‘It just feels like it's gonna be so very long?’ Exploring the resources used by university students in noticing, navigating, and resolving issues during math-intensive problem solving in chemistry†
Sofie
Ye
a,
Magnus
Jacobsson
b,
Maja
Elmgren
a and
Felix M.
Ho
*a
aDepartment of Chemistry – Ångström Laboratory, Uppsala University, 751 20 Uppsala, Sweden. E-mail: felix.ho@kemi.uu.se
bDepartment of Mathematics, Uppsala University, 751 20 Uppsala, Sweden
First published on 22nd January 2025
Abstract
Problem solving is a complex endeavour that requires students to understand concepts and procedures, as well as knowing when and how to apply them effectively. This study is part of a broader research project examining how university students engage with math-intensive problem solving in chemistry. Here, we focus specifically on the cognitive resources students use to notice, navigate, and resolve potential obstacles. We observed student pairs as they worked collaboratively on a task in chemical kinetics that involved deriving a rate law for a multi-step reaction. Through qualitative analysis of their discussions, we identified three categories of resources: implicit models, episodic memories, and standard procedures. Our findings suggest that implicit models and episodic memories play a key role in helping students navigate uncertainty by shaping their expectations, pointing to a connection between these resources and situational knowledge—a type of knowledge that is critical in enhancing students’ strategic flexibility and refining their intuitions. Overall, this work aims to provide insight into the role of intuitive reasoning in problem solving, emphasising the importance of integrating conceptual, procedural, and situational knowledge. It also opens up opportunities to help students foster expert-like problem-solving skills through directed learning activities that actively engage them in using and reflecting on these knowledge types and how these connect to their own intuitions.
Introduction
Problem solving in science, technology, engineering, and mathematics (STEM) education often poses significant challenges to students. These challenges stem not only from the need to grasp various concepts and procedures, but also from the difficulties in knowing when and how to apply them effectively (Lenzer et al., 2020). Such difficulties may be exacerbated when students are required to draw on knowledge from across different disciplinary domains.This study forms part of a larger research project aimed at investigating the cognitive processes and resources students engage in and use during problem solving at the interface of chemistry and mathematics. Empirical evidence suggests that chemistry students frequently struggle to understand concepts and solve problems in topics at this interface, such as in chemical kinetics and thermodynamics (Bain et al., 2014; Bain and Towns, 2016), and while it may be tempting to attribute these difficulties solely to a lack of mathematical competence, research has shown that the issue is more nuanced, involving factors beyond technical proficiency in mathematics (Uhden et al., 2012).
We previously explored problem solving in chemical kinetics through the lens of a mathematical modelling cycle, focusing on the cognitive processes and resources used by students (Ye et al., 2024). Here, we extend that work by further investigating a specific set of resources that were identified as particularly relevant when students faced uncertainty. More specifically, this study explores the cognitive resources students use to notice, navigate, and resolve potential issues during math-intensive problem solving in chemistry.
Review of the literature
As our research focuses on problem solving at the interface of chemistry and mathematics, we provide a brief review of the literature from the perspectives of chemistry education research (CER) and mathematics education research. Key concepts related to decision-making in moments of uncertainty are addressed in the Theoretical underpinnings.Problem solving in chemistry education
Chemistry covers a wide variety of subfields, each presenting its own unique problems to be solved. Across biochemistry, analytical, inorganic, organic, and physical chemistry, the nature of these problems can vary greatly, ranging from quantitative to qualitative, and from open-ended to highly structured. As a result, the literature on problem solving in chemistry education is substantial and multifaceted, with several comprehensive accounts published over the years (e.g., Gabel and Bunce, 1994; Bodner and Herron, 2002; Bodner, 2015; Tsaparlis, 2021).Research on problem solving in chemistry education can broadly be divided into two main areas: identifying different types of problems and investigating students’ approaches to solving them (Sevian et al., 2015; Broman, 2021).
One way to categorise problem types is by their level of structure. Johnstone (1993) proposed a model that places problems on a continuum from algorithmic (where the data, method, and outcome are all known) to open-ended (where the data may be incomplete, the method unfamiliar, and the outcome undefined). Research on students’ approaches and attitudes to solving open-ended problem shows that they find such problems challenging and often resort to algorithmic strategies. Still, open-ended problems seem to foster deeper engagement, especially in collaborative settings (e.g., Reid and Yang, 2002; Overton and Potter, 2011). This reliance on algorithmic strategies has also been noted in studies comparing the problem-solving behaviours of experts and novices (e.g., Overton et al., 2013; Randles and Overton, 2015; Lenzer et al., 2020). Experts engage in deep conceptual reasoning, evaluate their conclusions, and flexibly adapt their strategies based on context. Novices, on the other hand, tend to focus on surface-level features and follow algorithmic approaches, often without assessing their appropriateness. Research in both chemistry and physics education highlights the importance of domain-specific awareness and fluency in developing expertise (e.g., Airey and Linder, 2009; Lenzer et al., 2020). Awareness involves understanding what domain-specific resources are available, including their limits and values for the domain, while fluency adds the ability to know when and how to use these resources effectively.
Another way to categorise problem types is by their content. For example, problems in organic chemistry can be classified as mathematical or non-mathematical (Graulich, 2015). Because organic chemists rely heavily on qualitative reasoning to predict and understand reaction mechanisms, most of the educational research in this area focuses on non-mathematical problems, including structural analysis (e.g., Cartrette and Bodner, 2010; Connor et al., 2021; Fantone et al., 2024) and mechanistic problem solving (e.g., Bhattacharyya, 2014; Weinrich and Sevian, 2017; Webber and Flynn, 2018; Zotos et al., 2021; Eckhard et al., 2022; Dood and Watts, 2023).
In contrast, problem solving in physical chemistry is typically math-intensive. While it is widely recognised that the mathematical content significantly contributes to students’ difficulties with understanding concepts and solving problems in physical chemistry (Sözbilir, 2004; Fox and Roehrig, 2015), details of the specific causes remain to be identified and further investigated. In the past decade, several contributions have been made to gain more insight into this matter. Some have investigated how students integrate chemical and mathematical knowledge while solving problems in chemical kinetics (e.g., Rodriguez et al., 2018; Bain et al., 2019; Ho et al., 2019; Rodriguez et al., 2019) and thermodynamics (e.g., Becker and Towns, 2012; Bain et al., 2014). Others have examined students’ ability to construct mathematical models from data (e.g., Becker et al., 2017; Brandriet et al., 2018). How students engage in abstraction—that is, the act of extracting salient details, developing generalities, and connecting mathematical symbols to conceptual meaning—during problem solving in physical chemistry, has also been investigated more recently (e.g., Karch and Sevian, 2022). Taken together, these studies suggest that students tend to adopt algorithmic approaches to solving tasks they perceive as quantitative and that they often struggle to connect and translate between chemical concepts and mathematical relationships, particularly during problem solving. Efforts have been made to provide targeted learning materials that help students connect chemical concepts with mathematical relationships (e.g., Frodyma, 2020; Findley et al., 2024). Researchers have also explored ways to improve general problem-solving skills in chemistry by using scaffolding to develop students’ metacognitive abilities (e.g., Yuriev et al., 2017; Graulich et al., 2021; Vo et al., 2024).
Based on the well-documented challenges faced by students during math-intensive problem solving, it is clear that the issue goes beyond ‘just maths’, and that much remains to be learned about the specific factors contributing to these challenges. Recent studies in CER have begun to delve deeper into the intricacies of integrating chemical and mathematical knowledge (e.g., Karch and Sevian, 2022; Ye et al., 2024).
Mathematical modelling and the mathematical modelling cycle
In mathematics education research, problem solving at the interface of mathematics and other scientific disciplines is often framed through the lens of mathematical modelling—the bidirectional process of translating between “real world” phenomena and the abstract realm of mathematics (Blum and Borromeo Ferri, 2009). This translation is crucial for leveraging the power of mathematics to solve real-world problems and is widely recognised as a key objective in mathematics education (Niss and Blum, 2020).While many students can successfully apply mathematical models when these are explicitly given, research shows that constructing and understanding these models without external guidance is considerably more demanding (Jankvist and Niss, 2020). The book series from the International Community of Teachers of Mathematical Modelling and Applications (ICTMA) conferences, along with the proceedings from the International Congress on Mathematical Education (ICME), provide extensive exploration and syntheses of the research development in the teaching and learning of mathematical modelling (e.g., Stillman et al., 2015; Stillman and Brown, 2019).
The mathematical modelling cycle (MMC; Borromeo Ferri, 2006) is an analytical framework designed to capture the processes involved in mathematical modelling. In general, the classical MMC consists of six steps: understanding the real-world situation and constructing a mental model, simplifying the mental model into a real model, mathematising the real model into a mathematical model, performing mathematical work to obtain a mathematical result, interpreting the mathematical result in a real-world context, and validating the obtained real result against the mental model of the real-world situation.
Prior work and current research question
In a recent publication (Ye et al., 2024), we investigated the processes and disciplinary knowledge involved in mathematical modelling of chemical phenomena by following seven pairs of second-year undergraduate chemistry students as they collaborated in a think-aloud manner to solve a set of tasks in chemical kinetics. Deductive coding, using the classical MMC as our initial framework for analysis, combined with inductive coding, allowing for new themes and insights to emerge, resulted in the extended MMC.This empirically derived framework highlights that students engage in various sub-processes in addition to those found in the classical MMC, with deliberations, procedural checks and evaluations (through interpretation and validation) taking place throughout. The extended MMC further sheds light on a previously underexplored aspect of the MMC: the role and nature of extra-mathematical knowledge (Niss and Blum, 2020)—i.e., non-mathematical knowledge required to construct a mathematical model. Our findings indicated that students rely on extra-mathematical knowledge to:
(1) translate between the chemical and mathematical domains;
(2) define objectives, or goals, to work towards during mathematisation and mathematical work, thereby providing students with direction during their problem-solving attempts;
(3) set standards against which evaluation can be performed.
While extra-mathematical knowledge is often implicitly assumed to refer to non-mathematical disciplinary knowledge, such as knowledge in chemistry or physics, we found our students to utilise also other forms of non-mathematical knowledge. For example, there were instances where they deemed their obtained mathematical results as ‘too long’ or ‘too complex’. Such knowing is neither strictly chemical nor mathematical, but nevertheless appears to provide students with useful resources in line with the resources framework. The resources framework views knowledge as dynamic networks of small cognitive units, or resources (Hammer et al., 2005). The activation of resources is context-dependent and influenced by an individual's framing (or interpretation) of a given situation. Adopting a resources perspective allowed us to explore students’ reliance on extra-mathematical resources, including chemical and other resources. Chemical resources were defined as pieces of knowledge related to students’ understanding of chemistry, whereas other resources were defined as pieces of non-mathematical knowledge that did not correspond to students’ chemistry knowledge but that were nevertheless relied upon during the problem-solving activity. For clarity, other resources will be denoted in italics throughout the paper.
Three key findings from our earlier study have laid the groundwork for the present inquiry. Firstly, reliance on other resources often involve students making vague references to how they feel about a result or what they think should be the next step, without them necessarily justifying why. Although such references may appear haphazard, our findings demonstrated that other resources significantly influence students’ problem-solving approaches. Secondly, these references hint at a relationship between other resources and intuition, thus suggesting that a deeper understanding of this relationship could provide insights into students’ development of disciplinary intuition. Finally, students appear to rely more heavily on other resources in situations where they are unsure of how to proceed. Consequently, other resources often come into play during processes like deliberation, procedural checks and validation, indicating an interesting overlap between the use of other resources and metacognitive actions.
The overarching motivation of this study was to explore the nature and roles of other resources during problem solving in chemical kinetics that requires some degree of mathematical modelling. To this end, we formulated the following research question:
What actions do students take in problem-solving situations where they find themselves unable to proceed or notice something might be wrong?
Specifically:
(a) How do students become aware of potential issues?
(b) How do they attempt to resolve these issues?
(c) What resources are involved and what roles do they play?
Theoretical underpinnings
Given the observed tendency of our students to rely on other resources during moments of uncertainty (Ye et al., 2024), we find the topics of intuition and heuristics in disciplinary problem solving (chemistry and physics), as well as metacognition in mathematical modelling, to be relevant and particularly fruitful theoretical lenses for our study. In this section, we review some key aspects of the discussions in existing literature.Intuition and heuristics
What is intuition? The Oxford English Dictionary defines the use of the term in modern philosophy as ‘The immediate apprehension of an object by the mind without the intervention of any reasoning process; a particular act of such apprehension.’ Simply put, intuition describes a sense of ‘just knowing’ without necessarily being able to fully explain the origins of the knowing.The concept of intuition is closely intertwined with that of expertise, making it virtually impossible to explore one without also addressing the other. The different perspectives on the interplay between intuition and expertise can be thought of as a spectrum. At one end of the spectrum, intuition is seen as equivalent to expertise. With this view, intuition is understood as the immediate manifestation of an expert's holistic understanding of a given situation, suggesting that ‘experts don’t solve problems and don’t make decisions; they do what normally works’ (Dreyfus and Dreyfus, 1986, pp. 30–31). In other words, intuitive actions are considered the actions of experts, whereas deliberation, analytical thinking and rule-following typify non-experts. At the other end of the spectrum, intuition is instead associated with non-experts (Montero and Evans, 2011). From this perspective, experts are seen as fully rational in the sense that they should always be able to justify their actions. Between these two extremes, intuition is regarded as an integral component of expertise, though not equivalent to it (Newell and Simon, 1972; Gobet and Simon, 1996; Gobet and Chassy, 2009).
An important background for understanding the differences between these theories can be found in the research traditions from which they emerged and the interactions between these fields.
Within cognitive psychology, Newell and Simon (1972) proposed a model of intuition within their broader theory of human problem solving. At the core of their theory lies the concept of bounded rationality, suggesting that experts and novices grapple with the same cognitive constraints: we can only focus our attention on one thing at the time and we can only hold up to four items in our visual short-term memory (Simon, 1982). If human cognition, and our ability to be rational, is inherently bounded, how do we explain situations where individuals arrive at the correct solution in a shorter time than would be required through complete analytical thinking? According to Newell and Simon, this is because small pieces of knowledge can be ‘chunked’ into larger entities. In their chunking theory, intuition is synonymous with pattern recognition, while expertise is considered a combination of pattern recognition (intuition) and selective search through the states of a problem space.
In contrast to this cognitive approach to intuition, Dreyfus and Dreyfus (1986) aimed at providing a phenomenological description of intuition, emphasising the holistic nature of intuition. They criticised chunking theory for defining various types of ‘chunks’ in isolation from any contextual considerations, in contradiction to their holistic view of intuition and expertise.
In response to this criticism, cognitive psychologists proposed template theory (Gobet and Simon, 1996; Gobet and Chassy, 2009), suggesting that frequently recurring chunks within a specific environment can turn into larger cognitive structures known as ‘templates’. Overall, information processing within a template is fast since it consists of chunks and associations that have occurred many times before. By extending chunks to templates, the holistic nature of intuition was retained while maintaining the option to dissect intuition into smaller components (Gobet and Simon, 1996; Gobet and Chassy, 2009).
While most literature has focused on expert intuition, we aim to explore the notion of ‘novice intuition’ and how its evolution to expert intuition can occur and be supported. The literature does offer some (contrasting) insights into the implications of these theories for education. For instance, Dreyfus and Dreyfus's (1986) emphasis on the holistic nature of intuition means that the development of intuition in novices requires exposure to real-world scenarios, where they can accumulate procedural knowledge (know-how) and gradually transition into experts. This places a strong emphasis on learning through hands-on experience and solving problems in authentic contexts. By contrast, Montero and Evans (2011) conclude that it should be possible to extract heuristics from experts and then teach them to students through explicit instruction and feedback, since expert intuition, according to their view, is always rational and justifiable. Finally, advocates of template theory (Gobet and Simon, 1996; Gobet and Chassy, 2009) suggest a combination of these approaches. Experts leverage both their know-how and know-that to navigate during problem solving, and while know-that may be transmitted through explicit instruction, know-how often requires hands-on experience in specific contexts.
Within CER, the development of intuition has primarily been approached through the investigation of heuristics. Heuristics can be thought of as simple rules, allowing problem solvers to move forward without having to engage in full analytical problem-solving strategies. The role of heuristics in human reasoning has long been debated by researchers in the field of psychology, with the discussion being dominated by two views: dual-processing theory as proposed by Kahneman and Tversky (1972, 1982) and the adaptive toolbox as proposed by Gigerenzer and Todd (1999).
Dual-processing theory posits that the human mind operates through two types of processing: type 1 processing, which is fast and intuitive, and type 2 processing, which is slow and analytical. We rely on both types of processing to make sense of the world around us. However, our initial impressions are typically governed by fast, type 1, processing. While early research efforts emphasised the need for type 2 processing to correct the inaccuracies and biases caused by type 1 processing, more recent work focuses instead on understanding the circumstances under which each type of processing is most appropriate (e.g., de Neys, 2018).
Gigerenzer and colleagues criticised dual-processing theory because of the negative connotation given to heuristics as a kind of type 1 processing, proposing instead that we think of the human mind as a collection of heuristics that can be combined in different ways to solve certain tasks in specific environments. They refer to this collection of heuristics as the adaptive toolbox (Gigerenzer and Todd, 1999). Typically, a full characterisation of a heuristic consists of empirical evidence, a qualitative description and a statistical model.
Within CER, Talanquer and colleagues have investigated the heuristics employed by chemistry students in decision-making (McClary and Talanquer, 2010; Talanquer, 2014) and the implicit assumptions that constrain their cognition in ranking tasks (Maeyer and Talanquer, 2013). Additionally, Graulich and colleagues have studied students’ intuitive judgments and use of heuristics in the context of problem solving in organic chemistry (Graulich et al., 2010; Graulich, 2014). The research on heuristics within CER includes features from both dual-processing theory and the adaptive toolbox.
Today, researchers in chemistry and physics education research agree that the relationship between intuitive and analytical reasoning is mutual. To be able to navigate problem solving in uncertain situations, a problem solver should engage in both intuitive and analytical thinking. The challenge lies in finding a good balance, and experts have been shown to be better than novices at optimising which process to rely on when. Development of metacognitive competence has been suggested as a fruitful way to help students balance between intuitive and analytical strategies (e.g., McClary and Talanquer, 2010; Graulich, 2014; Kryjevskaia et al., 2021).
Metacognition in mathematical modelling
According to Flavell (1976): ‘“Metacognition” refers to one's knowledge concerning one's own cognitive processes and products or anything related to them (…) Metacognition refers, among other things, to the active monitoring and consequent regulation and orchestration of these processes in relation to the cognitive objects on which they bear, usually in the service of some concrete goal or objective.’ (at p. 232). For the kind of problem solving discussed herein, we find Garfalo and Lester's cognitive–metacognitive framework (1985) and Goos's concept of metacognitive ‘red flags’ (1998, 2002) especially relevant.Building on previous work in mathematics education research (Polya, 1945; Luria, 1973; Sternberg, 1980), Garofalo and Lester (1985) developed the cognitive–metacognitive framework as an analytical tool for examining the metacognitive aspects of mathematical performance. The framework consists of four main categories, representing different phases of solving a task: orientation (strategic behaviour to assess and understand a problem); organisation (planning of behaviour and choice of actions); execution (regulation of behaviour to conform to plans); and verification (evaluation of outcomes and decisions made). Notably, there are striking parallels between the sequence of phases put forward by the cognitive–metacognitive framework (i.e., orientation/organisation, execution and verification) and the sequence of recurring sub-processes in our extended MMC (i.e., deliberation, construction/calculation, interpretation and validation).
While analytical models like the cognitive–metacognitive framework acknowledge the importance of metacognitive actions, Goos (2002) has argued that they lack the necessary detail to guide these actions effectively. For instance, these models do not distinguish between different kinds of reflective activities, such as routine monitoring—automatic checks to confirm that ‘all is well’—and controlled regulatory responses, which involve deliberate efforts to manage one's reasoning and actions. To help identify situations where students need to engage in the latter, Goos introduced the concept of metacognitive red flags, proposing three kinds: lack of progress, error detection, and anomalous result. Goos also outlined possible outcomes that may result depending on whether a metacognitive red flag should be triggered (i.e. if something is actually wrong) and whether a metacognitive red flag is recognised (i.e. if students perceive a need to respond). The green boxes in Fig. 1 correspond to outcomes that Goos refer to as metacognitive success: one when no red flag is needed and one when a red flag is appropriately recognised. The yellow boxes in Fig. 1 represent two types of metacognitive failure: metacognitive mirage (responding when nothing is wrong) and metacognitive blindness (failing to respond when something is wrong).
 | ||
| Fig. 1 Scheme illustrating possible outcomes based on whether a metacognitive red flag should be triggered (i.e. if something is actually wrong) and whether a metacognitive red flag is recognised (i.e. if students perceive the need to respond). The green boxes represent outcomes related to metacognitive success, while the yellow boxes indicate outcomes related to metacognitive failure. | ||
Methods
Research setting, research participants and data collection
The findings of this study originate from further analysis of data gathered in a previously reported study (Ye et al., 2024), in which we observed university students enrolled in second-year physical chemistry courses as they worked on a set of tasks in chemical kinetics.In the present paper, we delve deeper into the details of a task that asked the students to derive the rate law of oxygen production based on the information provided by a multi-step reaction mechanism. Rate laws are differential equations that describe how the concentration of a certain molecular species changes with time. Ideally, a rate law should not contain any term that corresponds to the concentration of a reactive intermediate as these can be experimentally difficult to determine. To eliminate such terms, it is sometimes possible to employ the steady-state approximation and substitute the term corresponding to an intermediate concentration for terms corresponding to the concentrations of reactants and/or products. Importantly, although the total multi-step reaction in this task contained several molecular species serving as intermediates, only one acted as an intermediate in the formation of oxygen. Details about the task are outlined in the ESI.†
Seven student pairs were video-recorded during 75 minutes long problem-solving sessions.
During the first 45 minutes, students were asked to work together in a think-aloud manner, with the researcher taking a passive and unobtrusive role. Traditionally, think-aloud protocols are conducted with individual participants (Fonteyn et al., 1993). However, we posited that paired think-aloud would yield richer and more authentic data. When interacting with a peer, students are more likely to engage in dialogue, which may provide better access to their cognitive processes. Nonetheless, there are limitations to this approach, including the challenge of transcribing overlapping speech and the fact that only conscious, verbalised thoughts can be captured. Additionally, sociocultural factors, like students’ relationships and communication styles, may also influence the data, though these were not a focus of this study. It is interesting to note that the literature provides an example of using paired think-aloud (in a more structured manner) as a teaching method, aimed at leveraging students’ verbalisation of their thinking to help improve their problem-solving skills (Whimbey and Lochhead, 1986; Noh et al., 2005).
In the remaining time of the problem-solving session, the researcher interviewed the students about specific parts of their problem solving in order to gain a deeper understanding of their reasoning, while the students had the opportunity to ask questions about the task or the research study.
In addition to video recordings, a LivescribeTM smart pen was used to capture students’ written work in real time, synchronised with their spoken words. The video data was transcribed verbatim. For more details on the data collection, please consult Ye et al. (2024).
Data analysis
Data analysis was carried out by the first author (S. Y.), with supporting discussions with the other authors (primarily F. M. H.) concerning the development of the coding scheme and its implementation in data analysis. Transcribed video data from five out of seven student pairs were chosen through purposeful sampling, meaning that these particular student pairs exhibited the highest frequency and richest range of reliance on other resources during their problem solving.Initial coding was aimed at identifying instances where students seemed unsure of how to proceed and/or thought that something might be wrong. We employed a deductive approach and coded for moments of uncertainty, informed by literature on students’ decision-making under cognitive constraints (Talanquer, 2009; Maeyer and Talanquer, 2010) as well as metacognitive red flagsGoos (2002).
During the next stage of the analysis, we aimed to explore the cognitive resources activated by students during moments of uncertainty and in relation to the triggering of metacognitive red flags. Specifically, what resources they employed in noticing potential issues and in their attempts to resolve these issues.
Finally, to gain insight into the roles of these resources, we analysed our students’ problem-solving trajectories (Fig. 3–7). A problem-solving trajectory provides a chronological overview of how a problem-solving attempt unfolds (Fig. 2), offering insight into the events that occur prior and subsequent to a specific instance (as captured by the coding). Problem-solving trajectories thus allow for in-context analysis, making them particularly suitable for exploring the roles played by a certain resource in a given situation.
 | ||
| Fig. 2 A general problem-solving trajectory. The coloured bars represent the coding of the transcript, with the width of each bar reflecting the length of the corresponding coded section. Different colours indicate distinct phases of the mathematical modelling cycle (i.e., mathematisation, mathematical work, and evaluation), with different shades representing sub-codes within each phase. Quotes may be added to exemplify specific codes or highlight key moments and transitions. Note that bars can overlap where multiple codes apply to the same part of the transcript, such as when interpretation and validation take place together as described in Ye et al. (2024). | ||
In establishing our coding scheme (Table 1), we conducted inductive analysis focused specifically on the cognitive resources used by students during moments of uncertainty and instances where metacognitive red flags were triggered. However, in generating the problem-solving trajectories to gain insight into the roles of these resources, we applied the coding scheme to students’ entire problem-solving attempts. This broader exploration contributed to a more comprehensive understanding. Detailed descriptions of the codes are given in the Results and discussion.
| Code | Description | Examples |
|---|---|---|
| Moments of uncertainty | ||
| Hesitation | Students pause in their problem-solving attempt, often discussing how to proceed. | ‘I’m just thinking… These ones [concentrations of proton and perbenzoate] always show up together… We can try but I think we’ll have a problem to isolate [the concentration of] the proton…’ |
| Metacognitive red flag | Students are bothered by something, such as the lack of progress, the detection of an error, or an anomalous result. | ‘It feels like we’re gonna get a super long loop… That we’re just keep getting stuff we need to … substitute!’ |
| Resources activated in noticing, navigating, and resolving issues | ||
| Implicit model | Students comparing their implicit expectations of the problem-solving situation with the current situation as perceived by the students. | ‘This was a very complicated rate law [laughs] or are we making it too complicated…?’ |
| Episodic memory | Students referring to something that they have previously encountered. | ‘Although… Haven’t we gotten quite long [rate laws] in like … class?’ |
| Standard procedure | ||
| Not justified | Students employing a standard procedure without justifying its conceptual grounding. | ‘…I just realised that's not how you do it, so this should be OK.’ |
| Implicitly justified | Students employing a standard procedure with implicit or circumstantial justification of its conceptual grounding. | ‘So … what do we, what do we have … we want … Yeah, so what we want is [writes d[O2]/dt], right? (…) And we want it in way so that it's not described in terms of intermediates.’ |
| ‘I think that if we substitute this [mathematical term], things will start to cancel out.’ | ||
| Explicitly justified | Students employing a standard procedure with explicit justification of its conceptual grounding. | ‘I think we should take all reaction steps into account since the first two [rate constants] will affect how fast this [the intermediate] is produced, which affects how fast O2 is produced.’ |
| Modes of resource activation | ||
| Recall | Students actively searching for specific rules or problem-solving approaches in their memory. | ‘Then we had that f***ing symmetry that he talked about. That you can like add them together if you feel like it and things will get much easier…’ |
| Recognition | External stimuli triggering comparison between the stimuli and previously encountered examples (internal, often implicit, objects). The comparison can result in recognition or lack thereof. | ‘Or are we supposed to have such a long rate law?’ (lack of recognition) |
| ‘Yeah, but isn’t it now that we can like… Because now we’ll get an addition in the denominator (…) isn’t it then we can like ’Ah, under which conditions is this reaction of first order kinetics blablabla…’ (recognition) | ||
Ethical considerations
The ethical considerations for the study, including informed consent and adherence to the relevant regulatory frameworks in Sweden and the European Union are detailed in Ye et al. (2024).Results and discussion
We begin this section with a description of the deductive analysis carried out to organise our data and capture instances where students seemed troubled by potential issues. Following this, we describe the inductive analysis conducted to identify and categorise what resources our students employed in noticing and resolving these issues. We then highlight insights gained from analysing students’ problem-solving trajectories, offering an overview perspective on how students use their resources in navigating moments of uncertainty. Finally, we reflect on students’ development of intuition and expertise, and how the interplay between different resources relates to students’ ability to shift between disciplinary domains.Resources employed by students in noticing and resolving issues
To capture instances where students encountered issues in their problem solving, we employed a deductive approach, focusing on two key situations: (1) when students seemed uncertain about how to proceed (code: hesitation); and (2) when students demonstrated explicit awareness of potential issues (code: metacognitive red flag). The code metacognitive red flag was assigned only to situations in which students explicitly expressed concerns about the appropriateness of their approach (e.g., by voicing a lack of progress, commenting on an anomalous result, or mentioning the detection of an error; Goos, 2002). In contrast, the code hesitation was assigned to situations where students’ actions indicated uncertainty, such as silent pauses or hesitant deliberations. Essentially, these codes served as initial markers, mapping out critical points in our students’ problem-solving attempts, thus providing context for the next stage of analysis.We underscore that hesitation and metacognitive red flags are not mutually exclusive; rather, they often appear in tandem. That is, hesitation may involve, or be prompted by, a metacognitive red flag. Illustrative quotes are presented in Table 1.
Having identified situations where students seemed unsure of how to proceed or noticed something might be wrong, we shifted our focus to the resources activated in relation to these situations. Inductive analysis revealed various resources used by students in noticing and resolving issues: implicit models, episodic memories and standard procedures.
The following excerpt illustrates how Alice and Andrea use their implicit models and episodic memories to validate their obtained rate law.
Andrea: Ok, uhm… Or are we supposed to have such a long expression [rate law]?
Alice: I don’t know… It felt like we were supposed to get something much… But then again, haven’t we gotten quite long expressions in like … class? [emphasis added]
Andrea: Yeah, maybe we have? Sometimes you’ve been able to simplify a lot… But I… But I don’t know, I can’t see at all how you could do that here… [proceeds to the next phase]
In this exchange, Alice and Andrea are guided by what form they believe they can expect from the output of their mathematical work. They initially seem troubled by the appearance of their rate law, expressing that they had expected something simpler or, as they put it, less ‘long’. For them to arrive at this conclusion, they must have some idea of what a typical result ought to look like. Since Alice and Andrea did not explicitly articulate their expectations, we coded this instance as them activating an implicit model of the result. The implicit model ultimately raised a metacognitive red flag, alerting them to a potential issue, namely their result being anomalous. Later, Alice points out that they might actually have encountered rate laws of similar lengths in class (see the italicised sentence in the excerpt above). This occurrence was labelled as Alice articulating an episodic memory of something previously seen. Interestingly, this episodic memory assisted the students in resolving the issue, allowing them to move on to the next stage of the problem solving.
From the above, we conclude that Alice and Andrea's implicit models and episodic memories helped them notice and resolve an issue by shaping their expectations of the task, providing them with standards against which they could monitor their progress and evaluate their results. In this particular example, an implicit model alerted the students to a potential issue, which was later resolved by their recollection of an episodic memory. However, the roles could also be reversed; an episodic memory might trigger a metacognitive red flag and an implicit model could facilitate the resolution.
Inductive analysis further revealed that students relied on the recollection and realisation of standard procedures in their attempts to address metacognitive red flags and resolve issues. We categorised students’ reliance on these procedures into three types based on the extent of their justification: not justified—when students employed a known procedure without offering any explanation of what it entailed or why it was chosen; implicitly justified—when students provided indirect or circumstantial justification, meaning that although they might have verbalised what they planned to do, they did not explicitly explain why the chosen procedure was appropriate at a conceptual level, with their justification of the conceptual basis instead being inferable from their speech and behaviour in the studied context; and explicitly justified—when students relied on a procedure and articulated the conceptual grounding for its use, typically using causal terms such as ‘because’ or ‘since’.
Consider the quote below where Alice declares her recollection of a standard procedure (see italicised part):
Alice: I was wondering if we should have included this [reaction] step in our rate law but … I just realised that's not how you do it, so this should be OK [emphasis added].
Alice does not explain why she believes this particular procedure to be relevant, nor can it be inferred from the circumstances what might have led her to that conclusion. We therefore classified her reliance on this standard procedure as not justified. By contrast, the next quote illustrates a situation in which the student's reliance on a standard procedure was explicitly justified.
Andrea: I think we should take all reaction steps into account since the first two [rate constants] will affect how fast this [the intermediate] is produced, which affects how fast the oxygen is produced.
Here, Andrea deliberates on how to construct the mathematical model, making explicit reference to the chemical information provided in the reaction mechanism. Since she articulates the conceptual basis for choosing to include certain terms in their rate law, we coded this reliance on a known procedure as explicitly justified.
When reliance on a standard procedure was implicitly justified, the conceptual grounding was indirectly evident from how the students spoke or acted in response to the surrounding circumstances. In other words, while the students did not explicitly state or explain the conceptual rationale for using a particular procedure, it could be inferred from their behaviour. This implies that the justification may have been implicit even to the students themselves—they simply employed a known procedure because it seemed fitting at the time.
Upon closer examination, we noticed two standard procedures frequently used by students in this implicit manner. Since these procedures recurred across different student groups and were typically used as simplified rules for decision-making without reference to their conceptual basis, we classified them as heuristics within the studied context. The ‘no intermediates’ heuristic asserts that the final rate law should not contain any terms involving reactive intermediates and the ‘things usually cancel out’ heuristic suggests that mathematical terms often cancel each other out during extensive calculations. In the conversation that follows, Nelly and Noah are reviewing the first phase of their modelling attempt:
Nelly: We did the steady-state approximation in order to substitute the concentration of the intermediate with…
Noah: Right, you do not want to write the rate law with respect to the intermediate. [emphasis added]
Nelly: No, you want the stuff that's in the net reaction (…) the intermediates are not in there.
It is clear that Nelly and Noah are operating under the ‘no intermediates’ heuristic. They state what they consider appropriate to do (i.e., to avoid terms involving reactive intermediates) but do not explicitly explain why this approach is conceptually reasonable at this specific moment. They simply deem it appropriate given that they are employing the steady-state approximation for this task. The conceptual justification is therefore implicit, indirectly inferred from their choice to apply the steady-state approximation, with no explicit explanation of the underlying chemical events.
It is worth noting that, in practice, the explicitness of students’ justifications for using standard procedures falls along a spectrum. The boundaries between the three categories are not clear-cut, and contextualised interpretation of students’ actions and utterances are necessary to identify implicit justifications, as some cases fall near the borderline between categories. Nevertheless, the three categories provide a useful way to sort and clarify the data for analysis and gaining insights.
In summary, our findings suggest that students employ at least three kinds of resources in noticing and resolving issues. Implicit models and episodic memories provide navigational aid by shaping students’ expectations of a task, helping them notice as well as resolve issues. Moreover, standard procedures, some of which may be considered heuristics, primarily assist students in the resolution of issues by offering potential strategies for moving forward.
By explicitly distinguishing these memory retrieval processes from the resources themselves, we can examine the cognitive and metacognitive processes involved in problem solving without conflating the nature of a resource with the manner in which it is activated or used. To illustrate these distinctions, consider again the exchange between Alice and Andrea. Here, the obtained rate law acts as an external stimulus that initiates the process of recognition. First, Andrea compares the rate law to an implicit model (a resource). This comparison results in a lack of recognition, raising a metacognitive red flag that signals a potential issue. To resolve this issue, Alice uses active recall (a process) to access an episodic memory (a resource). This, in turn, spurs a comparison (a recognition process) between the current rate law and the recalled episodic memory, leading to recognition.
Insights from students’ use of resources during moments of uncertainty
In the following sections, we discuss how various resources assisted students in navigating moments of uncertainty. These findings were uncovered through the analysis of five problem-solving trajectories (Fig. 3–7). Note that these trajectories feature both the coding from our current analysis (upper part) and the coding from Ye et al., (2024; lower part). The latter was included to provide insight into the students’ locations within the extended MMC as they expressed the resources studied herein (Table 1). Because moments of uncertainty tend to span entire sequences of other codes, we include them in the problem-solving trajectories (depicted as grey boxes) only when they contribute with additional insights for our discussion. | ||
| Fig. 3 Problem-solving trajectory of David and Diana. M: mathematisation; MW: mathematical work; RF: metacognitive red flag. The brown text denotes instances where the students explicitly express an awareness of the limited time. The shaded coding stripes represent the coding from our prior work and have been included to indicate the students’ locations within the extended mathematical modelling cycle. Note that the x-axis of a problem-solving trajectory does not directly correspond to time as such; rather, it reflects the relative length (in text) of each coded segment. | ||
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| Fig. 4 Problem-solving trajectory of Jacob and Jonathan. M: mathematisation; MW: mathematical work; E: evaluation; RF: metacognitive red flag. The yellow circle with the letter K indicates instances in the transcript where Jacob or Jonathan mentioned the equilibrium constant. These instances are also highlighted in yellow in the corresponding quotes. The shaded coding stripes represent the coding from our prior work and have been included to indicate the students’ locations within the extended mathematical modelling cycle. Note that the x-axis of a problem-solving trajectory does not directly correspond to time as such; rather, it reflects the relative length (in text) of each coded segment. | ||
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| Fig. 5 Part of Nelly and Noah's problem-solving trajectory. M: mathematisation; MW: mathematical work; RF: red flag. This problem-solving trajectory sets off at a point in time where the students have already been stuck in the loop for some time. The shaded coding stripes represent the coding from our prior work and have been included to indicate the students’ locations within the extended mathematical modelling cycle. Note that the x-axis of a problem-solving trajectory does not directly correspond to time as such; rather, it reflects the relative length (in text) of each coded segment. | ||
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| Fig. 6 Problem-solving trajectory of Alice and Andrea. M: mathematisation, MW: mathematical work, E: evaluation; RF: metacognitive red flag. The shaded coding stripes represent the coding from our prior work and have been included to indicate the students’ locations within the extended mathematical modelling cycle. Note that the x-axis of a problem-solving trajectory does not directly correspond to time as such; rather, it reflects the relative length (in text) of each coded segment. | ||
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| Fig. 7 Problem-solving trajectory of Rebecca and Robin. M: mathematisation; MW: mathematical work; RF: metacognitive red flag. The shaded coding stripes represent the coding from our prior work and have been included to indicate the students’ locations within the extended mathematical modelling cycle. Note that the x-axis of a problem-solving trajectory does not directly correspond to time as such; rather, it reflects the relative length (in text) of each coded segment. | ||
David and Diana initially discuss how to derive the rate law with explicit reference to the chemical information provided by the reaction mechanism (Box 1, Fig. 3). However, in the next phase of the modelling cycle (the mathematical work), the connection to the chemical domain fades away and their efforts become predominantly guided by the ‘no intermediates’ heuristic (teal coding stripes) and the ‘things usually cancel out’ heuristic (purple coding stripes). The interplay between these two heuristics and the students’ mathematical resources traps them in an endless ‘loop’ of substituting one mathematical term for another. The dialogues in Boxes 2–4 (Fig. 3) illustrate this interplay.
The ‘no intermediates’ heuristic guides David and Diana's actions by providing them with a clear objective: to derive a rate law that does not contain any terms corresponding to intermediates. Consequently, each time David and Diana are to decide their next step, they choose actions they believe will bring them closer to this goal, such as substituting terms that correspond to intermediates. To their dismay, their mathematical resources alert them to a potential issue (Box 2, Fig. 3): they might be dealing with a circular dependency, trying to eliminate two mathematical terms that depend on each other (i.e., the perbenzoate concentration and the proton concentration). This error detection triggers a first metacognitive red flag, at which point the ‘things usually cancel out’ heuristic reassures David and Diana that they can persist with their current approach, since there is still a possibility that the mathematical terms they are trying to eliminate will vanish with continued effort. Adding to these factors influencing David and Diana's choices, are instances where David expresses concern about the remaining time for the task (brown text in Boxes 2 and 3, Fig. 3).
To clarify, the ‘no intermediates’ heuristic guides David and Diana towards the goal of eliminating all intermediates, despite their mathematical resources signalling this might not be mathematically feasible. Likewise, the ‘things usually cancel out’ heuristic encourage them to persist with their current approach, despite their awareness of the limited time advising against it. Together, these conflicting messages trigger metacognitive red flags throughout the entire problem-solving attempt.
Although David and Diana acknowledge several metacognitive red flags, for instance by explicitly articulating that they might be creating ‘a loop’ (Boxes 4 and 5, Fig. 3), they do not take any action to address this issue. Instead of altering their strategy and effectively tackle the red flags, they press ahead with their current approach (i.e., substituting all intermediates), citing the ‘things usually cancel out’ heuristic to temporarily dismiss, and resolve, the potential issues signalled by the red flags. It seems likely that the students’ heavy reliance on heuristics inhibited the activation of other crucial resources, preventing them from shifting their mindset and approach.
The case of Jacob and Jonathan (Fig. 4) is another example in which the students relied heavily on a heuristic. However, unlike other groups that were guided by heuristics associated with known standard procedures, they seemed unable to recall what had been taught in class. Instead, the episodic memory that an equilibrium constant should feature in the final expression became a heuristic that Jacob and Jonathan devised for themselves (Boxes 1 and 3, Fig. 4).
The problem-solving trajectory in Fig. 4 demonstrates how Jacob becomes increasingly fixated on the idea that the final rate law should contain an equilibrium constant. While this idea initially serves merely as a vague suggestion of something to pursue in the absence of other options, it eventually becomes the dominant objective of Jacob and Jonathan's problem solving. Even though they engage in evaluation of their work during the problem-solving process, their assessment of their progress against this inappropriate objective (i.e., obtaining the equilibrium constant) only reinforces their mistaken belief that they are on the right track. In this manner, the overly dominant episodic memory, and resulting heuristic, led them down the wrong path.
The application of heuristics demonstrated by these two student pairs aligns well with the discussion by Hjeij and Vilks (2023) on the nature and use of heuristics. They state that ‘heuristics are problem-solving methods that do not guarantee an optimal solution. The use of heuristics is, therefore, inevitable where no method to find an optimal solution exists or is known to the problem-solver, especially where the problem and/or the optimality criterion is ill-defined.’ Jacob and Jonathan, as well as David and Diana, seem to rely on their heuristics because they do not know what else to do. This is particularly evident with the case of Jacob and Jonathan, who even construct their own heuristic.
Note that we are not suggesting it is inherently problematic to rely on implicitly justified standard procedures. Certainly, heuristics and well-learnt procedures, even when disconnected from their conceptual groundings, can be efficient and productive when applied in the appropriate context. In such cases, heuristics serve as scaffolds that support cognitive processing and reduce cognitive overload, enabling students to navigate complex problems more effectively. Rather, the above-mentioned cases demonstrate the power of heuristics in driving progress while also cautioning against overreliance. In particular, our findings indicate that excessive reliance on heuristics may lead to strategic inflexibility, potentially hindering students’ ability to effectively address metacognitive red flags and adjust their approaches accordingly. A similar concern was raised by Kuo et al. (2017), who investigated undergraduate physics students working on a force problem. They found that scaffolding led to an increased reliance on standard procedures, which aided students in their problem-solving progression but also inadvertently limited their exploration of alternative approaches. Our observations also resonate with the broader literature on dual-processing theory, which recognises the utility of heuristics while also stressing the importance of complementing them with analytical reasoning (e.g., McClary and Talanquer, 2010; Graulich, 2014; Kryjevskaia et al., 2021).
Like David and Diana, Nelly and Noah are caught in a loop perpetuated by the two heuristics. After a while, they begin to question their approach (Boxes 1 and 2, Fig. 5). During their second moment of uncertainty (Box 3, Fig. 5), Nelly and Noah notice a discrepancy between what they expect the rate law to look like given their past experiences, and the rate laws they are currently obtaining with their mathematical work. This lack of recognition (pink coding stripes) ultimately alerts the students to an anomalous result, which triggers a metacognitive red flag.
Nelly: This was a very complicated rate law [laughs] or are we making it too complicated…?
Noah: Mm… [performs mathematical work] Okay, so… Now we still have to add more…
Nelly: Yeah, it's so much. Won’t this (…) it feels like we’re gonna get a super long loop… That we’re just keep getting stuff we need to substitute! (RF: metacognitive red flag)
Noah: Yeah, but now (…) where is the last intermediate (…) which ones do we have to get rid of?
Nelly: This, this, this and that (…) And in this [mathematical term] we’ve got the proton… We’re just adding new intermediates… [laughs]
Noah: Arghh… did we over-complicate stuff somewhere? (RF)
Nelly: It could also be that we, from the beginning, chose too many intermediates…
Noah: (…) It's like you say, if we just look … with respect to the reaction…
Nelly: Then the only intermediate… Actually, the proton is not part of this reaction so this [the perbenzoate] is our only intermediate.
In trying to formulate a potential explanation for this red flag, Nelly and Noah draw on their implicit models and episodic memories asserting that their obtained rate laws are ‘too complicated’ and attributing the complexity to their having ‘over-complicated stuff’. Despite these remarks being rather vague—e.g., ‘This is too much…’ and ‘I think we made it a bit complicated…’ (Box 4, Fig. 5)—they apparently provide valuable guidance, prompting Nelly and Noah to re-evaluate their model construction and revise their mathematical model with respect to their conceptual chemical resources.
Nelly and Noah's final moment of uncertainty occurs as they are about to validate their new result (Box 4, Fig. 5). Although their current version of the rate law is correct, Noah once again becomes troubled by the presence of a term corresponding to the proton concentration. This observation, combined with the ‘no intermediates’ heuristic, raises yet another metacognitive red flag. Interestingly, the issue is swiftly resolved by Nelly activating an episodic memory, leading to recognition (orange coding stripes, Fig. 5). Specifically, she recognises that their current rate law closely resembles examples they have previously seen in class, featuring a sum in the denominator, which the students associate with a familiar follow-up question. Altogether, the familiarity of the denominator and the follow-up question appears to provide Nelly and Noah with enough reassurance and confirmation for them to trust in their ongoing approach, preventing them from falling back into the loop.
To recapitulate: while both pairs tried to articulate the origin of their metacognitive red flags, Nelly and Noah's implicit models and episodic memories prompted the activation of their conceptual chemical resources. This activation provided them with a better understanding of the underlying issues and a clearer view of what lay ahead. The additional insight offered by their conceptual chemical resources seemingly helped them transition from implicit to explicit justification of their reliance on the ‘no intermediates’ heuristic, which, in turn, appeared to give them access to a broader range of strategies for resolving issues. In contrast, David and Diana did not (explicitly) activate any implicit models or episodic memories and were therefore left to rely solely on the heuristics to navigate their moments of uncertainty.
Another interesting case was that of Alice and Andrea. Similar to Nelly and Noah, they explicitly articulate a perceived disparity between their obtained rate law and their implicit model of what it should be, expressing that the rate law feels ‘too long’ (Box 3, Fig. 6). However, unlike Nelly and Noah, whose rate law was indeed ‘too complicated’ at that stage of their problem-solving attempt, Alice and Andrea's rate law is not too long. In fact, their concerns merely reflect a metacognitive mirage—a false perception of an issue where none exists (Fig. 1; Goos, 2002). The dialogue in Box 4 (Fig. 6) illustrates how the metacognitive mirage is dispelled when Alice recalls having seen similarly long expressions in class. The activation of Alice's episodic memory seems to widen the scope of the students’ implicit model, transforming the initial lack of recognition into recognition. This newfound sense of familiarity reassures Alice and Andrea of their current approach, encouraging them to activate their conceptual chemical resources and evaluate their rate law in relation to the reaction mechanism (Box 5, Fig. 6).
Essentially, our findings suggest that articulating the underlying conflicts of metacognitive red flags in terms of one's expectations and past experiences provides essential input and direction for navigating issues. Remarkably, such articulation does not have to be particularly specific, as long as it prompts the activation of other relevant resources. For example, Nelly and Noah, as well as Alice and Andrea, simply stated that their rate laws seemed more complicated and longer than what they expected based on their implicit models and episodic memories. Nevertheless, this enabled them to shift their focus and access resources that allowed them to re-evaluate their rate laws based on their conceptual understanding of chemistry. This ultimately helped them overcome their respective moments of uncertainty.
The trajectory analysis underscores an additional aspect of episodic memories. Unlike Alice and Andrea, who primarily used their episodic memories to address metacognitive red flags and resolve issues, Robin's recall of episodic memories served mainly as a way to gather his thoughts and explore possible strategies for moving forward, rather than in response to any specific concerns. While Robin's reliance on recall might seem haphazard, we would argue that this exploratory behaviour is akin to how experts approach new problems. Experts also rely on their previously gained knowledge to solve new problems. However, they are generally able to do this with better precision than novices. Their extensive experience and their capacity to identify underlying similarities and patterns allows them to quickly understand and assess unfamiliar contexts and circumstances. These factors all contribute to what is recognised as expertise and expert intuition. In this example, we observe similar behaviours among students: they draw on their past experiences in their attempts to devise strategies for tackling novel challenges, but since they have a more limited base of experiences to rely on, their application of prior knowledge is more inconsistent.
Reflection on the development of intuition and expertise
The findings presented in this paper offer a more detailed categorisation of the other resources identified in Ye et al. (2024). Specifically, our analysis revealed that implicit models and episodic memories frequently appeared as examples of other resources. Another example was when students applied standard procedures without any perceptible conceptual grounding. In examining how students relied on these resources, we observed several features indicative of intuitive behaviour. For example, such reliance was often expressed in terms of how students felt about a situation (e.g., ‘it feels like we’re gonna get a super long loop…’) or through assertions made without any mention of the conceptual rationale (e.g., ‘I just realised that's not how you do it…’). In other words, students’ use of their other resources often manifested as a sense of ‘just knowing’ without, seemingly, being able to fully explain the origins of that knowing—that is, as intuition.As previously discussed, we found that implicit models and episodic memories guide students’ problem solving in part by shaping their expectations of a task, thus making certain features of the task more noticeable. For instance, a discrepancy between an obtained result and an implicit model of the expected answer could raise a metacognitive red flag and direct students’ attention to specific aspects of the result (e.g. the length or complexity of a rate law). Likewise, a (dis)similarity between an obtained result and an episodic memory could either reassure students that they are on the right track or signal a need to reconsider their approach. The two heuristics—the ‘no intermediates’ heuristic and the ‘things usually cancel out’ heuristic—were also related to students’ expectations, with each hinting at anticipated outcomes for certain actions: if we do A, then B will probably occur. Within the literature on the teaching and learning of mathematical modelling, the crucial role of students’ expectations in navigating problem solving is captured by the concept of implemented anticipation—a modeller's ability to use their prior knowledge to foresee, and act on, potential issues—being considered an essential skill for students to become successful modellers (Niss, 2010).
Further analysis of implicit models and episodic memories indicated that these resources could be related to situational knowledge with respect to the roles they play. Situational knowledge involves understanding the typical features of problem-solving situations within a specific domain and influences a problem solver's selective perception and expectations for a given activity (e.g., de Jong and Ferguson-Hessler, 1996; Savelsbergh et al., 2002). From a resources perspective, resources that contribute to situational knowledge would be closely related to how students perceive, or frame, a situation. By shaping students’ ideas of ‘what is going on’ and what outcomes to expect, such resources would play vital roles both in determining which actions are considered relevant in a given moment and in monitoring progress. Put differently, resources that provide situational information are not about knowledge of scientific facts or skills in performing algorithms; rather, they concern how students view and navigate the problem-solving situation as a whole, influencing what details they find important and which actions they deem appropriate. In our data, we saw that students drew on internalised models and concrete past experiences for these purposes and would therefore argue that implicit models and episodic memories can be seen as contributors to building students’ situational knowledge. In this sense implicit models and episodic memories may also serve as important foundations for the development of domain-specific awareness and fluency (Airey and Linder, 2009; Lenzer et al., 2020).
It is worth noting that these resources do not necessarily have to be appropriate or relevant in order to influence students’ perceptions. Metacognitive red flags may be triggered for all the wrong reasons; still, their presence signifies what a problem solver deems noteworthy. Indeed, this is one of our key points: the gradual balancing and development of such resources (that inform students’ situational understanding) are integral to the maturation of disciplinary intuition and expertise.
What strategic (in)flexibility reveals about the interplay between conceptual, procedural and situational knowledge
We saw previously that Nelly and Noah, as well as David and Diana, got caught in ‘loops’ perpetuated by the same two heuristics. While Nelly and Noah eventually managed to escape the loop, David and Diana did not. In this section, we propose that the key difference between these pairs lay in the degree of strategic flexibility exhibited by the students and further reflect on possible contributing factors.The differences in strategic flexibility between these groups were evident in the ways in which they responded to their respective metacognitive red flags. In contrast to David and Diana, who continued relying on the heuristics, Nelly and Noah responded to their red flag by questioning their strategy. More specifically, they articulated the underlying assumptions of their problem-solving approach and re-evaluated their mathematical model. In doing so, they transitioned from implicit to explicit justification of their reliance on the two standard procedures. This ultimately helped them solve the task. From this, we infer that how students respond to their metacognitive red flags, to some extent, reflects their situational knowledge. Nelly and Noah were able to better perceive what was relevant in the domain-specific context of the task, which ultimately helped them activate useful concepts and procedures and select appropriate actions to overcome the present issue. They also took a step back and engaged in metacognitive reflection, activating their conceptual chemical resources to scrutinise their use of standard procedures. In all these ways, Nelly and Noah's situational knowledge contributed to their strategic flexibility. These contrasting examples can be used to discuss the interplay between situational, conceptual and procedural knowledge. The schematic representation in Fig. 8 attempts to capture the influence of situational knowledge on the application of conceptual and procedural knowledge.
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| Fig. 8 Schematic representation illustrating how our empirical findings map with different types of knowledge—procedural, conceptual, and situational. The light blue boxes contain our empirical findings, whereas the dark blue boxes represent different types of knowledge. This scheme aims to illustrate how situational knowledge can influence students’ strategic (in)flexibility, which, in turn, affects their engagement with conceptual and procedural knowledge. The grey shape in the back symbolises the absence of clear-cut boundaries between knowledge types when leveraged by a problem solver. | ||
Viewing the students’ problem-solving attempts through this lens, we infer that Nelly and Noah were able to leverage their conceptual and procedural knowledge more productively due to their stronger situational knowledge, helping them notice crucial aspects of the task and providing them with a higher degree of strategic flexibility. David and Diana, on the other hand, did not exhibit the activation of situational knowledge necessary to identify what was most relevant in the current situation. Their application of conceptual and procedural knowledge (as manifested in their heuristic approach) was insufficiently tempered by situational knowledge and this prevented them from escaping the ‘loop’. Jacob and Jonathan were instead led astray by their inappropriate application of a situational resource, specifically an episodic memory emphasising the importance of including the equilibrium constant in the final rate law. This memory became a dominant factor in their decision-making, overshadowing their comparatively weaker (or less present) procedural and conceptual knowledge.
In summary, while students clearly need conceptual and procedural knowledge to tackle problems effectively, this work underscores the importance of situational knowledge. Closely linked with intuition, situational knowledge shapes students’ expectations of a task and influences which concepts and procedures they see as relevant for solving it. These findings thus support prior research, indicating that expertise involves not just explicit knowledge of concepts and procedures, but also intuition regarding their applications (Airey and Linder, 2009; Lenzer et al., 2020). It is worth noting that this can be compared with the discussion in Ye et al. (2024) about students using knowledge-based as well as intuitive validation (Borromeo Ferri, 2006) to monitor and evaluate their work. We believe that students will rely on both types of validation regardless of our (and possibly their) preferences.
Adding to this, we emphasise the importance of balancing the interplay between situational, conceptual and procedural knowledge. The scheme in Fig. 8 aims to illustrate this interplay. The examples above reveal the pitfalls of relying too heavily on any one kind of knowledge, be it overreliance on standard procedures, or depending excessively on a certain episodic memory.
Strategic flexibility in navigating intuitions and resources at the interface of two disciplinary domains: a future research direction
In the context of this study, students were guided by their intuitions regarding both the chemistry and mathematics involved in the task. Our findings indicate that whether students regard a task as being about chemistry or mathematics largely depends on their implicit models and episodic memories, directing their attention towards specific task features, thus making some aspects more salient than other. Furthermore, and maybe more importantly, this framing restricts which conceptual, procedural and situational resources are accessible, such that intuitions from the prioritised domain may exert greater influence than those from the less emphasised domain. For instance, in a chemistry lab, students expect to solve chemistry-related tasks, and are therefore more prone to activate and rely on their chemical resources for navigation. By contrast, in tackling mixed problems, such as those encountered in mathematical modelling, the ability to shift flexibly between chemical and mathematical ‘reasoning modes’ becomes a central skill. The strong emphasis on mathematics observed in the case of David and Diana likely reflects this phenomenon. We propose that exploring how students acquire what we term strategic flexibility—the ability to shift between different domains or frames of reference—could be an interesting avenue for future research into the development of expertise and reasoning skills that transcend disciplinary boundaries. Training of such skills should also be a pedagogical priority.Implications for research and practice
The theoretical discussion about intuition in the beginning of this paper highlighted its undeniable relationship with expertise. At the very least, experts rely on their intuition in making choices about what to study or prove. In addition, it is well-known that experts use heuristics and various rules of thumb in their daily work. Expert intuition differs from that of a novice in several ways, especially regarding precision (experts’ intuitions are more often accurate) and conceptual grounding (experts are more often able to rationally justify their intuitions). However, it seems to us that the mechanisms by which experts’ intuitions operate are much the same as those observed for the novices studied herein, with implicit models (of what something is or of what to do) and episodic memories (of past experiences) serving as sources for analogical comparison.It seems reasonable to conclude that the development of expert intuition, by and large, occurs through a mechanism involving the construction of useful mental models derived from the abstraction of past experiences and the meaningful integration of complementary conceptual and procedural resources into these mental models. Our findings indicate that such mental processes and behaviours are present and evolving long before expertise is fully developed. This observation resonates with Brady et al. (2022), who explored intuition as the manifestation of tacit knowledge, emphasising that ‘at any expertise level, a person's tacit knowledge system can give rise to rapid interpretations of their surroundings, enabling them to act fluently and with confidence (even if these interpretations or actions are flawed)…’ Additionally, these findings align with modern perspectives in cognitive science, such as predictive processing, which emphasise the importance of habitual patterns and suggest that the primary role of conscious processing is to address deviations from expected patterns. Embracing this view, thereby recognising students’ apparently random and inconsistent choices and actions as both natural and necessary behaviours in their journeys towards expertise, rather than dismissing them as unhelpful and irrelevant, should yield valuable pedagogical and theoretical implications. For instance, it could lead to more inclusive learning environments where students feel confident and secure enough to openly engage with not only their well-established knowledge of how to solve a certain task but also their incomplete or evolving ideas related to it. This, in turn, should enable educators to present their students with unfamiliar tasks that require more expert-like inquiry, thereby helping students become more comfortable with the uncertainty inherent in such situations. This aligns with prior research, emphasising the importance of providing students with opportunities to practice open-ended problem solving in supportive, non-threatening environments, where they feel confident enough to take cognitive risks (Reid and Yang, 2002; Overton and Potter, 2011).
Based on these insights, we propose that considering intuition as including the sequential or concurrent activation of students’ implicit models and episodic memories, alongside their interaction with standard procedures, may offer a fruitful analytical lens for studying and understanding the transition from novice to expert intuition. As an example, educational researchers could conduct problem-solving sessions where students are asked to explicitly state their expectations of a given task, including what they perceive the task to be about and why. Encouraging students to articulate and evaluate their actions, as well as their reasoning behind these actions, enables a deeper exploration of how their implicit models, episodic memories, and knowledge of standard procedures interact as they tackle challenges.
In addition, engaging students in such reflective activities not only provides researchers with insights into students’ behaviour and thought processes, but also serve as opportunities for students to critically assess their own thinking. We emphasise the importance of encouraging students to reflect on the strengths and limitations of both intuitive and analytical reasoning. Through this process, they may come to view knowledge-based and intuitive validation as complementary strategies—a perspective that aligns well with dual-processing theory: if students understand the distinct roles of type 1 and type 2 processing, they are better equipped to determine when to rely on each.
As a final note, we propose that instructors should create opportunities for students to practice navigating uncertainty specifically at the interface of chemistry and mathematics. Building on our own findings and previous recommendations for fostering problem-solving skills and metacognition in STEM (e.g.Schoenfeld, 1992; Tanner, 2012; Overton et al., 2013; Graulich et al., 2021), a possible strategy could be the following. Firstly, practitioners could design learning activities that challenge students to engage more deeply in connecting their conceptual understanding of chemistry with mathematical relationships. This does not necessarily require designing entirely new tasks, which can be time-consuming. Instead, one could incorporate targeted prompts that encourage students to discuss the mathematical equations and results in terms of the chemical phenomena in question—this connects directly to the interpretation and validation steps of the extended MMC. We believe it is important to use a combination of structured and open-ended tasks so that students can learn to recognise and manage uncertainty in both familiar and unfamiliar situations. We also suggest prompting students to articulate their initial assumptions and identify relevant prior knowledge before engaging with a task. Such activities can enhance their awareness of the cognitive resources they rely on and help them understand how these resources may affect framing and problem-solving strategies. Additionally, guiding students to articulate their thought processes while solving the tasks can help them identify and address metacognitive red flags. After completing a task, group discussions and reflective journals—where students share how they adapted their problem-solving strategies and consider how to apply these insights in the future—may foster strategic flexibility and promote metacognitive growth. Together, these practices may help students become more comfortable in dealing with the uncertainty inherent in novel problem-solving situations and bringing them closer to becoming expert problem solvers.
Limitations
Since this study builds on previously collected data, many of its limitations (e.g. those related to data collection and teacher-student relationships) have been addressed in Ye et al. (2024).One limitation worth mentioning again concerns the task design, which was inspired by material that our students had already faced in their physical chemistry courses. Given this, we acknowledge that the challenges and strategies observed in this familiar context may differ from those that could emerge in more novel, unfamiliar problem-solving situations. That said, our study demonstrates that even in familiar settings, students still experience moments of uncertainty. We emphasise that this study serves as a starting point for understanding how students navigate uncertainty at the crossroads of chemistry and mathematics, and we are currently exploring this phenomenon with less familiar tasks in mathematical modelling.
Another challenge lies in our use of the resources framework. Although a resources perspective allowed us to capture the dynamic nature of students’ knowledge construction, the process of defining and categorising resources was not always straightforward. The flexibility of this framework is both a strength and a limitation, as noted in recent work within both chemistry and physics education research (e.g., Wittmann, 2018; Barth-Cohen et al., 2023; Rodriguez, 2024; Ye, 2024). On the one hand, it allows researchers to tailor the granularity and scope of resources to their specific inquiry. On the other hand, this flexibility places significant responsibility on researchers to clearly articulate how they have defined their resources and to remain mindful of how these choices will shape and influence both the analysis and the conclusions drawn from the study.
To address these challenges, researchers in physics education have shared guidelines for employing the resources framework in educational research (e.g., Richards et al., 2020; Barth-Cohen et al., 2023). Steps that can help ensure consistency and mitigate subjectivity include maintaining transparency about how resources were identified in the study, providing clear descriptions of coded resources in the codebook, and having multiple researchers independently code the same data using the codebook alone.
Conclusions
This study forms part of a broader research project investigating students’ reasoning during problem solving in chemical kinetics. Building on prior findings (Ye et al., 2024), we focused the present analysis on resources employed by students in navigating moments of uncertainty.Through a combination of deductive and inductive analysis, we identified three kinds of resources employed by students in noticing and resolving issues: implicit models, episodic memories, and standard procedures. Implicit models and episodic memories guide students’ problem solving by shaping their expectations of a task. While these resources provided students with objectives, which they could use in monitoring and evaluating their work, standard procedures equipped them with ready-made strategies for achieving those objectives. We found that students applied standard procedures with varying degrees of conceptual justification, ranging from none to explicit. Most often, however, the justification was implicit, and in such cases, the application of the standard procedure resembled heuristic reasoning.
Interestingly, our findings demonstrate that students, much like experts, simply do what they think will work to navigate moments of uncertainty. Importantly, this does not imply students possess fully developed expertise; although their behaviours may well resemble those of experts, key differences exist. For instance, experts tend to have richer repositories of different types of knowledge, greater metacognitive awareness, and more refined intuitions about various situations.
We previously concluded that mathematical modelling of chemical phenomena involves much more than ‘just maths,’ shedding light on the additional processes required beyond technical mathematical operations (Ye et al., 2024). While this prior work primarily focused on the division of knowledge by domain (e.g., chemical and mathematical knowledge), the current study also considered different types of knowledge. Analysing our data from this additional perspective built on our previous findings by specifying that effective navigation of such problem solving requires a combination of chemical and mathematical knowledge along with a balanced integration of conceptual, procedural, and situational knowledge.
Given these insights, we assert that teaching problem solving at the interface of chemistry and mathematics should go beyond imparting conceptual and procedural disciplinary knowledge; to aid our students in becoming expert problem solvers, we must also attend to problem-solving skills specifically. Our findings highlight the critical role of situational knowledge in this regard. Informed partly by implicit models and episodic memories, situational knowledge regulates the activation of additional resources (e.g. concepts and procedures) and contributes to students’ strategic flexibility (i.e., the ability to adapt problem-solving approaches based on context). In order to cultivate students’ situational knowledge, educators should design activities that challenge students to grapple with uncertainty in novel problem-solving contexts and prompt them to activate and integrate resources across disciplines, such as relating mathematical concepts to chemical ones. By further encouraging metacognitive reflection during these activities, we can help students turn their moments of uncertainty into valuable learning opportunities. This approach should make students more attuned to metacognitive red flags, as well as their use of various resources in addressing red flags, ultimately teaching them how to overcome obstacles while also fostering resilience and confidence—qualities that are, in and of themselves, integral for successful problem solving.
Taken together, our findings emphasise the vital role of situational knowledge in developing expertise, enhancing student’ strategic flexibility and sharpening their disciplinary intuitions. Because of this, we believe that aiding students in deepening their situational knowledge will not only promote individual learning but also advance science as a whole. This sentiment was famously captured by Rita Levi-Montalcini, Nobel Laurate in Physiology or Medicine (1986), who said: ‘I don’t believe there would be any science at all without intuition.’
Author contributions
Conceptualisation: all authors; methodology (development of task, methodology for data collection): S. Y., F. M. H.; investigation and data curation (data collection, transcription): S. Y.; formal analysis (detailed coding scheme development, detailed data analysis, and trajectory visualisation): S. Y.; validation (review and validation of data analysis): all authors; writing – original draft: S. Y.; writing – review and editing: all authors; funding acquisition: F. M. H., M. E., M. J.; supervision: F. M. H., M. E., M. J.Data availability
Due to ethical confidentiality requirements, the recorded and transcribed data have not been made publicly available. Our research participants have consented to share their data only with the researchers directly involved in this project.Conflicts of interest
There are no conflicts to declare.Acknowledgements
The financial support from the Faculty of Science and Technology, Uppsala University, Grant for discipline-based education research, is gratefully acknowledged. We also extend our gratitude to all student participants.References
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