We implemented a semester-long professional development workshop series to help Calculus I and II instructors implement active learning and equitable teaching practices. The workshops actively engaged instructors in approximations of practice (Grossman et al., 2009) in which they learned to enact various teaching practices like designing tasks; eliciting, noticing, and responding to evidence of student thinking; orchestrating class discussions; fostering inclusive social norms; and equitably managing student participation. After they participated in the workshops, we interviewed the instructors about their decisions to implement the teaching practices from the approximations of practice in their teaching. We analyzed the instructors’ resources, orientations, and goals that informed their decisions to enact those practices. We then identified the aspects of the approximations of practice that the instructors viewed as supporting their implementation of those teaching practices.
Researchers have exhibited ways in which prospective secondary mathematics teachers can connect content from abstract algebra with secondary mathematics in ways that reshape their understanding of secondary mathematics and thereby support their teaching of that content. However, there has been very little research done on what mathematics teacher educators can do in the classroom to help prospective teachers make connections between abstract algebra and secondary algebra in ways that support such reshaping. This study focuses on a mathematics teacher educator who guided a class of prospective teachers to connect properties of algebraic structures with equation-solving procedures by directing their focus towards intellectual needs for computation, structure, causality, certainty, and communication during whole-class discussions. Here, we describe illustrative classroom episodes around those targeted intellectual needs, while characterizing the teacher educator’s pedagogical moves used to provoke those needs and guide students toward the resolution of them. These seven moves included establishing shared terminology of mathematical properties, transforming spoken language into mathematical expressions, using a problem where the technique learned does not work, asking students to identify whether a step/procedure is valid or mathematically incorrect, prompting students to justify their claims, revising students’ work, and focusing students’ attention to the use of structural properties. We provide theoretical contributions on the teacher educator’s role in guiding prospective teachers to make connections between advanced and secondary mathematics.
This study investigated how undergraduate students comprehend misleading graphs published on public platforms such as news media and government websites. Nineteen students enrolled in a Quantitative Reasoning course participated in the study and interpreted four graphs, two of which were misleading. For each graph, the participants responded to three questions assessing their ability to read the data, read between and beyond the data. The participants provided written justifications of their responses to these three questions in order for the research team to understand how they reason mathematically when they are exposed to misleading graphs, and what differences exist between the mathematical reasoning of an individual who could successfully identify the inconsistencies present in a graph compared to someone who failed to do so. Students performed well reading data from graphs but struggled identifying relationships and making predictions from the misleading line graph. Analysis of students’ written responses to the misleading line graph revealed two distinct mathematical reasoning strategies: students who successfully identified its misleading elements relied more on numerical thinking, while those who did not relied primarily on the visual shape of the graph. Further, despite demonstrating mathematical competence, only two students spontaneously questioned the intent or credibility of misleading visualizations. Findings of the study suggest that mathematics teachers (i) encourage students to carefully consider the data present in graphs instead of being influenced by their shape, and (ii) cultivate critical questioning dispositions along with graph comprehension skills.
Fostering mathematical creativity requires intentional actions from instructors. In earlier work, we developed the Teaching for Mathematical Creativity Guide, which identified four types of teaching actions that students in creativity-based Calculus I courses associated with their creativity: Task-Related, Holistic Teaching, Active Learning, and Teacher-Centered.In this paper, we examine five affective outcomes students reported in response to these actions—Enjoyment, Confidence, Comfort, Reframing, and Negative Feelings—with Enjoyment, Confidence, and Comfort emerging most prominently. Enjoyment was most frequently linked to Holistic Teaching and Task-Related actions. We offer six concrete teaching actions surfaced from the analysis of teaching action-affect overlaps. These findings demonstrate how creativity-fostering teaching actions can cultivate not only students’ mathematical creativity but also their affective engagement with mathematics.
Students often struggle to apply calculus concepts, including definite integrals, in applied contexts such as physics. This study offers a novel analysis of how students construct the “product layer” ( f(x)· dx ) when modeling quantities in introductory physics. Building on the knowledge-in-pieces perspective, we introduce a modified dynamic conceptual blending diagram (MDCBD) that traces the activation and coordination of students’ knowledge resources during moment-to-moment reasoning. Five students enrolled in a calculus-based physics course participated in think-aloud interviews focused on tasks situated in physical contexts. Our analysis examines complex patterns of activation as students engage with multiplicative product layer reasoning in applied settings. We present two cases, one in which a student redirected their reasoning autonomously and one in which a student redirected reasoning in response to interviewer prompting. These findings deepen our understanding of the dynamic and situated nature of the activation and coordination of students’ knowledge resources when reasoning about definite integrals in applied problem contexts.
We investigate the meanings three mathematics education graduate students have for integrals, following participation in a calculus course centered around rate of change and accumulation. We conducted interviews on a task designed to elicit both antiderivative and accumulation considerations with possibly conflicting outcomes. We investigate this using a new approach to compartmentalization based on Thompson et al.’s recent theory of meaning (2024) and an associated novel methodology. Our findings suggest that for two of the three students, antiderivative and accumulation were only partially cognitively connected. This led to behavior indicating compartmentalization as well as cognitive conflict. We identify a more complex relation between compartmentalization and cognitive conflict than the literature identifies. Our results suggest that area, constant of integration, and integration bounds are potential ‘bridges’ between accumulation and antiderivative that could enable the formation of connections between them.
Implicit differentiation is a key topic in first-semester university calculus. Beyond being a useful technique, it is conceptually related to many topics in differential calculus, such as the chain rule and related rates. However, it has received limited research attention. Our study addressed this gap by exploring how students in a first-semester calculus course performed on an activity that emphasized the connection between the symbolic and graphical representations of implicit curves and their derivatives. During class, students worked in small groups on this activity, and later in the semester, responded to a similar question on the midterm exam. We analyzed students’ written responses to both the in-class activity and the exam question using an Implicit Differentiation Knowledge Components (ImDKC) framework. The results show that overall, students performed better on symbolic knowledge components than on graphical ones, indicating difficulties in coordinating these two modalities. The exam performance was significantly higher than during the in-class activity, especially on graphical-symbolic coordination and finding vertical tangent lines. Conversely, students’ performance on two symbolic knowledge components: chain rule and differentiation, significantly declined on the exam compared to the in-class activity.
Representations play a crucial role in the process of learning and communicating mathematics. Each representation carries its own inherent complexity, which can pose challenges for students as they engage with mathematical documents. The interest of the present case study is to explore how two engineering students attend to different representations (texts, figures, and formulae) of varying estimated complexity when engaging with the concept of double integrals. Eye-tracking methodology, combined with two interviews, was used to investigate students’ engagement with the topic. Their eye movement patterns were analysed in relation to the complexity of the representations, as judged by two experts, and triangulated with their self-reported experiences. In addition, a written test was administered to assess their understanding of the topic. The results suggest that both students’ engagement with figures was closely related to their estimated complexity, with longer fixations on more complex figures, though with some variations. In contrast, their engagement with texts and formulae did not align with the estimated complexity but instead reflected selective attention, as noted by the students in the interviews. Test results further revealed that this pattern of engagement was associated with a primarily procedural understanding of double integrals and difficulties conceptualizing their relation to Riemann sums. The study contributes to a more nuanced understanding of how students attend to mathematical representations of varying complexity, offering insights for both research and instructional practice.
Research on the teaching and learning of linear algebra has been ongoing for several decades, but it has recently been renewed in the context of non-routine tasks and technological ecosystems, including artificial intelligence. This article analyzes two key transitions: (a) from school-level geometry and algebra to linear algebra, and (b) between different paradigms of linear algebra. We propose a fundamental situation in which the goal is to find the vertices of a polygon given its midpoints. This problem can be modelled using systems of linear equations that may have a unique solution, infinitely many, or none. The situation was divided into three tasks and implemented in three universities in the Valparaíso region that train prospective mathematics teachers. Thirty pre-service teachers participated. Data collected included audio recordings, screen captures, and written productions. The analysis was conducted using the categories of the Mathematical Working Space and the paradigms of linear algebra. Results show that most students were able to transition from geometry to linear algebra, mainly within the first paradigm. The mathematical work was predominantly semiotic and instrumental, but a gradual increase in discursive work was observed all along the didactical situation. Various digital tools were used—mainly GeoGebra and MatrixCalculator, with some use of Wolfram Alpha and ChatGPT—serving to both find algebraic solutions and validate them geometrically. The findings suggest that a long sequence of several open tasks involving domain changes and diverse technologies can support the introduction of linear algebra in teacher education.
Building on prior work that identified students’ utilization schemes of short-answer questions (an interactive feature embedded in dynamic textbooks), we extend the analysis of students’ narrations about textbook viewing to further identify actions tied to specific goals students stated when they were using their textbooks. The data come from 366 students who were using one of three interactive university textbooks (in calculus, linear algebra, and abstract algebra). Using what we term “action paths,” we mapped sequences of student behaviors when using the textbooks for three goals: completing homework, preparing for examinations, and self-studying. We found variations in the sequences of actions depending on the stated goals, with preparing for examinations exhibiting more variety in the sequence of actions. We discuss implications of our findings for research on the study of dynamic textbook use.
This study focuses on students’ strategy flexibility when solving linear and quadratic equations in an introductory university mathematics course. In this setting, we relate results from a tri-phase flexibility test with course achievement. We find that strategy flexibility is related to course achievement for above average students. The results show statistically significant differences in flexibility between degree programs and high-school mathematics backgrounds. In addition, we compare the flexibility of the university students with that of high school students from an earlier dataset and find substantial differences in both strategy flexibility and equation solving accuracy.
The use of calculus in chemistry is much more than just for computation – it is an essential tool for the analysis, modelling and meaning-making of many chemical phenomena. Through discussions along three themes, this Commentary highlights various aspects of the interplay between chemistry and mathematics, as well as possible opportunities and synergies for a collaborative approach in undergraduate education to promote students’ deeper conceptual understanding of both disciplines.
This paper provides an overview of 154 measures (with some validity evidence) in use by RUME researchers over a twenty-year span (2000–2019). We share focal constructs, validity evidence, and reported usage of the measures. We assessed the strength of this validity evidence using the six categories from the AERA et al. (2014) standards: test content, internal structure, response process, relation to other variables, consequences of testing, and reliability. The most reported validity evidence was Cronbach’s Alpha, followed by factor analytic approaches and the expertise of the design team (or use of external experts). Among our identified instruments, only twelve addressed at least four categories of validity evidence. We advocate for more attention to validation using both quantitative and qualitative approaches to support claims for a measure’s intended use.
Proof validation is an essential mathematical practice in university mathematics. However, students are generally provided with limited opportunities to engage in proof validation during regular instruction, and research has consistently shown that they struggle with proof validation. This study explored the potential of ChatGPT to support students’ proof validation practices. Three university students who had similar experiences in proof validation and use of AI for mathematical learning, but differed in academic performance, participated in the study. Students individually validated a purported proof from number theory, subsequently consulted ChatGPT as needed, and then revised their initial validation accordingly. Student performance in proof validation was analyzed according to three rationality components (epistemic, teleological, and communicative) derived from Habermas’ construct of rationality. Findings indicated that all participants benefited from using ChatGPT, particularly in addressing the teleological rationality criteria, followed by the communicative and epistemic rationality criteria. However, differences emerged according to students’ academic performance levels: medium- and low-performing students demonstrated greater improvement, largely because they initially met fewer rationality criteria prior to consulting ChatGPT. All three students reported that ChatGPT was generally supportive during the proof validation processes and expressed a willingness to use it to assist future proof validation practices. These findings highlight ChatGPT’s capacity to serve as a teaching assistant in supporting students’ proof validation, and implications for using it to promote explicitly teaching proof validation are discussed.
Racialized gatekeeping in introductory mathematics courses is an international, systemic concern. However, research on instruction often focuses on classroom-level changes, which can provide limited guidance for the department-level reforms that are necessary to implement instructional change in coordinated, introductory courses. Furthermore, reform efforts can overlook historically dominant values that may be invisibly embedded within departmental structures and the discipline of mathematics, and can lead to incremental changes that fall short of providing equitable educational opportunities. To better understand how departmental values and dynamics may impede the advancement of equity-oriented reform in postsecondary mathematics departments, we present a case study of a mathematics department at a large U.S. university engaged in improving racial equity within its calculus program. Informed by critical whiteness studies, we take as given that whiteness is present throughout this equity-oriented initiative. We explore the nature of how whiteness is reproduced, and what role mathematics plays in the reproduction of whiteness. Findings exhibit how disciplinary values around rigor and quantification masked whiteness in the reform effort, and stymied the department’s stated equity-oriented goals. Implications are provided for equity-oriented change, both departmental and epistemological, in mathematics.
Abstract mathematical concepts underlie many procedures used by engineering students throughout their education. Understanding actual infinity is not only central to mastering advanced mathematical content but also crucial for developing the kind of abstract reasoning required in modern STEM education. This study examines the understanding of the concept of actual infinity among 269 mechanical engineering students. Three simple tasks related to cardinality, decimal representations, and infinite sums were used to assess students’ comprehension. Despite having completed a calculus course, students demonstrated persistent misconceptions, with accuracy rates ranging from 15.5
This summary provides commentary on the paper by White Brahmia and Thompson (this issue). We respond to issues raised by the paper and amplify several of them. In particular we highlight elements of practice in physics instruction that are not well aligned with what students encounter in their calculus courses and respond to the value, and limitations, of the proposed framework using the Fundamental Theory of Calculus as a unifying theme. Finally, we repeat the call for increased attention to interdisciplinary work.
The “rate of change” or “ratio of infinitesimal changes” meanings for derivatives are crucial to productively using calculus across STEM disciplines. While much research has examined the basic derivative concept itself, less work has examined how the derivative might be extended across a unit on derivatives in ways that are compatible with this rate/ratio meaning. In this paper, we propose a hypothetical learning trajectory (HLT) for coherently teaching the derivative extensions of the chain rule, implicit differentiation, and related rates under this quantitative paradigm. The HLT extends the covariational rate/ratio meaning for the basic derivative to a multivariational meaning in these three extension topics. We also present the results of a small-scale teaching experiment meant to test the plausibility of the HLT. Our results suggest the students developed the nested x→ y→ z relationship, and used it to construct the rate1 × rate2 = rate3 structure, which was often formulated through “ratios” of small changes. The results also imply that coherence was achieved in these lessons in that the students saw these three topics as related and connected under this same structure.
