Abstract
Implementation studies in mathematics education often cite teachers’ beliefs as a decisive factor in success or failure. While this perspective has yielded valuable insights, the construct of beliefs is frequently defined too broadly to offer precise guidance for implementation design. This paper explores an alternative approach by applying scheme theory to a re-analysis of Handal and Bobis’ (2004) study on thematic teaching. We demonstrate how findings attributed to beliefs can be understood as elements of teachers’ schemes, that is, organized structures of action comprising goals, rules of action, operational invariants, and inferences. This reframing highlights factors of influence beyond beliefs and offers a more operationalizable lens for analyzing implementation outcomes. We conclude that scheme theory, alongside other fine-grained perspectives, can generate actionable insights into factors that influence the outcomes of implementation projects, thereby strengthening the planning, support, and sustainability of innovation projects in mathematics education.
The impact sheet to this article can be accessed at https://doi.org/10.1163/26670127-bja10031 under Supplementary Materials.
1 Introduction
Mathematics education research is primarily an applied science that produces results on the teaching and learning of mathematics, intending to inform practical teaching. However, improving teaching practice with innovations based on research results has proven to be a challenging endeavor. The lack of success may be attributed to the difficulty in identifying the factors that determine why some initiatives succeed and others fail in detail.
Working with a systematic literature review on implementation projects in mathematics education (Ahl, Helenius et al., 2023), we observed that teachers’ beliefs about teaching and learning mathematics are frequently cited as obstacles to successful implementation (Ahl, Aguilar et al., 2023; Ahl, Helenius et al., 2023). However, we also found examples where teachers, despite believing in the innovation, were unsuccessful in implementing it. Swars and Chestnutt (2016) found that when examining teachers’ readiness to use the US Common Core State Standards, teachers tended to hold a decidedly optimistic view of the Standards, but still expressed very varied readiness to use them in practice. For example, one teacher identified a lack of mathematical subject knowledge as a barrier to implementing the Standards, which might make it difficult for them to build teaching based on using students’ work, as this would require identifying sound and unsound mathematical reasoning in students’ responses on the fly. This same phenomenon was identified by Manouchehri and Goodman (2000), as one of their case teachers was described as having the correct beliefs for teaching reform mathematics, yet was found to lack mathematical competence to adequately nourish productive classroom discussions, which led her to return to traditional teaching. In another of Manouchehri and Goodman’s (2000) cases, a teacher expressed that it was hard for her to see the sequencing of the mathematical content when using activities from the new reform-based textbook. Gainsburg (2012) examined the classroom practices that teacher education reform promotes and how student teachers bring these practices with them to the classroom as newly graduated teachers. While teachers’ beliefs may be the basis for the decision, Gainsburg found that the primary factor influencing whether new teachers intended to use specific reform-oriented practices was the extent to which they had had the opportunity to practice those same practices during their teacher education. We hypothesize that while research on teachers’ beliefs has provided essential insights into how teachers interpret and respond to reforms, we argue that additional theoretical tools are necessary to broaden our knowledge of the dynamics of classroom practice and the factors that influence the success of large-scale initiatives.
Our point of departure is that for implementation research to advance, detailed and transparent reporting of the factors of influence (FoI) in implementation projects is essential (cf. Century & Cassata, 2016). Ideally, implementation efforts should be guided by specified design logics or program theories, so that lessons can also be drawn from projects that do not fully succeed (Chen, 2012). It is well established that teachers’ beliefs play an essential role in shaping instructional practices and responses to proposed changes, and they have therefore been frequently identified as critical factors in implementation studies (Hannula et al., 2016; Prawat, 1992). Indeed, the study of teachers’ beliefs has a long tradition in mathematics education research, with roots extending back several decades (e.g., Ernest, 1989b). At the same time, the concept of belief is often described in broad or varied ways, which may limit its usefulness as a precise analytical tool for understanding implementation outcomes or for constructing program theories.
In this article, we suggest that in many cases it is possible to identify more specific and operationalizable factors beneath the general label of beliefs, thereby offering a complementary perspective for analyzing and informing implementation research. A study by the third author exemplifies the need for complementary theoretical perspectives when analyzing implementation projects.
In a teacher-researcher collaboration project led by the third author (Østergaard, 2023), both teachers in the study appeared to share beliefs aligned with the problem-solving and understanding-focused approach intended in the longitudinal intervention in two Danish middle school mathematics classes, as evidenced by their enthusiasm during the cooperative planning sessions. Yet, both teachers repeatedly omitted essential parts of the planned teaching, thereby creating a discrepancy between the intended outcome of the intervention and its actual realization. For a deeper understanding of the reasons behind this discrepancy, an exemplary lesson was analyzed using theoretical constructs from implementation research to identify possible influential factors, including characteristics of the teachers (such as their beliefs), attributes of the innovation, and the support strategies employed. In this lesson on the topic of probability, both teachers ended up skipping one of the four collaboratively planned phases, specifically the phase regarding students’ reflections — something the teachers did not even mention in their evaluation. Although the analysis pointed to the teachers’ non-conscious beliefs as an essential part of the reason for the gap between the intended and the realized outcome, this result does not necessarily provide input for a future design in relation to implementation. Other factors, such as social context, teachers’ experience and knowledge, school culture, and design of the intervention, seemed to influence not only the teachers’ enactment of the lesson, but also their beliefs. From an implementation perspective, teachers’ beliefs may thus be regarded as both explanatory and dependent on other influences, making it difficult to establish causality and provide directions for potential adjustments to the intervention. Hence, we believe that from an implementation perspective, some auxiliary analytical constructs could have helped suggest possible changes, for example, in support strategies, for a future re-implementation of a similar intervention. Among us authors, we have also examined similar effects in other relatively large-scale implementations and found some success in using scheme theory for trying to explain the teaching behavior in a way that could suggest improvements to the support strategies used together with the innovation (Ahl et al., 2022; Koljonen et al., 2023).
In this article, we argue that when evaluating implementation projects and generating insights for the design of future initiatives, it may be necessary to move beyond the construct of beliefs to capture the full range of factors influencing outcomes. As one alternative theoretical approach, among many potential ones, we suggest focusing on teachers’ schemes, defined as the “Invariant organization of behavior for a certain class of situations” (Vergnaud, 1998, p. 229), as a complementary perspective that can enrich existing implementation research. We ask: In what ways and to what extent may a focus on teachers’ schemes complement and enrich the perspective of teachers’ beliefs in relation to implementation research? To be clear, our goal is not to replace the construct of beliefs, or other closely related affective traits such as concerns, attitudes, views, and images, but rather to argue that scheme theory (Vergnaud, 1998, 2009) may contribute to designing and evaluating implementation projects in ways that support more effective and sustainable educational change.
2 Theoretical Elements
The theory section consists of three parts. First, we describe concepts from the field of implementation research to conceptualize various factors that influence the implementation of innovation, followed by a more in-depth description of the belief concept and its strengths and weaknesses. Finally, we present scheme theory, which we argue (and later demonstrate) offers greater analytical precision than beliefs for explaining teacher actions.
2.1 Factors of Influence for Innovation Implementation
One motivation for engaging in implementation research is to uncover the determinants that impact the effectiveness of innovation adoption, as highlighted by Century and Cassata (2016). These determinants, referred to as factors of influence, encompass characteristics associated with four distinct domains: the innovation itself, the users, the organizational setting, and the external environment (Century & Cassata, 2016). The specific attributes defining these factors of influence vary depending on the particular context being examined, such as within the realms of healthcare, sports organizations, manufacturing, or, in our case, educational institutions. Here, our primary focus is on mathematics education.
Drawing on Century and Cassata’s (2016) perspective, we define innovations as the central components targeted for modification or improvement. We are concerned explicitly with innovations stemming from research in mathematics education. Pertinent factors of influence intrinsic to the innovation include its adaptability and relevance to end-users, as well as the specific practices within which the innovation will be implemented.
The primary users are educators, particularly teachers, who play pivotal roles as change agents in schools and classrooms during educational reforms. Factors of influence for implementation projects involving teachers include their level of comprehension of the innovation, their expertise in mathematical and pedagogical aspects, previous experiences, organizational skills, classroom management style, and various affective traits like beliefs, values, attitudes, motivation, self-efficacy, and willingness to embrace novel approaches (Century & Cassata, 2016).
Organizational factors of influence are determined by decisions made by stakeholders within the educational institution. These factors encompass choices regarding class sizes, resource allocation, physical infrastructure, scheduling, and the overall organizational structure that governs teachers’ work. Additionally, administrative processes, management practices, and policy decisions related to the specific innovation under consideration constitute key aspects of organizational factors that influence it.
Factors of influence originating from the external environment comprise opportunities and constraints beyond the control of the educational institution’s stakeholders. Such external influences hold sway over the enactment of innovations and are typically beyond the sphere of influence of those within the school. Examples encompass economic conditions, infrastructural limitations, and shifts in political priorities. Tangible instances, such as poor network connectivity impeding professional development support to schools or changes in budget allocation following shifts in political leadership, are illustrative of external influences.
2.2 Beliefs Research in Mathematics Education
The concept of beliefs has been used ambiguously across the literature, an issue highlighted by several scholars (e.g., Furinghetti & Pehkonen, 2002; Leder, 2022; Rolka & Roesken-Winter, 2015). This variation in interpretation and application has significant implications for research in the field, influencing both the direction of studies and the reception of their findings (Di Martino & Zan, 2011). As noted by Pajares (1992) and Thompson (1992), and echoed in more recent scholarship, there is no universally accepted definition of beliefs nor a singular theoretical framework to guide their study.
Rokeach (1968) offers one of the earliest definitions of beliefs, succinctly stating that: “A belief is any simple proposition, conscious or unconscious, inferred from what a person says or does, capable of being preceded by the phrase ‘I believe that …’” (p. 113). Sigel (1985) sees beliefs as “mental constructs of experience — often condensed and integrated into schemata or concepts” (p. 351). Other definitions also encompass behavioral aspects, as well as the relationship to other affective concepts, such as attitudes and emotions. Schoenfeld (1992) describes beliefs as “… an individual’s understanding and feelings that shape the ways that the individual conceptualizes and engages in mathematical behavior” (p. 358), thereby including feelings and emphasizing the behavioral relevance of beliefs. Similarly, Eichler et al. (2017) conceptualize beliefs in terms of perception and action, stating that “… the term beliefs represents an individual’s conviction referring to a subject that represents a disposition of the ways of receiving information and acting in a specific situation” (p. 85). Grigutsch (1998) incorporates attitudes into a person’s mathematical worldview. He defines it as “a structure of attitudes which contains a wide spectrum of beliefs (cognitions), affections and intentions of actions concerning mathematics” (p. 171). Another perspective is provided by Richardson (1996), who synthesizes insights from anthropology, social psychology, and philosophy to define beliefs as “psychologically-held understandings, premises or propositions about the world that are felt to be true” (p. 103). Philipp (2007) adds a metaphorical dimension, suggesting that beliefs can be seen as “lenses through which one looks when interpreting the world” (p. 258). Based on a comprehensive review of literature on the beliefs of both teachers and students, he formulates the following definition, which is partly informed by Richardson’s earlier work, thus identifying beliefs as:
Psychologically held understandings, premises, or propositions about the world that are thought to be true. Beliefs are more cognitive, are felt less intensely, and are harder to change than attitudes. Beliefs might be thought of as lenses that affect one’s view of some aspect of the world or as dispositions toward action. Beliefs, unlike knowledge, may be held with various degrees of conviction and are not consensual. Beliefs are more cognitive than emotions and attitudes. (Philipp, 2007, p. 259)
Given that individuals hold a vast number of beliefs, these are structured within organized belief systems (Rokeach, 1968), which they continually develop and modify. Three dimensions are characteristic of belief systems (Green, 1971):
(1) Dependency: Beliefs are organized in belief systems with a quasi-logical structure, shaped by personal understanding rather than objective logic, unlike knowledge systems. Primary beliefs serve as foundational, while derivative beliefs are built upon them.
(2) Degree of conviction: Belief systems vary in psychological importance. Some beliefs are central, deeply held, and resistant to change, while others are peripheral and more flexible.
(3) Cluster structure: Beliefs are not held in isolation but exist in interconnected clusters. This structure enables individuals to hold seemingly contradictory beliefs, as long as they belong to separate clusters and are not activated simultaneously, thereby avoiding cognitive conflict.
The stability of beliefs is a source of debate within the research field (Liljedahl et al., 2012). While some claim that beliefs are stable and difficult to influence and change, requiring considerable time and dedication to alter (e.g., Pajares, 1992; Schoenfeld, 2015), others argue that it is both possible and relatively easy to change teachers’ beliefs. It seems to depend on what kind of belief it is. Some beliefs, especially central ones, are deeply ingrained and difficult to influence, while others are more compliant and receptive to new impressions (Liljedahl et al., 2012). For example, Eichler and Erens (2015) observed in their study of 51 secondary teachers that, although peripheral beliefs appeared to shift during the traineeship, central beliefs remained largely unaffected. Since beliefs exist within systems, changing one can affect others. Over time, beliefs tend to become more robust, even in the face of contradictory evidence. Central beliefs may also serve psychological functions, such as reinforcing identity or offering emotional protection (Op’t Eynde et al., 2002). A prerequisite for changes in beliefs is a conflict between existing beliefs and new information, and even then, change is often the last alternative (Pajares, 1992). As individuals often go to great lengths to preserve their existing beliefs, such as by reinterpreting or ignoring conflicting evidence, change is rarely the first choice.
Overall, there is consensus that teachers’ beliefs play a crucial role not only in mathematics teaching but also in shaping students’ beliefs. Ernest (1989a) identifies teacher beliefs, alongside social context and reflective thinking, as central to mathematics teaching practice. These beliefs encompass views on the nature of mathematics, mathematics teaching, and learning. Researchers have emphasized the distinction between espoused and enacted beliefs, which often diverge (e.g., Eichler, 2011; Furinghetti, 1996). For instance, teachers may adopt curriculum-driven beliefs superficially while holding deeper, conflicting, and often not conscious beliefs about teaching — what Furinghetti (1996) calls “ghosts in classrooms.” However, the relationship between beliefs and practice is not evident. Buehl and Beck (2014) display a lack of congruence by listing examples of research findings both suggesting that beliefs influence practice (e.g., Brown et al., 2012; Song & Looi, 2012; Wilkins, 2008), that practice influences beliefs (e.g., Rushton et al., 2011; Swain et al., 2012), that beliefs are disconnected from teachers’ practices (e.g., Jorgensen et al., 2010; Liu, 2011), and that the relationship between beliefs and practice is reciprocal (e.g., Kang 2008; Potari & Georgiadou-Kabouridis, 2009).
According to Ernest (1989a), there are two main reasons for the gap between espoused and enacted beliefs: the influence of social context and the teachers’ awareness of their own beliefs. Social context shapes beliefs through external pressures, including expectations from students, parents, and colleagues, as well as curriculum constraints, assessment systems, and broader educational traditions and culture. Likewise, Buehl and Beck (2014) point to several factors that may either support or hinder the enactment of teachers’ espoused beliefs, both internal, such as experience, knowledge, and self-awareness and reflection; and external, such as time, students’ ability, school culture, instructional resources, and curriculum standards.
Skott (2015) argues that beliefs are often studied without proper consideration of the context and suggests Patterns of Participation (PoP) as an alternative approach to classroom interaction:
[C]lassroom practices are viewed as social phenomena, not as an outcome of any individual’s actions. In PoP they are seen as constituted in a process during which each individual continuously makes symbolic interpretations of others’ actions as well as of others’ (possible) reactions to one’s own behaviour. (p. 15)
However, analyzing teachers’ beliefs can contribute to a deeper understanding of their practice, as well as challenges related to reforms. In their aforementioned study of 51 mathematics teachers, Eichler and Erens (2015) not only find evidence of the stability of central beliefs, but also enable identification of the structure of teachers’ belief systems through a qualitative interview design. Furthermore, the thorough study and analysis of not only interviews but also questionnaires, as well as classroom observations conducted over a half-year period, showed that teachers’ beliefs appear to vary across different domains of mathematics.
From the above, we learned that research on mathematics teachers’ beliefs often relies on broad characterizations, leaving little clarity about how beliefs shape practice in detail. At the same time, teacher development studies suggest that meaningful change occurs at the level of specific practices, yet few belief studies examine this grain size. Speer (2008) addresses this gap by analyzing fine-grained connections between beliefs and practices. Findings indicate that collections of beliefs offer a productive unit of analysis, particularly in relation to beliefs about student understanding and learning processes. Still, Speer’s approach is not mainstream, and none of the 103 papers in our literature review of implementation studies delved deeply into collections of beliefs as articulated by Speer (2008).
2.3 Understanding Teacher Behavior through Scheme Theory
Innovations in mathematics education that require a change in teacher behavior can be analyzed by understanding what drives teachers’ actions. A change in behavior towards reformed teaching will always include growth in professional competence. When discussing the development of professional competence, Eraut (1994) writes:
We have to develop implicit theories of action in order to make professional life tolerable. There are too many variables to take into account at once, so we develop routines and decision habits to keep mental effort at a reasonable level. To change the routine or to question the theory is to reverse the process, to draw attention once more to the myriads of additional variables, and to raise the possibility of paralysis from information overload and failing to cope. (p. 34)
One way to theorize the type of complexity Eraut discusses is to use scheme theory,1 first articulated by Piaget (1970) and later developed in mathematics education by Vergnaud (1998, 2009), who defines a scheme as the “Invariant organization of behavior for a certain class of situations” (Vergnaud, 1998, p. 229). Scheme theory is helpful because it provides a lens to analyze the processes underlying students’ mathematical activity, making it possible to design instruction that builds on and extends their current ways of reasoning. While scheme theory has most often been used to analyze students’ mathematical reasoning (e.g., Steffe & Olive, 2010), scholars have argued that it also provides a productive framework for studying teachers’ professional practice (e.g., Gueudet, 2019). It is well known that teachers, like students, draw on structured ways of acting when confronted with recurrent classroom situations (e.g., Sfard, 2023; Shulman, 1986). These schemes, which may include both explicit strategies and tacit forms of pedagogical knowledge, guide how teachers interpret student responses, select representations, and orchestrate mathematical discussions. Analyzing teachers’ schemes can reveal the implicit reasoning that underpins instructional decisions, highlight potential misalignments between intended implementation of innovation and enacted practices, and trace consistencies across different contexts. In this way, scheme theory provides a valuable lens for investigating the cognitive and practical structures that shape teaching, with implications for both research on teacher knowledge and the design of professional development initiatives.
Vergnaud describes schemes as having four components: goals and expectations, rules of action, operational invariants, and possibilities of inference. As we are interested in teaching situations related to implementation projects, we will provide some brief examples of how these components can be applied in a teaching context.
Goals and expectations are what direct the teaching. Goals can be what students are expected to learn in terms of mathematical content or competencies, but may also involve aspects such as classroom order and organization.
Rules of action involve routines, habits, or other more or less automated actions that can be invoked in a particular situation without much deliberation. A teacher may have a particular way of starting a lesson. In a mathematical discussion, one routine (that we have observed) is to give the word to a student who has begun to talk to a friend about something non-mathematical, to hopefully bring this student’s mind back to the mathematics and uphold classroom order.
Operational invariants are the concepts-in-action and the theorems-in-action that the subject uses when he or she acts (Vergnaud, 1998). The suffix in-action is used to indicate that the individual applies concepts and theorems with varying degrees of awareness. When Eraut discusses the implicit theory of action, the focus is not only on the action, in the sense of a routine, but also on the theory part, that is, the rationales for how things work and why, which might be implicit. Examples of theorems-in-action (implicit theories) include the idea that mathematical concepts and operations must be explained carefully before students are allowed to work on similar problems independently, as well as the opposite idea that students learn best by exploring mathematics themselves with the teacher as support when needed. Both of these theories can be true simultaneously and are likely more about the teaching style the teacher feels most comfortable with, the habits the teacher has in the classroom, and how she plans her teaching. Concepts-in-action, for example, involve building mathematical explanatory models, as well as how the teacher conceptualizes school activities as a whole.
Possibilities of inference include how observations of the effects of applying a scheme can be used to adapt the scheme or update it where it is applicable. A teacher generally has an intended lesson plan when she starts a lesson. In interactions with students, the intended plan will need to be adjusted through ad hoc decisions, resulting in a lesson that is more or less different from the original plan. All the decisions made in the moment are enabled by inference.
When teachers need to adjust their teaching practices as part of implementing innovations, it is primarily the implicit theories (Eraut, 1994) that underline their actions and shape their habits and routines that are important to capture. A fairly general property of teachers’ routines is that in the majority of cases, teachers’ actions are performed smoothly, without explicit deliberation about which move to perform next (Sfard, 2023). To successfully implement innovations, we must understand the routines and habits of teachers. How else can we influence them? One way to describe and analyze actions is by considering actions through scheme theory, where special focus is placed on the components involved: goals and expectations, rules of action, operational invariants, and possibilities of inference.
3 Method and Argumentation for the Key Case
The data consist of a key case (Thomas, 2011), chosen from the systematic mapping review of implementation research. The selected case is representative of the argument that teachers’ beliefs are a factor influencing the outcome of innovation implementation in practice, as noted in our review (Ahl, Helenius et al., 2023). Among the 103 articles, we identified 32 mentioning some kind of teacher-related factor of influence. Among these 32 articles, 20 reported that teachers’ beliefs were a factor influencing implementation outcomes.
Case study research involves in-depth exploration and analysis of a particular phenomenon or instance (Thomas, 2011). This approach is motivated by an interest in understanding the underlying reasons or mechanisms that cause something to occur or exist in a specific way. The definition guiding the use of case study in this work is drawn from Thomas (2011), who describes case studies as comprehensive investigations of individuals, events, decisions, periods, projects, policies, institutions, or systems examined through one or multiple research methods. The case under investigation represents an instance of a broader category or phenomenon, which serves as the analytical framework — or object — through which the case is explored and better understood.
Thomas (2011) emphasizes that case study methodology is fundamentally about using a particular case to illuminate a broader issue or phenomenon. This methodology involves distinguishing between the subject — the specific entity or instance being examined — and the object — the conceptual category or phenomenon being investigated through the case. While other approaches may refer to the person or entity being studied as the subject and the topic as the object, Thomas’s typology instead identifies the subject as the historically and contextually situated case, and the object as the analytically defined phenomenon made visible through a theoretical or conceptual lens.
The selection of the subject is guided by its potential to provide meaningful insights into the object. In this way, the case is not studied in isolation but rather as a way to explore and understand the broader class of phenomena it exemplifies. Using Thomas’s (2011) terminology for case study methodology, the object of study here is alternative explanatory models for factors that influence the enactment of innovation. We chose Handal and Bobis (2004) as our key case. Handal and Bobis’ article, “Teaching Mathematics Thematically: Teachers’ Perspectives,” reports on the implementation of the innovation thematic teaching. The innovation enactment of ten teachers for 14- to 16-year-old students was investigated using qualitative methods. The study bases its results on detailed interviews. A reason for choosing Handal and Bobis’ article as a key case is that it presents teachers’ beliefs as a quite fundamental factor of influence:
An examination of teachers’ beliefs regarding curricular factors affecting the teaching of thematic mathematics revealed a conflict between the present neo-behaviourist orientated educational context in which the Standard course is currently operating and the constructivist nature of the course itself. (Handal & Bobis, 2004, p. 11)
A second reason for choosing Handal and Bobis is that their results section is unusually clearly structured, a prerequisite for conducting a re-analysis, which we will do in the results section. The purpose of the reanalysis is not to suggest that the authors have made mistakes in their analysis. Instead, we intend to offer an alternative interpretation. We begin by presenting the context of the study.
The innovation in the case reported by Handal and Bobis (2004) was thematic teaching. As the concept is used here, thematic teaching primarily involves using real-world situations as themes to build mathematical concepts, thus progressing from the concrete to the abstract. The state of New South Wales in Australia made thematic teaching mandatory in the standard mathematics course for grades 9 and 10 in 1996. The Standard course is the simplest of the three mathematics courses available for these grades. In previous research, three instructional styles for thematic teaching were identified, with only one of these styles aligning with the curriculum reform’s intention (Handal & Bobis, 2003). The article we examine here represents an investigation into why teachers did not widely adopt thematic teaching. Referring to previous research, “teaching thematically requires a change of beliefs in the way teachers look at the curriculum, students, and teaching and learning” (Handal & Bobis, 2004, p. 4). The general finding of the article is that teachers prefer teaching organized by mathematical topics rather than thematically. The authors employ a framework from Memon (1997) and categorize the factors of influence into instructional, organizational, and curricular factors, with teachers’ beliefs classified as instructional factors. “Teachers’ beliefs about the effectiveness of thematic instruction as compared to other approaches is also a significant instructional factor influencing implementation” (Memon, 1997, p. 7). In addition to these categorizations, the authors report the teachers’ reports using various expressions, such as spoke about, perceived, expressed, mentioned, indicated, commented, agreed, noted, confirmed, declared, considered, and, of particular interest to us, believed.
Our method boils down to neutralizing the analysis of Handal and Bobis by not using their thematization and, secondly, treating all statements attributed to teachers as neutral reports of experienced states of affairs. Any statement attributed to teachers in the article is hence called a teacher report. We listed all these teacher reports and identified which ones were clearly attributed to beliefs by Handal and Bobis (2004). Among these belief-attributed teacher reports, we identified three themes: classroom order and student motivation, organization of mathematical content, and relationships to grading, testing, and assessment. We then related these themes to our four-dimensional categorization, which encompasses factors of influence related to the user, the innovation, the organization, and the external environment, and provided alternative explanations for these findings.
4 (Re)Analysis and Results
The first theme we will discuss concerns classroom order and student motivation. An illustrative example of items in this category is that teachers were attributed with “the belief that procedural-type mathematics instruction settles the students in class and they are thus easier to teach” (Handal & Bobis, 2004, p. 11). Typical for this theme are statements of the form that, because of certain characteristics of students in relation to certain characteristics of thematic teaching, thematic teaching will lead to a more disruptive classroom or students expressing less motivation. These statements are typically described as indications of teachers’ beliefs in Handal and Bobis (2004). We suggest relating them to a combination of goals, rules of action (e.g., routines), and invariants. In addition to other goals, maintaining classroom order is undoubtedly a key objective in most teaching and learning environments. Teachers’ typical topic-by-topic teaching routine over time has likely come to include elements to maintain classroom order and expressions of motivation at an acceptable level. Therefore, the procedural-type mathematics instruction that settles the students, which is attributed to a teacher’s belief, can just as well be described as a theorem-in-action, established by drawing inferences from the experience of teaching that is not procedurally oriented. When a central aspect of the teaching organization is changed, the other elements of the teaching routine are also disturbed. Therefore, it is reasonable to assume that teachers’ reports of thematic teaching hurting motivation and order are not an expression of personal beliefs but rather an empirical observation of what happens in the classroom when teaching thematically. An interpretation in terms of the innovation enactment causing routines to break down would lead to the conclusion that the innovation is simply unsuitable for the context, at least in the form it was implemented here. Moreover, it is reasonable to assume that, for the students, thematic teaching demands personal responsibility, collaboration, and a sense of organizing large amounts of information. It may not be suitable for students with a weak interest in school mathematics. Consequently, students’ characteristics would mean that the demands of a functioning thematic teaching routine would be extensive.
Our second theme relates to the organization of the mathematical content. “Most teachers believed teaching mathematics thematically was too fragmentary, repetitive, and lacked continuity when teaching basic skills” (Handal & Bobis, 2004, p. 10). Most expressions in this theme were related to the teaching becoming more fragmented or too repetitive, or that an appropriate progression was missing when teaching thematically. The items in this category are not systematically attributed to teachers’ beliefs, so it is unclear if Handal and Bobis intend to explain teachers’ complaints of fragmented mathematical progression by claiming it is a result of their beliefs. Nevertheless, in our analysis, it is reasonable for any teacher to have a goal of a working conceptual progression and repetition structure, and teachers have likely developed routines for organizing content in topic-by-topic teaching so that most students learn enough content reasonably, and introductions of operational invariants can follow one another without too much repetition. When they report that thematic teaching disrupts this content organization, it is likely a report of actual inferences drawn from trying to teach thematically. It is, therefore, reasonable that a factor influencing teachers not to adopt thematic teaching is a bad fit of the innovation, as it was implemented, to the teaching context, rather than a belief that this would be the case.
Our third theme concerns grading, tests, and assessment, which Handal and Bobis categorized under curricular factors. They write that “teachers’ beliefs regarding curricular factors affecting the teaching of thematic mathematics revealed a conflict between the present neo-behaviourist orientated educational context in which the Standard course is currently operating and the constructivist nature of the course itself” (Handal & Bobis, 2004, p. 11). To understand this theme, it is helpful to think of innovation as living in an environment with specific grading criteria (performance descriptors), yearly progress reports based on a particular rubric, and an external test (such as a school certificate test). Additionally, here, we can understand the teachers’ reports without making any assumptions about their beliefs. Teachers reported that the thematic approach to organizing the teaching made it more challenging to gather evidence for the progress reports, and the students were not adequately prepared for the certificate test, an external environmental factor of influence. We include parents in this theme because teachers reported that parents often worried about their child’s progress or test results and frequently contacted the teacher or the school. Seen through our theoretical framework, the testing regime and grading can be viewed as being part of the external environment, which is not directly related to innovation or the school organization. Therefore, the results described above could lead to the conclusion that a factor influencing teachers’ lack of thematic teaching could be a mismatch between the innovation and the external environment. In terms of teachers’ schemes, it is also helpful to view getting your pupils ready for external evaluation as something that is naturally part of teachers’ goal structures. Therefore, if new routines associated with thematic teaching make this goal more challenging to achieve, it is not surprising that teachers express criticism despite generally being positive about thematic teaching in principle.
Finally, it is worth mentioning that Handal and Bobis (2004) also report that teachers “… were unanimous in supporting the humanistic goals of thematic instruction because they felt that teaching in themes had the power of showing students the usefulness of school mathematics” (p. 9).
5 Discussion and Conclusions
This article argues that when evaluating implementation projects and generating insights for the design of future initiatives, it may be beneficial (or even necessary) to move beyond the construct of beliefs to capture the full range of factors influencing outcomes. We asked: In what ways and to what extent may a focus on teachers’ schemes complement and enrich the perspective of teachers’ beliefs in relation to implementation research?
While research on teachers’ beliefs has provided valuable explanatory models for teachers’ actions, this perspective may not be sufficient to advance our understanding of how to plan and implement effective interventions. One reason is that we are seeking factors that are not only explanatory but also actionable, thus allowing designers of pedagogical innovations to capitalize on them. For this, we use two approaches. By following Century and Cassata (2016) and examining the factors of influence that belong to the innovation, the organization, the external environment, and the user (in our case, the teacher), we have aimed to understand an implementation by referring to domains that have proven helpful in previous implementation research. Also, because the teacher is so central to implementation initiatives, we have sought to characterize the mental organization of teaching in terms of schemes. By utilizing the scheme theory components, goal structure, routine, invariants (theories-in-action), and possibilities of inference (Vergnaud, 1998), it was possible to a higher degree to understand reactions to the particular innovation in such a way that it suggested ways to make the implementation more successful.
First, our re-analysis of Handal and Bobis (2004) showed that the same data previously interpreted as teachers’ beliefs can also be interpreted as belonging to other domains of factors that influence implementation (Century & Cassata, 2016). In particular, we demonstrated that what is perceived as teachers’ beliefs regarding students in this course being weak and hence prone to reacting negatively to thematic teaching may, in fact, have been an actual consequence of teachers’ routines, their rules of action, for organizing the classroom, which were disrupted by the topic-by-topic teaching being replaced by thematic teaching. Even if teachers accept and believe in the goals of thematic teaching as sound in principle, they must also attend to maintaining classroom order. If a newly introduced thematic routine does not support this order, teachers are likely to conclude that the new routine is ineffective. If the new routine is far from the usual routine, the teacher may lose the ability to make inferences from the regular teaching to the new one. The gap becomes too large, and the teacher may be faced with such great demands for change that the old routines become unusable.
Second, we also argued that it is, in fact, much more challenging to organize teaching for a reasonable content progression when teaching thematically than when teaching topic by topic, and that this need not just be a belief that teachers hold. It is rational for teachers to maintain clear goals and expectations to ensure a smooth progression of content. Teaching that is not organized topic by topic requires teachers to draw on a more elaborate theoretical understanding of mathematics in the sense of a richer set of operational invariants, as well as to monitor and coordinate the mathematical content across lessons actively. Alternatively, well-organized material support can help teachers develop routines that handle both the thematization and content organization. This can be compared to the results of Gainsburg (2012), which also showed that a decisive factor for teachers using a particular practice was the support in terms of real practice in the preceding teacher education.
Third, addressing the relationship between innovation and the external environment, we described the clash in goals and expectations between the testing and grading regime and the thematic teaching as genuine, rather than merely a matter of holding a neo-behaviourist over a constructive set of beliefs. This factor can also be understood in terms of teachers’ goals, as it is entirely rational for teachers to have a goal that their students pass essential tests.
As we have noted, the belief construct has not only been described and widely accepted but also challenged. For example, the idea that beliefs are relatively stable and hard to change has been challenged by Liljedahl, who suggests that beliefs under the right circumstances can change in quite radical ways (Liljedahl, 2010) and that the apparent stability of beliefs is instead an artifact from the way some of the belief research on teachers has been conducted (Liljedahl et al., 2012). Beliefs may be a particularly unproductive selection of explanatory factors for what influences the implementation of innovations, precisely because they allow the people or organization responsible to use beliefs as a scapegoat. Looking outside of mathematics education, in program evaluation theory, it is strongly suggested that implementation programs should be designed with an underlying program theory — one that specifies the mechanisms through which the innovation is supposed to achieve its desired effects (Chen, 2012). If teacher beliefs are considered part of this mechanism, the program design should include ways to change these beliefs and reliably measure any changes, which is a challenging task (e.g., Pajares, 1992; Schoenfeld, 2015). As we have argued, a deeper understanding of teachers’ goal structures, routines, and theories in action can be a more helpful way of understanding teachers’ resistance to change. If the innovation to be implemented requires a complex change of teaching routines, it probably needs to come with quite elaborate support in terms of suggestions for new routines. Equally important is to address what changes in the goal structure can be expected and to provide explanations of why the suggested routines are supposed to work in relation to the overall goal structure.
In relation to scheme theory and in the context of teaching, beliefs may be understood as one of the components that shape the rules of action guiding instructional decisions. However, when conducting a broader analysis, we also considered goals and expectations, operational invariants, and possibilities of inference. The factors of influence shift from residing mainly in the teacher to residing in the relationships between the implemented innovation and the teacher or between the innovation and the external environment, as indicated in our re-analysis. There may also be an interplay between the innovation and the organization. Therefore, if we are not satisfied with simply pointing out that teachers have incorrect beliefs when innovations do not work, innovation designers and implementation program designers may need to admit when the innovation is simply not good enough, or at least not suitable for the context, or inadequate in some other way.
Another possible theoretical route to obtain the detailed information needed to inform future implementation projects about which factors contribute to the outcome of the implementation is to use fine-grained belief analysis, as proposed by Speer (2008). Speer’s notion of collections of beliefs can be fruitfully reinterpreted through Vergnaud’s (1998, 2009) scheme theory. In her fine-grained analysis, Speer demonstrates how different clusters of beliefs guide an instructor’s responses to classroom situations. From a scheme-theoretic perspective, these beliefs can be understood as distributed across the four components of a scheme. They shape the goals and anticipations teachers hold for their instruction, inform the rules of action they adopt when orchestrating classroom activity, and underlie the operational invariants that stabilize their practices. Beliefs also influence the inferences teachers draw from student responses, such as whether hesitation is interpreted as a lack of fluency or as evidence of sense-making. In this way, beliefs do not stand apart from practice but operate as integral elements of teachers’ schemes. Reframing Speer’s findings in terms of schemes can thus provide a more precise analytic lens, allowing us to connect teachers’ cognition and practice in ways that are particularly valuable for understanding and supporting implementation processes. In particular, scheme theory enables the operationalization of the role of beliefs within broader patterns of action, providing insights that can inform the design of more effective implementation strategies directly.
While scheme theory places high demands on the selection of data and the analysis of teachers’ actions, in the long run, it may benefit innovation designers and implementation program designers. To make successful implementation likely, program designers must carefully research the intended context to know what is at stake. Understanding context involves factors that may not initially seem related to the innovation. For example, teaching thematically may seem unrelated to maintaining classroom order. However, as we have shown, it is not. When an implementation effort is made on a larger scale, the contact between developers and users, specifically teachers, will necessarily be weaker (Aguilar et al., 2023); therefore, the design requirements to fit the implementation context become even more significant.
As a final note, the role of beliefs can also be challenged theoretically. Although contested (e.g., by Furinghetti & Morselli, 2011), a common argument is that beliefs guide our actions (e.g., Eichler et al., 2017; Pajares, 1992; Schoenfeld, 1992). But what if it is the other way round? In line with advancements in professional development theory (Clarke & Hollingsworth, 2002; Guskey, 1986), beliefs are a consequence of experiencing what works. We believe in what works for students, in organizations, and in relation to the external environment. If we accept that we believe in what works, then beliefs follow patterns of action rather than vice versa. Patterns of actions are what Sfard (2023) calls routines. Routines are not just behaviors but the organization of behaviors and complex mental constructs. However, unlike beliefs, routines are more closely connected to what is observable and describable.
In the introduction, we accounted for a project driven by one of the authors, Østergaard (2023). In the lesson design on probability that comprised the innovation, one central tenet was a lesson phase that involved a whole-class reflection. None of the two teachers in the study implemented this phase of the lesson design. Interestingly, in the follow-up, none of them even mentioned skipping it. Due to a desire not to challenge the sensitive teacher-researcher relationship, the researcher did not confront the teachers directly but instead focused on future collaboration on strengthening other aspects of the lesson design. This represents a missed opportunity to learn from the implementation, as several questions that could have helped advance it were left unanswered. What was it that made teachers avoid a particular teaching practice or lesson component? What were their arguments? How can the rationality behind the avoidance be analyzed? What kind of support can be offered to enable teachers to carry out the teaching practice or lesson component, based on the analysis? What will teachers’ feedback be after they have tried?
We are convinced that progress in implementation research would benefit from theoretical approaches that allow careful analysis of teachers’ existing practices and their responses to change, in ways that take into account their often complex goal structures and routines. Such analyses can help implementers suggest how new goals might interact productively with existing ones, and how routines can be adjusted or introduced without causing major disruptions. In this article, we have highlighted scheme theory. However, in the context of pedagogical innovations in mathematics education research, we propose that attention be directed toward factors influencing teachers’ routines and actions, while recognizing that alternative approaches, such as fine-grained beliefs analysis (e.g., Speer, 2008) or other theoretical perspectives, may also offer valuable insights. We argue that theoretical perspectives with sufficient analytic precision are necessary to inform the future design and redesign of implementation projects. Whether through scheme theory, fine-grained beliefs research, or other frameworks, innovations that require shifts in teachers’ practices must be adapted to the external environment, supported at the organizational level, and ultimately designed to enable students to learn more mathematics.
Acknowledgement
This paper is part of the 2020-04090 and 2022-04811 grants of the Swedish Research Council.
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The terms scheme theory and schema theory are sometimes confused but stem from different intellectual traditions. Scheme theory, as developed by Piaget (1970) and elaborated by Vergnaud (1998, 2009), refers to the invariant organization of action in recurrent situations. In mathematics education, schemes are used to analyze how learners (and teachers) structure their activity, including goals, rules of action, operational invariants, and possibilities for inference. By contrast, schema theory is a construct from cognitive psychology and information-processing models (e.g., Anderson, 2017), where schemas are mental structures that organize knowledge in memory. While both concepts highlight structured patterns in cognition, scheme theory emphasizes action and practice in context, whereas schema theory emphasizes knowledge representation in memory.
