Abstract
The development of mathematical understanding is widely recognized as a critical goal of mathematics education. Mathematical understanding can be both a product and a process. However, there are varied perspectives on what mathematical understanding is and how it may be fostered. This paper describes a framework that organizes and explains the ways that mathematics educators conceive of research and practice regarding mathematical understanding. Specifically, the framework describes five aspects of mathematical understanding (factual, procedural, conceptual, metacognitive, and affective) and five lenses (mathematical, cognitive, social, diagnostic, and instructional) that mathematics educators use when focusing on fostering students’ mathematical understanding. We illustrate the framework with examples drawn from work published in leading mathematics education research journals. We hope that researchers and practitioners can use this framework to shape their perspectives of different aspects and different lenses of understanding. The paper concludes with a discussion of future directions for research and practice on fostering mathematical understanding.
Keywords
There is wide agreement in the teaching and learning of mathematics on the importance of students’ mathematical understanding. However, mathematical understanding is also an unresolved topic that has been extensively studied, critically debated, and continuously developed by scholars across fields for nearly a century (e.g., Brownell, 1935; Cai & Ding, 2017). There is no consensus regarding what mathematical understanding really means and how teachers can foster students’ mathematical understanding in the classroom.
In the mathematics education field, researchers explore students’ understanding of mathematical topics and how to foster students’ mathematical understanding from multiple perspectives. 1 Although individual studies offer valuable insights into the nature of mathematical understanding and effective ways to promote it, an overarching framework that synthesizes these insights is still lacking. Without such a framework, researchers and practitioners do not have an effective tool to reflect on the successes and challenges associated with developing students’ mathematical understanding. Therefore, there is a pressing need to re-examine what we have accomplished as a field and to synthesize these insights into a usable framework that can guide practice and future research.
In this paper, we explore mathematical understanding from the perspectives of mathematics educators, with the aim of contributing to a conceptual framework on this topic. Although we are confident in asserting that mathematics educators are quite interested in ways to foster students’ mathematical understanding, we are also fully aware of the challenge we face in identifying the multiplicity of perspectives mathematics educators hold concerning mathematical understanding. To bolster the representativeness of the perspectives we present, we took the approach of examining publications related to mathematical understanding in two influential mathematics education journals—the Journal for Research in Mathematics Education (JRME) and Educational Studies in Mathematics (ESM). We expected these publications to provide valuable insights that would inform the development of a framework that reflects mathematics educators’ perspectives on mathematical understanding (Inglis & Foster, 2018).
Developing such a framework would not only provide a tool for organizing the kinds of mathematical understanding past studies have investigated but also provide future research with useful perspectives about the kinds of mathematical understanding involved in their studies and how to foster it. From a practical point of view, this paper addresses both the meaning of mathematical understanding and ways to foster students’ mathematical understanding.
This paper is structured in three sections. The first section presents our framework for aspects of mathematical understanding and lenses that mathematics educators have used in research on fostering mathematical understanding. The second section illustrates the framework with an example related to problem-posing-based learning (P-PBL), wherein we aim to present a holistic picture based on mathematics educators’ perspectives about mathematical understanding and ways to facilitate it. In the third section, we consider implications of our framework and how it may contribute to ongoing conversations and directions for future research.
Fostering Students’ Mathematical Understanding: A Framework
In the research on mathematical understanding, mathematics educators have long drawn on the cognitive domain taxonomy of Bloom et al. (1956), which organized the cognitive goals of education into a hierarchy of six categories: knowledge, comprehension, application, analysis, synthesis, and evaluation. This was later revised by Anderson and Krathwohl (2001) to reflect both a knowledge dimension and a cognitive process dimension. Wilson (2016) compared Bloom et al.'s original framework with Anderson and Krathwohl's revision, highlighting how the “nouns” in the former were reworded as verbs in the latter's cognitive process dimension (e.g., comprehension vs. understanding). In our prior work (Cai & Ding, 2017), we acknowledged the field's agreement on the following: (1) Understanding is conceptualized as both a process (“understanding” as a verb) and a result of that process (“understanding” as a noun); (2) understanding is both the making of connections and a result of connection making in a social environment; (3) understanding is a dynamic and continual process; (4) understanding may have different levels and kinds; and (5) the goal is to reach a deep level of mathematical understanding. To build on the existing literature, we propose a framework (Table 1) that captures five aspects of mathematical understanding that have been targets of research in mathematics education and five lenses that mathematics educators have used to shape their explorations of how to foster students’ mathematical understanding.
Fostering Students’ Mathematical Understanding: A Framework.
Fostering Students’ Mathematical Understanding: A Framework.
The development of the framework is a result of top-down and bottom-up approaches. The top-down approach involved initial sketches of categories based on our collective experience in mathematics education research over the years (e.g., Cai, 2017; Cai & Ding, 2017; Cai et al., 2017). Further refining those categories, the bottom-up approach is based on our examination of the papers from JRME and ESM. Rather than performing a systematic review of the literature on mathematical understanding, we aimed to identify a representative pool of publications that could enrich the development, validation, and presentation of our conceptual framework. Thus, as we discuss the elements of the framework below, we draw on the papers we found in JRME and ESM as well as papers from other research journals or books as needed to better illustrate each element.
Mathematical understanding is a complex construct that comprises multiple related aspects that each represents a (connected) part of one's understanding of mathematics. In research on mathematical understanding, mathematics educators have generally focused on five such aspects: factual, procedural, conceptual, metacognitive, and affective. The first four are rooted in Anderson and Krathwohl's (2001) revision of Bloom et al. (1956)'s original taxonomy of the cognitive domain. Although Bloom et al.'s original taxonomy was primarily hierarchical with knowledge 2 as the lowest level, the revision separated the cognitive domain into two dimensions: a knowledge dimension and a cognitive process dimension. Moreover, Anderson and Krathwohl expanded the knowledge dimension by adding metacognitive knowledge to the existing categories of factual, procedural, and conceptual knowledge, which were based on distinctions in Bloom et al.'s original framework.
Building on this seminal work, we provide operational definitions for five aspects of mathematical understanding that mathematics educators have studied. Beyond Anderson and Krathwohl's (2001) four categories, we also incorporate an affective aspect of mathematical understanding, which aligns with Krathwohl et al.'s (1964) affective domain.
The factual aspect of mathematical understanding is knowledge that is basic to mathematics. It includes essential facts, terminology, details, or elements students must know or be familiar with to understand mathematics or solve a mathematical problem (Anderson & Krathwohl, 2001; Wilson, 2016). The procedural aspect of mathematical understanding refers to very specific skills, algorithms, techniques, and particular methods in mathematics (Anderson & Krathwohl, 2001). The U.S. National Assessment of Educational Progress (NAEP) mathematics assessment refers to procedural knowledge as students’ ability to follow algorithms, perform calculations, and execute steps in a specific order to reach a solution (National Center for Education Statistics, 2021). The conceptual aspect of mathematical understanding refers to “interrelationships among the basic elements within a larger structure” (Wilson, 2016, p. 5), which includes knowledge of classifications, principles, generalizations, theories, models, or structures pertinent to mathematics. The NAEP mathematics assessment refers to conceptual understanding as students’ understanding of the underlying mathematical concepts and relationships rather than just the ability to perform calculations (i.e., procedural knowledge; National Center for Education Statistics, 2021). Students draw on conceptual understanding to explain the “why” behind mathematical operations and procedures. The metacognitive aspect of mathematical understanding is the awareness of one's own thinking and particular cognitive processes. It is strategic or reflective knowledge related to monitoring if solution strategies are appropriate and evaluating if answers are reasonable (Anderson & Krathwohl, 2001; Wilson, 2016). The affective aspect of mathematical understanding falls into a noncognitive dimension including but not limited to students’ attitudes, beliefs, motivation, identity, and emotion (Cai et al., 2017; Hannula et al., 2016; Schukajlow et al., 2023).
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Although these aspects are interrelated, for clarity, we discuss how mathematics educators have approached them separately.
Factual, Procedural, and Conceptual Aspects of Mathematical Understanding
In mathematics, there is an abundance of factual knowledge such as definitions, formulas, and concepts, from knowing the multiplication table to the meaning of areas and perimeters. The role of factual knowledge has been studied by mathematics educators in the development of students’ procedural fluency, problem-solving skills, and more complex mathematical understanding (e.g., National Research Council, 2001).
In contrast, the conceptual and procedural aspects of mathematical understanding have been widely debated topics in mathematics education. Hiebert and Lefevre (1986) defined conceptual knowledge as understanding the relationships between ideas and procedural knowledge as familiarity with rules and procedures. However, Star (2005) critiqued the overemphasis on conceptual knowledge in the field. In a survey of JRME articles, he found that only 11 of approximately 100 articles relevant to K-12 students’ mathematical content knowledge investigated students’ knowledge of procedures. Among various possible reasons for overlooking procedural knowledge, Star pointed to the potential influence of Hiebert and Lefevre's (1986) definition, which seems to differentiate the “type” and “quality” of knowledge, portraying procedural knowledge as less important. Consequently, Star called for a reconceptualization of procedural knowledge, advocating for the development of “deep procedural knowledge,” which emphasizes flexibility in applying procedures.
Baroody et al. (2007) countered that deep procedural knowledge inherently relies on conceptual understanding. According to these researchers, both procedural and conceptual knowledge can exist at either superficial or deep levels. However, developing either type depends on the application of the other. As Baroody et al. articulated, “it is unclear how substantially deep comprehension of a procedure can exist without understanding its rationale” and “likewise, deep conceptual knowledge depends on knowing the tools for applying and extending mathematical ideas” (2007, p. 119). These arguments emphasize the interconnections between procedural and conceptual knowledge—a perspective shared by Hiebert and Carpenter (1992). Indeed, definitions of conceptual and procedural understanding and the relationships between them have reached some level of consensus. In its influential publication, Adding It Up: Helping Children Learn Mathematics, the National Research Council (NRC, 2001) included both conceptual understanding and procedural fluency among the five strands that make up mathematical proficiency. Notably, they defined procedural fluency to include not only the ability to carry out procedures but also the skill to flexibly and appropriately use them.
Stein et al. (1996) provided a more practical way to understand factual, procedural, and conceptual knowledge based on their task framework that classifies instructional tasks into four categories: memorization, procedures without connections, procedures with connections, and doing mathematics. They categorized memorization tasks as focused on factual knowledge, tasks involving procedures without connections as focused on procedural knowledge, and tasks involving procedures with connections and doing mathematics as invoking conceptual knowledge. This task framework overlaps with (but does not entirely align with) both the NRC (2001) strands of mathematical proficiency and with our aspects of mathematical understanding. In particular, we see tasks involving procedures with connections as crossing both the conceptual and procedural aspects of mathematical understanding.
Given that mathematical understanding includes both procedural and conceptual aspects, it is not surprising that researchers (and teachers) have debated the sequence of building procedural and conceptual knowledge in the development of students’ mathematical understanding. Some researchers (e.g., Wearne & Hiebert, 1988) have argued that conceptual knowledge should precede procedural knowledge, whereas others (e.g., Rittle-Johnson et al., 2001) have suggested they should be developed iteratively. Despite the disagreement, researchers have agreed on the dynamic and continual process of developing mathematical understanding (Pirie & Kieren, 1994). In short, the complexity of these two types of understanding likely poses challenges for teachers in fostering students’ overall mathematical understanding in classrooms, underscoring the need for further research in this area.
The Metacognitive Aspect of Mathematical Understanding
Metacognition has long drawn the attention of mathematics educators. Garofalo and Lester (1985) pointed out that it is inadequate to analyze mathematical performance from a purely cognitive perspective. This is because metacognition plays a critical role in students’ learning. In fact, Cai (1994) found that the difference between “novice” and “expert” problem solvers lies in their metacognitive rather than their mathematical knowledge involved in the problem. Schoenfeld (1992) also highlighted the importance of the metacognitive aspect of understanding, reviewing the literature based on five aspects important for mathematical thinking and problem solving: the knowledge base, problem-solving strategies, monitoring and control, beliefs and affect, and practices. Among these, “monitoring and control” refers to self-regulation, which is a component of metacognition. Indeed, metacognition acts as a key part of understanding as a process, for example, as one engages in the process of exploring and understanding a complex mathematical problem.
Mathematics educators have conducted empirical studies on metacognitive knowledge at different grade levels, often involving other factors as well. For example, Carr et al. (1999) studied gender differences in first graders’ use of mathematical strategies and relevant metacognitive knowledge when solving addition and subtraction computational tasks. Moreover, they explored how the metacognitive instruction provided by parents and teachers contributed to gender differences in children's strategy use. Data collected from teachers and parents indicated that boys were more influenced than girls by adults’ strategies and actions.
Another example is Magiera and Zawojewski's (2011) study of ninth graders’ metacognitive activities during mathematical problem solving. They used a three-aspect framework—awareness, regulation, and evaluation—to characterize the situations and contexts in which spontaneous metacognitive activity occurred. Through in-depth qualitative analysis, they identified the types of situations that tended to elicit both social- and self-based metacognitive activity. For instance, interpreting diverse perspectives and engaging in explanations were common in social-based contexts, and seeking personal satisfaction was common in self-based contexts. These findings provide insights for developing students’ metacognitive knowledge, thereby enhancing their mathematical understanding.
The Affective Aspect of Mathematical Understanding
Research in mathematical understanding should consider both cognitive and noncognitive learning outcomes (Cai et al., 2017). Affective constructs such as students’ attitudes, beliefs, identity, and motivation (Hannula et al., 2016; Schukajlow et al., 2023) shape students’ relationship with mathematical learning and their cognitive skills. Indeed, the affective aspect of mathematical understanding is essential for students to develop productive dispositions towards mathematics—dispositions that allow students to understand mathematics as a useful discipline that makes sense and is amenable to their exploration and effort (NRC, 2001).
Despite the important relationship between students’ affect and their mathematics achievement (Middleton & Spanias, 1999), students’ negative affect (e.g., mathematics anxiety, fear of mathematics, low motivation) remains an unresolved issue, and mathematics education researchers have explored ways to address this challenge. For instance, Uhlig (2002), recognizing students’ challenges with mathematical proof in an elementary linear algebra course, emphasized the importance of fostering both an appreciation for proofs and “a student-felt need, as well as an intellectual understanding of the necessity and beauty of proofs in the students’ minds” (p. 337). Lesh et al. (2008) also stressed “a science of need” in designing tasks to engage students in modeling complex data, discussing how students should be able to see the utility of mathematics and “buy into the premise that mathematics has relevance and importance in their own lives” (p. 127).
Research on mathematical affect, including emotion and motivation, has grown since 2000 (Schukajlow et al., 2023). A review of 95 papers published in JRME and ESM in 2021 showed that 44% examined students’ mathematical affect (Schukajlow et al., 2023). Such increased attention to students’ emotions and motivation suggests that educational research should expand beyond immediate academic outcomes to consider broader lifelong effects to help ensure that research supports students’ mathematical learning and development across multiple dimensions, including affect.
Five Lenses for Exploring How to Foster Mathematical Understanding
Mathematics educators approach the issue of how to foster students’ mathematical understanding in multiple ways, employing mathematical, cognitive, social, diagnostic, and instructional lenses that each focuses their attention on a different set of concerns. Because the process of developing students’ mathematical understanding is inherently complex and multidimensional, when exploring questions of how to foster that understanding, mathematics educators must adopt lenses suitable to the strategy, concern, pathway, or mechanism that is the primary focus for that exploration. More generally, any research in mathematics education is both situated and oriented by a choice of theoretical stance and structure as well as ways of interpreting phenomena that match the purpose and nature of the research questions (Hiebert et al., 2023). Research on how to foster mathematical understanding is no exception. The five lenses we discuss here represent relatively broad categories. They are not mutually exclusive but, like the aspects of mathematical understanding, they are interrelated and often complementary for examining how to foster mathematical understanding. For example, Hiebert (1992) coordinated mathematical, cognitive, and instructional lenses in his analysis of the understanding of decimal fractions and how it may be fostered. Below, we elaborate on how mathematics educators have employed these five lenses, drawing on examples from JRME and ESM alongside additional elucidation from other publications.
The Mathematical Lens
One distinctive feature of research on fostering mathematical understanding is the analysis of the mathematics itself and its practices and processes. Researchers have sought to unpack specific mathematical ideas and tease out what it means to understand that idea mathematically (Carpenter et al., 2014; Hiebert, 1992). For example, mathematically, decimal fractions represent a confluence of whole numbers and common fractions. Thus, the major task for students in acquiring a meaningful understanding of decimals is to connect the written symbols and rules of decimal fractions with the quantities they represent (Hiebert & Carpenter, 1992).
As another example, consider the mathematical idea of the average. As a statistic (the arithmetic mean), it is used to describe and make sense of a data set. As a tool, it is used with standard deviation to summarize and compare data sets. It is also a computational algorithm, and understanding the average as an algorithm involves both understanding its meaning and developing fluency in applying the algorithm (Cai, 2000). Students should know the averaging algorithm as well as when and how to correctly apply it to problem situations in daily life or statistical analysis. In summary, understanding the arithmetic average involves (1) a procedural understanding of the idea as an algorithm, (2) a conceptual understanding of the idea as an algorithm, and (3) a conceptual understanding of the concept as a statistic (the mean) to describe and make sense of and compare data sets (Cai, 2000; Mokros & Russell, 1995; Watson & Moritz, 2000).
A further example concerns mathematical proof. Although students often struggle with proofs, it is frequently observed that mathematical rigor is sacrificed in classrooms, which limits students’ opportunities to develop their understanding of proofs. Dawkins and Weber (2017) presented a framework for understanding proofs through the perspective of mathematical values and norms that support them (e.g., using decontextualized reasoning). They argued that the norms and values upheld by mathematicians in proofs should not be sacrificed in mathematics classrooms as students’ acceptance of these values plays a critical role in their apprenticeship into the practice of proving; those who fail to embrace these values will likely struggle with adhering to the norms and ultimately the practice of proving.
The Cognitive Lens
Mathematics educators have also employed a cognitive lens to explore ways to better develop students’ mathematical understanding. This approach differs from that of cognitive or educational psychologists, who often use mathematics primarily as a case to develop models and theories of cognition. For example, Byers and Erlwanger (1985) discussed the role of memory in mathematics understanding, given that many mathematics educators at the time viewed memory in a negative manner (e.g., mathematics does not require deliberate memorization). However, classroom teachers did not ignore memory, instead emphasizing “practice and repetition.” Being aware of the gap between memory research and its usage in classrooms, Byers and Erlwanger (1985) discussed various aspects of memory (e.g., retention, structure) and its different uses in mathematical learning (e.g., relational, instrumental understanding), concluding that “the crucial question is not whether memory plays a role in understanding mathematics but what it is that is remembered and how it is remembered by those who understand it—as well as by those who do not” (p. 261).
Some researchers have also conducted empirical studies applying cognitive theories to students’ learning of specific mathematical topics. For example, Wearne and Hiebert (1988) tested the cognitive process theory by examining middle school students’ learning of decimal numbers, focusing on the semantic phase of the cognitive process, specifically connecting and developing. The connecting process involves constructing links between symbols (e.g., decimals) and familiar referents (e.g., Dienes, or base-ten, blocks). The developing process refers to using familiar referents to develop procedures for working with symbols (e.g., when adding decimals, one combines the blocks of the same size). It was found that a semantic approach to instruction effectively supported students’ learning, especially for those who had not been previously exposed to syntactic rules. This study underscores the importance of introducing initial representations to develop meanings for symbols before practicing syntactic routines.
Similarly, Kotsopoulos and Cordy (2009) explored the potential of imagination as a cognitive space for middle school students’ mathematical learning. They developed a framework and applied it in a geometry classroom, finding that shared visualizations and problem-posing activities, along with the use of imagination as a cognitive learning tool, played a significant role in fostering students’ mathematical understanding.
The Social Lens
Mathematics is a social activity, one in which students can learn together through interacting with each other and teachers. In fact, social activity is a process for not only developing mathematical understanding but also for fostering social and emotional growth. Yet, researchers have often chosen a cognitive lens to explore mathematical understanding and learning. For example, in their framework, Hiebert and Carpenter (1992) assumed that knowledge is represented internally and that these internal representations are structured. Consequently, they focused on the cognitive aspects of understanding mathematics to discuss a range of related issues. In contrast, others (e.g., Cobb, 1994) have offered a sociocultural lens of mathematical understanding. The use of the social lens in mathematics education has gained increasing attention in recent decades, reflecting the “social turn” identified by Lerman (2000) and later echoed by Inglis and Foster (2018). According to Lerman, there was a shift from theories centered on individuals’ thought processes to those that view meaning, thinking, and reasoning as products of social activity.
One study that illustrates the shift from a cognitive to a social lens is Albert (2000), which, although focusing on students’ cognitive processes, explored them using a social lens that highlighted the role of written communication in students’ thought processes. Students in this study were put in small groups to collaboratively solve problems. After discussing solutions and strategies with peers, they worked independently to write statements (including diagrams) explaining their strategies and solution procedures. In the next class, students reconvened to compare solutions. Based on an analysis of seven students’ problem-solving strategies, it was found that students’ thought processes, initially developed in oral conversation with peers (outside-in process), were further advanced through their written communication (inside-out process). Albert (2000) concluded that “the social nature of mathematical communication must be an integral and substantial part of the learning process” (p. 138).
Research on fostering mathematical understanding that employs a social lens frequently involves language, such as mathematical discourse (e.g., Herbel-Eisenmann & Wagner, 2010) and classroom discussions (Pirie & Schwarzenberger, 1988). Other research topics amenable to a social lens include students’ collaboration (Mueller et al., 2012) and engagement (Lesh et al., 2008). Across these types of studies, researchers have examined how social factors can be leveraged to promote students’ mathematical understanding. Themes like student authority, agency, and equity were often explored along with affective factors like attitudes and motivation.
For example, Herbel-Eisenmann and Wagner (2010) analyzed language use in 148 transcripts of secondary classrooms, finding that teachers’ recurring language choices could structure mathematics classroom discourse and reinforce teacher authority, thereby limiting student agency. They argued that it is essential for the field to engage in discussions about authority as these issues are closely related to the development of mathematical understanding within classroom discourse.
The Diagnostic Lens
Mathematics educators are cognizant of the important role that assessment plays in supporting mathematics teaching and student learning. In both teaching and research, mathematical understanding frequently needs to be assessed. There are three kinds of assessment related to classroom instruction based on when the assessment is conducted: before the lesson to diagnose students’ existing understanding, during the lesson to inform teachers about students’ understanding so they can adjust their teaching, and after the lesson to evaluate and draw conclusions about students’ mathematical understanding. From the perspective of mathematics educators, all three types of assessment are important. Instruction needs to build on what students know and what they do not know. Indeed, in the current mathematics education community, formative assessment is sometimes referred to as assessment for learning (Nieminen et al., 2023; William, 2006).
Some empirical studies adopting a diagnostic lens have conducted fine-grained assessments of students’ mathematical understanding. Building on a theory that differentiates two stages in students’ learning—participatory and anticipatory—Tzur (2007) conducted a whole-class teaching experiment on fractions with third graders. Students’ responses to the assessment tasks showed that differentiating between these two stages was useful in guiding teachers’ task selection to address students’ mathematical thinking. It was also found that many students who appeared to “understand” a concept were only at the participatory stage. To minimize such situations, Tzur proposed three levels of assessment rigor—rough, intermediate, and sophisticated—aimed at supporting teachers’ classroom instruction.
Säfström et al. (2024) developed a framework for diagnosing students’ reasoning difficulties during mathematical problem solving in primary and secondary schools. This framework aimed to address one main question (What specific difficulty does the student need help with?) supported by three diagnostic subquestions designed to elicit evidence of students’ difficulties (What does the task ask you to do? What have you done so far? And why did you do that?). Based on an analysis of 285 difficulties collected from 152 lessons across Grades 1–12, the framework was finalized to include four phases: interpret, explore, create solution ideas, and utilize solution ideas. Two different types of difficulties—students being stuck and students needing help—were proposed across these four phases, with corresponding indicators. Overall, the framework provides potential support for teachers to elicit and interpret evidence of students’ thinking during problem solving. Other studies have explored teachers’ assessment processes during teaching (Wildgans-Lang et al., 2020) and the use of technology in mathematical assessment (Weigand et al., 2024).
In addition to empirical studies, mathematics educators have attempted to conceptualize assessment research. For instance, Nieminen et al. (2023), based on a critical review of 127 studies on mathematics classroom assessment, identified five types of assessment in which students were positioned differently through discourse: measurement, cognition, empowerment, monitoring, and performance. In the first two types, students were merely passive recipients of assessment. These assessments either portrayed classroom assessment through quantification or primarily referred to psychocognitive processes. In contrast, the next two types positioned students as active agents, aligning with the assertion of “assessment for learning.” In these assessments, students were actively engaged, empowered, monitored, and responsible for their learning. An example of active assessment is computerized formative assessment, which monitors students’ progress and provides automatic feedback. Finally, performative assessment intertwines both active and passive discourse, allowing these two seemingly opposing perspectives to complement one another. For example, self-assessment, where students can self-correct and reflect, was found to contribute to their final exam performance. The above conceptualization of different types of assessment reflects, in some sense, the broader debate between cognitive and social lenses of research in mathematics education as well as their potential middle ground.
The Instructional Lens
Another very distinctive lens that mathematics educators employ focuses on how to use instruction to facilitate students’ mathematical understanding. From this perspective, researchers often consider learners’ characteristics when exploring instructional approaches that may better facilitate students’ understanding of a mathematical topic. Notably, other lenses—mathematical, cognitive, social, or diagnostic—are often involved.
Consider, for example, the aforementioned challenging topic of proof. Uhlig (2002) compared two instructional approaches taken in elementary linear algebra courses: DLPTPC (Definition–Lemma–Proof–Theorem–Proof–Corollary) and WWHWT (“what, why, how, ‘what is true here’”; p. 338). DLPTPC is a formal approach that mathematicians often take to communicate their work—however, because students in elementary linear algebra have little experience with proof, this approach is often ineffective in supporting students’ learning. With WWHWT, students are encouraged to explore the subject intuitively, considering questions like “What happens if?” “Why does it happen?” “How do different cases occur?” and “What is true here?” This approach is exploratory and emphasizes students’ intuitive mathematical reasoning in a manner similar to that of mathematicians prior to the 1850s. 4 Uhlig suggested that such a sequence of exploratory questions can lead to students’ conceptual understanding, mentally and emotionally preparing students for the formal DLPTPC sequence.
Regarding the above-mentioned topic of average, Cai and Ding (2017) surveyed Chinese teachers on how to develop students’ understanding. Many teachers emphasized the importance of understanding the essence of the concept and engaging students in the process of knowledge development. One teacher shared that she would use rectangular bars with different lengths to represent a real-life data set and then draw a line to show the approximate location of the average number. Such an instructional approach attends to representations, especially concrete modeling, which aims to facilitate students’ learning experiences.
Similar suggestions related to topics in the early grades have been advocated. Ding and Li (2014), focusing on the distributive property, explored how the instructional environment can better support students’ learning. These researchers analyzed a total of 319 instances of the distributive property in one Chinese elementary textbook series (Grades 1–6), finding a consistent pattern in representational sequence, shifting from concrete to abstract. For instance, the fourth-grade lesson that formally introduced the distributive property started with a real-world context of shopping (a word problem accompanied by pictures), which students were guided to solve in two different ways, resulting in two numerical solutions. The textbook then suggested that students compare these two solutions, which leads to one equation, an instance of the distributive property. Next, the textbook asked students to pose more examples of this sort, leading to the revelation of the distributive property in a formal way, (a + b)c = ac + bc. This instructional approach aligns with the “concreteness fading” method advocated by educational psychologists (e.g., Fyfe et al., 2015) and echoes the semantic approach of Wearne and Hiebert (1988) from the cognitive perspective.
Some studies on classroom instruction incorporate social perspectives. McClain and Cobb (2001) reported on a classroom-based teaching experiment in a first-grade classroom in which the teacher taught lessons on addition and subtraction. An analysis of the classroom lessons indicated that the teacher played a critical role in supporting students’ mathematical learning and the development of sociomathematical norms. For instance, by symbolizing students’ suggested solutions, she established norms regarding acceptable explanations, mathematical differences, and efficient, simple, or easy solutions in her classroom. These findings highlight how teachers can create learning opportunities that foster students’ mathematical understanding and dispositions in dynamic classroom environments.
Summary
In this section, we presented five aspects of mathematical understanding and five lenses for research and practice related to fostering students’ mathematical understanding. Although we discussed them separately, we view these aspects of understanding and lenses as interconnected: The five lenses represent perspectives on phenomena that work together to produce different aspects of mathematical understanding.
A key feature of our framework lies in the intersections between aspects and lenses. Collectively, these intersections provide a kind of map of the terrain of mathematical understanding in our field. That is, each intersection represents a particular way of conceptualizing and investigating mathematical understanding and how it may arise. Our intent is for this map to serve as a useful resource when reflecting on the field's efforts, whether in terms of identifying particular foci of research and practice related to mathematical understanding, considering whether and how different lenses might shed new light on familiar aspects of understanding, considering where more work is still needed, or even looking for connections across intersections. Given that it is unlikely that all aspects and lenses can be addressed in an equally balanced way in a given study or classroom setting, we also hope that both researchers and practitioners can draw on this framework to examine and reflect on how their choices of lenses direct their gaze and which aspects of understanding they are attending to and actively supporting.
In what follows, we present a holistic example that illustrates how elements of our framework can inform an analysis of fostering students’ mathematical understanding. We then turn our attention to the implications of our framework for both mathematics education practitioners and researchers.
Fostering Students’ Mathematical Understanding: P-PBL as an Example
Mathematics educators have strived to enhance the impact of research on practice (Silver, 2017). The ultimate goal of research on mathematical understanding is to foster students’ mathematical understanding. In this section, we draw on the example of P-PBL to discuss how to foster students’ learning (Cai, 2022), illustrating the five aspects of mathematical understanding and five lenses for exploring how to foster students’ mathematical understanding.
Problem posing is the process of formulating or reformulating problems by generating new questions based on available information and prior knowledge (Silver, 1994). Unlike problem solving, where students work with predefined problems, problem posing encourages learners to create their own problems, allowing for greater flexibility and deeper engagement with the subject matter (Cai et al., 2015; Kopparla et al., 2019). Indeed, formulating problems is a key element of strategic competence, another of the strands of mathematical proficiency (NRC, 2001). Moreover, a strong correlation between problem-solving and problem-posing skills has been documented (Cai et al., 2015). The P-PBL approach integrates students’ problem posing into the learning process, emphasizing the creation and exploration of problems alongside problem solving (Cai, 2022). The following two cases illustrate P-PBL in practice and provide the basis for a brief analysis using our framework (see Table 1).
The first case is a Grade 1 problem-posing lesson discussed in Cai (2022) involving a review of the addition and subtraction of two-digit whole numbers. In this topic, students often depend on naïve keyword associations to know which operation to perform (e.g., “all together” must mean addition) rather than understanding the meaning of addition and subtraction, potentially reflecting a focus on procedures that is disconnected from other aspects of mathematical understanding, such as conceptual or metacognitive aspects. In the problem-posing lesson, the teacher began by asking the students to write down their favorite two-digit numbers and then describe them, seeing if other students could guess their numbers by the description alone. For example, one student said, “The number I like is the one between 26 and 28.” The class correctly guessed 27, and the teacher followed up by asking the class to describe the number 27 in other ways. Students generated statements like “The number that is 7 more than 20.” One student drew a picture with 2 big hearts (representing 10s) and 7 little hearts (representing 1s). Here, the teacher turned the mathematics learning into a game to further students’ number sense and understanding of place value. The teacher then asked, “How many real-life problems can you pose and solve using the addition equation 27 + 39 = 66?” Students posed various problems, like “There are 27 red flowers and 39 yellow flowers. How many flowers are there in total?” Notably, the students were able to pose and solve real-life problems involving addition and subtraction without naïvely reducing the operations to a default set of “key words” which explicitly suggest addition and subtraction. Thus, using problem posing to help students understand relationships benefited their understanding of subtraction and addition.
The second case is a lesson on linear functions described in Hwang et al. (2024). The middle school teacher in this case found a problem-solving-based lesson in her curriculum unsatisfactory and subsequently redeveloped it to better target her students’ learning. Specifically, the teacher modified the lesson to have students pose problems based on the two equations y = 1200 + 3x and y = 1200 − 3x. This activity was designed to foster students’ understanding of the concept of slope, including both positive and negative slopes. She also replaced the original problem-solving task with a problem-posing gym membership task (see Figure 1).
Gym membership problem used in Hwang et al. (2024).
The teacher gave the students time to think about and pose problems for the gym situation. The students posed a variety of problems, such as “After n visits, how much money would she pay under each plan?” Through this activity, the teacher was able to elicit the ideas of the y-intercept, slope, and so forth, connecting them to the symbolic representation.
The five lenses for exploring how to foster mathematical understanding can be applied to analyze what is being attended to or centered in both of these cases. First, the mathematical lens is exhibited in students’ understanding of place value and addition and subtraction as well as word problem solving in the first case and linear functions in the second case. Their cognitive understanding can be seen in their processing of posing various problems. Socially, the students in both cases engaged in these tasks in the classroom, with their activity necessitated by interactions with each other. Moreover, the tasks could be used as diagnostic tools for the teacher to gauge students’ understanding of respective mathematical ideas. Finally, the tasks are instructional tools used to further the learning goals of the lessons, such as the task increasing students’ engagement with linear functions.
The five aspects of mathematical understanding can also be fostered through the P-PBL approach, which has increasingly been the target of research exploring students’ mathematical understanding. Ran et al.'s (2025) meta-analysis found a significant positive effect of the impact of problem-posing interventions on learners’ cognitive learning outcomes (Hedges’ ḡ = 0.53). The outcome measures in the studies included factual, procedural, and conceptual knowledge. The meta-analysis also found a large and statistically significant effect on learners’ affective outcomes (Hedges’ ḡ = 0.67) as measured by learners’ noncognitive performance, such as their attitudes toward, interest in, and beliefs about mathematics. Cai et al. (2023) reviewed studies that not only evaluated the cognitive effects but also the affective processes of problem-posing processes. Studies have also examined metacognitive behaviors in mathematical problem posing (e.g., Baumanns & Rott, 2023). A study by Akben (2020) suggested that structured, semi-structured, and free problem-posing activities improve students’ problem-solving skills and metacognitive awareness. In sum, P-PBL clearly provides a promising avenue through which to foster students’ mathematical understanding along all five aspects of mathematical understanding, and research on P-PBL has made use of all five lenses.
This paper has examined the complex construct of mathematical understanding that mathematics educators draw on as represented by publications in the flagship journals JRME and ESM. Our goal has been to contribute to the broader conversation about how to foster students’ mathematical understanding. We outlined five aspects of mathematical understanding—factual, procedural, conceptual, metacognitive, and affective—and discussed five lenses employed by mathematics educators—mathematical, cognitive, social, diagnostic, and instructional—when considering how to foster students’ mathematical understanding. We hope this framework provides an integrated view of mathematical understanding and inspires ongoing research in this area, and we hope the framework will be refined through the careful scrutiny of other researchers. We now turn to implications for research and practice on fostering mathematical understanding.
Implications for Research
As with instructional practice, we suggest that our framework can inform reflection on the state of mathematics education research on fostering mathematical understanding. As a field, we can use this framework to examine existing literature, including cross-cultural studies, and identify the successes and challenges we have documented regarding promoting students’ mathematical understanding and ways to better support them. Building on the intersections in our framework, we think it can serve as a guide not only for individual research (as well as practice) but also for the field as a whole as we work collectively to study and deepen students’ mathematical understanding.
From mathematics educators’ perspectives, mathematical analysis is critical for illuminating mathematical understanding. There is a need to systematically unpack different mathematical ideas and lay out what it means to understand a given mathematical idea. Research is also needed on how to build on mathematical, cognitive, social, and diagnostic analyses to develop and design instruction. Indeed, researchers have frequently highlighted issues related to mathematical instruction in classrooms—specifically, the selection or design of instructional tasks (National Council of Teachers of Mathematics, 1991)—and discussed the importance of understanding the nature of instructional tasks that foster students’ mathematical understanding (Cai et al., 2024; Hiebert & Wearne, 1993; Stein et al., 1996). However, research is needed on how teachers can select, design, or modify textbook tasks for classroom instruction, taking into consideration the five aspects and five lenses. One such effort under investigation is the use of problem-posing tasks to help students learn mathematics and foster their mathematical understanding (Cai & Hwang, 2023).
Current research on mathematical understanding tends to mainly focus on factual, procedural, and conceptual aspects of mathematical understanding. Fostering mathematical understanding involves helping students not only know facts and procedures but also develop their ability to make connections between mathematical ideas and make sense of the underlying principles and theories behind them. In addition, the metacognitive aspect has been increasingly recognized as essential to the development of students’ thinking and plays an important role in students’ monitoring their own thinking when solving and posing mathematical problems. Unfortunately, research has less often focused on the affective aspect of students’ mathematical understanding. However, this aspect is beginning to receive increased attention (e.g., Cai et al., 2017; Schukajlow et al., 2023). Cai and Leikin (2020) pointed out that affective dimensions have been used in mathematics education research in three ways: (a) as a factor to understand how affective aspects are related to other aspects of students’ thinking, (b) as a learning outcome, and (c) as an instructional intervention to foster students’ mathematical understanding including developing their cognitive and affective knowledge. Thus, research is needed on the kind of instruction capable of producing positive changes in students’ affect. 5
Furthermore, learning is an individual and cognitive as well as a sociocultural process. For example, students may solve mathematical problems individually in different ways, and different students may use different strategies and draw on different kinds of knowledge to solve problems. In the classroom context, however, students can present and share their different solution strategies, reaching shared mathematical understanding through discussion. Thus, the classroom is fundamentally a social setting that fosters students’ mathematical understanding. Recently, cognitive scientists (e.g., Chi & Boucher, 2023) have suggested that co-constructing knowledge through an interactive mode in classrooms can help students reach the highest levels of learning and cognitive engagement. But it is not yet clear how the classroom setting may best be organized to leverage this productive social interaction that invites students to communicate to foster mathematical understanding.
Finally, as noted at the beginning of the paper, besides mathematics educators, there is another group of researchers—cognitive psychologists—who are interested in many of the same issues related to mathematics learning and teaching. However, these two groups have collaborated and interacted very little on a grand scale. Their research on mathematics learning and teaching is typically conducted from different perspectives, but we believe that these perspectives are complementary rather than conflicting. For example, Mix and Battista (2018) and Norton and Alibali (2019) laid out the two groups’ perspectives to initiate more direct dialogue with respect to the teaching and learning of mathematics in general and mathematical understanding in particular. Future studies could continue this line of research to explore mathematical understanding from joint perspectives.
Implications for Practice
The multiple aspects of mathematical understanding and perspectives on how to foster such understanding represented in our framework can inform curriculum design, pedagogical decisions, and classroom practice. As noted above, the intersections in our framework can be seen as a type of map that identifies different approaches to fostering various aspects of mathematical understanding. Such a map can also help identify which aspects of mathematical understanding may be missing or underemphasized in practice (e.g., the affective aspect is often overlooked) or in curriculum materials and suggest lenses that could be employed to recognize and strengthen these underrepresented aspects. For example, when a mathematics classroom emphasizes the mathematical and cognitive aspects without similar emphasis on other aspects, students’ ownership of learning may be diminished, potentially feeding into mathematical anxiety and negative attitudes toward mathematics. In such cases, social and diagnostic lenses may help identify pathways to enhance instruction by supporting aspects of understanding that have been neglected.
Indeed, given that a major focus in mathematics education is to design instruction that fosters students’ learning with understanding, instruction should build on mathematical analysis, diagnostic information, students’ cognitive difficulties, and the social dynamics of the classroom. Thus, mathematics educators working with an instructional lens focus on what learning opportunities classroom teachers or the relevant learning environment can provide to students. In doing so, they can draw on the other four lenses to provide fundamental perspectives on designing instruction. In particular, two of the most important components of instruction are selecting or designing instructional tasks and orchestrating classroom discourse (Hiebert & Wearne, 1993; National Council of Teachers of Mathematics, 1991). Although it may be easy to see how a mathematical and cognitive lens might focus and influence task selection and design (e.g., considering how the target mathematical concepts are mathematically and conceptually connected to other concepts students have already encountered), these are clearly professional activities that should also be informed with a gaze through a social lens (e.g., attending to how discourse should be elicited and guided so that all students have opportunities to communicate mathematically and interact productively with their peers around the target mathematical ideas) and a diagnostic lens (e.g., considering how tasks might serve a formative assessment role that provides the teacher pedagogically relevant information).
Fundamentally, the five aspects of mathematical understanding in our framework (factual, procedural, conceptual, metacognitive, and affective) represent learning outcomes we aim to observe in students, and these aspects are interconnected. This is consistent with the intertwining of elements of other frameworks, such as the five strands in the NRC's (2001) characterization of mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. In that framework, each strand of mathematical proficiency supports the other strands in the development of mathematical proficiency, just as the aspects of mathematical understanding that we highlight in our framework rely on one another. Although the categories in the two frameworks do not map directly (e.g., we might consider adaptive reasoning as including elements of both the conceptual and metacognitive aspects of mathematical understanding), they both reflect the highly connected qualities of both mathematical understanding and mathematical proficiency. Both constructs depend on high-quality learning opportunities that strengthen the connections. Thus, classroom practice can be informed by our framework in a way that supports students to build a more robust mathematical understanding in which each aspect is lifted up.
Ultimately, we believe that mathematics educators will benefit from carefully evaluating how extant approaches to fostering mathematical understanding have (or have not yet) addressed each of the aspects of mathematical understanding. Moreover, mathematics education as a field has benefitted from the multiplicity of lenses that researchers and practitioners have employed, as each lens provides a complementary perspective on mathematical understanding. Especially given the rapidly evolving technological landscape, as new phenomena such as artificial generative intelligence arise, it will be critical to use these lenses to focus future research on how such technologies can support the development of each aspect of mathematical understanding in practice.
Footnotes
Acknowledgements
The preparation of this manuscript was supported by a grant from the National Science Foundation (DRL-2101552). Any opinions expressed herein are those of the authors and do not necessarily represent the views of the National Science Foundation.
Author Contributions
Jinfa Cai contributed to conceptualization and writing. Meixia Ding contributed to conceptualization and writing. Stephen Hwang participated in writing. All authors read and approved the final manuscript.
Funding
The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the U.S. National Science Foundation Division of Research on Learning in Formal and Informal Settings (grant number DRL-2101552).
Declaration of Conflicting Interests
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.