Härledda enheter ligger i området proportionalitet, vilket matematikdidaktisk forskning betraktar som både svårt att undervisa och svårt att lära sig. I ett regionalt skolutvecklingsprojekt omformade vi högskolematematikens enhetsanalys (dimensionsanalys) till att passa åk 7-gy då grundidén är att betrakta en storhets enhet, exempelvis kr/kg, och dra slutsatser om hur storheten (kvot eller produkt) ska beräknas. I projektet utvecklade vi både övningsproblem och undervisningsmetoder för detta.

Abstract

In its elementary form, the idea of dimensional analysis is to look at the unit of a magnitude, for example speed, and conclude that speed measured in m/s is a quotient having meters in the numerator and seconds in the denominator. However, despite dimensional analysis being a not un-common theme for research in mathematics education at tertiary level teaching, it seems in practice absent below that level. This is surprising since dimensional analysis is useful in mathematical problem solving involving proportionality. Even more so since, from halves and doubles in preschool to scalar multiplication in upper secondary school, proportionality permeates school mathematics and is well-researched (Lamon, 2007). Hence, the authors initiated a research project on how to teach dimensional analysis in school years 6-12. The novelty of this area made us choose didactical engineering (Artigue, 2015) as a theoretical framework suitable for developing the teaching of dimensional analysis. Accordingly, in a cycle of a priori and a posteriori analyses, the authors are developing an empirically based teaching model for dimensional analysis including the topic of constructing exercises for students. We present results from this on-going development project. 

References

Artigue, M. (2015). Perspectives on Design Research: The Case of Didactical Engineering. In A. Bikner-Ahsbahs, C. Knipping, N. Presmeg. (Eds.). Approaches to Qualitative Research in Mathematics Education. Dordrecht: Springer.

Lamon, S. J. (2007). Rational numbers and proportional reasoning: Towards a theoretical framework. In F.K. Lester (red) Second handbook of research on mathematics teaching and learning, vol 2 (pp. 629–668). Charlotte, NC: NCTM.

The point of departure in this paper is my previous research in which I analysed how the idea of problem solving is recontextualised into the mathematics curriculum for upper secondary school in Sweden and how this increases the risk for excluding lower SES students from future power. I discuss how this research could be followed up through a suggestion of how problem solving could be viewed in three different ways: as an ideology, a competence and an activity. Bernstein's pedagogic device and dichotomy of vertical and horizontal discourses are crucial in this suggestion. By seeing problem solving as an activity, and connect this to what Bernstein labels the evaluative field, I thereby tie the whole pedagogic device together by taking an overall view of problem solving as global policy-speak.

Similar to other curricula, the Swedish mathematics curriculum emphasises problem solving both as an end in itself and as a means to becoming a competent citizen. Thus, a goal for mathematics is the creation of a problem-solving citizen. In this paper, I explore how critical discourse analysis, and parts of social activity theory can be used to operationalise Bernstein’s pedagogic device in relation to the construction of the problem-solving citizen. It is proposed that critical discourse analysis can be used for a linguistic analysis of official documents, like the curriculum whilst social activity theory’s different domains, mainly public and esoteric, can be used to analyse the national tests. These tests assess students’ problem solving both as a means and an end in two different senior secondary programmes, an academic orientated one and a vocational one. In this exploration, two examples are given to show how these methodological tools can be used.  

In this article, Bernstein’s vertical and horizontal discourses are used to identify tensions that arise from recontextualising global policyspeak about problem solving into the mathematics curriculum for upper secondary schools in Sweden. The new curriculum in Sweden, launched in 2011, is used as an exemplar of a wider global trend within mathematics education which emphasises competences and especially problem-solving competence. An interdiscursive analysis of the curriculum is carried out together with an analysis of the national tests where problem-solving tasks are categorised as pure or applied tasks originating from the vertical and horizontal discourses, respectively. It is argued that an equal distribution of and access to the vertical discourse is crucial for the goals of schooling to be achieved, but the present curriculum is likely to contribute to students from low socioeconomic background having fewer opportunities to gain entry into this discourse. Therefore, there is a risk that class difference and social inequality are reinforced, and reproduced via mathematics education, despite proclamations of contrary intentions. 

The present thesis is made up by three articles and in all of these the mathematics curriculum for upper secondary school in Sweden is analysed. The main focus is the citizen and citizenship and the point of departure is problem solving as a competence. Besides an investigation of the connection between citizenship and the curricu- lum or the role the citizen have in the curriculum, questions about what tensions appear when problem solving is recontextualised in- to the curriculum are posed. Following an international trend in (mathematics) education, the mathematics curriculum in Sweden stresses demands made on the students and citizens instead of rights that the students or citizens have. Demands that everyone must become problem-solving citizens. By the use of Bernstein’s theories about the pedagogic device and his division of different knowledge forms into a vertical and a horizontal discourse, I inves- tigate possible effects of these demands. Despite intentions that all should be included, I show that there is a risk for exclusion instead. Bernstein suggested that school reproduces social inequity. In this thesis I discuss how this is done in the curriculum. My conclusion points at a risk of segregation and exclusion of lower socio- economic groups from influence, power and control. Furthermore, the reproduction of social inequity is build more solidly into the system with the new curriculum as although it is unclear whether the purpose of the changes to the curriculum was really to divide groups and exclude some from power.

In the poster, I describe how different agents (curriculum makers, national test makers and teachers as test makers) make use of the different kinds of problems (pure/applied) in relation to the goal of producing problem solving citizens. I also report on a pilot study in which students were interviewed on their perceptions about problems and problem solving. My initial analysis shows that discussions of problem solving, as well as descriptions of the kinds of problems that are or should be used in policy documents are ambiguous. Applied problems seem more commonly used for teaching and assessing problem solving. The pilot study suggests that students can see the difference between pure and applied problems and recognise that this difference is important in regard to their future as problem-solving citizens.

In this article, the mathematics needed for citizenship is discussed in relation to the Swedish curriculum. The article considers two approaches for discussing mathematics as demanded by, or developed within, a society: mathematical literacy and ethnomathematics. These approaches provide an alternative un-derstanding for school mathematics in relation to citizenship. In reconsidering the expectations upon the future citizen produced from implementing the cur-riculum, an argument is made for the curriculum to include elements from critical and socially responsible mathematics education, which include ele-ments of ethnomathematics and mathematical literacy. Such reconsideration is necessary because the transfer of mathematics from school to the outside world is not a straightforward matter. Therefore, it is essential that more focus is directed at citizens in the curriculum, and the transitions they undertake during their trajectories in life, to and from school.