ interview utterances (RQ 1)

Altogether, 1,463 utterances were identified to indicate metacognitive knowledge. Table 1 cross-tabulates the interview data by means of the conceptualisations of Schraw and Moshman (1995) and Flavell (1979). The vertical row categorises the qualitative level of knowledge as declarative, procedural or conditional. The horizontal columns identify different contexts of knowledge, including person, task and strategy.

Looking at the Table 1 from qualitative level viewpoint, procedural metacognitive knowledge dominated, and from a contextual viewpoint, strategic knowledge was most frequently analysed from the data. The individual cells of the table reveal that students were not able to present their person-related conditional and task-specific procedural metacognitive knowledge. We will next discuss in more detail how the three contexts are interpreted qualitatively in relation to the three qualitative levels.

The majority of person-related utterances focused on declarative knowledge (f = 246 utterances). Declarative knowledge is centred on describing one’s competence or the relevance of mathematics on a general level. For example, one student described the relevance of mathematics along these lines: ‘It [mathematics] is important if you think about some future job opportunities’ (723). Meanwhile, another student analysed the weaknesses as follows: ‘Probably geometry, because there are some details I don’t understand. I don’t know why it is so difficult, but I can’t envisage it’ (9111).

The person-related procedural knowledge (f = 180 utterances) encapsulated a learner’s view of mathematics at practical level by identifying, for example, the diversity of learning and varied phenomena affecting one’s own learning. To exemplify the previously mentioned, on a procedural level, the students briefly analysed their views of mathematics, talked about their current viewpoints towards mathematics and often justified their views through the relevance of mathematics: ‘I’m passionate about math, it’s always nice to learn something new. Math is useful in many professions, e.g. my mother works in a bank’ (6216).

Conditional person-related knowledge (f = 63 utterances) focused on either the change in one’s relationship with mathematics and the factors influencing it or diverse learners and addressing ways to support them. The first of the aforementioned is corroborated by the following utterance made by a ninth grader: ‘I’ve always liked math, or maybe I didn’t always, but it has not been something I disliked. In any case, I liked math when I was the sixth grader or in secondary school. The teacher has influenced me a lot. The elementary school teacher taught us a wide range of things, so I have a good foundation for math in upper secondary school. It [math] is concrete. There are so many ways to do it, therefore it’s logical. It cannot be influenced by my own opinions’ (9114).

In addition, the students analysed the diversity of learning with respect to individuals’ varying needs and skill levels: ‘I think it is important to pay some attention to everyone as an individual. You need to progress based on the needs of students. So, those, who are the worst, do not have to go at the same pace as the others – if they don’t want to. If someone wants to do tasks alone, let them do it. If someone wants more collaborative teaching, then give them that. By working together, everyone would learn and enjoy it there’ (725).

The qualitative nature of knowledge was evident in several students’ responses when they described person-related knowledge. While many students were able to articulate declarative knowledge, the frequency of other types of knowledge, such as procedural and conditional knowledge, decreased as the knowledge level of complexity increased.

Task-related knowledge of the declarative level (f = 141 utterances) contained ways to solve math tasks on a general level. For instance, one student explained that ‘a lot depends on the situation, you think about that topic and then make some decisions based on it [knowledge]’ (723), showing the responses focused on how to solve the task, not, for instance, on how the necessary skills were acquired. Concurrently, another student stated, ‘One thing at a time. Little by little, it’s progressing slowly, but surely’ (9173).

At the procedural level (f = 80 utterances), students’ verbalisations revealed task-specific knowledge as they elaborated on their practical approaches employed to tackle tasks. Typically, these responses included descriptions of how students usually approach solving mathematical tasks, such as: ‘I always read carefully, you know, what the question is, what is being asked. I cover all the numbers and their relations carefully’ (623) and ‘I try to come up with multiple solutions and choose the one that best matches the task. I spend a lot of time considering various options’ (7251).

Instead, knowledge at the conditional level (f = 142 utterances) stressed an accurate description of how the student solved a certain task. Such utterances primarily included a description of how a student solved the given mathematical problem: ‘It is incorrect because if you multiply 0.8 ∙ 120, it is 96’ (929).

The qualitative aspect of knowledge was manifested differently in person-related knowledge compared to task-related knowledge; while person-oriented knowledge focused on declarative knowledge, declarative and conditional knowledge related to the task were equally prevalent. However, the description of task-related procedural knowledge emerged as a more intricate and demanding form of knowledge.

Strategy-related knowledge at the declarative level (f = 131 utterances) consisted of a general description of learning strategies. The students were able to analyse different learning strategies in general. To illustrate, a ninth grader generally described learning strategies as follow: ‘Basically, you read something from a book and do some exercises’ (922).

The most recognised metacognitive knowledge in the data was strategy-related knowledge at the procedural level (f = 357 utterances). This kind of knowledge involves more practical knowledge than declarative knowledge. Such utterances contained knowledge about how to use specific learning strategies, as indicated by the following quotations: ‘Well… If one calculates on the whiteboard, then you make a similar calculation. You mark the numbers in the same place based on the example. In a way, you can copy it at first, then it sticks in your mind’ (7180) and ‘I learn through the book or actually through instructions, so I just look at the instructions in the book to learn how to solve exercises if I didn’t know how to do a particular task before‘(622).

The third qualitative level (i.e. conditional level) was demonstrated by a significant number of students (f = 123 utterances). This type of knowledge offers insights into learning strategies, providing detailed guidance on the circumstances and rationale behind a learner’s choice to employ a specific learning strategy to attain their desired learning outcomes. For example, one student stated, ‘My goal is always to make sure that I would know the formulas we use in math. If I knew how the formulas work, I would know how to solve all the tasks in math. I must write all the interphase in a notebook. When you repeat it a few times, it will stick in your mind’ (9217).

As a sum, the qualitative differences in the data yielded intriguing results. The students demonstrated greater proficiency in describing procedural strategic knowledge. The prevalence of both conditional and declarative knowledge was approximately equal, remaining less noteworthy than procedural knowledge.

Observed differences in metacognitive knowledge by students’ grade level (RQ 2)

Out of the overall amount of metacognitive knowledge (f = 1,463 utterances), 394 utterances were from 6th grade students, 515 from 7th grade students, and 554 from 9th grade students. There were no statistically significant differences (p > 0.05) observed between the grade levels regarding students’ metacognitive knowledge (Table 2).

Table 2. Cross-tabulation of the qualitative level and context of students’ metacognitive knowledge by grade level (f = 1,463 utterances).

Grade level	6th graders			7th graders			9th graders
Context/Qualitative level	Person	Task	Strategy	Person	Task	Strategy	Person	Task	Strategy
Declarative	73	44	30	66	43	44	107	54	57
Procedural	49	17	101	72	33	145	59	30	111
Conditional	18	37	25	16	49	47	29	56	51

Nevertheless, Table 2 provides indications considering similarities among 6th, 7th and 9th graders. First, it indicates that regardless of grade, the most recognised metacognitive knowledge was procedural and strategic. This means that students were more likely to focus on the tangible, hands-on aspects of their learning process. Second, the considerably high number of utterances regarding declarative person-related metacognitive knowledge indicated that students would either depict their own experiences with math or illustrate the significance of mathematics.

On the other hand, the prevalence of conditional person-related knowledge was quite rare among the students. That is to say, the students least often depicted in detail their relationship with math or different types of learners and their support.

Based on the chi-square test (χ² = 19.37, df = 4, p < 0.001), utterances indicating as declarative were found less than expected in grade 7 (f = 153 utterances), while utterances containing procedural were more than expected among 7th graders (f = 250 utterances). The reverse pattern was observed in grade 9 (declarative: f = 218 utterances; procedural: f = 200 utterances). Cramer’s V was 0.72 (p < 0.001), indicating a very strong association between students’ grade level and their metacognitive knowledge.

Discussion and conclusion

This study aimed to identify metacognitive knowledge possessed by students and to explore grade-based differences in metacognitive knowledge among the 6th, 7th and 9th graders. The study used two conceptualisations of metacognitive knowledge to provide a multi-perspective understanding of it: the qualitative level designed by Schraw and Moshman (1995) and the context designed by Flavell (1979). The combined conceptualisation enabled the analysis of students’ metacognitive knowledge from a general descriptive level up to procedurally oriented strategic descriptions.

Regardless of the grade, observed students’ metacognitive knowledge was mostly strategy-related and procedural in nature (i.e. their reflections were practical and related to descriptions about learning). There was a clear increase in this type of knowledge from 6th to 7th grade. However, the increase levelled off by the students in 9th grade.

Among this data, the most challenging knowledge linked person-related conditional knowledge and task-related procedural knowledge. It appears that the factors that influence an individual’s perspective on mathematics are still not fully discernible and that there is a lack of awareness regarding specific task-related knowledge. The acquisition of task-related procedural knowledge is crucial, as it constitutes relevant knowledge that is essential for solving mathematical tasks effectively (Desoete et al., 2001). On the other hand, person-related conditional knowledge is also important to develop because it supports the development of students’ metacognitive skills, facilitating their ability to monitor their own learning progress (Hargrove & Nietfeld, 2015).

The present study highlighted the critical role of students’ metacognitive knowledge in the context of mathematical problem-solving. Most of the utterances about the problem-solving task included direct references to task knowledge. However, person and strategy knowledge also encompassed important knowledge that could aid in a problem-solving task, such as understanding different learning strategies and one’s own view of mathematics. For instance, truly critical conceptions related to oneself as a mathematics learner can potentially hinder an individual’s capacity to solve mathematical tasks (Efklides & Vlachopoulos, 2012). The study done by Efklides and Vlachopoulos (2012) proposed that developing students’ overall knowledge in mathematical problem-solving is of great importance.

In agreement with the results of a meta-analysis conducted by Sercenia and Prudente (2023), incorporating metacognition into the design of mathematics learning materials is beneficial for mathematics achievement. Based on our findings, learning materials can support mathematics learning by considering the researched differences in metacognitive knowledge. For example, exercise books with reflective questions or prompts encourage students to think about their problem-solving strategies and evaluate their own understanding, as also Salinitri et al. (2015) have suggested. Moreover, in line with the thoughts of Rieser et al. (2016) instructional videos or interactive tutorials could be developed to incorporate metacognitive scaffolding, guiding students through the problem-solving process and encouraging them to reflect on their own thinking.

The findings revealed that 9th grade students were better in articulating their problem-solving strategies in comparison to their 6th and 7th grade counterparts. However, 7th graders displayed the rationale behind their approach and the appropriate timing for implementing a particular strategy better than 9th graders. The age difference between 9th and 7th graders is 36 months at its most, which indicates that the observed differences in procedural knowledge may also be due to training and not solely determined by age. The prior experiences and training of students may have influenced their manifestation of metacognitive knowledge, as both Efklides and Vlachopoulos (2012) and Schneider and Artelt (2010) suggested. On the one hand, the problem presented to the students may have been more challenging for 7th graders than for 9th graders, which was reflected as a greater exhibition of conditional knowledge among 7th graders. Meanwhile, the 9th graders may have perceived the task as too easy, and they felt it was unnecessary to provide highly detailed conditional knowledge.

On the other hand, statistically significant age-related differences were not prominently observed. This suggests that age, alongside its associated cognitive development, is merely one contributing factor behind the development of metacognitive knowledge. The finding suggests that the framework we have developed provides a more precise alternative to the developmental, grade-specific perspective for examining metacognitive knowledge. The synergy of frameworks enhanced the understanding of metacognitive knowledge and supports to identify the areas within students’ metacognitive knowledge where individual support is needed. This kind of knowledge is crucial to foster the development of metacognitive knowledge among diverse learners.

In this study, problem-solving appeared to reveal students’ metacognitive knowledge and the variance within it. Recent research done by Mršnik et al. (2023) indicates a growing appreciation among teachers for problem-solving, a sentiment that echoes the Gagnean perspective and highlights the importance of integration problem-solving within education (Hiebert et al., 1996). When supporting students’ metacognitive knowledge, teachers can benefit from paying individual attention to their students’ metacognitive awareness, rather than relying solely on general or age-related patterns. However, it is essential to note that the study did not aim to investigate the factors underlying these differences. Longitudinal studies would be necessary in the future to investigate the evolution of students’ metacognitive knowledge over time. Additionally, the absence of an initial measurement in this study implies that an intervention-type study would provide interesting insights into the contexts and processes of development of metacognitive knowledge.

A comprehensive understanding of metacognitive knowledge is crucial in supporting learners’ mathematical proficiency. By discerning the differences in the qualitative level and context of metacognitive knowledge, targeted interventions can be designed to support the development of students’ individual metacognitive knowledge in mathematics. This will allow learners to become more self-aware of their learning process and strategic in their learning approach (Baten et al., 2017) to foster mathematics learning and promote continued academic success (Nelson & Fyfe, 2019).

Recommendations

In light of the findings of this study, we suggest that targeted pedagogical interventions at the classroom level must be designed to support the development of individual metacognitive knowledge in mathematics. Educators play a significant role in promoting their students’ metacognitive knowledge through training programs that focus on developing metacognitive knowledge and applying the procedures to their classroom practice.

This study highlights the need for further research to investigate the factors underlying differences in metacognitive knowledge. A closer examination by means of a longitudinal study is necessary in the future to investigate the evolution of students’ metacognitive knowledge over time. Furthermore, the absence of a pre-test in this study implies that an intervention-type study is necessary to figure out the changes and developments in different qualitative levels of metacognitive knowledge. Also, incorporating teachers’ perspectives could provide valuable insights into how metacognitive knowledge is fostered in classroom settings.

Moreover, considering the complexity of problem-solving, it is also essential to incorporate multiple factors that might impact problem-solving abilities. This includes not only metacognitive aspects but also affective and motivational components that can influence how students approach and solve mathematical problems. Hence, future research should aim to include a holistic view of the factors contributing to problem-solving success.

Disclosure statement

The authors report there are no competing interests to declare.

Data availability statement

The participants of this study did not give written consent for their data to be shared publicly, so due to the sensitive nature of the research supporting data is not available.

Additional information

Notes on contributors

Susanna Toikka

Susanna Toikka is a doctoral researcher at the School of Applied Educational Science and Teacher Education, University of Eastern Finland. Her research interests are math education and thinking skills, including reflection and metacognition. In her doctoral study, Toikka is studying on thinking skills of comprehensive school students in a mathematical problem-solving situation, with the goal of identifying the nature of these skills and potential differences across grade levels.

Lasse Eronen

Lasse Eronen (PhD) works as a university lecturer, specializing in mathematics education, at the University of Eastern Finland, School of Applied Educational Science and Teacher Education. He has worked as a mathematics and science teacher and teacher educator for 25 years. His current research interests concern the student-centered learning environments in mathematics education and integrative approaches to teaching and learning.

Päivi Atjonen

Päivi Atjonen is professor (emerita since 1.1.2024) of education in the Philosophical Faculty of the University of Eastern Finland. Her area of research expertise has been developmental assessment in learning and teaching. During more than 40 years’ academic career at two different universities, Päivi has taught numerous courses on versatile topics of educational sciences, conducted several research and developmental projects on assessment, and worked as an active educator for in-service teachers regarding assessment, pedagogical ethics, and curriculum development.

Sari Havu-Nuutinen

Sari Havu-Nuutinen is a Professor of Education, specializing in early years education. She works as a head of School of Applied Educational Science and Teacher Education, and she has totally 30 years of experience in teacher education in Finland. As teacher educator she contributes to minor studies in pre-primary education, an international master’s degree program, and doctoral studies in education. With nearly, Havu-Nuutinen has played a significant role in curriculum development in teacher education as well as in basic education. Her international collaborative research focuses on young children’s teaching and learning in science education. She has actively participated in science education research networks, European Union funded projects and developmental projects internationally. Currently, she leads (WP leader and PI) Finnish Academy funded project about science capital and the international project about the sustainability in teacher education in Africa. She has served as the senior Fulbright scholar at Ohio University, U.S.