Search

Simon Fraser University in Vancouver, Canada offers two introductory calculus courses designed for students enrolled in science and engineering programs. Students identified as needing additional support based on their admission grades take the version of the course where students meet weekly for four hours instead of three. A new approach for the fourth hour was introduced in Fall 2019 splitting the class into smaller sections of 20-40 students to work in small groups on vertical whiteboards under the supervision of the instructor, complemented by peer tutors. This format introduces an active learning element that improves student performance and attitude. The project was first implemented in Fall 2019, although interrupted due to COVID-19. After returning to the in-person classroom, the whiteboard seminar program was reinstated. In Fall 2022 and Spring 2023, survey feedback was collected from both students and peer tutors for a more holistic perspective on the program. In this study, the results of these surveys are analyzed, focussing on four areas of measurement: students' satisfaction, learning calculus concepts and problem-solving techniques, the contribution of the instructor, and the contribution of the peer tutors. Exploratory factor analysis revealed three latent variables underlying student survey responses. The survey results suggest that interactions with the instructor, peer tutors, and active learning activities all contributed to students' enjoyment of the course and their perceptions of their own learning. We compared students' final course grades from Fall 2018, where the fourth hour was devoted to lecture before the program was implemented, to Fall 2019 and 2022. The data shows a decrease in the percentage of students failing the course and an increase in those earning A's and B's since the program's implementation. Additionally, survey results show that replacing a typical lecture-style hour with a whiteboard seminar improved students' attitudes and perceptions of the course.

The contents of calculus, known for their complexity, present significant challenges for students, particularly in mastering multiple integrals and effectively visualizing related concepts. The transition to distance learning prompted by the COVID-19 pandemic has further complicated the learning process in multiple integrals. In line with this and considering the potential of computer-supported collaborative learning (CSCL), in this research we explore the impact of teaching in a CSCL environment on student achievements, focusing on students from the computer science study program. Through data analysis using ANOVA and Bonferroni post-hoc tests, it was found that students exposed to collaborative learning in GeoGebra environment demonstrated higher levels of theoretical and practical knowledge compared to peers who acquired knowledge without using GeoGebra. Additionally, this group of students achieved results comparable to those of students who attended traditional in-person teaching, showing noticeable improvements in solving complex tasks. Our findings show the effectiveness of CSCL approach in context of distance learning and highlight potential of collaborative environments enhanced with technology in facilitating student understanding and achievements in calculus education.

The article analyzes professional collaboration in a lesson study involving university mathematics professors, with the objective of contributing to professional development. The research was guided by the question, What principles promoted professional collaboration among university mathematics professors participating in a lesson study? The research focused on a lesson study involving university professors of mathematics and mathematics education over twelve weekly meetings of two hours each. The empirical material included the researchers' field notes and the transcripts of the recordings of the lesson study sessions. A qualitative analysis, based on content analysis, revealed that collaboration involved: "professional sharing" and "decision-making." The research shows that by fostering collaboration the lesson study mobilized different principles of teaching professionalism, such as knowledge that is basic to teaching, values and ways of conducting university teaching, a social and moral commitment to teaching, and teaching autonomy, contributing to the professional development of higher education professors.

Educators are faced with the sudden infiltration of AI, including artificially intelligent tools that generate content far more sophisticated than any prior technological advancement. In this study, the researchers investigated the use of ChatGPT (currently the most used generative AI tool) as a means of learning Calculus. The study examined student perceptions toward the use of ChatGPT in their learning of mathematical concepts in Calculus and their perceptions of the advantages and disadvantages of its use in the classroom. The participants in this study were 64 undergraduate students enrolled in a Calculus course. Participants completed a 5-point Likert-type questionnaire with two open-ended questions. The collected data were analyzed both quantitatively and qualitatively. Results of the study indicated that most students supported the use of ChatGPT in learning Calculus; they believed that ChatGPT had a positive impact on their learning, and even increased their classroom participation and engagement.

This paper aims to study the effects that asynchronous and synchronous instructional methods have on student perceptions and attitudes towards online education. We analyze both qualitative and quantitative survey responses from 496 students in three large (greater than 200 enrolled students) first-year calculus classes at the University of Guelph. The survey results show significant differences among the three online instructional formats in students' attitudes toward asking questions, the similarity of each instructional modality to face-to-face delivery, technology issues encountered, students' perceived ability to keep up with course content, how collaborative the course felt, and student preferences for mode of delivery. There were no significant differences reported in relation to time-management skills or communication with instructors. Qualitative analysis of student comments also revealed perceived strengths and weaknesses of each mode of instruction. The results are used to make suggestions for improving student experience in online, post-secondary mathematical teaching.

There is a plethora of textbooks to choose from. While most textbooks contain the same content, teachers need to understand the textbook they have chosen to support their students' learning, as textbooks can significantly impact student learning and teacher instruction. When searching for a way to understand the various textbooks, my research question became, How do calculus textbooks tell the story of definite integrals? My research is in response to the never-ending search for how teachers can select and use their resources. Also, this study contributes to understanding the relationships between stories and knowledge construction using Dietiker's (2015) narrative framework to analyze the written calculus curriculum. I wrote the specific stories of introducing the definite integral as told by five calculus textbooks. Analyzing how the stories developed disclosed the similarities and differences among the texts in teaching definite integrals. The analysis also revealed a variety of avenues to introduce and teach definite integrals, including a focus on the area under the curve, Riemann sums, or the Fundamental Theorem of Calculus, and some implications to the context of physics related to these avenues. These insights provide teachers with a deeper understanding of textbook lessons and the variances that can potentially alter student learning.

During the pandemic and lockdown situations of the last few years, we transitioned to online, physical, and hybrid education approaches at various times. We had difficulty teaching multiple calculus and analytical geometry units online. In light of the current situation, a graphical user interface (GUI) teaching tool has been developed to teach university students many plane curve modules through a flipped classroom. With a few inputs and a single click as output, these executable spreadsheets can assist physics and mathematics students in practising several principles of analytical geometry. Spreadsheets can readily draw planes in three-dimensional geometry due to their widespread use. Some online vector GUIs present the results; however, this GUI conveniently illustrates the plot. The well-designed user interface also helps create new questions with answer keys and gives access to the teachers of physics and mathematics, which can make online teaching very easy. The effectiveness of this GUI was checked through Kolmogorov-Smirnov and Shapiro-Wilk tests, and a good agreement has been found in the results.

This paper presents a modified approach to subject-matter didactics, in which the focus is not on the content itself, but on the students' view of the content. The introduction deals with an overview of subject-matter didactics and the notion of beliefs used in this paper. The main portion of the paper deals with presenting the concepts of a book and a seminar based on the student-centered subject-matter didactics approach. For the first qualitative evaluation, selected reflections of students are analyzed. Finally, initial findings are summarized and an outlook is provided.

Early identification of at-risk students for timely intervention is critical to prevent non-completion of study programs. This article proposes a flipped class framework to support the academic success of at-risk students in an undergraduate Calculus course. It comprises three main components of setting, conduct, and monitoring. A flipped support class was implemented as periodic sessions throughout the learning semester over six consecutive semesters for the selected total of 560 at-risk students. At-risk students who attended the flipped support class reported a higher passing percentage than those who did not, in each of the six semesters. A strong mathematics foundation contributed to the likelihood of passing the course; however, it can be further increased by attending more hours of the flipped support class.

It is evident that more qualitative research on Advanced Placement [AP] classrooms for students from marginalized backgrounds is needed to uncover how AP practices and pedagogies might better serve students who had been historically excluded from the AP Program. I asked my former calculus students (32 total - 22 participated) from my last three years of teaching to complete a survey that was developed with open and closed-ended questions to explore the student experience in their AP Calculus AB course. I aimed to investigate the specific practices and encouragements that are most impactful in helping students meet the demands of the AP Calculus curriculum. Using this AP program as a phenomenological study, my intention was to gain insights that could support educators in helping students meet the demands of a challenging course. I used a six-phase thematic data analysis to make sense of and create a report on the corpus of data collected. After conducting the thematic analysis of the coded excerpts, seven themes and a variety of subthemes emerged: Participants felt supported to meet the demands of the course; they reported that the extended class time provided the necessary duration to master the course concepts; they expressed value in the variety of In-Class Tasks; they found the Out-of-Class-Time Tasks challenging but helpful to their learning; they conveyed great benefit from the overall organization of the course; they expressed gains in transferable skills; and they described a variety of struggles or barriers to success in the course. The findings are presented in conjunction with representative excerpts that were extracted verbatim from the survey and indicate that an AP course should provide targeted support to be successful on the AP Exam, but it should also provide opportunities to develop deep understanding of concepts and procedures through a variety of modalities to positively impact students' future success in college. The results are discussed in connection to relevant mathematics education literature and to implications and future directions of related research.

Students are required to have the ability to implement mathematics in solving everyday life problems. A good solving process will produce an excellent solving ability. The existing problem-solving stages cannot be used in solving problems with the Higher Order Thinking Skills (HOTS) category. The aims of this research are 1) to know students' problem-solving process in solving HOTS category questions; 2) to design the stages of problem-solving that can be used to solve HOTS category questions. Twenty-four students who took the differential calculus course in mathematics education at one of Indonesia's private universities made up the study group. The solving process is divided into high, medium, and low. The descriptive and qualitative research method describes the Polya stages' problem-solving approach. Researchers used four sets of problem-solving by making nine indicators. Based on the analysis results, it can be concluded that several indicators of the problem-solving stages have not been appropriately implemented, including I4, I8, and I9. In addition, the steps at the Polya problem-solving stage of devising a plan and looking back must be improved. Based on the findings above, research recommendations design the stages of problem-solving consisting of 6 steps. Future research will develop steps of problem-solving with the characteristics of the HOTS category questions.

The present study falls into the efforts to improve practices for addressing errors produced by learners in various situations involving the calculation of integrals. We attempt to clarify as precisely as possible the types of errors that secondary school students produce when using integrals in algebraic and graphical frames. Based on the synthesis of several works dealing with errors specific to integral calculus, we have been able to outline a typology of possible errors that can be produced by students in secondary school. We determine some subcategories for the three known categories of errors: conceptual, procedural, and technical. After administering a test to a random sample of secondary school students and conducting a principal component analysis, we were able to deduce that in the algebraic frame, certain conceptual and procedural subcategories dominate, with a notable advance for errors due to failure to recognize the integrand function. In the graphical frame, errors related to technical subcategories represent a major source of the erroneous productions of the students tested.

Special tutorials both online and off-line were experimented in order to provide extra support for the senior pre-service mathematics teachers at an Australian regional university to improve their learning experience and achieve the best possible learning outcomes in an advanced mathematics course focusing on solving ordinary differential equations and applying mathematical modelling. Two types of special tutorials were offered to the students, the progressive tutorials on solving the same problem with different methods according to the learning progression and student's instant requests, and the targeted tutorials to address the common problems shared by many students in attempting questions in the formal assessments. The experiments on these special tutorials indicated that the targeted tutorials were immensely useful for the students to either expand the scientific knowledge related to a real-world scenario described by words so as to begin problem solving with correct setting-ups or streamline multiple mathematical processes together to solve a complicated real-world problem described in words. This approach motivated most students to achieve their best possible learning outcomes. The progressive tutorials were effective in addressing student's curiosity of solving the same problem by multiple techniques and hence improving student's mathematical thinking and problem-solving skills in general. This exploratory study also found that there were common problems with a lack of general science knowledge and retention of the previously learnt mathematical techniques among most students. There also existed a portion of students who showed no interest in engaging with learning regardless of how much extra learning support provided to them.

The objective of this research is to study if the use of the calculator decreases students' anxiety in the subject of differential and integral calculus. Specifically, the research is carried out with 30 engineering students. Auzmendi anxiety factor questions are used to measure anxiety. The study is carried out in two moments; in the first the calculator is not used and in the second if they use it. In the second case, the calculator is used in problems involving the application of derivatives and integrals. The results indicate that the students reduce their anxiety when they use the calculator. Thus, in seven of the nine questions, students who use the calculator obtain a higher mean, it helps to reduce anxiety. In general, t-student test indicates that the moment students use the calculator they have less anxiety, and it helps to reduce errors in the mathematical process. Teachers should consider the calculator as a resource that motivates the student and helps reduce anxiety. In addition, the use of the calculator helps to reduce mathematical errors in basic operations and in the application of derivatives and integrals.

The anthropological theory of didactics (ATD) provides a lens to view mathematics education by placing mathematical practices within socio-cultural and historical contexts. The significance of institutions, in what regards with educational establishments, societal structures, cultural norms, and historical contexts, influences the perception and practice of mathematics in both students and teachers. This perspective is relevant for understanding the complexities of teaching multivariate calculus. Traditional approaches to this subject have been lecture-based, but with the evolution of educational paradigms, active pedagogies have emerged as vital. These include among others inquiry-based learning, collaborative learning, the flipped classroom approach, technology-enhanced active learning, and project-based learning. The challenges students face in multivariate calculus range from conceptual complexities to visualization challenges and over-reliance on memorization. In this work, we propose a holistic approach for defining class activities in multivariate calculus with emphasis in integrating visualization techniques, real-world applications, collaborative activities, and feedback mechanisms. Accordingly, four class activities, grounded in the principles of ATD and tailored to address challenges in multivariate calculus, are proposed. These activities aim to foster significant understanding, relevance, and engagement in students.

This study investigates the challenges faced by second-year undergraduate engineering students in understanding Stokes' theorem in vector calculus, focusing on the misconceptions found in interconnected concepts that form its foundation. Stokes' theorem involves the application of line integrals, surface integrals, the curl of a vector field, and the flux of a vector field, which are essential for a thorough understanding of the theorem. This article reports on a study conducted to identify these misconceptions through the qualitative and quantitative analysis of the test papers from 47 students at the University of Cape Town who were studying vector calculus. The results reveal difficulties in grasping line integrals, curve parametrization, vectors, curl of a force field, and the projection factor in surface integrals. Our study concludes that proficiency in these underlying concepts is crucial for students to effectively understand Stokes' theorem, highlighting the need for targeted teaching approaches that address these known difficulties.

Contribution: This article identifies possible ruptures between the ways fundamental notions of exact differential and exact differential equations (EDEs) are employed in mathematics courses and professional engineering disciplines. Background: Engineering students often experience difficulties with mathematics courses which may even lead to dropout from engineering programs. Students also face problems applying acquired mathematics knowledge in professional courses. Research is needed to understand how fundamental mathematics concepts are used in advanced engineering courses. Research Questions: How are the notions of exact differential and EDEs used in mathematics and engineering courses? What potential learning difficulties originate from different institutional practices and how can they be addressed? Methodology: The anthropological theory of the didactic is employed to analyze how six different STEM disciplines approach fundamental concepts of exact differential and EDEs. Distinctions in praxeologies associated with different institutions reveal possible learning difficulties students face relating new knowledge in engineering disciplines to that acquired in mathematics courses. Findings: Student learning can be facilitated by bridging the way exact differentials are introduced in Calculus and Differential Equations. Student conceptual understanding can be facilitated through the cross-disciplinary collaboration between mathematicians and engineers in the development of new courses and study programs.

We collected data on students' self-assessment behavior from four sections of a Calculus II course. Students were asked to write their expected scores on each of the weekly in-class quizzes and problems in the exams, immediately after they completed them. They were then asked to justify their expectation in writing. One-on-one interviews were conducted with a purposefully selected sample of students. During the interviews, they were asked to explain their perceived reasons for their self-assessment behaviors. While the results from quantitative analysis seemed to partially reinforce the findings of existing research that low performers generally overestimate, high performers underestimate their performance, and those in-between performers were (almost) accurate predictors, results from qualitative analysis provided additional insights into their self-assessment behaviors. After analyzing qualitative data, we identified five categories of student behavior: knowing about knowing (KK), not knowing about knowing (NKK), knowing about not knowing (KNK), knowing something is not known but not sure what (KBNKW), and not knowing about not knowing (NKNK). The quantitative analysis showed that students exhibited greater accuracy in assessing their performance when they belonged to the categories KK, KNK, and KBNKW, while their accuracy was lower when they fell into the categories NKNK and NKK. In other words, students who had greater awareness of their level of knowledge were more accurate in predicting their scores compared to their peers, irrespective of their actual performance levels. The logistic regression model revealed a substantial increase in the likelihood of underperforming students overestimating their performance compared to their high-performing counterparts.

Nowadays, mathematics teachers in K-12 strive to promote their students' mathematical knowledge and computational thinking (CT) skills. There is an increasing need for effective CT-embedded mathematics learning material and a better understanding of students' views toward them. In this work, we present the results of a research study, which included the design of a six-lesson learning activity aimed at fostering 16- to 17-year-old secondary students' CT skills in calculus lessons using the dynamic mathematics software "GeoGebra." Our goal was to investigate how students experienced the CT-embedded calculus lessons with "GeoGebra" and what challenges they faced during their interaction with the learning material and software. We collected and analyzed data from students' code in "GeoGebra," workbooks, semi-structured interviews, and questionnaires. Our findings suggest that most students mastered using CT concepts in calculus activities to a satisfactory degree and could reason about their computational solutions using "GeoGebra" and the generated graphs. Students' understanding of the mathematical content knowledge introduced was essential to complete the lesson series successfully and unnoticed gaps in prior knowledge emerged. Our study shows that students appreciate the CT-embedded calculus lessons and "GeoGebra's" exploratory approach to mathematics problems when provided with appropriate support. We conclude that an integrated approach to mathematics education and CT is viable and can contribute not only to fostering CT but also to increasing interest in mathematics.

It is widely accepted that the classroom teaching practices of in-service secondary mathematics teachers can benefit from the advanced mathematics they learned during their university education. However, pinpointing explicit instances of how this advanced knowledge informs teaching practices has proven challenging for both teachers and mathematicians alike. This paper aims to address this challenge by examining three statements related to Calculus and demonstrating how advanced mathematical knowledge can be utilized to generate counterexamples. By using results encountered in university-level courses, we construct functions that counter these statements. In conclusion, we identify a thought process that can be generally followed in assessing whether a given statement is true, highlighting the explicit connections of advanced mathematics in generating counterexamples via these statements. Similar endeavors of carefully generating counterexamples using advanced mathematics in other areas of mathematics can contribute to the designing of courses for teacher development that aim at explicitly highlighting connections between university mathematics and school mathematics.

Physical experiments in classrooms have many benefits for student learning, including increased student interest, participation and knowledge retention. While experiments are common in engineering and physics classes, they are seldom used in first-year calculus, where the focus is on solving problems analytically and, occasionally, numerically. In this paper, we detail a three-pronged lesson introducing differential equations using analytical, numerical and experimental approaches in a large first-year differential calculus course. Presenting the three approaches in succession allows students to evaluate advantages and disadvantages. The lesson incorporates software and programming and provides opportunities for active, experiential, team-based learning.

Calculus continues to be an important topic of discussion among mathematics education researchers given how it often acts as a gatekeeper for students in STEM. In their extensive 2017 review of calculus literature, Larsen and colleagues identified two main areas of applied research that had largely been neglected: research related (1) to efforts to improve calculus teaching and learning and (2) to equity and social justice. In this review we investigate how scholars have answered this call by reviewing recent literature related to these two themes. First we identified some promising intervention studies that investigated changes at the course level (e.g., calculus courses intended for engineering students) and at the level of specific calculus topics (e.g. using digital tools to help students understand the Fundamental Theorem). Second, we identified several studies on diversity, equity, and inclusion. We found that some studies in this collection still approached this research through traditional methods (e.g., so called achievement gaps) but we also identified promising new directions for research in which scholars utilize critical theories and provide counter-narratives that highlight the strengths of calculus students from historically marginalized groups. We conclude our review by discussing future directions we hope to see in the field that we argue will strengthen current work.

The concept of intellectual need, which proposes that learning is the result of students wrestling with a problem that is unsolvable by their current knowledge, has been used in instructional design for many years. However, prior research has not described a way to empirically determine whether, and to what extent, students experience intellectual need. In this paper, we present a methodology for identifying students' intellectual need and report the results of a study that investigated students' reactions to intellectual need-provoking tasks in first-semester calculus classes. We also explore the relationship between intellectual need, affective need, and students' learning. Although the overall percentage of students who reported experiencing an intellectual need was low, we found positive associations between intellectual need and learning.

Inquiry-Based learning (IBL) is a fairly well-known term in the United States (US) describing a range of student-centered or active pedagogical approaches in mathematics. However, the 'big tent' definitions of IBL mean that there is much variation in IBL users' instructional practices, variation which we set out to codify. Cluster analysis of self-reported data from a survey of postsecondary calculus instructors across the US reveals three instructional profiles among self-declared users of IBL: (a) heavy users of group work; (b) users of a variety of mixed approaches; and (c) heavy users of didactic lecture. The instructional profile of this third group is indistinguishable from that of calculus instructors who report never having heard of IBL. We further investigate the relationship of these instructional profiles to certain beliefs about teaching and learning. All groups agree that inquiry supports learning; the groups who spend a minority of time in didactic lecture disagree with statements that there are benefits to lecturing. Implications for research and adoption of IBL are discussed.

The recent push toward active learning -- engaging students in the learning process -- is meant to benefit students. Yet there is still much to learn about students' perceptions of this phenomenon. We share results from an interview study of students' perceptions of features of two active learning models institutionalized at a large doctoral-granting university -- a model for teaching foundational mathematics courses and an early field experience model for teaching preservice secondary students to teach mathematics. These models were implemented simultaneously in a single precalculus course. Interviews were conducted with both student populations (i.e., precalculus students and preservice teachers) to understand which in-class features of the models students noticed and identified as beneficial to their learning. Precalculus students identified specific opportunities related to active learning in the undergraduate mathematics teaching model -- working in groups on mathematics tasks that engaged students in sensemaking and interacting with their instructor around mathematics. Preservice teachers identified specific opportunities related to three features of the university field experience model -- observing a mathematics instructor enacting ambitious instructional practices, planning and teaching a "real" lesson, and observing student thinking and practicing teaching moves during groupwork. We conclude with pedagogical recommendations about particular features of the models that may help mitigate student resistance to active learning.

In Thai mathematics classrooms, there is a lack of attention to support students' mathematical problem-solving skills by working from real-world contexts that make sense to students. This study aimed to investigate how preservice mathematics teachers' problem-solving can be explored in their content course, intervening with technology and Realistic Mathematics Education (RME) through the Mean Value Theorem (MVT) lesson. The study included nine pre-service mathematics teachers purposively selected from a public university in Thailand who attended a Calculus course. Data was collected from classroom artifacts, observation notes, and interviews. It was found in this study that the intervention of technology and RME in pre-service mathematics teachers' content courses has the potential to build pre-service mathematics teachers' problem-solving abilities. It was also discussed that the intervention could use RME to conceptualize mathematics theorem and cultivate Polya's problem-solving steps. The findings provide light on the efficacy of using technology and RME in enhancing problem-solving skills among other content courses and could be used to inform the creation of mathematics curricula and instructional strategies in undergraduate content courses for mathematics education programs.

This is an exploratory study considering instructional methods in mathematics education. The study examined misconceptions arising from selected assignments undertaken by sixty-five students. The students were attending a calculus mathematical course unit on limits of functions, in different academic areas offered at selected tertiary institutions in Western Uganda. In this study, forms of expressing ideas on limits of functions are considered as far as they facilitate access to underlying mathematical principles. We explored the use and application of tools for representing mathematical ideas to enhance students' conceptual understanding and problem solving skills. A small-scale pilot of interventions using readily available GeoGebra dynamical computer software was applied. The diagnostic assignment on limits of functions was used for data collection. The main objective was to examine whether or not students preferred the application of multiple representations to the analytic approach. The semi-structured interview protocol was also conducted to probe further and correlate students' responses and their problem-solving abilities. The results showed that multiple representations-based instructions significantly changed students' understanding of the limits of functions. The results promise better access and understanding of more abstract mathematical concepts and may support students' problem-solving abilities. The semi-structured interviews conducted indicated that multiple representations supported and, therefore, enhanced students' understanding and solving of the limits of functions. This study highly recommends that mathematics educators should adapt multiple representations-based instructions to enhance students' critical thinking, and problem-solving.

Using the tools of praxeological analysis and didactical transposition analysis, the treatments of the Fundamental Theorem of Calculus in one Norwegian, Grade 13 textbook is analysed, with a particular focus on the development of the logos block of the FTC. The terms structure, functioning and utility, first introduced by Chevallard in 2022, is further to describe different dimensions of the mathematical object at stake. Through the analysis, a lack in the logos relating to the concept of integrability is identified in the textbook, and consequences of this is explored in relation to a set of tasks found in the book.

We report a characterization of the reflections made by high school mathematics teachers when discussing their resolutions of three numerical tasks related to the concept of function understood as a mathematical procept. Thematic analysis was used to code and categorize the reflections. Six teachers with at least 5 years of professional experience and an average age of 30 years participated. The results show that the teachers reflected on their actions in the tasks; their content knowledge; their teaching practice; and their professional training. As a result of these reflections, they recognized the need to improve their knowledge and practice of teaching functions in a more proceptual sense. The results contribute to deepen in how to make teachers reflect and incorporate a much more structural vision of functions in their teaching practices and, at the same time, incorporate the importance and role of their professional training in their reflections.

A recent review of literature in topic of rate of change mostly highlighted students and teachers' difficulties in handling conceptual questions. This initiated the researcher to investigate the Malaysian pre-service mathematics teacher's subject matter knowledge in rate of change through a case study design. A task-based interview was conducted with Chong, a pre-service mathematics teacher that majored in mathematics. There are 6 tasks given, comprising topics of rate of change and derivatives. The result shows that Chong has difficulties dealing with derivatives caused by his lacking knowledge in rate of change conception. His interpretation of rate of change under multiple contexts were also inaccurate, even for motion graphical context. His weakness in graphical representation also reflected his insufficient knowledge in rate of change as he was unable to convey the first principle accurately. The overall result suggested that Chong's subject matter knowledge of rate of change is insufficient, especially in meaning of rate of change, describing the occurrence of rate of change, and distinguishing between rate of change and instantaneous rate of change.

This study examines the perceptions of first-year, mathematics graduate teaching assistants (GTAs) participating in a six-week summer course designed to deepen their knowledge of collegiate mathematics teaching practices before being assigned to instruct undergraduate students in the subsequent fall semester. Through hybrid deductive-inductive thematic analysis, GTAs' written reflections were analyzed and matched against changes in their self-efficacy assessed along two conceptual dimensions: self-improvement and stimulation of student learning. Results suggest that GTAs' conceptualization of teaching practice informs changes in self-efficacy along both conceptual dimensions, with efficacy in self-improvement changing more than that of their ability to stimulate student learning. Therefore, the strength of the efficacy changes may be moderated by features of the professional development course. We explore defining features of the professional development course employed in this study, describing implications for the education and development of novice collegiate mathematics instructors, and the potential to optimize change along both conceptual dimensions of self-efficacy.

This paper presents a case study conducted by two universities seeking to explore Open Education Resources (OER) in their precalculus course. Students not only gained access to their textbook for free on the first day of class, but also Lumen OHM, an online mathematics assessment platform. The majority of students involved in the study were satisfied with the textbook, characterizing it as useful, accessible, and affordable, and agreeing that they would be very likely to register for a future course with an online text like that used in the course. This study also found that when comparing OER to non-OER settings using the final exam scores, for the most part, there was no significant difference observed in the learning outcomes. Based on these findings, the researchers believe that there is no need to continue teaching with costly physical textbooks when students can achieve the same academic level results with OER.

Over the last decade, there has been a national effort to prepare a more diverse student population for success in science, technology, engineering, and mathematics (STEM) careers. While there are a variety of best teaching practices in undergraduate STEM education, student engagement has been widely used to improve student learning. Specifically, it is important for students to have opportunities to practice or apply the theory they have learned in courses to realworld problems. Implementing active learning is one model that can be used to achieve this objective. However, due to the COVID-19 pandemic, professors have had to offer courses in online formats and determine how to continue to implement effective teaching strategies during this unprecedented time. Professors are anticipating the possibility of the need to continue offering courses in fully online and hybrid formats even in the future. With an emphasis on a STEM gateway course sequence, the aim of this paper is to highlight pre-COVID-19 research studies in online STEM education at Historically Black Colleges and Universities (HBCUs), explore how COVID-19 impacted students, teaching, and learning in undergraduate calculus courses, and outline opportunities for continuing to implement teaching techniques that work effectively for in-person and online calculus instruction.

Standards-based grading is an alternative grading system that gives students multiple opportunities to demonstrate proficiency in course standards or objectives. It has the potential to provide flexibility and reduce student anxiety, which are particularly important during difficult times such as the COVID-19 pandemic. In this study, students in a Precalculus course using standards-based grading between Spring 2021 and Spring 2022 were surveyed about how the pandemic had affected their learning and the benefits and challenges of standards-based grading. The students who took the course in Spring 2021, whose previous learning was least affected by the pandemic, were most likely to find standards-based grading beneficial. However, in all three semesters, students on average reported that compared with exams in previous math courses, assessments in the standards-based graded course were lower stakes and would better help them understand the material and to know what to study.

Mathematics plays a vital role in education; however, Filipino students encounter difficulties, evident in their performance on both national and international assessments. However, current literature suggests that leveraging technology, such as calculators and AI, in mathematics education can enhance students' proficiency, particularly in the pre-calculus domain. Thus, this research study examined the effectiveness of Photomath in enhancing Filipino students' self-reliance, interest, and performance in pre-calculus. A quasi-experimental one-group pretest-posttest design was used, with 49 Grade 11 students from NEUST Laboratory High School participating. Descriptive and inferential statistical analyses were performed to examine the pre-and post-intervention data. Results revealed a significant increase in students' self-reliance, interest, and performance in pre-calculus following the use of Photomath. The mean scores on all three measures increased significantly from the pretest to the posttest. The results of the t-tests further confirmed the effectiveness of Photomath in enhancing students' self-reliance, interest, and performance in pre-calculus. These results suggest that Photomath can be a valuable tool for educators to incorporate into pre-calculus teaching to enhance students' confidence and interest, foster intrinsic motivation, and ultimately improve their performance.

The outbreak of COVID-19 and the consequent lockdown took the world by surprise, forcing the trimming of the school curriculum and institutions of higher learning to urgently adopt online platforms to continue teaching and learning. This article is a case study that explores the use of a social media platform, WhatsApp, to improve peer learning and peer support in a South African context. The study sought to explore the nature of peer-to-peer learning that took place, and how peer support contributed positively to a student's conceptual understanding of the mathematics concepts. It involved a cohort of 180 entry-level students of an undergraduate engineering mathematics course. Guided by social constructivism, we analysed the data qualitatively using grounded theory. The results suggest that WhatsApp served as a viable platform for communication and peer-to-peer learning of engineering mathematics concepts. The study also found that the WhatsApp postings peer support alleviated the negative effects of remote learning. Some implications of the study as well as recommendations are suggested.

Articulated pedagogical and didactic strategies allow improving academic quality and strengthen individual and institutional cognitive processes such as student retention. Mathematical modelling, articulated with complementary actions, contributes to teaching and learning processes within situations that go beyond the classroom since they are permeated by social, political, and cultural environments that globally define pedagogical work. Therefore, the objective of this article was to visualize the progress of the implementation of the mathematical modelling of integrated Science-Technology-Engineering-Mathematics (STEM) education and classroom teaching to improve the academic quality of Linear Algebra and Vector Calculus courses at Universidad de Bogotá Jorge Tadeo Lozano (Utadeo). Through a mixed methodology composed of three phases one qualitative, one implementation, and one quantitative the theory and practice of academic work were guided, and the strategy, classroom implementation, and work materials were designed. The results obtained, in terms of appreciation of the work in the classroom and publications, evidence the articulation of various efforts and institutional, disciplinary, personal, technological, and theoretical resources in the context of changing realities.

Various studies have concluded that many students have difficulty understanding concepts related to function derivatives. One of the concepts in function derivatives is higher-order derivatives, primarily the nth derivative pattern. This study aims to: (1) identify various types of learning obstacles experienced by prospective mathematics teacher students on high-level derivative topics; (2) design alternative didactic designs based on the findings of learning obstacles; and (3) identify the effectiveness of the didactic design developed for the learning obstacle solution. This study used a qualitative method with a Didactic Design Research (DDR). Involves 41 students who have received Differential Calculus courses and 43 second-semester students at one of the universities in Indonesia. Data from test results, interviews, and document studies were analyzed through identification, clarification, reduction, and verification techniques and presented narratively. The results showed that: (1) students were still experiencing epistemological, didactical, and ontological types of conceptual and instrumental learning obstacles; (2) the development of a didactic design based on learning obstacle findings refers to the Theory of Didactical Situation combined with the Socratic Questioning technique; and (3) the alternative didactic design is quite effective in overcoming the initial learning obstacle, but a new obstacle appears, namely the ontological psychological type obstacle. The results of this study indicate that the phenomenon of learning obstacle findings and the developed didactic design can be used to improve the quality of learning on an ongoing basis.

Some researchers have questioned whether there is a causal connection between Advanced Placement (AP) STEM coursetaking and the choice of a STEM college major and a STEM occupation. Their research findings strongly suggest that if prior interest in STEM as well as other possible confounders are taken into account, the relationships of taking AP mathematics to expecting to be in a STEM major and/or a STEM occupation are small to nonexistent. Results from the current study, which uses data from a national longitudinal sample--actual measures of who majors in STEM--and prior interest in STEM along with other possible confounders, indicate that taking AP mathematics and AP science courses is significantly related to majoring in STEM and/or expecting to be in a STEM occupation at age 30--the sole exception being the taking of AP statistics.

The preparation gap [Salehi et al., Phys. Rev. Phys. Educ. Res. 15, 020114 (2019)] refers to gaps in students' prior knowledge that can negatively affect their learning as they engage in introductory physics courses. To better characterize the gap, the current study distinguished the impact of various prior knowledge components on learning gains. Measured components came from within the course domain (e.g., energy and force, angular kinematics) and outside it (e.g., algebra, vectors, calculus, and scientific reasoning). We conducted the study in two different institutional contexts: An algebra-based course offered at a Northeastern State University (NESU) and a calculus-based course offered at a Midwestern Private University (MWPU). Furthermore, we defined three levels of physics learning outcome measures with increasing difficulty. Multiple regression analysis was used to predict learning gains with the various prior knowledge components as predictor variables. The results indicate that greater prior knowledge from both within and outside the domain predicted higher learning gains and explained 30%-50% of the variance in outcome measures. Predictive, in-domain prior knowledge was the same for both groups--i.e., prior knowledge of energy and force, as measured by the Mechanics Baseline Test [Hestenes and Wells, Phys. Teach. 30, 159 (1992)]. Predictive, outside-domain prior knowledge differed between the groups. Better scientific reasoning was highly predictive of learning in the NESU (algebra-based) group but did not predict learning in the MWPU (calculus-based) group. Math prior knowledge predicted learning in both groups, although different topics within the math domain. These results suggest that measuring distinguishable components of prior knowledge will better characterize the preparation gap in ways that can be informative to educators. Specifically, measuring multiple, distinct types of prior knowledge can indicate "which types" are leading to a preparation gap for some students, putting them at a disadvantage for learning, whereas measuring a single type of prior knowledge or measuring prior knowledge too coarsely (without distinguishing among types) cannot provide sufficient diagnostic power.

This case study explored Grade 12 learners' misconceptions in limits of functions. The study aimed at understanding problems that learners face in learning the concepts of limit by analyzing misconceptions that learners have and identifying the possible sources of these misconceptions to take remedial action. An exploratory research design was adopted, and purposive sampling employed to select 35 Grade 12 learners (21 females and 14 males) who wrote a test on limits. From the 35 learners who wrote the test, five learners were interviewed guided by their written responses. Theoretically, the study was guided by the constructivism theory and used a systematic error conceptual framework to categories the learners` misconceptions according to extrinsic and intrinsic calculus misconceptions. The study sought to answer the questions: (a) What type of misconceptions do Grade 12 learners exhibit in responding to limit questions? (b) What are the possible sources of these misconceptions? The study found that Learners exhibited misconceptions on the limit concept and related symbolism. Learners who had a flawed understanding of algebraic concepts struggled to apply limits correctly. The weak foundation on algebraic skills impacted negatively on their learning of new concepts on limits. The study recommends that, educators should always check and make sure that learners have appropriate prior knowledge before the teaching of new concepts in calculus. It is recommended that that teaching and learning approaches need to be adjusted to give equal weight to both the procedural and the conceptual understanding of mathematical knowledge in learners.

An onto-semiotic analysis of the mathematical connections established by one in-service mathematics teachers and university students when solving a problem about launching a projectile using the derivative was carried out. Theoretically, this research was based on the articulation between the Extended Theory of Mathematical Connections and the Onto-semiotic Approach. The methodology was qualitative-descriptive where data was collected through interviews based on a task. Subsequently, following the joint analysis method of both theories, the mathematical activity of the participants when they solved the task was analyzed. The results show that, teacher and students established a system of connections of feature type, different representations, meanings, part-whole, procedural and implications in terms of practices, processes, objects, and semiotic functions that relate them. However, some students presented difficulties caused by some incorrect mathematical connection such as stating that the maximum height of the projectile is the time obtained with the critical number, errors in performing arithmetic calculations when evaluating the function, graphically representing the quadratic function as a straight line and use the general formula in an inappropriate way that prevents the procedural connection from being made.

This research explores the link between achievement emotions and covariational reasoning, a type of mathematical reasoning involving two variables. The study employs a case study approach, focusing on a high school calculus student named Valeria, and develops a theoretical framework based on the control-value theory and levels of covariational reasoning. The results reveal that Valeria's ability to coordinate variables and her perceived importance of solving mathematical problems influence her experience of achievement emotions, including enjoyment and frustration. The case study enables the creation of a hypothetical model that explains how students feel achievement emotions when tackling mathematical tasks that require covariational reasoning. The study highlights the significance of comprehending the interplay between emotions and mathematical reasoning to cultivate advanced cognitive and emotional abilities. This research is essential in bridging the gap between emotions and mathematical reasoning, a topic that has been previously overlooked in research on the affective domain.

We use the Survey of Thermodynamic Processes and First and Second Laws-Long, a research-based survey instrument with 78 items at the level of introductory physics, to investigate introductory and advanced students' difficulties with internal energy, work, and heat transfer. We present analysis of data from 12 different introductory and advanced physics classes at four different higher education public institutions in the U.S. in which the survey was administered in person to more than 1000 students. We find that not only introductory but also advanced physics students have many common difficulties with these introductory thermodynamic concepts after traditional lecture-based instruction in relevant concepts. We utilize a wide variety of problem types and contexts and our sample includes large numbers of introductory algebra-based, calculus-based, and advanced students. Some of our findings are consistent with prior research in this area, but others expand upon them and reveal previously unreported aspects of students' thinking. Findings related to common difficulties of students before and after traditional lecture-based instruction in college physics courses can help instructors of these courses plan instruction and curricula to improve student understanding. These findings can also be valuable for developing effective research-based curricula and pedagogies to address student difficulties and help students develop a functional understanding of these fundamental thermodynamic concepts.

In the field of physics education research, numerous studies have been dedicated to investigating the relationship between gender identity and physics learning. However, these studies have predominantly employed binary gender measurement methods, which may limit the range of research questions that can be explored and impede the discovery of crucial insights. In this study, we adapted gradational measures from prior research to investigate students' self-identified femininity, masculinity, and androgyny, as well as their reflected appraisal of femininity, masculinity, and androgyny (i.e., perceptions of how others perceive them) in both algebra-based and calculus-based introductory physics courses. The use of gradational measures revealed significant variation in students' self-identified femininity, masculinity, and androgyny within the binary categories of women and men, providing new insights into gender dynamics in physics. We found that self-identified women in the calculus-based courses, where they are underrepresented, tend to perceive themselves as more masculine and less feminine than how they believe others perceive them. Similarly, students of color are also more likely than White students to perceive themselves as more masculine than they believe others perceive them. Using structural equation modeling, we found that students' gender stigma consciousness plays an important role in mediating the effects of identifying as women and students of color on the observed discrepancies. Additionally, we found that women also exhibit a tendency to perceive themselves as more androgynous than they believe others perceive them in both algebra-based and calculus-based physics courses, and this phenomenon is also related to gender stigma consciousness. Moreover, our analyses revealed that students in the calculus-based courses tend to have a higher level of gender stigma consciousness even after controlling for gender and race. Our findings underscore the potential of gradational gender measurements in deepening our understanding of gender-related issues in physics education, shedding light on the complex interplay between students' gender identity, perceptions from others, and their educational experiences in the field.