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The purpose of this paper is to explore students' mathematization through a flow of lessons using the Thailand Lesson Study Incorporated Open Approach (TLSOA) to improve the excellence of instruction.

A total of 16 Grade 4 students were selected because they have been taught using the TLSOA model for four years. Six Lesson Study (LS) team members participated, and two instruments were utilized, namely student worksheets, and field notes. An ethnographic research design was employed.

The results revealed that the students' mathematical ideas were developed from the real world to the mathematical world through a flow of lessons based on the four phases of the Open Approach (OA).

Firstly, the students demonstrated their ability to represent the real world independently when the teacher posed an open-ended problem. Secondly, the students demonstrated their ability to use semi-concrete aids to develop their ideas while self-learning. Thirdly, the students showed how they developed their ideas to solve the open-ended problem using relevant objects or related concepts as part of a whole-class discussion and comparison exercise. Finally, the students demonstrated their abilities to represent the mathematical world using numbers and symbols to communicate their ideas when they were required to make a summary by connecting their mathematical ideas.

This study adds new insight to the literature on students' mathematization using the TLSOA model.

Attempts to throw some light on the sensible use of mathematics in economic theory. Argues that mathematics is a valuable and useful tool which economists should and must apply as long as its use is economically sensible. The dangers of going beyond the “frontier” of what is economically sensible occur when economists depart from the actual (empirical) subject matter because of the applied mathematical instruments, when the underlying value judgements are not, or only insufficiently, taken into consideration, when the recording and measurement of empirical magnitudes as an economic problem is underestimated or is even subordinate under the requirements of the formal language, and when the process of mathematization is considered as a substitute for the process of Verstehen. Concludes that although mathematical reasoning is one way of logical deduction, which secures a style of logical consistency in reasoning, it is a fallacy to believe that mathematical reasoning alone can secure logical, consistent reasoning. Mathematization for the sake of mathematization is useless.

The purpose of this chapter is to study the mathematisation of finance – excessive use of mathematical models in finance – which has been widely blamed for the recent financial and economic crisis. We argue that the problem might actually be the financialisation of mathematics, as evidenced by the gradual embedding of branches of mathematics into financial economics. The concept of embeddedness, originally proposed by Polanyi, is relevant to describe the sociological relationship between fields of knowledge. After exploring the relationship between mathematics, finance and economics since antiquity, we find that theoretical developments in the 1950s and 1970s lead directly to this embedding. The key implication of our findings is the realization that it has become necessary to disembed mathematics from finance and economics, and proposes a number of partial steps to facilitate this process. This chapter contributes to the debate on the mathematisation of finance by uniquely combining a historical approach, which chronicles the evolution of the relation between mathematics and finance, with a sociological approach from the perspective of Polyani’s concept of embedding.

Argues preliminarily that quantitative‐mathematical social science, including economics, is not possible because it applies a method useful in other areas to a field to which it cannot be applied and because the truth claim of science so conceived is self‐referential to begin with. The argument is primarily based on the classic Gadamerian hermeneutic critique of the natural sciences and on the conception of the social sciences as related to phronésis.

The authors investigate the use and potential of a theoretical combination of Realistic Mathematics Education (RME) and the Theory of Didactic Situation (TDS) to support Lesson Study (LS) in upper secondary mathematics.

Case study performed by university researchers, based on theoretical analysis and case studies based on documents and observation from lesson studies.

Even within a project lasting just about three years, teachers (with no preliminary experience of lesson study) engaged in lesson design based on the combination of theoretical perspectives from TDS and RME in ways that confirm the potential of that combination to enrich and focus teachers' professional development within the framework of LS . It is not clear to what extent the intensive and continued engagement of university researchers has been or would be essential for similar and longitudinal realizations of these potentials.

As current European frameworks seek to engage researchers and teachers in collaboration and exchange across countries, networking of major paradigms of research (like TDS and RME) and uses of them as supports for teachers' inquiry (like demonstrated in this paper) is of considerable institutional interest and potential impact on schools.

Teachers' Inquiry in Mathematics Education (TIME) is a prerequisite for the development of Inquiry Based Mathematics Education, which in turn is required in many countries across the world, with the aim of fostering critical and competent citizens.

This combination of (major) mathematics education theories to support and enrich LS has not previously been investigated. While several aspects of adapting to LS Western contexts have been investigated in the past, including the inclusion of perspectives and tools from academic research, the role of university researchers is also quite open. While authors do not offer a systematic study of this role, authors examine how this role may involve development of new practical combinations of different, complementary theoretical tools, which indeed hold potential to support lesson study in a European context.

Seeks to define the proper role for mathematics to play in economic theorizing by spelling out its limits. Specifically, has the mathematization of economics contributed to its narrowing of scope since the anti‐classical reaction of the 1870s? If so, is mathematics inherently narrow? Or does the modern, and often other‐worldly, treatment follow from the particular way in which mathematics has been applied? To describe “bad” mathematical economics, one must state what kinds of problems its formulations tend to exclude. This paper focuses on “equilibrium” analysis that rules out such real‐world phenomena as instability and economic polarization, thereby diverting attention from the actual structural problems at work.

Econometrics labours under the same limitations as economics: it rests on unrealistic hypotheses (and non‐operational concepts) and is isolated from other sciences. It should try to test economic hypotheses and estimate relationships that constitute theory, notwithstanding the poor available data. Many econometrists are no longer interested in measurement, but in art for art’s sake: econometrics becomes abstract mathematisation.

In this chapter, the interplay between the development of the discipline, the development of the field of study, and the emergence of professional fields is examined using the example of mathematics. In connection with the formation of the modern research university, mathematics has emerged as an independent scientific discipline and as an independent field of study. In the process, mathematics attains a high degree of formalization and internal coherence. This is the basis for the penetration of mathematicians into more and more professional fields, even outside science. Real problems or real facts are reduced to aspects that are amenable to mathematical modeling by treating them as quantifiable parameters. As mathematics expands as a field of study, more and more professional sectors become applications of mathematical models. As a consequence, more mathematical fields of study are differentiating themselves, specializing in these application fields. This chapter analyzes this dynamic and its preconditions.

The aim of this paper is to explore the lessons offered by the financial crisis about the appropriate epistemological approaches to apply in the study of human affairs in general, and of the financial markets in particular.

The paper applies a qualitative and historical approach invoking debates in the philosophy of social science to dominant themes and concepts in modern quantitative finance. It is argued that underlying the theory and practice of modern quantitative finance is a commitment to an empiricist epistemology modeled on the natural sciences.

In the financial crisis, modern social science, with its positivist/quantitative orientation, was put to the test in a way that it had never been before. That it failed this test is one of the chief lessons of the financial crisis. Mathematical techniques are inherently incapable of accounting for human behaviour. The crisis serves to underline that a fundamental divide exists between the natural and human realms.

While the mathematical‐positivist techniques continue to hold some promise in the study of finance, it has become obvious that this dominant approach needs be enhanced by more qualitative techniques.

The paper shows that although in popular media outlets the mathematization of finance has been singled out as a cause of the crisis, the broader implications for the analysis of human activity have not yet been probed. Nor, in the wake of the crisis, has there been a systematic, philosophically informed, critique of the positivist‐quantitative orientation buttressing academic research into the financial markets.

The purpose of this contribution is the investigation and the assessment of François Perroux’s book – Unités Actives et Mathématiques Nouvelles – Révision de la théorie de l’équilibre économique général (Perroux, 1975a) – in connection with the research program he initiated before and during the Second World War. Concerning the analytical relevance of this book there is no consensus. According to some economists it has to be considered as the masterpiece of Perroux’s intellectual project which provides an alternative approach to the usual theory of general economic equilibrium (GEE). Others think that today the book is almost entirely forgotten and it does not help really to improve Perroux’s general research project. We try to set here our own view combining two ways of proceeding in the investigation about the relevance of the book. The first one develops an analysis of Unités Actives as dispassionate as possible, avoiding taking into account the personality as well as the social and political views of François Perroux. The second favors a retrospective approach relating the 1975 book to the previous contributions of the author. The first three sections of our contribution are dedicated to Perroux’s contributions written before his attempt to “revise” the usual version of the GEE theory. The five following sections are devoted to the direct contents of the “revision” of this theory and tries to propose a global assessment of it.