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The purpose of this paper is to explore three different types of digital environments for mathematics learning that may support mathematical play and the failure and feedback mechanics present in each.

Interaction analysis and the lenses of failure, feedback and mathematical play are used to analyze the mathematical interactions afforded by three different digital environments.

Each digital environment supports or restrains the potential for mathematical play through mathematical representations, failure and feedback.

The primary contribution of this paper is to highlight different ways in which digital failure and feedback designs can influence the emergent experience of mathematical play.

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.

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.

Current ethnographic research is marked by an expanding variety of approaches that indicates not only the infusion of paradigmatic proliferation into the field but also the expansion of technologies mediating ethnographic exploration as well as the growth in research agendas of educational research. Current researchers therefore have much to draw upon but the diversity of approaches indicates and presents major challenges (Walford, 2002). In particular, the double crisis of representation and legitimation that has become apparent in qualitative research has problematized the very possibility of valid/trustworthy/authentic/useful research. Thus, before an ethnographer even steps one foot, gingerly, in the research site, she would have had to grapple with weighty issues, such as the transparency of language, that have considerable bearing on the formulation of research questions and on the construction of an initial frame for her study. More importantly, the agency of the researcher, as well as the agency of her research subjects – vis-à-vis the structural constraints of epistemic access to the nature of reality – has been cast into doubt, as has the capacity of the researcher and the research subjects to produce a meaningful, dare we say truthful, account of their ways of knowing.

The purpose of this paper is to examine how lesson study can lead to teacher learning and improvement of instruction by developing teachers’ mathematical-task knowledge and by supporting teachers’ selection, modification, and implementation of mathematical tasks. Mathematical-task knowledge includes knowledge needed to use tasks that require a high level of thinking and reasoning.

A qualitative case study design was used to explore the learning and instruction of three teachers as they went through the process of lesson study, developed knowledge around mathematics tasks, and made changes to their instruction. Methods included direct observation of lessons, semi-structured interviews, and participant observation.

This lesson study project supported teachers in developing mathematical-task knowledge and in making change to instructional practice. The teachers discussed in this paper added to their understanding of mathematical tasks and changed how they implemented tasks after lesson study. The teachers began to challenge students to go beyond memorizing or executing procedures to deepen the students’ understanding of mathematical concepts. Teachers developed key insights and understandings of mathematical tasks, triggering shifts in their thinking, and changes to instruction. Collaboration and reflection altered the selection, modification, and implementation of mathematical tasks.

This study reveals the connections between features of lesson study and the pathways that lead to learning and improvements to instruction. Limitations included the demands of the school district’s pacing guide and curriculum, and a limited number of interviews and observations were conducted after lesson study.

There are many different ways lesson study has been implemented in the USA, yet the effectiveness of many lesson study projects is still unclear. This study reveals more about the lesson study process, what features are important, and how these features lead to development of knowledge and practice. This study examines how teachers within the same lesson study group added to their knowledge and practice and how different features of lesson study prompt them to make changes.

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 paper is to apply the exp‐function method to construct exact solutions of nonlinear wave equations. The proposed technique is tested on the (2+1) and (3+1) dimensional extended shallow water wave equations. These equations play a very important role in mathematical physics and engineering sciences.

In this paper, the authors apply the exp‐function method to construct exact solutions of nonlinear wave equations.

In total, four forms of the extended shallow water wave equation have been studied, from the point of view of its exact solutions using computational method. Exp‐function method was employed to achieve the goal set for this work. The applied method will be used in further works to establish more entirely new solutions for other kinds of nonlinear wave equations. Finally, it is worthwhile to mention that the proposed method is straightforward, concise, and it is a promising and powerful new method for other nonlinear wave equations in mathematical physics.

The algorithm suggested in the paper is quite efficient and is practically well suited for use in these problems. The method is straightforward and concise, and its applications are promising.

The equation of unified knowledge says that S = f (A,P) which means that the practical solution to a given problem is a function of the existing, empirical, actual realities and the future, potential, best possible conditions of general stable equilibrium which both pure and practical reason, exhaustive in the Kantian sense, show as being within the realm of potential realities beyond any doubt. The first classical revolution in economic thinking, included in factor “P” of the equation, conceived the economic and financial problems in terms of a model of ideal conditions of stable equilibrium but neglected the full consideration of the existing, actual conditions. That is the main reason why, in the end, it failed. The second modern revolution, included in factor “A” of the equation, conceived the economic and financial problems in terms of the existing, actual conditions, usually in disequilibrium or unstable equilibrium (in case of stagnation) and neglected the sense of right direction expressed in factor “P” or the realization of general, stable equilibrium. That is the main reason why the modern revolution failed in the past and is failing in front of our eyes in the present. The equation of unified knowledge, perceived as a sui generis synthesis between classical and modern thinking has been applied rigorously and systematically in writing the enclosed American‐British economic, monetary, financial and social stabilization plans. In the final analysis, a new economic philosophy, based on a synthesis between classical and modern thinking, called here the new economics of unified knowledge, is applied to solve the malaise of the twentieth century which resulted from a confusion between thinking in terms of stable equilibrium on the one hand and disequilibrium or unstable equilibrium on the other.