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The five papers published by Hodgkin and Huxley in 1952 are seminal works in the field of physiology, earning their authors the Nobel Prize in 1963 and ushering in the era of membrane biophysics. The papers present a considerable challenge to the novice student, but this has been partly allayed by recent publications that have updated the reporting of current and voltage to reflect the modern convention and two books that describe the contents of the papers in detail. A disadvantage is that these guides contain hundreds of pages, requiring considerable time and energy on behalf of the reader. We present a concise guide to the Hodgkin and Huxley papers that includes only essential content, with the data presented in a linear and logical manner. We have color-coded figures for ease of understanding and included boxes that summarize key information for easy reference. It is our expectation that this article will act as an accessible introduction for students to the work of Hodgkin and Huxley and hopefully foster an appreciation for a fascinating story that repays in-depth study.

Computer-based testing is an expanding use of technology offering advantages to teachers and students. We studied Calculus II classes for science, technology, engineering, and mathematics majors using different testing modes. Three sections with 324 students employed: paper-and-pencil testing, computer-based testing, and both. Computer tests gave immediate feedback and allowed multiple submissions and pooling. Paper-and-pencil tests (PPTs) required work and explanation allowing inspection of high cognitive demand tasks. Each test mode used the strength of its method. Students were given the same lecture by the same instructor on the same day and the same homework assignments and due dates. The design is quasi-experimental, but students were not aware of the testing mode at registration. Two basic questions examined were as follows: (a) Do paper-and-pencil and computer-based tests (CBTs) measure knowledge and skill in science, technology, engineering, and mathematics Calculus II in a consistent manner? (b) How does the knowledge and skill gained by students in a fully computer-based Calculus II class compare to students in a class requiring pencil-and-paper tests and hence some paper-and-pencil work. These results indicate that CBTs are as consistent with PPTs as CBTs are with themselves. Results are also consistent with classes using PPTs having slightly better outcomes than fully computer-based classes using only computer assessments.

Collaborative learning is well-established as a method to improve student learning and retention in engineering classrooms. One problem with collaborative learning is difficulty maximizing engagement of all students during group activities. We tested a change in implementation of collaborative problem solving sessions (Workshops) in a first-year Calculus for Engineers course. Previously, students sat at tables and worked together in small groups to solve provided problems, with each student ending the class period with her or his own written solution. The innovation in this study had students solving the same problems in the same small groups, but working together on the chalkboards, with each group ending the class period with a mutually agreed upon solution. Data collection to assess the innovation included student surveys, TA feedback, and observation and video recording of Workshop sections. The data support this innovation as a way to increase student collaboration and engagement during collaborative problem solving.

Assessments are usually thought of as ways for instructors to get information from students. In this work, we flip this perspective and explore how assessments communicate information to students. Specifically, we consider how assessments may provide information about what faculty and/or researchers think it means to know and do physics, i.e., their epistemologies. Using data from students completing assessment questions during one-on-one think aloud interviews, we explore how assessment features did (or did not) impact student engagement with the assessment problems. We analyze video recordings and transcripts to infer the epistemological framings and resources students use while completing introductory-level physics problems. Students' framings tended to be fairly stable, but when shifts occurred, they were triggered by a shift in epistemological resource, which can be activated by assessment feature. This work extends existing work on epistemological framing into the realm of assessment and allows us to consider the effects of assessments on our students' understanding of physics teaching and learning.

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.

Online assessment in mathematics is becoming more prominent as mathematics and the internet become more compatible. This paper is based on a case study of a web-based calculus course in which a combination of paper and online assessment is used for assessing various activities. The paper first investigates what student assessment preferences are and shows that there is a leaning towards a preference for online assessment, yet students still show a preference for a combination of online and paper assessment for term tests. Secondly, the paper shows that standards are maintained when incorporating an equally weighted online component into the assessment model for the case under consideration. The investigations are conducted from a background of having been involved in teaching online mathematics courses for the last four years.

The idea of cutting out a curve while looking at the direction of cut of the scissors is used to explore concepts related to curves, tangents and derivatives, especially tangent line, concavity and points of inflection, curvature, and a curve as envelope of tangent lines. (Contains 6 figures.)

Homework gives students an opportunity to practice important college-level physics skills. A switch to Web-based homework alters the nature of feedback received, potentially changing the pedagogical benefit. Calculus- and algebra-based introductory physics students enrolled in large paired lecture sections at a public university completed homework of standard end-of-the-chapter exercises using either the Web or paper. Comparison of their performances on regular exams, conceptual exams, quizzes, laboratory, and homework showed no significant differences between groups; other measures were found to be strong predictors of performance. This indicates that the change in medium itself has limited effect on student learning. Ways in which Web-based homework could enable exercises with greater pedagogical value are discussed. (Contains 2 notes, 7 tables, and 4 figures.)

Mathematical modelling has great potential to motivate students towards studying mathematics. This article discusses several different approaches to integrating research work with a second-year undergraduate, mathematical modelling subject. I found sourcing papers from the areas of epidemiology and ecology to be a fruitful source area, particularly models involving only two or three coupled differential equations. These models were amenable to students as well as interesting and relevant to students because they came from real research papers. I will describe the use of scenarios and case studies in lectures, and group projects for assessment. The scenarios and case studies were published in a textbook that I wrote. Scenarios, case studies and projects provided an opportunity to expose students to some novel applications of differential equations. One example is developed here as a "Classroom Note": modelling the dynastic cycles in Chinese history.

In this paper, we present the analysis of the students' achievements in learning calculus in a dynamic software environment during the COVID-19 crisis. Two groups of students, the experimental and the control one, were monitored. Blended learning was applied to the students in the experimental group, with the help of "Microsoft Teams" and dynamic software "GeoGebra," in autumn 2020. All students in the control group learned in the classroom without using "GeoGebra" in 2019. The comparison between these two groups of first-year students, regarding their calculus test results, is described in this paper. It is interesting that the results of the experimental group were significantly better than the results of students in the control group, despite the fact that the students from the experimental group learned during the COVID-19 crisis.

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.

Computer-based testing is an expanding use of technology offering advantages to teachers and students. We studied Calculus II classes for science, technology, engineering, and mathematics majors using different testing modes. Three sections with 324 students employed: paper-and-pencil testing, computer-based testing, and both. Computer tests gave immediate feedback and allowed multiple submissions and pooling. Paper-and-pencil tests (PPTs) required work and explanation allowing inspection of high cognitive demand tasks. Each test mode used the strength of its method. Students were given the same lecture by the same instructor on the same day and the same homework assignments and due dates. The design is quasi-experimental, but students were not aware of the testing mode at registration. Two basic questions examined were as follows: (a) Do paper-and-pencil and computer-based tests (CBTs) measure knowledge and skill in science, technology, engineering, and mathematics Calculus II in a consistent manner? (b) How does the knowledge and skill gained by students in a fully computer-based Calculus II class compare to students in a class requiring pencil-and-paper tests and hence some paper-and-pencil work. These results indicate that CBTs are as consistent with PPTs as CBTs are with themselves. Results are also consistent with classes using PPTs having slightly better outcomes than fully computer-based classes using only computer assessments. [ABSTRACT FROM AUTHOR]

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.

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.

The goal of the paper is to pay attention to some important techniques and approaches including adequate designations as a tool for unambiguous understanding and a key to success in solving problems, vivid visual images as a mnemonic techniques, and special formulas as a universal tool for solving typical problems, when teaching medical students of mathematics. The motivation for this paper is to help non-mathematics students understand complicated mathematical topics in an easy, natural, and simple way.

Literature has established that some learners encountered difficulties solving first order ordinary differential equations (ODEs). The use of error analysis in teaching ODEs is believed to make essential contribution towards calculus knowledge development. This paper therefore focuses on analyzing pre-service teachers' (PSTs) errors and misconceptions apropos of first order ODEs. The paper analyzed the nature of errors made in a test which was written by PSTs on the above topic. The test comprised various types of first order differential equations such as ODEs with separable variables, exact ODEs, ODEs that needed integrating factors, linear ODEs, and homogeneous ODEs. The purpose was to investigate the challenges faced by PSTs in various types of ODEs and the nature of misconceptions that they had in each particular type. This is a qualitative study that involved 63 PSTs who wrote a test on ODEs after being taught the topic for two weeks. The authors marked the work in order to ascertain the misconceptions and errors exhibited by the participants in the test. The PSTs' performance in the test was analyzed using the SOLO taxonomy and the Newman's theory mistake analysis. The study established that the topic was rather difficult for PSTs due to various reasons that included, among others, knowledge gaps in integration rules, algebraic computations and, in rare cases, differentiation, as well as misapplication of the rules of natural logarithms. This research therefore recommends that mathematics teacher educators ought to rather focus on the concept of integration and basic algebra before introducing the topic on ODEs to teachers on training.

This study was conducted to identify mathematical errors and mistakes made by preservice elementary mathematics teachers in the calculus course. To that end, the document analysis method of qualitative research models was used in the research. The sample of the research included a total of 75 preservice teachers who were attending the Department of Elementary Mathematics Education in the Faculty of Education at a public university and taking the calculus I course during the fall term of 2016-2017 academic year. Accordingly, written documents including the participant preservice teachers' papers of interim exams, practice exams, and general exams constituted the data source. The exam papers were scanned with a scanner and transformed into the electronic environment. A content analysis was performed with the data by using MAXQDA 12 qualitative data analysis program, and "data coding" of data analysis techniques was utilized. It was concluded that the preservice teachers made procedural and conceptual errors and mistakes, mathematical errors and mistakes such as recalling generalizations incompletely or incorrectly. Some recommendations were made in light of these results.

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.