Eigentheory concepts are central in mathematics and physics; they serve multiple functions, such as symbolizing physical phenomena and facilitating mathematical computations. Words and meanings associated with eigentheory develop and vary over time, as do their associated symbols. In this study, we investigate how "eigen" develops over…

The concept of function is critical in mathematics in general and abstract algebra in particular. We observe, however, that much of the research on functions in abstract algebra (1) reports widespread student difficulties, and (2) focuses on specific types of functions, including binary operation, homomorphism, and isomorphism. Direct, detailed…

In many advanced mathematics courses, comprehending theorems and proofs is an essential activity for both students and mathematicians. Such activity requires readers to draw on relevant meanings for the concepts involved; however, the ways that concept meaning may shape comprehension activity is currently undertheorized. In this paper, we share a…

This paper reports on students' conceptions of minima points. Written assignments and individual interviews uncovered salient, concept images, as well as erroneous "mis-out examples" that mistakenly regard examples as non-examples and "mis-in examples" that mistakenly grant non-examples the status of examples. We used Tall and…

Recent research on integration has shown the importance of quantities-based meanings for integrals. However, this research body is still in need of detailed empirical accounts of "how" students develop such understandings across an entire unit on integration. This paper contributes by providing one such account, based on a…

Industrial-scale models require considerable setup time; hence, once built, they are used in myriad ways to consider closely related cases. In practice, the code for these models frequently evolves without appropriate notational choices, largely as a result of the lengthy development time of, and the number of individuals contributing to, their…

In the early 1980s, the "theory of didactic situations" put the analysis of mathematical activity at the core of the didactic analysis, thus initiating a new research programme in mathematics education: the "epistemological programme." This paper describes and interprets some of the contributions of the "anthropological…

Functions are an essential concept in mathematics. The studies that have examined functions in advanced contexts have primarily focused on students' reasoning about specific types of functions (such as binary operations and isomorphisms) but not on the core characteristics of well-definedness and everywhere-definedness. Here, we report on a study…

This study investigates when and how university students in first-semester introductory calculus interpret multiple representations of the same function. Specifically, it focuses on three tasks. The first task has students give their definitions of 'function sameness', the results of which suggests that many students understand a function as being…

Imaginary latent variables are variables with negative variances and have been used to implement constraints in measurement models. This article aimed to advance this practice and rationalize the imaginary latent variables as a method to detect possible latent deficiencies in measurement models. This rationale is based on the theory of complex…

This paper reports on five secondary school mathematics prospective teachers' conceptions of "extreme point." The analysis of the data addressed students' definitions, examples, and evaluation of given examples, with special attention to the related domain. Written assignments and individual interviews uncover salient, erroneous concept…

We have introduced a novel approach to competency-based education in mechatronics from the undergraduate to the postgraduate level. What distinguishes this approach is the integration of modeling and control of sampled systems right from the beginning of the undergraduate education. It is achieved by changing the structure of the first-semester…

Mathematical ideas are developed and spread during argumentative whole class discussions between a teacher and her students. The goal of the current study is to characterize how ideas about quadratic functions emerge and are spread during a whole-class discussion among ninth graders. To this end, we recorded both discussions between pairs of…

While many aspects of the teaching and learning of specific advanced mathematics courses have been studied, limited work has examined mathematical themes like sameness or its instantiations across disciplines. In this paper, we explore algebraists' collective example space for mathematical sameness. We used qualitative methods to analyze survey…

A derivation of the conservation equations for fixed bed tubular reactors in cylindrical coordinates is presented and a differential operator for the substantial derivative in porous beds is introduced. The emphasis on the distinction between the functions of the void and volume fractions in the derivation of the conservation equations sets the…