In this work we critically examine a mnemonic designed for the pedagogy of first-order ordinary differential equations. The particular mnemonic takes the form of the SHIELDS acronym. We perform a critical analysis on mnemonics, outlining some of their benefits and limitations from the literature. As a result, we propose a general mnemonic model that has the potential to move mnemonics from being viewed as a trick where learners parrot back information they do not understand towards a deeper, more conscious experience where learners are fostered to think beyond the mnemonic. The SHIELDS acronym is designed for learners and teachers to encode, store and recall the different types of first-order differential equations. We critically compare SHIELDS against our proposed mnemonic model. We see that the SHIELDS acronym is not a panacea that will magically solve differential equations. Rather it is designed to act as a trigger for students to initially classify the equations — with the onus on the student — to then solve these equations via appropriate mathematical techniques. To further engage in the debate on SHIELDS as a pedagogical tool, we shared the acronym with two types of audiences: those within the traditional university classroom through face-to-face lectures and a more global and diverse assemblage who accessed a short digital video online that was openly available. We examined their perspectives, reactions and comments associated with SHIELDS. After triangulation of data, we discovered that the vast majority of feedback received was positive. Feedback themes included perceptions of SHIELDS as a tool to counter anxiety and boost engagement and levels of interest, suggestions for SHIELDS to act as an instrument of the student within examinations, the efficacy of SHIELDS as a mnemonic and identifying the boundaries of SHIELDS.
{"title":"On mnemonic instruction and the SHIELDS acronym in the pedagogy of first-order differential equations","authors":"Christopher C Tisdell","doi":"10.1093/teamat/hry001","DOIUrl":"https://doi.org/10.1093/teamat/hry001","url":null,"abstract":"In this work we critically examine a mnemonic designed for the pedagogy of first-order ordinary differential equations. The particular mnemonic takes the form of the SHIELDS acronym. We perform a critical analysis on mnemonics, outlining some of their benefits and limitations from the literature. As a result, we propose a general mnemonic model that has the potential to move mnemonics from being viewed as a trick where learners parrot back information they do not understand towards a deeper, more conscious experience where learners are fostered to think beyond the mnemonic. The SHIELDS acronym is designed for learners and teachers to encode, store and recall the different types of first-order differential equations. We critically compare SHIELDS against our proposed mnemonic model. We see that the SHIELDS acronym is not a panacea that will magically solve differential equations. Rather it is designed to act as a trigger for students to initially classify the equations — with the onus on the student — to then solve these equations via appropriate mathematical techniques. To further engage in the debate on SHIELDS as a pedagogical tool, we shared the acronym with two types of audiences: those within the traditional university classroom through face-to-face lectures and a more global and diverse assemblage who accessed a short digital video online that was openly available. We examined their perspectives, reactions and comments associated with SHIELDS. After triangulation of data, we discovered that the vast majority of feedback received was positive. Feedback themes included perceptions of SHIELDS as a tool to counter anxiety and boost engagement and levels of interest, suggestions for SHIELDS to act as an instrument of the student within examinations, the efficacy of SHIELDS as a mnemonic and identifying the boundaries of SHIELDS.","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":"38 2","pages":"74-84"},"PeriodicalIF":0.8,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/teamat/hry001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50259652","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}
We report on an effort tomeasure the effect of replacing traditional lecture-based teachingin calculus with a flipped-classroom approach.We base the comparison between the twoteaching models on dat ...
{"title":"Improved engagement and learning in flipped-classroom calculus","authors":"M. Cronhjort, L. Filipsson, Maria Weurlander","doi":"10.1093/TEAMAT/HRX007","DOIUrl":"https://doi.org/10.1093/TEAMAT/HRX007","url":null,"abstract":"We report on an effort tomeasure the effect of replacing traditional lecture-based teachingin calculus with a flipped-classroom approach.We base the comparison between the twoteaching models on dat ...","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":"37 1","pages":"113-121"},"PeriodicalIF":0.8,"publicationDate":"2018-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/TEAMAT/HRX007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46096651","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":"CATO--A Guided User Interface for Different CAS.","authors":"Hans-Dieter Janetzko","doi":"10.1093/TEAMAT/HRW023","DOIUrl":"https://doi.org/10.1093/TEAMAT/HRW023","url":null,"abstract":"","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":"36 1","pages":"232-239"},"PeriodicalIF":0.8,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/TEAMAT/HRW023","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48924829","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}
Mark Prendergast, Fiona Faulkner, C. Breen, M. Carr
This paper details an initial analysis of the transition of a second level curriculum reform to higher education in Ireland. The reform entitled ‘Project Maths’ involved changes to what second level students learn in mathematics, how they learn it and how they are assessed. Changes were rolled out nationally on a phased basis in September 2010. Students who were taught and assessed through the new curriculum first entered third level education in September 2012. It is important that third level mathematics lecturers are aware of the changes to the curriculum since certain topics such as vectors and matrices are no longer taught at second level. Hence third level courses may need to be adapted accordingly. This study investigates mathematics lecturers’ awareness of Project Maths and whether they have made any adaptions to their course content, teaching and assessment approaches as a result of the new curriculum being introduced. The findings, from a return rate of 23% of eligible respondents, show that although many lecturers are mindful of the concept of Project Maths, they are not aware of the changes in full and how it affects their own course content, teaching and assessment strategies. Accordingly, the gap between second and third level education remains. This study highlights that more needs to be done to ensure there is coherent and uniform approaches to the teaching, learning and assessment of mathematics in the transition from second to third level education.
{"title":"Mind the Gap: An Initial Analysis of the Transition of a Second level Curriculum Reform to Higher Education","authors":"Mark Prendergast, Fiona Faulkner, C. Breen, M. Carr","doi":"10.1093/TEAMAT/HRW024","DOIUrl":"https://doi.org/10.1093/TEAMAT/HRW024","url":null,"abstract":"This paper details an initial analysis of the transition of a second level curriculum reform to higher education in Ireland. The reform entitled ‘Project Maths’ involved changes to what second level students learn in mathematics, how they learn it and how they are assessed. Changes were rolled out nationally on a phased basis in September 2010. Students who were taught and assessed through the new curriculum first entered third level education in September 2012. It is important that third level mathematics lecturers are aware of the changes to the curriculum since certain topics such as vectors and matrices are no longer taught at second level. Hence third level courses may need to be adapted accordingly. This study investigates mathematics lecturers’ awareness of Project Maths and whether they have made any adaptions to their course content, teaching and assessment approaches as a result of the new curriculum being introduced. The findings, from a return rate of 23% of eligible respondents, show that although many lecturers are mindful of the concept of Project Maths, they are not aware of the changes in full and how it affects their own course content, teaching and assessment strategies. Accordingly, the gap between second and third level education remains. This study highlights that more needs to be done to ensure there is coherent and uniform approaches to the teaching, learning and assessment of mathematics in the transition from second to third level education.","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":"36 1","pages":"217-231"},"PeriodicalIF":0.8,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/TEAMAT/HRW024","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47541831","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}
Marinos Anastasakis;Carol L. Robinson;Stephen Lerman
Currently, there is a dearth of studies exploring the kind of tools that undergraduates use when studying mathematics with previous approaches focusing mostly on digital or institutionally provided resources. In this article, we present and discuss the results from our investigations into the different types of tools that a cohort of second year engineering undergraduates uses. Our survey (N = 201) showed that although to some extent students use resources external to their university, their practices are dominated by tools that their institution provides to them. Analysis of six follow-up interviews revealed that when students use the most popular resources they aim mostly for exam-related goals. This suggested that students in our sample choose to use certain tools because these enable them to pursue their exam-driven goals.
{"title":"Links between students’ goals and their choice of educational resources in undergraduate mathematics","authors":"Marinos Anastasakis;Carol L. Robinson;Stephen Lerman","doi":"10.1093/teamat/hrx003","DOIUrl":"https://doi.org/10.1093/teamat/hrx003","url":null,"abstract":"Currently, there is a dearth of studies exploring the kind of tools that undergraduates use when studying mathematics with previous approaches focusing mostly on digital or institutionally provided resources. In this article, we present and discuss the results from our investigations into the different types of tools that a cohort of second year engineering undergraduates uses. Our survey (N = 201) showed that although to some extent students use resources external to their university, their practices are dominated by tools that their institution provides to them. Analysis of six follow-up interviews revealed that when students use the most popular resources they aim mostly for exam-related goals. This suggested that students in our sample choose to use certain tools because these enable them to pursue their exam-driven goals.","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":"36 2","pages":"67-80"},"PeriodicalIF":0.8,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/teamat/hrx003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50265445","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}
Researchers and teachers often recommend motivating exercises and use of mathematics or statistics software for the teaching of basic courses in probability and statistics. Our courses are given to large groups of engineering students at Lund Institute of Technology. We found that the mere existence of real-life data and technology in a course does not automatically increase students’ motivation or enhance their learning. Careful integration and alignment between learning activities and assessment are also needed to obtain enhanced learning. Over a number of several years we have developed a learning environment for these basic courses where engaging applied problems and technology are used in an aligned way. We give several examples of the application of the alignment concepts in teaching basic probability and statistics inference. One example describes how applications and real-life data sets, in a consulting setting, give a more positive attitude to the subject. Others concern digital resources, where online exercises and tests used in an aligned way or interactive scripts in Matlab to investigate the theory improve result in the final exam and are perceived as very helpful by the students.
{"title":"Applied problems and use of technology in an aligned way in basic courses in probability and statistics for engineering students–a way to enhance understanding and increase motivation","authors":"Lena Zetterqvist","doi":"10.1093/teamat/hrx004","DOIUrl":"10.1093/teamat/hrx004","url":null,"abstract":"Researchers and teachers often recommend motivating exercises and use of mathematics or statistics software for the teaching of basic courses in probability and statistics. Our courses are given to large groups of engineering students at Lund Institute of Technology. We found that the mere existence of real-life data and technology in a course does not automatically increase students’ motivation or enhance their learning. Careful integration and alignment between learning activities and assessment are also needed to obtain enhanced learning. Over a number of several years we have developed a learning environment for these basic courses where engaging applied problems and technology are used in an aligned way. We give several examples of the application of the alignment concepts in teaching basic probability and statistics inference. One example describes how applications and real-life data sets, in a consulting setting, give a more positive attitude to the subject. Others concern digital resources, where online exercises and tests used in an aligned way or interactive scripts in Matlab to investigate the theory improve result in the final exam and are perceived as very helpful by the students.","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":"36 2","pages":"108-122"},"PeriodicalIF":0.8,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/teamat/hrx004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48354943","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}
This paper is based on an on-going project for modernizing the basic education in mathematics for engineers at the Norwegian University of Science and Technology. One of the components in the project is using a computer-aided assessment system (Maple T.A.) for handling students’ weekly hand-ins. Successful completion of a certain number of problem sets in Maple T.A. is required for being admitted to the final examination. This also gives partial credit towards the final grade. In this paper, I will look at possible influence Maple T.A. may have on the ways students engage with the mathematical problems.
{"title":"Influence of computer-aided assessment on ways of working with mathematics","authors":"Frode Rønning","doi":"10.1093/teamat/hrx001","DOIUrl":"https://doi.org/10.1093/teamat/hrx001","url":null,"abstract":"This paper is based on an on-going project for modernizing the basic education in mathematics for engineers at the Norwegian University of Science and Technology. One of the components in the project is using a computer-aided assessment system (Maple T.A.) for handling students’ weekly hand-ins. Successful completion of a certain number of problem sets in Maple T.A. is required for being admitted to the final examination. This also gives partial credit towards the final grade. In this paper, I will look at possible influence Maple T.A. may have on the ways students engage with the mathematical problems.","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":"36 2","pages":"94-107"},"PeriodicalIF":0.8,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/teamat/hrx001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50259612","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}
So-called ‘Mathematics-anxiety’ can be a key inhibitor for some adult learners considering higher education. The Institute of Technology Tallaght (ITT) in Dublin hosts the ‘Centre of Expertise for Adult Numeracy/Mathematics Education’ mathematics research group which is a hub of EPISTEM, formerly known as the National Centre for Excellence in Mathematics and Science Teaching and Learning. Members of this group identified the key issues that informed the design of enabling-mathematics courses for adults returning to higher education; how to structure their re-engagement, and how to adapt the re-engagement process to practical time constraints. This article outlines how insights from the successful Primer Mathematics module for mature students taking the FLexible AccesS to Higher Education Higher Certificate in Electronic Engineering at ITT were used to identify the key elements of a model to shape such courses. How these key elements were used to design and implement a new module in preparatory mathematics for students entering all modules in ITT, as part of Certificate in Preparatory Study for Third Level, is explained. The positive impact of this new module on student learning is evaluated using student feedback. The article concludes with a discussion of the emergent strong, positive correlation between favourable progression rates for those who completed the module, compared with those who did not.
{"title":"Models of re-engaging adult learners with mathematics","authors":"Ciaran O’Sullivan;Paul Robinson;John Keogh;John O’Neill","doi":"10.1093/teamat/hrw027","DOIUrl":"https://doi.org/10.1093/teamat/hrw027","url":null,"abstract":"So-called ‘Mathematics-anxiety’ can be a key inhibitor for some adult learners considering higher education. The Institute of Technology Tallaght (ITT) in Dublin hosts the ‘Centre of Expertise for Adult Numeracy/Mathematics Education’ mathematics research group which is a hub of EPISTEM, formerly known as the National Centre for Excellence in Mathematics and Science Teaching and Learning. Members of this group identified the key issues that informed the design of enabling-mathematics courses for adults returning to higher education; how to structure their re-engagement, and how to adapt the re-engagement process to practical time constraints. This article outlines how insights from the successful Primer Mathematics module for mature students taking the FLexible AccesS to Higher Education Higher Certificate in Electronic Engineering at ITT were used to identify the key elements of a model to shape such courses. How these key elements were used to design and implement a new module in preparatory mathematics for students entering all modules in ITT, as part of Certificate in Preparatory Study for Third Level, is explained. The positive impact of this new module on student learning is evaluated using student feedback. The article concludes with a discussion of the emergent strong, positive correlation between favourable progression rates for those who completed the module, compared with those who did not.","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":"36 2","pages":"81-93"},"PeriodicalIF":0.8,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/teamat/hrw027","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50259613","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":"Special issue: Contributions from the SEFI Working Group on Mathematics Conference 2016","authors":"Daniela Velichová;Tommy Gustafsson","doi":"10.1093/teamat/hrx010","DOIUrl":"https://doi.org/10.1093/teamat/hrx010","url":null,"abstract":"","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":"36 2","pages":"65-66"},"PeriodicalIF":0.8,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/teamat/hrx010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50259614","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}
The goal of this article is to introduce the revised Conceptual Understanding Weighting System (CUWS) Greene & Shorter (2012, Trans. Dial.), a tool for categorizing and weighting a mathematical task in order to identify the level of conceptual understanding being assessed by the task. In addition, there is discussion of how this weighting system relates to other frameworks that are currently used to classify mathematical tasks and student reasoning. Examples are provided to demonstrate how the revised CUWS can be used to calculate a conceptual understanding score for a particular exam, to create mathematical tasks that assess understanding at different levels and to analyse student understanding within those different levels. This in turn allows instructors to identify concepts with which students are struggling and quickly adapt classroom activities to improve students’ understanding. The ability to use the CUWS to create mathematical tasks (not just categorize them) is one of the important goals of the CUWS and one that is not regularly emphasized when working with other frameworks to categorize mathematical tasks.
{"title":"Conceptual Understanding Weighting System: a targeted assessment tool","authors":"Mairead Greene;Paula Shorter","doi":"10.1093/teamat/hrw003","DOIUrl":"https://doi.org/10.1093/teamat/hrw003","url":null,"abstract":"The goal of this article is to introduce the revised Conceptual Understanding Weighting System (CUWS) Greene & Shorter (2012, Trans. Dial.), a tool for categorizing and weighting a mathematical task in order to identify the level of conceptual understanding being assessed by the task. In addition, there is discussion of how this weighting system relates to other frameworks that are currently used to classify mathematical tasks and student reasoning. Examples are provided to demonstrate how the revised CUWS can be used to calculate a conceptual understanding score for a particular exam, to create mathematical tasks that assess understanding at different levels and to analyse student understanding within those different levels. This in turn allows instructors to identify concepts with which students are struggling and quickly adapt classroom activities to improve students’ understanding. The ability to use the CUWS to create mathematical tasks (not just categorize them) is one of the important goals of the CUWS and one that is not regularly emphasized when working with other frameworks to categorize mathematical tasks.","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":"36 1","pages":"1-17"},"PeriodicalIF":0.8,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/teamat/hrw003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50274530","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}
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