A Galilean Approach to the Galileo Affair, 1609–2009

Galileo’s telescopic discoveries of 1609–1612 provided a crucial, although not conclusive, confirmation of the Copernican hypothesis of the earth’s motion. In Galileo’s approach, the Copernican Revolution required that the geokinetic hypothesis be supported not only with new theoretical arguments but also with new observational evidence; that it be not only supported constructively but also critically defended from objections; and that such objections be not only refuted but also appreciated in all their strength. However, Galileo’s defense of Copernicanism triggered a sequence of events that climaxed in 1633, when the Inquisition tried and condemned him as a suspected heretic. In turn, the repercussions of Galileo’s condemnation have been a defining theme of modern Western culture for the last four centuries. In particular, the 20th century witnessed a curious spectacle: rehabilitation efforts by the Catholic Church and anti-Galilean critiques by secular-minded left-leaning social critics. The controversy shows no signs of abating to date, as may be seen from the episode of Pope Benedict XVI’s attitude toward Paul Feyerabend’s critique of Galileo. Nevertheless, I have devised a framework which should pave the way for eventually resolving this controversy, and which is modeled on Galileo’s own approach to the Copernican Revolution.     

The Pendulum: From Constrained Fall to the Concept of Potential

Kuhn underlined the relevance of Galileo’s gestalt switch in the interpretation of a swinging body from constrained fall to time metre. But the new interpretation did not eliminate the older one. The constrained fall, both in the motion of pendulums and along inclined planes, led Galileo to the law of free fall. Experimenting with physical pendulums and assuming the impossibility of perpetual motion Huygens obtained a law of conservation of vis viva at specific positions, beautifully commented by Mach. Daniel Bernoulli generalised Huygens results introducing the concept of potential and the related independence of the ‘work’ done from the trajectories (paths) followed: vis viva conservation at specific positions is now linked with the potential. Feynman’s modern way of teaching the subject shows striking similarities with Bernoulli’s approach. A number of animations and simulations can help to visualise and teach some of the pendulum’s interpretations related to what we now see as instances of energy conservation.     

Lawson’s Shoehorn, Reprise

Lawson’s (Lawson, A.: 2004, Science & Education, 13, 155–177) analysis of the meteorite hypothesis of dinosaur extinction exhibits flaws similar to his earlier (2002) analysis of Galileo’s discovery of Jupiter’s moons (Allchin, D.: 2003, Science & Education, 12, 315–329).     

The Practices of Mathematicians: What do They Tell us About Coming to Know Mathematics?

In 1997, an interview-based study of 70 research mathematicians was undertaken with a focus on how they ‘come to know’ mathematics, i.e. their epistemologies. In this paper, I discuss how these mathematicians understand their practices, locating them in the communities of which they claim membership, identifying the style which dominates their organisation of research and looking at their lived contradictions. I examine how they talk about ‘knowing’ mathematics, the metaphors on which they draw, the empiricist connections central to the work of the applied mathematicians and statisticians, and the importance of connectivities to the construction of their mathematical Big Picture. I compare the stories of these research mathematicians with practices in mathematics classrooms and conclude with an appeal for teachers to pay attention to the practices of research mathematicians and their implications for coming to know mathematics. This revised version was published online in July 2006 with corrections to the Cover Date.     

Christoph Clavius’ “Ordo Servandus in Addiscendis Disciplinis Mathematicis” and the Teaching of Mathematics in Jesuit Colleges at the Beginning of the Modern Era

The Ordo servundum in addiscendis disciplinis mathematicis is a milestone in the history of the teaching of mathematics. Conceived by Christoph Clavius for the Jesuit Colleges, it was not only a syllabus for mathematical studies for the students at Jesuit colleges but also an instrument for training mathematics teachers. Its coherence and its skilful, innovative, modular format have made it a masterpiece of pedagogic technique. This marks the end of a long and arduous journey throughout which Clavius’ patient and persistent persuasion convinced the Company that mathematics should be an integral part of the syllabus in Jesuit colleges.     

Natural nano-machines

In the first part of this article, we read about Alice’s guided tour through the cellular microfactory. In the second part, I introduce the methods of studying the materials and mechanisms of the molecular machines through dialogues. The three participants in this discussion are Alice, her elder brother Alex and her father Albert. The style of presentation here is adapted from Galileo’s Dialogue Concerning the Two Chief World Systems. Albert, a professor of biophysics, emphasizes the crucial differences between the mechanisms of the natural nano-machines and those of their macroscopic counterparts. He also points out some practical applications of this interdisciplinary research in biomedical science and nano-technology. Debashish Chowdhury is a professor of Physics at IIT Kanpur. He is a theoretical physicist and the areas of his research interest are statistical and biological physics. He plays with toy models using mathematical formulae and simulates nature with computers.     

Teaching students to think critically about science and origins

In David Long’s article, Scientists at Play in a Field of the Lord, he studies the discourse between a network of regional scientists, atheists, activists and evolutionists at the opening of The Creation Museum on Memorial Day, 2007. This review essay examines the teaching of evolution through the teacher’s ‘lens of empathy’ and also considers a ‘pupil centeredness’ approach. As a practicing science educator, I have found it paramount to take into consideration my students’ backgrounds and their families’ beliefs in order to understand their preconceived notions about the origins of life. By teaching evolution as ‘a theory with both facts and fallacies’ only then does it become an opportunity for critical thinking that fosters growth and risk taking in a safe environment. Most times students hear evolution preached as a one-sided lecture by teachers who believe it’s “my way or the highway” and leave little or no room for dialogue. I believe that a teacher’s job is to stay updated with current research on the theory of evolution and then present all the information to students in a way that creates personal opportunities for them to adjust their existing schema without demeaning them, their ideas, or their faith or belief system. This not only shows value, compassion and tolerance for them as thinking humans, but also allows them opportunities to develop critical thinking, which helps to shape whom they become as adults.     

Integrating History of Science in Science Education through Historical Microworlds to Promote Conceptual Change

This paper proposes a new way to integrate history of science in science education to promote conceptual change by introducing the notion of historical microworld, which is a computer-based interactive learning environment respecting historic conceptions. In this definition, “interactive” means that the user can act upon the virtual environment by changing some parameters to see what ensues. “Environment respecting historic conceptions” means that the “world” has been programmed to respect the conceptions of past scientists or philosophers. Three historical microworlds in the field of mechanics are presented in this article: an Aristotelian microworld respecting Aristotle’s conceptions about movement, a Buridanian microworld respecting the theory of impetus and, finally, a Newtonian microworld respecting Galileo’s conceptions and Newton’s laws of movement.     

Mathematical thinking of kindergarten boys and girls: similar achievement, different contributing processes

The objective of this study was to examine gender differences in the relations between verbal, spatial, mathematics, and teacher–child mathematics interaction variables. Kindergarten children (N = 80) were videotaped playing games that require mathematical reasoning in the presence of their teachers. The children’s mathematics, spatial, and verbal skills and the teachers’ mathematical communication were assessed. No gender differences were found between the mathematical achievements of the boys and girls, or between their verbal and spatial skills. However, mathematics performance was related to boys’ spatial reasoning and to girls’ verbal skills, suggesting that they use different processes for solving mathematical problems. Furthermore, the boys’ levels of spatial and verbal skills were not found to be related, whereas they were significantly related for girls. The mathematical communication level provided in teacher–child interactions was found to be related to girls’ but not to boys’ mathematics performance, suggesting that boys may need other forms of mathematics communication and teaching.     

Maternal mediation in book reading, home literacy environment, and children’s emergent literacy: a comparison between two social groups

The researchers addressed two questions: (1) Does maternal reading mediation and family home literacy environment (HLE) relate to children’s emergent literacy (EL) level? and (2) Do the relationships among these variables differ as a function of socioeconomic strata (SES) level. A total of 94 5–6-year-old children, 47 from low SES (LSES) and 47 from high (HSES) families, and their mothers participated. Mother–child interactions while reading an unfamiliar book were videotaped and their verbal expressions were coded for extracting maternal mediation level. Children’s independent EL level was assessed prior to the interaction. Compared with the LSES group, HSES children showed higher EL levels and their homes had a richer literacy environment. Maternal mediation level differed by SES: LSES mothers paraphrased text more often; HSES mothers’ higher mediation level included a discussion of the written system and making connections beyond the text. In the HSES group, maternal mediation level and HLE related to children’s EL; no such relationships appeared in the LSES group. Results are discussed in terms of children’s socio-economic background and their reading experiences. Implications for researchers and educational practices about the relationships between children’s literacy development, SES, HLE, and parental mediation are discussed.     

Weeping for．My Smoking Daughter的介入资源分析

运用评价理论的介入子系统框架。结合文化语境与情景语境分析课文Weeping for my smoking daughter中介入资源的修辞效果。受语篇的批驳本质影响,文中收缩资源远多于扩展资源,“否定”、“反驳”与“宣布”手段交替出现展开冲突性对话,“容纳”手段使作者观点不武断或强调记忆的不确定性,“支持”与“归属”手段导入了相对立的语篇外声音。介入资源导入各种显性或隐性评价,形成作者与语篇内其它声音之间、作者与读者之间的意义协商与互动。     

Why and how mathematicians read proofs: an exploratory study

In this paper, we report a study in which nine research mathematicians were interviewed with regard to the goals guiding their reading of published proofs and the type of reasoning they use to reach these goals. Using the data from this study as well as data from a separate study (Weber, Journal for Research in Mathematics Education 39:431–459, 2008) and the philosophical literature on mathematical proof, we identify three general strategies that mathematicians employ when reading proofs: appealing to the authority of other mathematicians who read the proof, line-by-line reading, and modular reading. We argue that non-deductive reasoning plays an important role in each of these three strategies.     

The Tacit-explicit Dimension of the Learning of Mathematics: An Investigation Report**

This paper reports on study that investigated the tacit-explicit dimension of the learning of mathematics. The study was carried out in a secondary school and consisted of an episode analysis related to a class discussion about the difference between plane figures and spatial figures. The data analysis was based on integration between some aspects of Polanyi’s theory on tacit knowledge and Ernest’s model of mathematical knowledge, with reference to its mainly explicit and mainly tacit components. This integration has involved not only the types of knowledge – mainly explicit or mainly tacit – the students used in a psychological way to perform a mathematical task involving conversation, but also and particularly how much the projection of those types of knowledge on the task were manifest tacitly or formalized by the students. Among the results of the research, a strong finding was that the lack of correspondence between the students’ utterances and their original understandings is directly related to the manner in which the tacit co-operates with the explicit in the process of articulation.This paper is a development of three short papers presented at ICME-10, Copenhagen, Denmark, 4–11 July 2004, and PME-28, Bergen, Norway, 14–18 July 2004.     

Grasping the Unimaginable: Recent Holocaust Novels for Children by Morris Gleitzman and John Boyne

This discussion explores the role that storytelling and stories might have in leading children towards an awareness of uncertainty and ambiguity in relation to Holocaust representation. It focuses on Morris Gleitzman’s Once (2006), its sequel Then (2008), and John Boyne’s The Boy in the Striped Pyjamas (2006) to consider the narrative techniques used to draw young readers into an understanding of the Holocaust. In particular, the discussion examines the role of silence within these narratives to suggest that a meaningful dialogue with silence is a crucial aspect in communicating the fractured nature of Holocaust history. Literature aimed at a young audience engages explicitly with the oft-cited injunction not to forget the Holocaust by setting out to inform a new generation of readers about the horrors of the Nazi genocide. In my analysis of these texts, however, I want to consider whether we should assume that such works do necessarily perform a progressive educative role. The article argues that the blunt didacticism of Boyne’s text might close down possibilities for the child reader’s imaginative engagement with the ungraspable nature of the Holocaust. In contrast, Gleitzman’s novels confront the child reader with a complex set of ideas about the relationship between narrative and subjectivity.     

Mathematical biography and key rhetoric

The intention of this article is to consider the use of narrative and rhetorical inquiry as research methods. I construct a mathematical biography of one elementary teacher trainee, Sirpa, who had performed well in her advanced mathematics course in upper secondary school. I describe her development as a mathematics teacher during teacher education from her second-year methods course and teaching practice to fourth-year teaching practice. Narrative inquiry is more than a case study: Sirpa’s mathematical biography is a story that describes how she constructs her mathematical identity. I also analysed the key rhetoric that Sirpa used in her talk, key rhetoric here referring to the strategy by which she constructed continuity and coherence.     

MATHEMATICS EDUCATORS’ VIEWS ON THE ROLE OF MATHEMATICS LEARNING IN DEVELOPING DEDUCTIVE REASONING

This study examines the views of people involved in mathematics education regarding the commonly stated goal of using mathematics learning to develop deductive reasoning that is usable outside of mathematical contexts. The data source includes 21 individual semi-structured interviews. The findings of the study show that the interviewees ascribed different meanings to the above-stated goal. Moreover, none of them said that it is possible to develop formal logic-based reasoning useful outside of mathematics, but for different reasons. Three distinct views were identified: the intervention–argumentation view, the reservation–deductive view, and the spontaneity–systematic view. Each interviewee’s view was interrelated with the interviewee’s approach to deductive reasoning and its nature in mathematics and outside it.     

Yes…But: Footnotes To Sage Advice

In this commentary, I assume the role of a footnoter to the contributions in this special issue devoted to the development of educational psychologists. In those footnotes, insights and recommendations forwarded by the contributors are extended, clarified, or occasionally countered. Topics of dialogue include advice on defining a research program, supporting the professional development of women and underrepresented populations, achieving balance in one’s life, discussing reasons why academics publish, building collaborative relationships, and considering the mentor–mentee relationship.     

Using culture as a resource in mathematics: the case of four Mexican–American prospective teachers in a bilingual after-school program

This paper explores Mexican–American prospective teachers’ use of culture—defined as social practices and shared experiences—as an instructional resource in mathematics. The setting is an after-school mathematics program for the children of Mexican heritage. Qualitative analysis of the prospective teachers’ and children’s interactions reveals that the nature of the mathematical activities affected how culture was used. When working on the “binder activities,” prospective teachers used culture only in non-mathematical contexts. When working on the “recipes project,” however, culture was used as a resource in mathematical contexts. Implications for the mathematics teacher preparation of Latinas/os are discussed.     

Yup’ik Cosmology to School Mathematics: The Power of Symmetry and Proportional Measuring

This article shows how Yup’ik cosmology, epistemology, and everyday practice have implications for the teaching of school mathematics. Math in a Cultural Context (MCC) has a long–term collaborative relationship with Yup’ik elders and experienced Yup’ik teachers. Because of this long–term ethnographically–oriented relationship, the authors – both insiders and an outsider – have been able to understand the mathematical implications of everyday Yup’ik practice. As the article demonstrates, body proportional measuring and symmetry/splitting are two generative solution strategies used by Yup’ik elders in solving everyday problems. We argue that proportional measuring coupled with symmetry/splitting can provide school mathematics with an alternative pathway to the teaching of some aspects of geometry and rational number reasoning.     

What about choice for a change?

Through the lens of exile, this conceptual study explores the complex associations of choice and change, and their implications for education, in a range of areas from sociology through psychology and physiology. In bringing together and synthesising diverse thoughts about choice and change, it inquires into how educators and their professional practices can support the choices of young people in creating significant change and the hopeful future through knowledge economies. It concludes that, if the moral purposes of change are to be social and communal development, mental models should be chosen and knowledge created to accomplish those purposes. At stake is the legacy of today’s choices for young people’s futures. Dedicated to my Dad, Pieter, 19 April 1922–18 April 2004, ‘Recte faciendo neminem timeas’.