| 1. |
- Gerlee, Philip, 1980, et al.
(författare)
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Scientific Models : Red Atoms, White Lies and Black Boxes in a Yellow Book
- 2016
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Bok (övrigt vetenskapligt/konstnärligt)abstract
- A zebrafish, the hull of a miniature ship, a mathematical equation and a food chain - what do these things have in common? They are examples of models used by scientists to isolate and study particular aspects of the world around us. This book begins by introducing the concept of a scientific model from an intuitive perspective, drawing parallels to mental models and artistic representations. It then recounts the history of modelling from the 16th century up until the present day. The iterative process of model building is described and discussed in the context of complex models with high predictive accuracy versus simpler models that provide more of a conceptual understanding. To illustrate the diversity of opinions within the scientific community, we also present the results of an interview study, in which ten scientists from different disciplines describe their views on modelling and how models feature in their work. Lastly, it includes a number of worked examples that span different modelling approaches and techniques. It provides a comprehensive introduction to scientific models and shows how models are constructed and used in modern science. It also addresses the approach to, and the culture surrounding modelling in different scientific disciplines. It serves as an inspiration for model building and also facilitates interdisciplinary collaborations by showing how models are used in different scientific fields. The book is aimed primarily at students in the sciences and engineering, as well as students at teacher training colleges but will also appeal to interested readers wanting to get an overview of scientific modelling in general and different modelling approaches in particular.
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| 2. |
- Gerlee, Philip, 1980, et al.
(författare)
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Scientific Models
- 2016
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Bok (övrigt vetenskapligt/konstnärligt)abstract
- A zebrafish, the hull of a miniature ship, a mathematical equation and a food chain - what do these things have in common? They are examples of models used by scientists to isolate and study particular aspects of the world around us. This book begins by introducing the concept of a scientific model from an intuitive perspective, drawing parallels to mental models and artistic representations. It then recounts the history of modelling from the 16th century up until the present day. The iterative process of model building is described and discussed in the context of complex models with high predictive accuracy versus simpler models that provide more of a conceptual understanding. To illustrate the diversity of opinions within the scientific community, we also present the results of an interview study, in which ten scientists from different disciplines describe their views on modelling and how models feature in their work. Lastly, it includes a number of worked examples that span different modelling approaches and techniques. It provides a comprehensive introduction to scientific models and shows how models are constructed and used in modern science. It also addresses the approach to, and the culture surrounding modelling in different scientific disciplines. It serves as an inspiration for model building and also facilitates interdisciplinary collaborations by showing how models are used in different scientific fields. The book is aimed primarily at students in the sciences and engineering, as well as students at teacher training colleges but will also appeal to interested readers wanting to get an overview of scientific modelling in general and different modelling approaches in particular.
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| 3. |
- Pejlare, Johanna, 1976
(författare)
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On Axioms and Images in the History of Mathematics
- 2007
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This dissertation deals with aspects of axiomatization, intuition and visualization in the history of mathematics. Particular focus is put on the end of the 19th century, before David Hilbert's (1862–1943) work on the axiomatization of Euclidean geometry. The thesis consists of three papers. In the first paper the Swedish mathematician Torsten Brodén (1857–1931) and his work on the foundations of Euclidean geometry from 1890 and 1912, is studied. A thorough analysis of his foundational work is made as well as an investigation into his general view on science and mathematics. Furthermore, his thoughts on geometry and its nature and what consequences his view has for how he proceeds in developing the axiomatic system, is studied. In the second paper different aspects of visualizations in mathematics are investigated. In particular, it is argued that the meaning of a visualization is not revealed by the visualization and that a visualization can be problematic to a person if this person, due to a limited knowledge or limited experience, has a simplified view of what the picture represents. A historical study considers the discussion on the role of intuition in mathematics which followed in the wake of Karl Weierstrass' (1815–1897) construction of a nowhere differentiable function in 1872. In the third paper certain aspects of the thinking of the two scientists Felix Klein (1849–1925) and Heinrich Hertz (1857–1894) are studied. It is investigated how Klein and Hertz related to the idea of naïve images and visual thinking shortly before the development of modern axiomatics. Klein in several of his writings emphasized his belief that intuition plays an important part in mathematics. Hertz argued that we form images in our mind when we experience the world, but these images may contain elements that do not exist in nature.
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| 4. |
- Olteanu, Constanta, 1960-
(författare)
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"Vad skulle x kunna vara?" : andragradsekvation och andragradsfunktion som objekt för lärande
- 2007
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Algebraic equations and functions play an important role in various mathematical topics, including algebra, trigonometry, linear programming and calculus. Accordingly, various documents, such as the most recent Swedish curriculum (Lpf 94) for upper secondary school and the course syllabi in mathematics, specify what the students should learn in Mathematics Course B. They should be able to solve quadratic equations and apply this knowledge in solving problems, explain the properties of a function, as well as be able to set up, interpret and use some nonlinear functions as models for real processes. To implement these recommendations, it is crucial to understand the students’ way of experiencing quadratic equations and functions, and describe the meaning these have for the students in relation to the possibility they have to their experience of them. The aim of this thesis is to analyse, understand and explain the relation between the handled and learned content, which consists of second-degree equations and quadratic functions, in classroom practice. This means that content is the research object and not the teacher’s conceptions or knowledge of, or about this content. This restriction implies that the handled and learned contents are central in this study and will be analysed from different perspectives. The study includes two teachers and 45 students in two different classes. The data consist of video-recordings of lessons, individual sessions, interviews and the teachers’/researcher’s review of the individual sessions. The students’ tests also constituted an important part of the data collection. When analysing the data, concepts relating to variation theory have been used as analytical tools. Data have been analysed in respect of the teachers’ focus on the lesson content, which aspects are ignored and which patterns of dimensions of variations are constituted when the contents are handled by the teachers in the classroom. Also, data have been analysed in respect of the students’ focus when they solve different exercises in a test situation. It can be shown that the meaning of parameters, the unknown quantity in an equation and the function’s argument change several times when the teacher presents the content in the classroom and when the students solve different exercises. It can also be shown that the teachers and the students develop complicated patterns of variation during the lessons and that the ways in which the teachers open up dimensions of variation play an important role in the learning process. The results indicate that there is a convergent variation leading the students to improve their learning. By focusing on some aspects of the objects of learning and create convergent variations, it is possible for the students to understand the difference between various interpretations of these aspects and thereafter focus on the interpretation that fits in a certain context. Furthermore, this variation leads the students to make generalisations in each object of learning (equations and functions) and between these objects of learning. These generalisations remain over time, despite working with new objects of learning. An important result in this study is that the implicit or explicit arguments of a function can make it possible to discern an equation from a function despite the fact that they are constituted by the same algebraic expression.
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| 5. |
- Dyrvold, Anneli, 1970-
(författare)
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Difficult to read or difficult to solve? : The role of natural language and other semiotic resources in mathematics tasks
- 2016
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- When students solve mathematics tasks, the tasks are commonly given as written text, usually consisting of natural language, mathematical notation and different types of images. This is one reason why reading and interpreting such texts are important parts of being mathematically proficient, at least within the school context. The ability utilized when dealing with aspects of mathematical text is denoted in this thesis as a mathematical reading ability; this ability is useful when reading mathematical language, for example, in task text. There is, however, a lack of knowledge of what characterizes this mathematical language, what students need to learn regarding the mathematical language, and exactly which mathematical language that tests should preferably assess. Therefore, the purpose of this thesis is to contribute to the knowledge of aspects of difficulty related to textual features in mathematics tasks. In particular, one aim is to distinguish between a difficulty that has to do with a mathematical ability and another that has not. Different types of text analyses are utilized to capture textural features that might be demanding for the students when reading and solving mathematics tasks. Aspects regarding vocabulary are investigated both in a literature review and in a study where corpora are used to analyse word commonness. Other textual analyses focus on textual features that concern mathematical notation and images, besides natural language. Statistical methods are used to analyse potential relations between the textual features of interest and both task difficulty and task demand on reading ability. The results from the research review are sparse regarding difficult vocabulary, since few of the reviewed studies analyses word aspects separately. Several of the analysed textual features are related to aspects of difficulty. The results show that tasks with more words that are uncommon both in a mathematical context and in an everyday context, may favour students with good reading ability rather than students with good mathematical ability. Another textual feature that is likely to be demanding for students, is if the task texts contains many meaning relations, for example, when several words refer to the same or similar object. These results have implications for the school practice both regarding textual features that are important from an educational perspective and regarding the construction of tests. The research does also contribute to an understanding of what characterizes a mathematical language.
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| 6. |
- Viirman, Olov, et al.
(författare)
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Different views – some Swedish mathematics students’ concept images of the function concept
- 2010
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Ingår i: Nordisk matematikkdidaktikk. - Göteborg : NCM. - 1104-2176. ; 15:4, s. 5-24
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- This study analyses what kind of concept images a group of engineering and teacher students have of the function concept, and how these concept images are related to the historical development of this concept. The study was conducted using questionnaires, and 34 students at a Swedish university participated. It is found that the students primarily rely on operational conceptions of the function concept, with only a minority of students possessing structural conceptions. The definitions given by the students mostly resemble an 18th or 19th century view of functions. The study also indicates that the character of the definitions given in the textbooks used by the students affect their concept images.
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| 7. |
- Muhrman, Karolina
(författare)
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Inget klöver utan matematik : En studie av matematik i yrkesutbildning och yrkesliv
- 2016
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Syftet med avhandlingen är att öka förståelsen för aspekter som inverkar på relationen mellan yrkeselevers kunskaper och de kunskaper som behövs i yrkeslivet. Detta görs genom att undersöka olika aktörers perspektiv på yrkeslivets behov av matematikkunskaper, matematikundervisning på yrkesprogram och utformningen av läroplanen Gy11. Studien görs inom gymnasiets naturbruksprogram med inriktning mot lantbruksutbildning och inom lantbruksyrket. Det empiriska materialet består av kvalitativa intervjuer med yrkesverksamma lantbrukare, yrkeslärare och matematiklärare samt intervjuer och enkäter med elever. Resultaten från den empiriska studien har analyserats i relation till läroplanen Gy11. För analysen används framförallt Bernsteins läroplansteoretiska perspektiv med begreppen pedagogiska koder och diskurser, i vissa delar tillsammans med D’Ambrosios etnomatematiska perspektiv. Resultaten visar att lantbruksyrket kräver goda matematikkunskaper, men att det i vissa fall finns ett diskursivt gap mellan skolans matematikundervisning och behovet av matematikkunskaper i yrkeslivet. Trots ämnesplanens inriktning mot en matematik som är relaterad till elevernas yrkesinriktning, är matematikundervisningen ofta starkt knuten till en matematikbok utan relation till det yrke eleverna utbildas för. Undervisningens organisering styrs av en mängd ramfaktorer som både kan handla om tid, gruppsammansättningar och schema, men också om lärarnas syn på kunskap. De nationella proven utgör en betydande ramfaktor som i många fall styr undervisningen i högre grad än vad innehållet i ämnesplanen gör. Undervisningens svaga koppling till elevernas yrkesinriktning gör dem omotiverade att lära sig matematik eftersom de har svårt att se hur de ska kunna använda sina kunskaper i sitt kommande yrke. En del elever har också svårigheter med att rekontextualisera sina matematikkunskaper från skolkontexten till yrkeslivets kontext vilket bland annat kan försämra deras anställningsmöjligheter.
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| 8. |
- Westin, Jonas, et al.
(författare)
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Rollspelsövningar för undervisning av matematiska modeller på universitetsnivå
- 2022
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Ingår i: Högre Utbildning. - : Cappelen Damm Akademisk. - 2000-7558. ; 12:1, s. 38-46
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- Matematiska modeller används i många sammanhang för att beskriva system, undersöka frågeställningar och skapa beslutsunderlag. Samtidigt ställer denna typ av matematiskt tänkande och problemformulering ofta höga krav på användarens matematiska förmågor. På kurser med inslag av matematisk modellering kan därför en del studenter ha svårt att tillgodogöra sig och förstå abstrakta matematiska modeller och deras kopplingar till den omvärld som modellerna avser att beskriva. I denna artikel beskrivs och diskuteras resultatet av ett pedagogiskt utvecklingsprojekt med syfte att utveckla och utvärdera rollspel som metod för undervisning i matematik och statistik på universitetsnivå. En slutsats från projektet är att det framförallt är studenternas reflektioner och analyser efter själva övningen som skapar en djupare förståelse för de bakomliggande koncepten och idéerna rollspelet avser att spegla genom att länka samman matematiska modeller och teorier med studenternas upplevelser från rollspelsövningen. Användning av rollspelsövningar vid undervisning i högre matematik kan både vara ett verktyg för att repetera kursinnehåll och för att sänka abstraktionsnivån. En tidig version av artikeln har presenterats vid den universitetspedagogiska konferensen vid Umeå universitet 2017.
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| 9. |
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| 10. |
- Liu, Yang, 1960-
(författare)
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Syllogistic Analysis and Cunning of Reason in Mathematics Education
- 2013
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This essay explores the issue of organizing mathematics education by means of syllogism. Two aspects turn out to be particularly significant. One is the syllogistic analysis while the other is the cunning of reason. Thus the exploration is directed towards gathering evidence of their existence and showing by examples their usefulness within mathematics education.The syllogistic analysis and the cunning of reason shed also new light on Chevallard's theory of didactic transposition. According to the latter, each piece of mathematical knowledge used inside school is a didactic transposition of some other knowledge produced outside school, but the theory itself does not indicate any way of transposing, and this empty space can be filled with the former.A weak prototype of syllogism considered here is Freudenthal's change of perspective. Some of the major difficulties in mathematics learning are connected with the inability of performing change of perspective. Consequently, to ease the difficulties becomes a significant issue in mathematics teaching. The syllogistic analysis and the cunning of reason developed in this essay are the contributions to the said issue.
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