Time out of mind: Subben's checklist revisited: A partial description of the development of quantitative OR papers over a period of 25 years

This short paper aims to investigate some of the historical developments of one classic, well-cited and highly esteemed scientific journal in the domain of quantitative operations research - namely the INFORMS journal Operations Research - over a period of 25 years between 1981 and 2006. As such this paper, and the journal in question, represents one representative attempt to analyze - for the purpose of possible future generalization - how research production has evolved, and evolves, over time. Among the general developments that we think we can trace are that (a) the historical overviews (i.e., literature surveys) in the articles, as well as the list of references, somewhat counter-intuitively shrink over time, while (b) the motivating and modelling parts grow. We also attempt to characterize - in some detail - the appearance and character, over time, of the most cited, as well as the least cited, papers over the years studied. In particular, we find that many of the least cited papers are quite imbalanced. For example, some of them include one main section only, and the least cited papers also have shorter reference lists. We also analyse the articles' utilization of important buzz words representing the constitutive parts of an OR journal paper, based on Subben's checklist (Larsson and Patriksson, 2014, 2016). Based on a word count of these buzz words we conclude through a citation study, utilizing a collection of particularly highly or little cited papers, that there is a quite strong positive correlation between a journal paper being highly cited and its degree of utilization of this checklist

Rizanesander’s Recknekonsten or “The art of arithmetic” – the oldest known textbook of mathematics in Swedish

The present paper considers an important cultural treasure in the early Swedish history of mathematics education. After the reformation in the 16th century it became possible to study mathematics in Sweden. The first printed textbook in Swedish on arithmetic appeared in 1614, but already in 1601, the oldest known manuscript in Swedish on arithmetic was written by Hans Larsson Rizanesander. In this paper we investigate Rizanesander’s manuscript in its historical context.publishedVersio

Exploring student teachers’ instrumental genesis of programming

As a result of digital competence becoming more important in society, programming was introduced in the mathematics curriculum for Swedish upper secondary school in 2018. In the mathematics courses – where it is included – programming is described as a strategy for mathematical problem solving. By integrating programming and mathematical problem-solving, new opportunities for mathematics education in upper secondary school are opened. Nevertheless, the teaching of programming also implies new challenges and potential pitfalls. Since programming heretofore has only to a minor extent been part of the mathematics teacher education in Sweden, the new content of the curriculum also sets higher demands on the development of the teacher education program. The present study is part of a larger project, aiming at contributing to the research on the ongoing integration of programming into secondary school mathematics, through investigating in what way programming can offer additional possibilities for learning mathematics, compared to a more traditional education. In this study we investigate student teachers’ work with programming, in a calculus course in their first year of the secondary school teacher education program. To theoretically frame the study, we use the instrumental approach, in order to study students’ instrumental genesis, i.e., the process where an instrument is formed from an artefact, when using programming as a tool in mathematics (Trouche, 2004). The instrumental genesis consists of two important processes: the instrumentalization, which is the process where the user gets to know the tool, and the instrumentation, which is the process that allows the user to develop an activity within some boundaries. Most students enrolled at the secondary teacher education program at the University of Gothenburg do not have any, or have very little, prior experience in programming. In the calculus course the students take part in a computer lab where they – with the help of Python – are asked to explore Riemann-sums of continuous functions. Through observations during the students’ work with exercises, and through a follow-up questionnaire, we explore the potentials for learning mathematics through programming. In particular, we investigate what difficulties regarding programming and mathematical content the students encounter during the beginning of their instrumental genesis. A majority of the students answering the questionnaire argued that the programming part of the lab was difficult, but that it helped them to gain a deeper understanding of Riemann-sums. Some students argued that, to be able to construct a correct program, they had to decompose the concept of Riemann-sums in order to understand how they are structured.Reference Trouche, L. (2004). Managing the complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning, 9(3), 281–307

Student teachers\u27 learning and teaching mathematics with programming

Sweden’s school curriculum was revised to include programming as a mathematical content from the year 2018. However, teacher education included using IT-tools for teaching, but not specifically programming. To correct this situation, we have developed a new programming strand in the training of secondary school mathematics teachers at the university of Gothenburg. The aim of this study is to observe how student teachers towards the end of the program use their knowledge to plan uses of programming in their own mathematics teaching. It is part of a larger research and development project on introducing programming in secondary school mathematics education. Two theoretical frameworks are relevant here: instrumental genesis, i.e., the process where an instrument is formed from an artefact, when students use programming as a tool in mathematics (Trouche 2004), and the theory of didactic transposition, to frame the student teacher\u27s transformation of their own knowledge into knowledge to be taught (Chevallard 2006). Each mathematics course in the teacher education program for secondary teachers at the university of Gothenburg contains a few computer lab sessions. About half of them are focused on using IT-tools (e.g. Geogebra) to learn mathematics. The other half of the computer labs use programming to highlight mathematical concepts. There is no course in programming per se. In the first mathematics course, the students build a first block program in Scratch to draw regular polygons, using loops and variables. In the second course, in Calculus, they estimate integrals with Riemann sums, using loops in Python. The strand then continues in Python, with data analysis in Statistics and prime numbers and cryptography in Number theory. Four school practice periods are spread throughout\ua0the program, and in the third of these, student teachers plan a mathematics lesson with programming, to use with their school class. In this study we analyze the student’s lesson plans and discussion in a seminar where the students discuss their lesson design. Of interest is both the design itself and the students’ attitude, self-efficacy and reflections regarding their teaching of mathematics through programming. In particular, we are interested in the students’ argumentation on how their planned lessons may help the pupils to achieve the learning goals. Preliminary results indicate that students struggle to meet the double goal of introducing programming and supporting the mathematics in the curriculum. References: Chevallard, Y. (2006). Steps towards a new epistemology in mathematics education. In Bosch, M. Proceedings of the 4th Conference of the European Society for Research in Mathematics Education (CERME 4) (pp. 21–30). Trouche, L. (2004). Managing the complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning, 9(3), 281–307

Writing the History of Mathematics: Interpretations of the Mathematics of the Past and Its Relation to theMathematics of Today

In the present chapter, interpretations of the mathematics of the past are problematized, based on examples such as archeological artifacts, as well as written sources from the ancient Egyptian, Babylonian, and Greek civilizations. The distinction between history and heritage is considered in relation to Euler’s function concept, Cauchy’s sum theorem, and the Unguru debate. Also, the distinction between the historical past and the practical past, as well as the distinction between the historical and the nonhistorical relations to the past, are made concrete based on Torricelli’s result on an infinitely long solid from the seventeenth century. Two complementary but different ways of analyzing\ua0the mathematics of the past are the synchronic and diachronic perspectives, which may be useful, for instance, regarding the history of school mathematics. Furthermore, recapitulation, or the belief that students’ conceptual development in mathematics is paralleled to the historical epistemology of mathematics, is problematized emphasizing the important role of culture

The role of Swedish school algebra in a historical perspective

Solid knowledge of algebra is vital to manage mathematics at upper secondary and university level. Traditionally, algebra has been postponed until adolescence because of former assumptions that young children would not be cognitively capable of thinking algebraically. However, recent research reveals that it is possible and even beneficial to start working with algebra already in early grades (Blanton et al., 2015). This has influenced school mathematics where many countries have revised their syllabuses in order to incorporate algebra in primary school. In Sweden algebra is a part of mathematics that causes pupils major difficulties. In international evaluations, Swedish pupils have performed below the international average in algebra since the 1960s. Although there has been various attempts to improve school algebra teaching the results in algebra have not improved. The overall purpose of the present study is to contribute to the international research field regarding the complex issue of implementing algebra in early school mathematics by investigating the Swedish case. More specifically, we examine how algebra is traditionally treated in the last five Swedish syllabuses for grades 1–9 from 1962, 1969, 1980, 1994, and 2011. The study is part of a broader research project aiming at characterizing Swedish school algebra on both formulation and realization arenas (Hemmi et al., 2018). The project is theoretically embedded in Bernstein’s theory about classification and framing of educational knowledge.In order to characterize the algebraic content as well as to investigate what role algebra plays in school mathematics we have conducted a qualitative content analysis where Blanton et al.’s (2015) five big ideas of algebra have been applied as an analytical tool. The big ideas are: Expressions and equations, Generalized arithmetic, Functional thinking, Variables, and Proportional reasoning.The results show both similarities and differences between the syllabuses. For instance, in the 1980 syllabus algebra represents a very small part of the mathematical content, especially compared with the 2011 syllabus where algebra is emphasized already from earlier grades. All five syllabuses emphasize the importance of everyday mathematics and the practical use of mathematics in contexts relevant for the students. However, there are differences regarding which role algebra plays in everyday life. The 1980 syllabus states that algebra is less important in everyday life and students only need a “certain orientation” of algebra, which is probably a reaction to the great focus on abstract mathematics of “New math” in the 1969 syllabus. In the 2011 syllabus everyday life appears frequently within the algebraic content. A common feature of all five syllabuses is the weak emphasis on the big idea generalized arithmetic.References:Blanton, et al. (2015). The development of children’s algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46(1), 39–87.Hemmi, et al. (2018). Characterizing Swedish school algebra – initial findings from analyses of steering documents, textbooks and teachers’ discourses. In Papers of NORMA 17, The Eighth Nordic Conference on Mathematics Education (pp. 299-308). Stockholm

The role of Swedish school algebra in a historical perspective

Solid knowledge of algebra is vital to manage mathematics at upper secondary and university level. Traditionally, algebra has been postponed until adolescence because of former assumptions that young children would not be cognitively capable of thinking algebraically. However, recent research reveals that it is possible and even beneficial to start working with algebra already in early grades (Blanton et al., 2015). This has influenced school mathematics where many countries have revised their syllabuses in order to incorporate algebra in primary school. In Sweden algebra is a part of mathematics that causes pupils major difficulties. In international evaluations, Swedish pupils have performed below the international average in algebra since the 1960s. Although there has been various attempts to improve school algebra teaching the results in algebra have not improved. The overall purpose of the present study is to contribute to the international research field regarding the complex issue of implementing algebra in early school mathematics by investigating the Swedish case. More specifically, we examine how algebra is traditionally treated in the last five Swedish syllabuses for grades 1–9 from 1962, 1969, 1980, 1994, and 2011. The study is part of a broader research project aiming at characterizing Swedish school algebra on both formulation and realization arenas (Hemmi et al., 2018). The project is theoretically embedded in Bernstein’s theory about classification and framing of educational knowledge.In order to characterize the algebraic content as well as to investigate what role algebra plays in school mathematics we have conducted a qualitative content analysis where Blanton et al.’s (2015) five big ideas of algebra have been applied as an analytical tool. The big ideas are: Expressions and equations, Generalized arithmetic, Functional thinking, Variables, and Proportional reasoning.The results show both similarities and differences between the syllabuses. For instance, in the 1980 syllabus algebra represents a very small part of the mathematical content, especially compared with the 2011 syllabus where algebra is emphasized already from earlier grades. All five syllabuses emphasize the importance of everyday mathematics and the practical use of mathematics in contexts relevant for the students. However, there are differences regarding which role algebra plays in everyday life. The 1980 syllabus states that algebra is less important in everyday life and students only need a “certain orientation” of algebra, which is probably a reaction to the great focus on abstract mathematics of “New math” in the 1969 syllabus. In the 2011 syllabus everyday life appears frequently within the algebraic content. A common feature of all five syllabuses is the weak emphasis on the big idea generalized arithmetic.References:Blanton, et al. (2015). The development of children’s algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46(1), 39–87.Hemmi, et al. (2018). Characterizing Swedish school algebra – initial findings from analyses of steering documents, textbooks and teachers’ discourses. In Papers of NORMA 17, The Eighth Nordic Conference on Mathematics Education (pp. 299-308). Stockholm

Characterizing Swedish school algebra – initial findings from analyses of steering documents, textbooks and teachers’ discourses

The paper reports the first results of an ongoing research project aiming at characterizing Swedish school algebra (grades 1-9). Both diachronic and synchronic studies are conducted to identify the specific teaching tradition developed in Sweden and different theoretical approaches are applied in the overall project in order to obtain a rich picture of the Swedish case. The results reported here are based on the analyses of mathematics curriculum, textbooks and focus group interviews with teachers in seven schools. The initial results indicate that, since 1980s, algebra is vaguely addressed in the steering documents and the progression of algebraic thinking is elusive in teachers\u27 discourses. Moreover, certain important ideas, such as generalized arithmetic, are largely missing in the curriculum and mathematics textbooks for grades 1-6. We discuss the implications of the initial findings for our project

Exploring students’ procedural flexibility in three countries

BackgroundIn this cross-national study, Spanish, Finnish, and Swedish middle and high school students’ procedural flexibility was examined, with the specific intent of determining whether and how students’ equation-solving accuracy and flexibility varied by country, age, and/or academic track. The 791 student participants were asked to solve twelve linear equations, provide multiple strategies for each equation, and select the best strategy from among their own strategies.ResultsOur results indicate that knowledge and use of the standard algorithm for solving linear equations is quite widespread across students in all three countries, but that there exists substantial within-country variation as well as between-country variation in students’ reliance on standard vs. situationally appropriate strategies. In addition, we found correlations between equation-solving accuracy and students’ flexibility in all three countries but to different degrees.ConclusionsAlthough it is increasingly recognized as an important construct of interest, there are many aspects of mathematical flexibility that are not well-understood. Particularly lacking in the literature on flexibility are studies that explore similarities and differences in students’ repertoire of strategies for solving algebra problems across countries with different educational systems and curricula. This study yielded important insights about flexibility and can push the field to explore the extent that within- and between-country differences in flexibility can be linked to differences in countries’ educational systems, teaching practices, and/or cultural norms around mathematics teaching and learning