International Journal of Mathematical Education in Science and Technology最新文献

{"title":"Love stories in a differential equations classroom","authors":"Yanping Ma, Gail Tang","doi":"10.1080/0020739x.2023.2251016","DOIUrl":"https://doi.org/10.1080/0020739x.2023.2251016","url":null,"abstract":"AbstractWe believe that developing cultural competencies can help students learn mathematics and conversely that learning mathematical content can help students learn about themselves and others. Using frameworks introduced by Rendón (Citation2009) and Gutiérrez (Citation2018), we present a five-part bundle of activities for undergraduate differential equations course instructors, including one pre-activity reflection assignment, three modeling activities, and a final project. The Pre-Activity Assignment engages students to draw on their own personal and/or cultural experiences with the concept of love. Three activities focus on developing revision skills in mathematical modeling and practicing methods of solving systems of ordinary differential equations (ODEs). In these activities, students would collaborate to construct a relationship model consisting of a system of first-order linear ODEs and solve different model iterations with the characteristic polynomial, matrix form, or Laplace transform method. The final project connects the reflections in Pre-Activity Assignment with the skills developed in the three activities by inviting students to create a relationship scenario, model, revise, solve it, and present the conclusions. By engaging in this set of assignments, students connect personal and cultural experiences with the concept of love and perceive themselves and their peers in the curriculum, fostering a sense of belonging and relevance.Keywords: Mathematical modelingculturally relevant pedagogysystems of linear ordinary differential equationscharacteristic polynomial methodLaplace transform methodmatrix form Data availability statementDetailed solutions, sample MATLAB live scripts, and grading rubrics are available upon request. We purposefully do not include them here in case students attempt to search for solutions online.AcknowledgementsThe authors would like to express our gratitude to the guest editors and reviewers for providing valuable feedback and insightful suggestions. The authors also appreciate the organizations of SIMIODE EXPO 2022, where the idea of this project started.Disclosure statementNo potential conflict of interest was reported by the authors.Notes1 This section tests future SAT questions. These unscored sections look just like the scored sections of the SAT so students do not know which sections are scored or unscored.","PeriodicalId":14026,"journal":{"name":"International Journal of Mathematical Education in Science and Technology","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135815337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The supportive role of active learning in a calculus course on low precalculus proficiency students","authors":"Charity N. Watson, Pablo Duran, Adam Castillo, Edgar Fuller, Geoff Potvin, Laird Kramer","doi":"10.1080/0020739x.2023.2255189","DOIUrl":"https://doi.org/10.1080/0020739x.2023.2255189","url":null,"abstract":"AbstractCollege calculus plays an important role in STEM students’ degree and career aspirations. One of the key factors considered in assessing a student’s ability to be successful in calculus is their proficiency in topics from prior mathematics courses such as algebra and precalculus. This study set out to examine the impact of students’ precalculus proficiency on their achievement in introductory calculus based on their classroom environment. Results from the implementation of the Modeling Practices in Calculus (MPC) model, an innovative, active learning approach, are presented. Using a randomized-controlled trial research design, students were randomly assigned to MPC and traditional, lecture-based calculus sections. The Precalculus Concept Assessment inventory was administered to gauge students’ precalculus proficiency. We found that students exposed to the MPC model were more likely to be successful in their calculus course, even if they began with low precalculus proficiency. Also, students enrolled in the MPC sections saw significant growth in their precalculus proficiency from the beginning to the end of the semester. Additionally, we observed this model providing support for students in key demographics (low proficiency, female, first and second year undergraduates) in terms of the development of their proficiency that they may not receive in traditional classrooms.KEYWORDS: Active learningcalculusprecalculus proficiencycalculus achievementSubject classification code: 97D40 AcknowledgementsWe thank the instructors, students, and Learning Assistants involved in this study for their support during the administration and collection of data. We also thank our institution and the State of Florida for the initial support that made this project possible. Any opinions, findings, and conclusions in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.Disclosure statementNo potential conflict of interest was reported by the authors.Additional informationFundingThis work was supported by the US National Science Foundation [grant number: 1832450].","PeriodicalId":14026,"journal":{"name":"International Journal of Mathematical Education in Science and Technology","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136061584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Investigating how lower secondary school students reason about quadrilaterals emerging in dynamic constructions","authors":"Lukáš Vízek, Libuše Samková, Jon R. Star","doi":"10.1080/0020739x.2023.2255184","DOIUrl":"https://doi.org/10.1080/0020739x.2023.2255184","url":null,"abstract":"This contribution focuses on reasoning about quadrilaterals provided by lower secondary school students when working with pre-prepared dynamic constructions. On this topic, we present an exploratory empirical qualitative study carried out within a GeoGebra Classroom environment, and our diagnostic instrument consists of a set of dynamic constructions of quadrilaterals that are based just on a composition of lines and circles. The dynamic constructions consist of the same construction steps as with a straightedge and a compass on paper, without any relational or measurement information provided by the software. The hierarchy in the dynamic constructions is tied to properties of diagonals and takes on various levels and structure (one level, two consecutive levels, two parallel pairs of consecutive levels). For each of the constructions, the participants of the study reasoned which shapes could be found in the construction and why. Various levels of reasoning as well as various levels of students’ understanding of quadrilaterals appeared in data. The findings indicate that, at least for the lower secondary school students, the combination of dynamic manipulations and geometric constructions could form a significant space for scaffolding the identification of the features of quadrilaterals and the comprehension of the inclusive relations between them.","PeriodicalId":14026,"journal":{"name":"International Journal of Mathematical Education in Science and Technology","volume":"2016 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136314848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Model, analyse and prevent fatal aircraft manoeuvres","authors":"Yves Nievergelt","doi":"10.1080/0020739x.2023.2249473","DOIUrl":"https://doi.org/10.1080/0020739x.2023.2249473","url":null,"abstract":"AbstractOn 24 June 1994 at Fairchild Air Force Base, during practice for an air show, a low-flying B-52H aircraft banked its wings vertically and crashed. Emphasizing the activity of modeling drag and gravity, these notes examine the possibility of recovery with several models. First, with algebra, historical data lead to a model where in a free fall near Earth's surface, the distance fallen is proportional to time squared. Calculus then gives an ordinary differential equation to model free fall near Earth's surface. Second, with calculus, historical data lead to various models of drag on objects moving through air. Third, combining models of drag and gravity with Newton's Laws of Motion leads to ordinary differential equations that model free fall in air. Fourth, predictions from such models with ordinary differential equations are consistent with the aircraft crash. Further models examine the possibility of increasing engine thrust to regain the vertical component of lift. All models fit in a first course in differential equations, without requiring any computational machinery. However, numerical experiments show how uninformed use of professional software can produce rounding errors to cause the modelled aircraft to plunge to the bottom of the deepest oceans and shoot up into space.KEYWORDS: Accelerationderivativedraginitial conditionsintegralpositionvelocity AcknowledgmentsThis work was supported in part by a Professional Leave from Eastern Washington University.Disclosure statementNo potential conflict of interest was reported by the author.","PeriodicalId":14026,"journal":{"name":"International Journal of Mathematical Education in Science and Technology","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136314855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Topic-specific characteristics of proof-related reasoning","authors":"Andreas Bergwall","doi":"10.1080/0020739x.2023.2255190","DOIUrl":"https://doi.org/10.1080/0020739x.2023.2255190","url":null,"abstract":"Students’ difficulties with proofs are well documented. To remedy this, it is often recommended that reasoning and proving be focused on in all grades and content areas of school mathematics. However, proofs continue to have a marginal place in many classrooms, or are only given explicit attention in courses in Euclidean geometry. Geometry is also the most common topic for educational research on reasoning and proving. This paper compares what four other topics in secondary school mathematics – logarithms, primitive functions, definite integrals, and combinatorics – can offer in terms of opportunities to learn proof. The types and natures of reasoning in expository sections and students’ tasks in 11 Swedish and Finnish textbooks are analysed in search of similarities and differences between these topics. The results are accounted for with special focus on opportunities for reasoning about general cases. Finally, the findings are discussed in relation to mathematical aspects of the four analysed topics.","PeriodicalId":14026,"journal":{"name":"International Journal of Mathematical Education in Science and Technology","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136314463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A simpler elementary proof that <i>e</i> is irrational","authors":"F. M. S. Lima","doi":"10.1080/0020739x.2023.2255868","DOIUrl":"https://doi.org/10.1080/0020739x.2023.2255868","url":null,"abstract":"AbstractIn this short note I present an elementary proof of irrationality for the number e, the base of the natural logarithm. It is simpler than other known proofs as it does not use comparisons with geometric series, nor Beukers' integrals, and it does not assume that e is a rational number from the beginning.Keywords: Irrationality proofEuler's numberMaclaurin seriesalternating seriesMathematic Subject classifications: 41-0197-0111J72 AcknowledgmentsThe author thanks M. R. Javier for some hints on how to simplify and reduce his initial proof without losing the mathematical rigour. Thanks are also due to the anonymous reviewers for their suggestions and hints on additional references.Disclosure statementNo potential conflict of interest was reported by the author.Notes1 The above conclusion that 1/e is not an integer makes unnecessary to check the case q=1.","PeriodicalId":14026,"journal":{"name":"International Journal of Mathematical Education in Science and Technology","volume":"96 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136374223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Modifying an existing SIMIODE project to create an in-depth project requiring a written report","authors":"Forest Mannan","doi":"10.1080/0020739x.2023.2251503","DOIUrl":"https://doi.org/10.1080/0020739x.2023.2251503","url":null,"abstract":"AbstractThis article considers starting with an existing SIMIODE modeling scenario [Winkel, B. (2015). 1-031-CoolIt-ModelingScenario. SIMIODE (Version 2.0). QUBES Educational Resources. https://doi.org/10.25334/3WG8-EC31] that develops Newton's law of cooling by considering data on the cooling of a beaker of water in a room, and expanding upon it to create a longer project modeling the temperature of a building with an internal heat pump that is also subject to a sinusoidal varying outside temperature. The proposed project was undertaken early on in the course, since little background was required, and a longer project format was utilised so that students could return to the project and see direct applications of new concepts, such as autonomous vs. nonautonomous ordinary differential equations, slope fields, and Euler's method as they were introduced. The project prompt is provided as well as a detailed discussion of the motivation behind the questions. Finally, the pros and cons of requiring the students to submit a written lab report for the project are reflected upon and a sample rubric is provided.KEYWORDS: Ordinary differential equationsmodelingwritten reportsrubricscoolingMATHEMATICAL SUBJECT CLASSIFICATION: 97M10 AcknowledgmentsParts of this project were motivated by the SIMIODE modeling scenario COOL IT. The real-world data on the temperature of the beaker of water was also taken from this source.Disclosure statementNo potential conflict of interest was reported by the author.","PeriodicalId":14026,"journal":{"name":"International Journal of Mathematical Education in Science and Technology","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135059540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Literate programming for motivating and teaching neural network-based approaches to solve differential equations","authors":"Alonso Ogueda-Oliva, Padmanabhan Seshaiyer","doi":"10.1080/0020739x.2023.2249901","DOIUrl":"https://doi.org/10.1080/0020739x.2023.2249901","url":null,"abstract":"AbstractIn this paper, we introduce novel instructional approaches to engage students in using modelling with data to motivate and teach differential equations. Specifically, we introduce a pedagogical framework that will execute instructional modules to teach different solution techniques for differential equations through repositories and notebook environments during real-time instruction. Each of these teaching modules employs a literate programming approach that uses the notebook environment to explain the concepts in a natural language, such as English, interspersed with snippets of macros and traditional source code on a web browser. The pedagogical approach employed is reproducible and leads to openaccess material for students to motivate and teach differential equations efficiently. We will share examples of this framework applied to teaching advanced concepts such as machine learning and neural network approaches for solving ordinary and partial differential equations as well as estimating parameters in these equations for given datasets. More details of the work can be accessed from https://aoguedao.github.io/teaching-ml-diffeq.Keywords: Literate programmingdifferential equationsmachine learning AcknowledgmentsThe authors are also very grateful to the anonymous reviewers whose feedback was very useful.Disclosure statementNo potential conflict of interest was reported by the authors.Additional informationFundingThis work is partially supported by the National Science Foundation [grant numbers DMS-2031029 and DMS-2230117].","PeriodicalId":14026,"journal":{"name":"International Journal of Mathematical Education in Science and Technology","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135016248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An optimal control problem for resource utilisation by microorganisms","authors":"Glenn Ledder, Stefano Manzoni","doi":"10.1080/0020739x.2023.2254314","DOIUrl":"https://doi.org/10.1080/0020739x.2023.2254314","url":null,"abstract":"AbstractDecomposition of organic matter controls the flow of carbon and nutrients in terrestrial and aquatic ecosystems. Several kinetic laws have been proposed to describe decomposition rates, but they neglect adaptation of the microbial decomposer to environmental conditions. Here we formalise decomposition as an optimal control problem by assuming that microorganisms regulate the uptake rate of a substrate to maximise their growth over the period of decomposition. The result is an optimal control problem consisting of two differential equations and auxiliary conditions that determine the optimal value of the control variable (the uptake rate), the remaining substrate at any given time, and the optimal completion time. This problem serves as a case study to illustrate the solution of differential equations and optimal control problems for students in undergraduate courses. The mathematical analysis of the problem requires rewriting the differential equations in reverse time along with the solution of a nonhomogeneous linear first order differential equation. We then return to modelling with some biologically motivated questions about how the parameters of the model representing environmental conditions and microbial functional traits affect the outcome. Finally, we discuss alternative ways to use the material with students.Keywords: Optimal controlresource utilisationmicrobial decomposition Disclosure statementNo potential conflict of interest was reported by the authors.Notes1 We are using capital T for time to reserve the usual symbol t for the dimensionless version of the problem.2 In a different context, with G a function of X rather than U, this is the Monod function, which can be derived from first principles (Liu, Citation2007).3 See (Ledder, Citation2023), for example, for a derivation of this function from first principles (where it is presented as the Holling type 2 function for a consumer-resource system). Note that the function is approximately αU/β when U is small.4 With a change of sign from the usual statement, the Lagrange multiplier rule can be recast as a necessary condition that the vector u that maximises a function g(u) subject to a constraint f(u)=0 must maximise the combined function H(u)=g(u)+λf(u) for some constant λ.Additional informationFundingThis project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 101001608).","PeriodicalId":14026,"journal":{"name":"International Journal of Mathematical Education in Science and Technology","volume":"137 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135395238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Teaching differential equations through modelling: hot water heater","authors":"Viktoria Savatorova, Aleksei Talonov","doi":"10.1080/0020739x.2023.2249899","DOIUrl":"https://doi.org/10.1080/0020739x.2023.2249899","url":null,"abstract":"AbstractWe present an example of one of the modelling projects we assign to students in our differential equations classes. Students are asked to determine how to run a cost-efficient hot water heating system. We consider a cylindrical tank filled with water and heated by a heating element immersed in it. Together with students we discuss physical laws governing the process and make assumptions necessary to build a model. One of the goals of the modelling activity is to compute the time needed for water to reach the desired temperature given the power of the heating element and the size of the water tank. Several different scenarios can be discussed. Assuming the cost of heating the water is directly related to the time for which the power is ON, students can choose the most cost-efficient scenario. We provide the description of several modelling activities and obtained results.Keywords: Mathematical modellingordinary differential equationshot water heater Disclosure statementNo potential conflict of interest was reported by the authors.","PeriodicalId":14026,"journal":{"name":"International Journal of Mathematical Education in Science and Technology","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135395227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}