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

{"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}

{"title":"Modelling labs for a course on differential equations","authors":"Ala' J. Alnaser, Justin Hoffmeier","doi":"10.1080/0020739x.2023.2251480","DOIUrl":"https://doi.org/10.1080/0020739x.2023.2251480","url":null,"abstract":"AbstractDifferential equations are widely used tools for modelling the world around us, making a course in differential equations a natural place for students to connect concrete mathematical applications to abstract concepts. Since students grasp the concepts better by applying them, introducing differential equations through modelling becomes essential. However, students often struggle to create and understand these models in the course. The purpose of this article is to share and examine a scaffolded sequence of labs implemented at regular intervals throughout a semester intended to train students in construction and analysis of mathematical models using differential equations. In these labs students learn basic principles of modelling that better prepare them for many of the high-quality modelling projects already available in differential equations. The course in which these activities were developed is taught using the SIMIODE text, Differential Equations: A Toolbox for Modelling the World, by Kurt Bryan. The labs are split into two components, with the first completed during class time and the second as out of class assignments. Furthermore, the course was re-organised to allow for the time needed to introduce the labs in class. We include classroom materials for labs, analysis, results on student perceptions and pedagogical recommendations.Keywords: Ordinary differential equationsmodellinglabspedagogyMathematics Subject Classification: 97M10 AcknowledgmentsWe would like to thank Alexandra Desaulniers for tirelessly working to support the creation and review of the labs. In addition, we thank Kurt Bryan for creating a textbook and many helpful resources to help us teach modelling in our differential equations course.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":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134911928","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 set theory analysis of the relationship between active learning and equitable and inclusive teaching","authors":"Amy Been Bennett, Karina Uhing, Molly Williams, Nancy Kress","doi":"10.1080/0020739x.2023.2255183","DOIUrl":"https://doi.org/10.1080/0020739x.2023.2255183","url":null,"abstract":"AbstractAlthough research on active learning suggests strong connections to equitable and inclusive teaching, few studies have explored the relationship between the two concepts. In this qualitative study, we examined how 13 participants in an equity-focused workshop described the relationship between active learning, and equitable and inclusive teaching. We used set theory as an analytic tool to categorise their descriptions. We found that all participants saw active learning, and equitable and inclusive teaching as related. The two relationships that were most strongly supported by participants’ statements were that active learning, and equitable and inclusive teaching are related yet independent sets, or that equitable and inclusive teaching is a subset of active learning. We discuss implications of these findings and pose questions for further exploration.KEYWORDS: Active learningequitable and inclusive teachingset theoryundergraduate mathematics AcknowledgmentsData were collected through an NSF-funded project (#1624639, #1624643 - SEMINAL). We would like to thank SEMINAL for designing the larger project from which these data were collected. All conclusions and opinions in this paper are from the authors and do not represent the opinions of the National Science Foundation.Disclosure statementNo potential conflict of interest was reported by the authors.Additional informationFundingThis work was supported by National Science Foundation [grant number 1624639, 1624643].","PeriodicalId":14026,"journal":{"name":"International Journal of Mathematical Education in Science and Technology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135783516","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":"Cooling the perfect cup with Laplace","authors":"R. C. Harwood","doi":"10.1080/0020739x.2023.2250337","DOIUrl":"https://doi.org/10.1080/0020739x.2023.2250337","url":null,"abstract":"AbstractAfter waiting in a long line for your favourite cup of coffee, you finally sit down with your mug and find that the coffee is still scalding hot! How long do you need to wait before you can enjoy it? Once it cools enough, how much time do you have to enjoy it? Are there ways to speed up the process? These questions motivate the presented modelling scenario about tracking the temperature of a cup of coffee as it cools. Students are put in the role of an inquisitive coffee enthusiast who does their due diligence in preventing burns and carefully experimenting on their coffee so that they, and others to come, can enjoy that perfect cup. They identify their assumptions and interventions before developing model differential equations for each case, which force discontinuities on the derivative and even on the solution itself. Being familiar with basic Laplace transforms and learning key properties of the unit step and unit impulse functions, they solve these differential equations and compare the interval of time when the coffee will be at its peak level of enjoyment. This paper includes an implementation guide, grading rubric, example solutions, and example assessment questions.Keywords: Newtonlaw of coolingLaplace transformmodellingtemperaturecoffeeMathematics Subject Classifications: 00-0101-0134-0134A25 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":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784187","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}