Teaching Mathematics and Its Applications最新文献

{"title":"Construction of the concept of pairability in comparing infinite sets in the individuals’ mind: an APOS approach","authors":"Mohadaseh Mahdiyan;Ali Barahmand","doi":"10.1093/teamat/hraa008","DOIUrl":"https://doi.org/10.1093/teamat/hraa008","url":null,"abstract":"This study focuses on the concept of pairability as a fundamental metaphorical concept in the Cantorian set theory regarding comparison of infinite sets. In this theory, pairability appears in a hierarchical manner of generating and developing as a one-to-one correspondence by arrows in finite, and then in infinite states, a bijection map through an explicit formula and the concept of cardinality. Adapting these hierarchical components of pairability and the APOS theory, about description of constructing and understanding a mathematical concept in a hierarchical manner, this study examines the construction of the concept of pairability in the individuals’ mind. In this way, their imaginations of pairability and practical performances in different situations will also be surveyed. In so doing, a total of 20 mathematics teachers and university lecturers holding at least an M.Sc. degree in mathematics were chosen. To collect the data, interviews were conducted in which the participants not only answered questions about the concept of the various types of pairability but also compared infinite countable sets in different situations. Our findings revealed that a bijection map via an explicit formula was the individuals’ dominant conception of pairability and most of the incorrect answers were related to unsuccessful attempts to recall a formula or method as the only possible way, and the encapsulated concept of cardinality was used less frequently in practice. Therefore, there was not a total schema of actions, processes and objects of the concept of pairability in the individuals’ mind.","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/teamat/hraa008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50346340","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":"Questioning corona—a study and research path","authors":"Mette Jensen;Carl Winsl⊘w","doi":"10.1093/teamat/hrab003","DOIUrl":"https://doi.org/10.1093/teamat/hrab003","url":null,"abstract":"We present a so-called ‘study and research path’, designed for advanced high school teaching in the context of a master's thesis project, along with an outline of the mathematical and didactic background. The aim of the paper is to present and discuss an adaptation of Chevallard's ‘paradigm of questioning the world’ to currently prevailing institutional conditions and in particular to show how mixed mathematics may be revived at a time of crisis, such as the coronavirus outbreak during which the specific design was developed.","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/teamat/hrab003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50346341","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":"Mathematics learning support centres in Germany—an overview","authors":"Mirko Schürmann;Lara Gildehaus;Michael Liebendörfer;Niclas Schaper;Rolf Biehler;Reinhard Hochmuth;Christiane Kuklinski;Elisa Lankeit","doi":"10.1093/teamat/hraa007","DOIUrl":"https://doi.org/10.1093/teamat/hraa007","url":null,"abstract":"Mathematics learning support centres (MLSC) are widely established and evaluated in English-speaking countries (such as the UK, Ireland and Australia). In most of these countries, several national surveys on MLSCs exist. They give an overview of the number of MLSCs as well as their characteristics in these countries. In Germany, there is a lack of studies on MLSCs and the landscape of MLSCs has not been described yet. This article presents basic information concerning counts of MLSCs and their characteristics at universities. Based on a three-step approach of analysing university homepages and additional personal contact via email or phone calls, we gathered typical MLSC features (e.g. staff quantities and qualification, opening and support hours, supported study programmes). We analysed 190 universities from a web-based register on study programmes. In total, we found 61 MLSCs located at 51 German universities. Another 16 support centres were specialized on mathematics didactics, which means they focussed on didactical and methodological support for preservice teacher students and often provided different teaching materials. Thirty-eight centres were located at universities (62.3%) and 23 MLSCs at universities of applied sciences and comparable universities (37.7%). The MLSCs were different in their sizes of staff and opening hours, and both the numbers of staff and the service hours differed greatly. The student groups MLSCs at German universities focus on differ concerning characteristics like study programme or semester. We will provide the main categories describing these groups. We seek to answer research questions concerning the characteristics of MLSCs in Germany and discuss the results compared to international findings. This information is useful for further international collaborative research, for example a standardized international survey. From a national perspective, these findings support networking and collaborations between the MLSCs as well. As some German MLSCs are facing financial cuts, these results might help in gaining additional funding.","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/teamat/hraa007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50423628","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":"Harder things will stretch you further: helping first-year undergraduate students meaningfully engage with recent research papers in probability and statistics","authors":"Nicholas Grindle;Elinor Jones;Paul Northrop","doi":"10.1093/teamat/hraa001","DOIUrl":"https://doi.org/10.1093/teamat/hraa001","url":null,"abstract":"Undergraduate research increasingly features in university mathematics degrees. Despite this, research papers are used infrequently in mathematics teaching, and this is especially the case for first-year undergraduates. Mathematical subjects are more likely than other STEM disciplines to pinpoint cognitive difficulty as the principal reason for not exposing undergraduate students to research papers. In this paper, we test whether first-year students can engage effectively with research papers. We describe an intervention that exposes first-year, first term undergraduate students to current research in probability and statistics by asking them to read a research paper and summarize it for a general readership following an interview with the paper’s author. Our findings show that the activity introduced students to new fields of knowledge and helped to develop a clearer understanding of scientific process, leading to a heightened sense of personal satisfaction at engaging closely with current research. We argue that structured reading of research papers can lead to productive and rewarding engagement with difficult content, recent and current research and with research processes and that this should make us reconsider the role of research papers in the undergraduate mathematics curriculum.","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/teamat/hraa001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50346335","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":"Digital technologies as a means of accessing powerful mathematical ideas. A study of adults with low schooling in Mexico","authors":"Santiago Palmas;Teresa Rojano;Rosamund Sutherland","doi":"10.1093/teamat/hraa004","DOIUrl":"https://doi.org/10.1093/teamat/hraa004","url":null,"abstract":"This paper derives from a study which main purpose was to investigate how a group of adults with low schooling can have access to powerful mathematical ideas when working with activities that involve the use of technology resources and that take into account the adults’ previous experience with mathematics. Specifically, adults’ previous experience with area calculation was considered. Principles of the Theory of Didactical Situations (TDS) formulated by Brousseau guided the study design, and Pick’s theorem was recreated in a dynamic digital setting, with which it is possible to calculate the area of regular and irregular polygons. In this approach, intuitive notions of area and perimeter are resorted to, seeking to promote the experience with powerful ideas such as ‘the generality of a method’, ‘realizing the existence of different methods used for one and the same end’ and ‘realizing that each method possesses advantages and limitations’. Analysis of interview protocols from three noteworthy cases (which include both adults’ work in the digital setting and their discussions with the researcher) suggests the presence of powerful underlying mathematical ideas, such as the idea of generality and the power of a method and the features of the constituent elements of a geometric figure that are involved in calculating its attributes, attributes such as area.","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/teamat/hraa004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50346336","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":"Transition to university: contributions of a specialist mathematics school","authors":"Jennie Golding","doi":"10.1093/teamat/hraa005","DOIUrl":"https://doi.org/10.1093/teamat/hraa005","url":null,"abstract":"School-university transition in mathematics is of global concern, with multiple cognitive, social and affective disjunctures evidenced. Access to, successful participation in and retention on, competitive mathematically intensive degree courses remain particular challenges in England, especially for disadvantaged young people with high mathematical aptitude. One response has been to establish mathematics specialist schools aimed at such students aged 16–18. Early cohorts have achieved encouraging progression to and through such university courses, but more qualitative and longitudinal outcomes have been less well evidenced. The reported study harnessed a student lens and documentary scrutiny to analyse the contribution to building for successful transition of the particular approaches used. Data suggest that the model adopted has initially supported transition to target degree courses well. I relate the findings to known transition challenges in the global issue of successful passage into and through university mathematics education. I argue many of those are in principle transferable to other post-16 contexts. The study therefore offers evidence suggesting broadly applicable specific strategies that can begin to address widely problematic disjunctures in transition.","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/teamat/hraa005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50346337","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":"More students doing more maths? Student attitudes to mathematics and perceptions of its teaching in a new post-compulsory course","authors":"Matt Homer;Rachel Mathieson;Innocent Tasara;Indira Banner","doi":"10.1093/teamat/hraa006","DOIUrl":"https://doi.org/10.1093/teamat/hraa006","url":null,"abstract":"In England, a relatively new set of post-16 qualifications has been developed under the umbrella term ‘Core Maths’, with a focus on the application of mathematics in context, including the kinds of mathematics needed to support other subjects, to provide a sound basis for the mathematical demands of higher education and employment and to develop problem-solving skills for use in life. The UK government has an ambition that all post-16 students should be studying some mathematics, and Core Maths was designed in part with this aspiration in mind. In this paper, longitudinal questionnaire data from over 100 Core Maths students in 13 case study institutions are analysed to measure students’ views of teaching as transmissionist, their mathematical dispositions and self-identification and how these change over a year of studying Core Maths. We find some evidence that pedagogy in lessons is perceived as being less transmissionist than it was in school mathematics pre-16. There is also some evidence of a negative change in students’ mathematical dispositions over a year of Core Maths. We conclude that supporting teachers in embedding new pedagogical approaches remains a challenge and that this issue could inhibit the growth of new qualifications like Core Maths.","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/teamat/hraa006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50346338","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 alternative route to the Mandelbrot set: connecting idiosyncratic digital representations for undergraduates","authors":"Richard Miles","doi":"10.1093/teamat/hraa003","DOIUrl":"https://doi.org/10.1093/teamat/hraa003","url":null,"abstract":"Mathematics undergraduates often encounter a variety of digital representations which are more idiosyncratic than the ones they have experienced in school and which often require the use of more sophisticated digital tools. This article analyses a collection of digital representations common to undergraduate dynamical systems courses, considers the significant ways in which the representations are interconnected and examines how they are similar or differ from those students are likely to have experienced at school. A key approach in the analysis is the identification of mathematical objects corresponding to manipulative elements of the representations that are most essential for typical exploratory tasks. As a result of the analysis, augmentations of familiar representations are proposed that address the gap between local and global perspectives, and a case is made for greater use of isoperiodic diagrams. In particular, these diagrams are proposed as a new stimulus for students to generate their own explorations of fundamental properties of the Mandelbrot set. The ideas presented are expected to inform the practice of teachers seeking to develop visually rich exploratory tasks which pre-empt some of the issues of instrumentation that mathematics undergraduates experience when introduced to new digital tools. The overarching aim is to address significant questions concerning visualization and inscriptions in mathematics education.","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/teamat/hraa003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50346339","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 mathematical modelling: a framework to support teachers’ choice of resources","authors":"Peter K Dunn;Margaret F Marshman","doi":"10.1093/teamat/hrz008","DOIUrl":"https://doi.org/10.1093/teamat/hrz008","url":null,"abstract":"Mathematics teachers are often keen to find ways of connecting mathematics with the real world. One way to do so is to teach mathematical modelling using real data. Mathematical models have two components: a model structure and parameters within that structure. Real data can be used in one of two ways for each component: (a) to validate what theory or context suggests or (b) to estimate from the data. It is crucial to understand the following: the implications of using data in these different ways, the differences between them, the implications for teaching and how this can influence students’ perceptions of the real-world relevance of mathematics. Inappropriately validating or estimating with data may unintentionally promote poor practice and (paradoxically) reinforce in students the incorrect idea that mathematics has no relevance to the real world. We recommend that teachers approach mathematical modelling through mathematizing the context. We suggest a framework to support teachers’ choice of modelling activities and demonstrate these using examples.","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/teamat/hrz008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50405525","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 mathematical backgrounds of undergraduates from England","authors":"Jeremy Hodgen;Michael Adkins;Anthony Tomei","doi":"10.1093/teamat/hry017","DOIUrl":"https://doi.org/10.1093/teamat/hry017","url":null,"abstract":"Participation in any kind of mathematical study during upper secondary education in England is significantly lower than in other educational systems. As a result, many English students enter university at age 18 or 19 having not studied mathematics for 2 years or more and relatively large proportions of students entering numerate degree programmes do not have a qualification in advanced school mathematics. To date, the mathematical preparation of those university entrants who do not have an advanced school mathematics qualification has not been documented. This study addressed this by analysing a large dataset formed from the combination of two large national databases in England: the National Pupil Database and the Higher Education Statistical Agency Database (N = 253,557). This dataset provided the school mathematics qualifications of undergraduates from England across all degree subjects, who took General Certificate of Secondary Education (GCSE) Mathematics in 2008 and entered a UK university between 2010 and 2012. The study found that approximately 10% of undergraduates did not have a C grade at GCSE Mathematics, which is commonly assumed to be a minimum requirement for entry to university. In general, degree subjects with more mathematical demand recruited students with stronger mathematical backgrounds: 64% of undergraduates in subjects with high mathematical demand had an advanced school mathematics qualification compared to 24% in subjects with medium mathematical demand and 12% in subjects with low mathematical demand. For some university subjects with high and medium mathematical demand, for example Electronic and Electrical Engineering, there were substantial proportions of students with weak school mathematics backgrounds. There was considerable variation across universities with undergraduates in the high-status Russell Group institutions having stronger school mathematics qualifications within the same degree subject.","PeriodicalId":44578,"journal":{"name":"Teaching Mathematics and Its Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/teamat/hry017","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50301877","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}