Development of the schema for the differential of a function of several variables

This chapter is a discussion of the design and implementation of instruction using APOS Theory. For a particular mathematical concept, this typically begins with a genetic decomposition, a description of the mental constructions an individual might make in coming to understand the concept (see Chap. 4 for more details). Implementation is usually carried out using the ACE Teaching Cycle, an instructional approach that supports development of the mental constructions called for by the genetic decomposition.

Este artículo presenta un modelo de la cognición y la epistemología, una descomposición genética (DG), del concepto de la diferencial de una función en varias variables (DFVV) del pensamiento matemático avanzado (PMA), en términos de las estructuras mentales de Acción, Proceso, Objeto y Esquema (APOE) y de los mecanismos para la construcción de éstas estructuras por abstracción reflexiva. La DG se constituye en un referente para el diseño de estrategias de enseñanza y como instrumento para dar cuenta del comportamiento observable del sujeto en la comprensión del concepto.

The study aimed to explore and assess students' conceptual understanding of partial derivatives in functions of several variables. A case study research design was employed, with four classes randomly selected for participation. Qualitative data were collected through interviews and open-ended questions developed by the researchers. The analysis was conducted using APOS theory, based on the proposed genetic decomposition. The findings reveal that the majority of students’ understanding falls under action conception, interpreting partial derivatives similarly to single-variable derivatives. Additionally, many students struggled to extend their knowledge from single-variable derivatives to partial derivatives and faced difficulties in converting between algebraic/symbolic and graphical representations. Based on these results, the researchers recommend that educators adopt instructional strategies that foster deeper conceptual understanding, rather than relying on traditional teaching methods. Keywords: APOS Theory; conceptual understanding; function of several variables; partial derivatives

Reliability and risk analysis data base development: an historical perspective

The aim of this research is to determine the trends of education technology researches on Constructivist Learning Approach, which were published on database of ScienceDirect between 2010 and 2016. It also aims to guide researchers who will do studies in this field. After scanning the database, 81 articles published on ScienceDirect’s data base from January 2010 to December 2016 were examined considering the following criteria in order to serve the purpose of the study; the concepts they deal with, instruments, software, education techniques used in the research, research methods, research designs, data collection instruments, types and sizes of samples, data analysis, the country where it takes place, number of authors, common studies done by researchers from different countries and number of sources. According to the research results, computer-supported education was the most addressed subject in the articles. It was found out that the most frequently used tool was the computer and the most frequently used software was learning management systems in the studies. Cooperative learning technique was the most preferred in terms of the used educational techniques. Quantitative method and survey design came into prominence in the studies. It was figured out about the data collection procedures that questionnaire as a data-collecting tool, university students as the subjects or the participants and descriptive analysis as data analysis method were mostly preferred. It is thought that results obtained will lead other future studies. Keywords: Constructivist learning, constructivist approach, education technology, content analysis

In this paper, we show that the Laplace transform of the expansion <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h(t) = \sigma_{n = 0}^{\infty} c_{n} g_{n} (t)</tex> for some important sets <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g_{n} (t)</tex> is equivalent to a power-series expansion. Techniques based on this result are presented for obtaining the coefficients c. as those of a power series; also, methods are presented for obtaining the functional series inverse. The set of Laguerre functions is discussed in detail and, using the power-series equivalence, the truncation error is obtained. The application of the power-series equivalence to the summing of series is shown and illustrated with the Neumann series. Finally, the extension of the power-series equivalence to the expansion of functions of several variables is given. The areas for which the techniques developed are relevant include the analysis and design of signals and the identification and synthesis of processes and systems.

Constructing a schema: The case of the chain rule?

Literature Reviews: Key Considerations and Tips From Knowledge Synthesis Librarians.

Education in reproductive science is operating from an outdated paradigm of teaching and learning. Traditionally, reproductive education follows the pattern where students read a textbook, listen to instructor presentations, re-read the textbook and class notes and then complete a test. This paradigm is inefficient, costly and has not incorporated the potential that technology can offer with respect to increases in student learning. Further, teachers of reproductive science (and all of science for that matter) have little training in the use of documented methods of instructional design and cognitive psychology. Thus, most of us have learned to teach by repeating the approaches our mentors used (both good and bad). The technology now exists to explain complex topics using multimedia presentations in which digital animation and three-dimensional anatomical reconstructions greatly reduce time required for delivery while at the same time improving student understanding. With funding from the Small Business Innovation Research program through the U.S. Department of Education, we have developed and tested a multimedia approach to teaching complex concepts in reproductive physiology. The results of five separate experiments involving 1058 university students and 122 patients in an OB/GYN clinic indicate that students and patients learned as much or more in less time when viewing the multimedia presentations when compared to traditional teaching methodologies.

Students’ understanding of the concepts involved in one-sample hypothesis testing

This article reports on the use of the Action-Process-Objects-Schema (APOS) theory as a theoretical framework to investigate first-year students' understanding of the chain rule at a University of Technology in South Africa. Instructional design as part of APOS, based on the genetic decomposition, was used to teach the first-year students differentiation involving the concept of the chain rule and its use. A questionnaire based on functions, the composition of functions, derivatives and the structure of the integrand was used to monitor the development of the chain rule schema. Later, worksheets based on the use and application of the chain rule were designed to foster collaborative learning. A sample of 30 students participated in the study. In this manner differentiation of each function in the composite function was accomplished. Students either operated in the Inter- or Trans-stage of the Triad. It was found that students who had inadequate understanding of the composition of functions, performed well in the application of the chain rule. The analysis led to a formulation of three techniques for using the chain rule, namely: (1) the straight form technique; (2) the link form technique; and (3) the Leibniz form technique.

The study investigated the perceived influence of counseling-driven value reorientation in addressing moral decadence among undergraduate students in universities in Benue State. The study adopted a survey research design. The population comprises 36,326 undergraduate students in public Universities in Benue State. The sample size for the study was 396 undergraduate students who were selected using proportionate stratified and accidental sampling techniques. The instrument for data collection was a structured questionnaire titled “Counselling-Driven Value Reorientation and Moral Decadence Questionnaire” (CDVRMDQ). The questionnaire was validated by experts. The reliability of the questionnaire was established using Cronbach Alpha method which yielded a coefficient of 0.83. Data collected for the study were analyzed using Means and Standard Deviation to answer the research questions while Chi-Square Goodness of Fit was used to test the hypotheses at 0.05 level of significance. The study’s findings revealed that academic dishonesty, drug and substance abuse, sexual misconduct, criminal activities, examination malpractice, lack of civic responsibility, dishonesty, cultism, gang activity, among other issues, were the prevalent forms of moral decadence among undergraduates in public universities in Benue State. Furthermore, the key factors contributing to moral decadence were identified as peer influence, socioeconomic challenges, parental and family influence, inadequate counseling and guidance, academic pressure, and concerns related to unemployment, among others. Additionally, the study indicated that undergraduate students expressed a strong willingness to actively participate in counseling-driven value reorientation programs. Moreover, it was established that counseling-driven value reorientation programs had a perceived positive influence in addressing moral decadence among undergraduate students in universities within Benue State. In conclusion, the study emphasizes the significant role that counseling-driven value reorientation programs can play in addressing and mitigating moral decadence among undergraduate students in public universities located in Benue State. To address these concerns effectively, the study recommends that universities should integrate value reorientation and moral education as core courses within their curriculum, providing students with a robust foundation in ethics and civic responsibility. Furthermore, universities should continuously assess and adapt counseling-driven value reorientation programs to ensure their ongoing effectiveness in addressing moral decadence among undergraduate students.

The aim of the study is to analyze the best angular approximations of functions of many variables and to estimate these approximations from above. This allows one to better understand the applicability limits of these methods and optimize their use in problems of mathematical modeling and data analysis. In this paper, the properties of functions of several variables were studied and approaches to their approximation using angular approximations were proposed. Generalized Liouville–Weyl derivatives, which are used instead of classical mixed Weyl derivatives, are considered. The concept of general monotone sequences plays an important role in the study. The methodology includes the use of functional analysis, approximation theory, and analysis of parameters affecting the accuracy of approximation. The main results of the work are to obtain upper bounds for angular approximations for generalized Liouville–Weyl derivatives depending on the characteristics of the approximated functions. The results obtained are of interest to theorists working in the field of multivariate data analysis and mathematical modeling. The topic discussed in the article is closely related to the articles by A.A. Konyushkov, S.B. Stechkin, M.F. Timan, M.K. Potapova, B. Simonov, S. Tikhonov.

Aim: To find out variation in flexibility, strength and balance of university and school students on their dominant and non-dominant sides. Study Design: A Cross section study. Place and duration of study: Data was collected from Lahore grammar school (LGS), Al’Ala school and university of management and technology (UMT), Lahore. After approval of synopsis (RE-092-2021) the data collection was done from the duration of 10th November 2021 to 15th February 2022. Methodology: A cross section study conducted on healthy school and university students. The data was collected by non-probability convenient sampling. After verbal Informed consent the data was obtained from school students of grade 7-10 from Lahore Grammar school and Al’Ala international Islamic school and undergraduate’s students of UMT by using convenient sampling. Physical performance tests for measuring strength (sit-ups), flexibility (Zipper, sit and reach, dorsiflexion) and balance (Flamingo test) were demonstrated and were asked to perform. Readings were taken of each test. Results: Sit ups excellent results were reported in majority of university students (36%). Sit and reach test was found better in university (32%), zipper test positive results were better in school students. Dorsiflexion test normal values were more in university (18%). Flamingo test showed 45% university students had good balance. There was statistically significant difference in zipper (P=0.004), flamingo (P=0.036) on non-dominant side and sit and reach dominant (P=0.006) and non-dominant side (P=0.031). Practical implication: The normative data of physical performance measures can help prevent injuries, as these can be used as a reference to detect poor health and lethargy at school and university level. These measures can be used as a reference for primary prevention among university going and school going students. Furthermore, to increase awareness about physical fitness among young population. Conclusion: Both groups showed good physical performance measures. Flexibility test showed better results in School students. Strength test showed better result in university students whereas, balance test reported better in school students. Keywords: Balance, Flexibility, Limb dominance, musculoskeletal injuries, Physical activity, Strength,

Aim: To find out variation in flexibility, strength and balance of university and school students on their dominant and non-dominant sides. Study design: A Cross section study. Place and duration of study: Data was collected from Lahore Grammar School (LGS), Al’Ala school and University of Management and Technology (UMT), Lahore. After approval of synopsis (RE-092-2021) the data collection was done from the duration of 10th November 2021 to 15th February 2022. Methodology: A cross section study conducted on healthy school and university students. The data was collected by non-probability convenient sampling. After verbal Informed consent the data was obtained from school students of grade 7-10 from Lahore Grammar school and Al’Ala international Islamic school and undergraduate’s students of UMT by using convenient sampling. Physical performance tests for measuring strength (sit-ups), flexibility (Zipper, sit and reach, dorsiflexion) and balance (Flamingo test) were demonstrated and were asked to perform. Readings were taken of each test. Results: Sit ups excellent results were reported in majority of university students (36%). Sit and reach test was found better in university (32%), zipper test positive results were better in school students. Dorsiflexion test normal values were more in university (18%). Flamingo test showed 45% university students had good balance. There was statistically significant difference in zipper (P=0.004), flamingo (P=0.036) on non-dominant side and sit and reach dominant (P=0.006) and non-dominant side. Practical implication: The normative data of physical performance measures can help prevent injuries, as these can be used as a reference to detect poor health and lethargy at school and university level. These measures can be used as a reference for primary prevention among university going and school going students. Conclusion: Both groups showed good physical performance measures. Flexibility test showed better results in School students. Strength test showed better result in university students whereas, balance test reported better in school students. Keywords: Balance, Flexibility, Limb dominance, musculoskeletal injuries, Physical activity, Strength,