| 1. |
- Birky, Geoffrey, et al.
(author)
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One problem, nine student-produced proofs
- 2011
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In: The College Mathematics Journal. - : Informa UK Limited. - 0746-8342 .- 1931-1346. ; 42:5, s. 355-360
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Journal article (peer-reviewed)abstract
- The context of this paper is a project aimed at meeting the challenges of teaching an introduction-to-proof course at our own institutions. These courses are typically taken by second year students intending to major in mathematics or computer science and are taught in sections of 15 to 24 students. These students benefit from being given the freedom to explore different methods of proof without the presumption that they will use a certain method. This is the story of how what could have been a routine, closed problem became an open one.
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| 2. |
- Levenson, Esther, et al.
(author)
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Young children's aesthetic development in the context of mathematical explanation
- 2017
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In: Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME10). - Dublin : Dublin City University. - 9781873769737 ; , s. 1861-1868
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Conference paper (peer-reviewed)abstract
- Mathematicians routinely report that beauty is both a reward and a motivation for the work they do. However, how and to what extent children can appreciate mathematical beauty is an open question. This exploratory study looks at young children (ages 6-12, with a focus on the younger years) as they evaluate different explanations of claims about even numbers and triangular numbers. While our results are fairly speculative, we provide case studies which illustrate possible kinds of aesthetic reactions, and some of the factors which might impact on those reactions.
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| 3. |
- Raman, Manya, 1971-, et al.
(author)
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Key ideas and insights in the context of three high school geometry proofs
- 2006
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In: Mathematics Teacher. - 0025-5769 .- 2330-0582. ; 99:9, s. 644-649
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Journal article (peer-reviewed)abstract
- How the concept of "key idea" can be used in high school geometry. Article discusses connecting students' intuitive informal arguments with rigorous formal mathematical proofs. Includes three examples.
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| 4. |
- Raman-Sundström, Manya, 1971-
(author)
-
A pedagogical history of compactness
- 2015
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In: The American mathematical monthly. - : Informa UK Limited. - 0002-9890 .- 1930-0972. ; 122:7, s. 619-635
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Journal article (peer-reviewed)abstract
- Compactness is a central notion in advanced mathematics, but we often teach the concept without much historical motivation. This paper fills in many of the gaps left by the standard textbook treatment, including what motivated the definition, how did the definition evolve, and how can compactness be expressed in terms of nets and filters.
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| 5. |
- Raman Sundström, Manya, 1971-
(author)
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An Exercise on Limits
- 2016
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In: Journal of Humanistic Mathematics. - : Claremont Colleges Library. - 2159-8118. ; 6:2
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Journal article (pop. science, debate, etc.)abstract
- A poem about different kinds of limits, not all mathematical.
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| 6. |
- Raman Sundström, Manya, 1971-, et al.
(author)
-
Beauty as fit : A metaphor in mathematics?
- 2013
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In: Research in Mathematics Education. - : Informa UK Limited. - 1479-4802 .- 1754-0178. ; 15:2, s. 199-200
-
Journal article (peer-reviewed)abstract
- Beauty, which plays a central role in the practice of mathematics, is almost absent in discussions of school mathematics. This is problematic, because students will decide whether or not to continue their studies inmathematics without having an accurate picture of what the subject is about. In order to have a discussion about how to introduce beauty into the school mathematicscurriculum, we need to have a clear idea about what beauty means. That is the aim ofthis study, with a focus on characterising beauty in mathematical proof.
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| 7. |
- Raman Sundström, Manya, 1971-
(author)
-
Beauty as Fit : An Empirical Study of Mathematical Proofs
- 2012
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In: Proceedings of the British Society for Research into Learning Mathematics. ; , s. 156-160
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Conference paper (peer-reviewed)abstract
- Beauty has been discussed since ancient times, but discussions of beautywithin mathematics education are relatively limited. This lack ofdiscussion is surprising given the importance of beauty within the practiceof mathematics. This study explores one particular metaphor of beauty,that of beauty as fit, as a way to distinguish between proofs that areconsidered beautiful and those that are not. Several examples areexamined, supported by empirical data of mathematicians andmathematics educators who judged and ranked different proofs in aseminar on mathematical beauty.
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| 8. |
- Raman-Sundström, Manya, 1971-
(author)
-
Beauty in Mathematics Textbooks
- 2011
-
Conference paper (other academic/artistic)abstract
- This paper poses two questions:Should mathematics textbooks try to convey the beauty of mathematics?If so, how?These questions are difficult, and subjective, and will not be settled in this paper, but the purpose in raising them is to discuss if we agree that the answer to (1) is yes, then how can we make progress on question (2).
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| 9. |
- Raman Sundström, Manya, 1971-
(author)
-
Dedekind-like
- 2020
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In: The Mathematical intelligencer. - : Springer. - 0343-6993 .- 1866-7414. ; 42:4, s. 42-43
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Journal article (other academic/artistic)abstract
- Published in The Mathematical Intelligencer.
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| 10. |
- Raman Sundström, Manya, 1971-
(author)
-
Elements : a love song
- 2011
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In: The Mathematical Intelligencer. - : Springer Science+Business Media B.V.. - 1866-7414.
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Other publication (pop. science, debate, etc.)
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| 11. |
- Raman Sundström, Manya, 1971-
(author)
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Excavation
- 2012
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Other publication (pop. science, debate, etc.)
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| 12. |
- Raman Sundström, Manya, 1971-, et al.
(author)
-
Mathematical fit : A first approximation
- 2015
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In: Proceedings of the ninth conference of the Euorpean Society for research in Mathermatics education (CERME9). - Prague : Charles University. - 9788072908448 ; , s. 185-191
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Conference paper (peer-reviewed)abstract
- We discuss here the notion of mathematical fit, a concept that might relate to mathematical explanation and mathematical beauty. We specify two kinds of fit a proof can have, intrinsic and extrinsic, and provide characteristics that help distinguish different proofs of the same theorem.
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| 13. |
- Raman-Sundström, Manya, 1971-, et al.
(author)
-
Mathematical fit : a case study
- 2018
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In: Philosophia mathematica. - Cary : Oxford University Press. - 0031-8019 .- 1744-6406. ; 26:2, s. 184-210
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Journal article (peer-reviewed)abstract
- Mathematicians routinely pass judgments on mathematical proofs. A proof might be elegant, cumbersome, beautiful, or awkward. Perhaps the highest praise is that a proof is right, that is that the proof fits the theoremin an optimal way. It is also common to judge that a proof fits better than another, or that a proof does not fit a theorem at all. This paper attempts to clarify the notion of mathematical fit. We suggest six criteria that distinguish proofs as being more or less fitting, and provide examples from several different mathematical fields.
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| 14. |
- Raman Sundström, Manya, 1971-, et al.
(author)
-
The case of Brandon : the dual nature of key ideas in the classroom
- 2009
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In: Nordisk matematikkdidaktikk. - 1104-2176. ; 14:2, s. 29-47
-
Journal article (peer-reviewed)abstract
- This paper looks at proof production in the midst of classroom interaction. The setting is a collegelevel geometry course in which students are working on the following task: Prove that two paralleltransported lines in the plane are parallel in the sense that they do not intersect. A proof of this statement istraced from a student's idea, through a small group discussion, to a large class discussion moderated by ateacher. As the proof emerges through a series of increasingly public settings we see ways in which the keyidea of the proof serves to both open and close class discussion. We look at several examples of openingand closing, showing how not only the key idea, but the warrants and justifications connected to it, play animportant role in the proof development.
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| 15. |
- Raman-Sundström, Manya, 1971-, et al.
(author)
-
The Nature and Experience of Mathematical Beauty
- 2016. - 1
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In: Journal of Humanistic Mathematics. - Claremont, CA, USA : Claremont Colleges Library. - 2159-8118. ; 6:1, s. 3-7
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Journal article (peer-reviewed)
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| 16. |
- Raman Sundström, Manya, 1971-
(author)
-
The Notion of Fit as a Mathematical Value
- 2016
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In: Mathematical Cultures. - Basel : Birkhäuser Verlag. - 9783319285825 ; , s. 271-285
-
Book chapter (peer-reviewed)abstract
- Fit, or the lack of it, might occur in a large number of different settings, from visual art, to music, to relationships between people, and so on. This paper explores the metaphor of beauty as fit as it occurs in mathematics. The central claim is that, be it instantiated in the relationship of a type of tree to its natural shape or of a particular proof to a theorem, fit is what brings about the feeling that that tree or that theorem is beautiful. The greater the degree of fit, the greater our sense of beauty (given that we have the requisite understanding to appreciate the fit.) This claim will not be fully defended, but we will set up some ground work for doing so. In particular, the paper will illustrate two distinct types of fit in mathematics via contrasting examples of proofs typically held to be beautiful or not beautiful.
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| 17. |
- Raman Sundström, Manya, 1971-, et al.
(author)
-
Two beautiful proofs of Pick's theorem
- 2011
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In: Proceedings of Seventh Congress of European Research in Mathematics Education. - : University of Rzeszow Publishing House. - 9788373386839 ; , s. 224-232
-
Conference paper (peer-reviewed)abstract
- We present two different proofs of Pick’s theorem and analyse in what ways might be perceived as beautiful by working mathematicians. In particular, we discuss two concepts, generality and specificity, that appear to contribute to beauty in different ways. We also discuss possible implications on insight into the nature of beauty in mathematics, and how the teaching of mathematics could be impacted, especially in countries in which discussions of beauty and aesthetics are notably absent from curricular documents.
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| 18. |
- Stanley, Dick, et al.
(author)
-
Extended analyses : finding deep structure in standard high school mathematics
- 2007
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In: Journal of Mathematics Teacher Education. - : Springer Science and Business Media LLC. - 1386-4416 .- 1573-1820. ; 10:4-6, s. 391-397
-
Journal article (peer-reviewed)abstract
- This paper describes an approach for pre-service or in-service mathematics education thatteaches sophisticated mathematics using only the tools of high school mathematics. The idea is to start witha standard problem from high school mathematics and let the solution to this problem serve as a platform forasking good mathematical questions and searching for deeper mathematical structure beyond the obvious"answer" to the question. Surprisingly, it turns out that high school mathematics is remarkably open to thissort of analysis, and the results are more interesting mathematically than one might initially think.
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| 19. |
- Sundström (Raman), Manya, 1971-
(author)
-
Coordinating informal and formal aspects of mathematics : student behavior and textbook messages
- 2002
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In: Journal of Mathematical Behavior. - 0732-3123 .- 1873-8028. ; 21:2, s. 135-150
-
Journal article (peer-reviewed)abstract
- In this paper I illustrate difficulties students have coordinating informal and formal aspects of mathematics. I also discuss two ways in which precalculus and calculus textbooks treat mathematics that may make this coordination difficult: emphasizing the informal at the expense of the formal and emphasizing the formal at the expense of the informal. By looking at student difficulties in light of textbook treatments, we see evidence that student difficulties are not merely developmental. Students are not given many opportunities to make the kinds of connections which, while difficult, are an essential component of mathematical thinking.
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| 20. |
- Sundström (Raman), Manya, 1971-
(author)
-
Epistemological messages conveyed by three high school and college mathematics textbooks
- 2004
-
In: Journal of Mathematical Behavior. - : Elsevier BV. - 0732-3123 .- 1873-8028. ; 23:4, s. 389-404
-
Journal article (peer-reviewed)abstract
- Mathematics textbooks embody a particular set of assumptions about mathematics or the mathematics intended for students at a particular level. Thus, an epistemological analysis of textbooks can provide some context for understanding, for example, the difficulties many students encounter when moving from high-school to collegiate mathematics. In this study, we consider how typical pre-calculus, calculus, and analysis texts treat the topic of continuity. We find that these texts send conflicting messages regarding the status and purpose of mathematical definitions.
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| 21. |
- Sundström (Raman), Manya, 1971-
(author)
-
Key ideas : What are they and how can they help us understand people view proof?
- 2003
-
In: Educational Studies in Mathematics. - 0013-1954 .- 1573-0816. ; 52:3, s. 319-325
-
Journal article (peer-reviewed)abstract
- This paper examines the views of proof held by university level mathematics students and teachers. A framework is developed for characterizing people's views of proof, based on a distinction between public and private aspects of proof and the key ideas which link these two domains.
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| 22. |
- Wistedt, Inger, et al.
(author)
-
Quality and equity in mathematics education : a Swedish perspective
- 2010. - 1
-
In: Mapping equity and quality in mathematics education. - Dordrecht : Springer Netherlands. - 9789048198023 - 9789048198030 ; , s. 341-350
-
Book chapter (other academic/artistic)abstract
- Discussions about quality and equity in education around the world often focus on the disadvantaged, the students for whatever reason do not have the resources to make the most of public education, or to get an education at all. In Sweden, where there is a fairly even economic playing field, the discussion -- which is not always made public -- addresses another type of inequity, namely that bright and talented students are not given sufficient education, particularly in math and science, to help them reach their potential. We describe some of the historical and cultural background for this inequity, and offer a modest, but not unbiased, account of what we claim is a false dilemma involved in promoting excellence in a democratic society.
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