Bridging History of the Concept of Function with Learning of Mathematics: Students’ Meta-Discursive Rules,Concept Formation and Historical Awareness

In this paper we present a matrix-organised implementation of an experimental course in the history of the concept of a function. The course was implemented in a Danish high school. One of the aims was to bridge history of mathematics with the teaching and learning of mathematics. The course was designed using the theoretical frameworks of a multiple perspective approach to history, Sfard’s theory of thinking as communicating, and theories from mathematics education about concept image, concept definition and concept formation. It will be explained how history and extracts of original sources by Euler from 1748 and Dirichlet from 1837 were used to (1) reveal students’ meta-discursive rules in mathematics and make them objects of students’ reflections, (2) support students’ learning of the concept of a function, and (3) develop students’ historical awareness. The results show that it is possible to diagnose (some) of students’ meta-discursive rules, that some of the students acted according to meta-discursive rules that coincide with Euler’s from the 1700s, and that reading a part of a text by Dirichlet from 1837 created obstacles for the students that can be referenced to differences in meta-discursive rules. The experiment revealed that many of the students have a concept image that was in accordance with Euler’s rather than with our modern concept definition and that they have process oriented thinking about functions. The students’ historical awareness was developed through the course with respect to actors’ influence on the formation of mathematical concepts and the notions of internal and external driving forces in the historical development of mathematics.     

New Avenues for History in Mathematics Education: Mathematical Competencies and Anchoring

The paper addresses the apparent lack of impact of ‘history in mathematics education’ in mathematics education research in general, and proposes new avenues for research. We identify two general scenarios of integrating history in mathematics education that each gives rise to different problems. The first scenario occurs when history is used as a ‘tool’ for the learning and teaching of mathematics, the second when history of mathematics as a ‘goal’ is pursued as an integral part of mathematics education. We introduce a multiple-perspective approach to history, and suggest that research on history in mathematics education follows one of two different avenues in dealing with these scenarios. The first is to focus on students’ development of mathematical competencies when history is used a tool for the learning of curriculum-dictated mathematical in-issues. A framework for this is described. Secondly, when using history as a goal it is argued that an anchoring of the meta-issues in the related in-issues is essential, and a framework for this is given. Both frameworks are illustrated through empirical examples.     

Gains From Training in Phonological Awareness in Kindergarten Predict Reading Comprehension in Grade 9

The effects of a kindergarten training program in phonological awareness with 209 Swedish-speaking children were followed up until the end of Grade 9. Initial levels of letter knowledge and phonological awareness were positively associated with the level of decoding skill in Grade 3 but not with its growth afterward. The intervention group performed significantly better in decoding in Grade 3, and the difference was maintained until Grade 6. The trained children also scored higher in Grade 9 reading comprehension. Although the results give empirical support for a connection between early phonological awareness training, later word decoding development, and still later reading comprehension, the theoretical explanation for the link between especially word decoding and reading comprehension is far from clear.     

Developing Students’ Reflections on the Function and Status of Mathematical Modeling in Different Scientific Practices: History as a Provider of Cases

Mathematical models and mathematical modeling play different roles in the different areas and problems in which they are used. The function and status of mathematical modeling and models in the different areas depend on the scientific practice as well as the underlying philosophical and theoretical position held by the modeler(s) and the practitioners in the extra-mathematical domain. For students to experience the significance of different scientific practices and cultures for the function and status of mathematical modeling in other sciences, students need to be placed in didactical situations where such differences are exposed and made into explicit objects of their reflections. It can be difficult to create such situations in the teaching of contemporary science in which modeling is part of the culture. In this paper we show how history can serve as a means for students to be engaged in situations in which they can experience and be challenged to reflect upon and criticize, the use of modeling and the significance of the context for the function and status of modeling and models in scientific practices. We present Nicolas Rashevsky’s model of cell division from the 1930s together with a discussion of disagreement between him and some biologists as one such episode from the past. We illustrate how a group of science students at Roskilde University, through their work with this historical case, experienced that different scientific cultures have different opinions of the value of a model as an instrument for gaining scientific knowledge; that the explanatory power of a model is linked not only to the context of its use, but also to the underlying philosophical and theoretical position held by the modeler(s) and the scientists discussing the model and its use. The episode’s potential to challenge students to reflect upon and criticize the modeling process and the function of models in an extra mathematical domain is discussed with respect to the notions of internal and external reflections.     

Coordinating quality practices in Direct Trade coffee

Over the past few decades, many food niches have emerged with a specific focus on quality. In specialty coffee, micro roasters have brought about Direct Trade coffee as a way of organising an alternative around new tastes and qualities through ongoing and ‘direct’ relations to farmers and cooperatives. But Direct Trade also involves exporters. We ask, how do exporters and roasters work together in these new coffee relations, and what do they work on? We observe and participate in a situation where Colombian coffee exporters visit Danish roasters. They tour the roasting facilities and taste a number of coffees. Often, the term power is used to analyse such value chain interactions, but we argue that the term coordination better opens up these interactions for exploration and analysis. What emerges is a coordination of quality. Through touring and tasting, issues emerge and differences are laid out. We learn that quality is a continuous achievement. There is friction between the ways in which the roasters and exporters do quality, but these are not done away with through power. They are made known and discussable through the work of coordination. The activity of tasting quality is a coordination device that allows for bringing out differences in how quality is done in practice. Coffee, in this event, is not a fixed object, but shifts as issues of quality are brought up in tasting. This suggests a decentering of the object on the issue of quality.     

Beyond motivation: history as a method for learning meta-discursive rules in mathematics

In this paper, we argue that history might have a profound role to play for learning mathematics by providing a self-evident (if not indispensable) strategy for revealing meta-discursive rules in mathematics and turning them into explicit objects of reflection for students. Our argument is based on Sfard’s theory of Thinking as Communicating, combined with ideas from historiography of mathematics regarding a multiple perspective approach to the history of practices of mathematics. We analyse two project reports from a cohort of history of mathematics projects performed by students at Roskilde University. These project reports constitute the experiential and empirical basis for our claims. The project reports are analysed with respect to students’ reflections about meta-discursive rules to illustrate how and in what sense history can be used in mathematics education to facilitate the development of students’ meta-discursive rules of mathematical discourse.