Papers by Anatoli Kouropatov
Analogical Models, Intuition, and Knowledge Construction: The Case of a System of Algebraic Equations
WTM-Verlag eBooks, 2024
While there are parallels between what has been called, in the literature, Basic Mental Models (B... more While there are parallels between what has been called, in the literature, Basic Mental Models (BMMs, Grundvorstellungen) and what has been called Personal Meanings, there are fundamental differences between them. In this paper, we work out some of these differences, using the notion of integral as example. Roughly summarized, our findings are that BMMs, including individual ones, are epistemological whereas Personal Meanings are cognitive. Here epistemological refers to a content analysis, often from a didactic point of view, and hence is informative, for example, for curriculum developers; cognitive refers to individual students' personal conceptions, and hence is of interest, among others, to teachers.
The results of a quantitative survey of 104 in-service mathematics and science teachers who taugh... more The results of a quantitative survey of 104 in-service mathematics and science teachers who taught remotely as a result of the COVID-19 pandemic pointed to an increase in their knowledge regarding the existence of e-learning environments. During the pandemic the teachers integrated more digital technology and their goals for using computer tools changed, such that after the pandemic these tools were used mostly to create a learning environment suitable for distance learning. Most teachers experienced difficulties in technology integration, citing not enough preparation time, logistics problems, lack of technological knowledge, and insufficient technical conditions as the reasons. The results suggest that teachers need adequate knowledge and technical support to be able to integrate computerized technologies successfully into their teaching.
The terms Analysis and Calculus are widely used in mathematics. It seems that the professional co... more The terms Analysis and Calculus are widely used in mathematics. It seems that the professional community dealing with research on the teaching and learning of analysis and calculus is gradually realizing that Mathematical Analysis and Calculus are not one and are not the same subject, no matter how closely related they are. We agree that there is a substantial difference between them, leading to genuine didactical challenges. The study reported below provides empirical evidence supporting this claim.
Constructing the integral concept on the basis of the idea of accumulation: suggestion for a high school curriculum
International Journal of Mathematical Education in Science and Technology, Jul 1, 2013
ABSTRACT Students have a tendency to see integral calculus as a series of procedures with associa... more ABSTRACT Students have a tendency to see integral calculus as a series of procedures with associated algorithms and many do not develop a conceptual grasp giving them the desirable versatility of thought. Thus, instead of a proceptual view of the symbols in integration, they have, at best, a process-oriented view. On the other hand, it is not surprising that many students find concepts such as the integral difficult when they are unable to experience these processes directly in the classroom. With a view towards improving this situation, constructing the integral concept on the basis of the idea of accumulation has been proposed (Educ Stud Math. 1994;26:229–274; Integral as accumulation: a didactical perspective for school mathematics; Thessaloniki: PME; 2009. p. 417–424). In this paper, we discuss a curriculum that is based on this idea and a design for curriculum materials that are intended to develop an improved cognitive base for a flexible proceptual understanding of the integral and integration in high school. The main focus is on how we (mathematics teachers and mathematics educators) might teach the integral concept in order to help high school students to construct meaningful knowledge alongside acquiring technical abilities.
Mathematics in Computer Science, Jul 10, 2018
HAL (Le Centre pour la Communication Scientifique Directe), Feb 2, 2022
The concept of Rate of Change (RoC) is often presented in an Extra-Mathematical Context (EMC) whi... more The concept of Rate of Change (RoC) is often presented in an Extra-Mathematical Context (EMC) which evokes subjective judgments due to interpretations of the described real-life situation and of the missing information in the problem. In this study, we investigated different learners' interpretations of several EMC problems involving RoC, with the aim to examine which aspects of the notion of RoC are prone to subjective reasoning, due to their structure or due to missing information, and which aspects are objective. We found that while the problems raised subjective thoughts for different learners, analysis of both the EMC and the mathematical concepts can help predict which aspects of the mathematics are prone to be subjective, and which are not.
HAL (Le Centre pour la Communication Scientifique Directe), Feb 2, 2022
In this paper we report case study findings regarding two forms of the collapse metaphor and thei... more In this paper we report case study findings regarding two forms of the collapse metaphor and their implications on learners' personal meanings about accumulation and the integral. We found that students can interpret adding up lines as both adding up their lengths and adding up their areas. These conceptions appear to be related to personal meaning and context.
Teaching Mathematics and Its Applications, Mar 25, 2022
The learning of calculus concepts is considered challenging for students. This claim is actual fo... more The learning of calculus concepts is considered challenging for students. This claim is actual for calculus in general and for specific concepts in particular. In this paper, we focus on the concept of the inflection point. We argue that one of the roots of this problem is the lack of a useful and productive meaning of the concept-the understanding of the inflection point as the point where the behaviour of a curve (graph of function) changes in relation to the tangent line. With the purpose of helping students to construct this meaning we developed a specific digital tool: a teaching unit based on the interactive diagrams framework. Does this tool help students to achieve this meaning (i.e., to construct and consolidate new knowledge)? To answer this question, we conducted an empirical feasibility experiment (in the form of a case study with two first-year students from the Industrial Engineering College) and analysed the gathered data using the framework of abstraction in context as the theoretical and methodological basis. Our findings show that the designed tool (the interactive digital teaching unit) has potential for helping students to make the above-mentioned meaning for this mathematical concept and can serve as a useful basis to continue the investigation of designing tools that support the meaningmaking of advanced mathematical concepts.
Teaching Mathematics and Its Applications, Jun 1, 2022
HAL (Le Centre pour la Communication Scientifique Directe), Feb 1, 2017
We show how a combination of two theories, Abstraction in Context and Proceptual Thinking, served... more We show how a combination of two theories, Abstraction in Context and Proceptual Thinking, served as basis for design decisions in the framework of a research study about learning the integral concept in high school via constructing knowledge about accumulation.
Learning the integral concept by constructing knowledge about accumulation
Zdm – Mathematics Education, Feb 23, 2014
ABSTRACT We propose an approach to the integral concept for advanced high school students and pro... more ABSTRACT We propose an approach to the integral concept for advanced high school students and provide evidence for the potential of this approach to support students in acquiring an in-depth proceptual view of the integral. The approach is based on the mathematical idea of accumulation. A ten-lesson unit has been implemented with four pairs of students. The students’ learning processes were micro-analysed using the methodological–theoretical framework of Abstraction in Context. In this paper, we focus on the lessons in which the notions of approximation and accumulation are introduced. The work of one student pair is analysed in detail, and the work of the other pairs is summarized. Our results show that most of the students reached a proceptual understanding of the integral that prepared them for the next step in the curriculum, namely the Fundamental Theorem of Calculus.
Eurasia Journal of Mathematics, Science and Technology Education
This quantitative retrospective study examines a non-probable convenience sample of 104 in-servic... more This quantitative retrospective study examines a non-probable convenience sample of 104 in-service mathematics and science teachers with the aim to understand their changes in attitudes toward the integration and use of information and communications technology (ICT) as a result of the COVID-19 pandemic. Data included a self-report questionnaire. The results indicate an increase in teachers’ knowledge regarding the e-learning environments available at their schools. The teachers revealed that before the pandemic, they mostly used computer tools to support struggling students or carry out basic calculations. However, during, because they had to plan distance learning environments, they introduced adaptive pedagogical use of ICT tools for all their students. The findings reveal that most teachers experienced difficulties emanating from lack of preparation time, technological knowledge, and/or technical conditions, thus suggesting that policymakers must decide how to provide adequate k...
Le Centre pour la Communication Scientifique Directe - HAL - Université Paris Descartes, Feb 2, 2022
In this paper we report case study findings regarding two forms of the collapse metaphor and thei... more In this paper we report case study findings regarding two forms of the collapse metaphor and their implications on learners' personal meanings about accumulation and the integral. We found that students can interpret adding up lines as both adding up their lengths and adding up their areas. These conceptions appear to be related to personal meaning and context.
Le Centre pour la Communication Scientifique Directe - HAL - Inria, Feb 2, 2022
The concept of Rate of Change (RoC) is often presented in an Extra-Mathematical Context (EMC) whi... more The concept of Rate of Change (RoC) is often presented in an Extra-Mathematical Context (EMC) which evokes subjective judgments due to interpretations of the described real-life situation and of the missing information in the problem. In this study, we investigated different learners' interpretations of several EMC problems involving RoC, with the aim to examine which aspects of the notion of RoC are prone to subjective reasoning, due to their structure or due to missing information, and which aspects are objective. We found that while the problems raised subjective thoughts for different learners, analysis of both the EMC and the mathematical concepts can help predict which aspects of the mathematics are prone to be subjective, and which are not.
Le Centre pour la Communication Scientifique Directe - HAL - Archive ouverte HAL, Feb 1, 2017
A unit on dynamic geometric construction was included in a professional development course for in... more A unit on dynamic geometric construction was included in a professional development course for inservice mathematics teachers. As a final task in that unit 28 teachers were required to construct a rhombus based on their own choice of given objects and tools, using the dynamic geometry software GeoGebra. Their responses were analysed according to: the choice of given objects; the choice of tools; the explanation and validity test; and the number of different rhombuses they claimed to have obtained. The teachers were found to have different concept images of a rhombus and different conceptions of what constitutes a valid geometric construction. While many claimed to have obtained an infinite number of different rhombuses, differences were observed in the "type of infinite". Recommendations are given for improving the task design to strengthen teachers' mathematical and pedagogical knowledge.
Teaching Mathematics and its Applications: An International Journal of the IMA, 2022
Teaching Mathematics and its Applications: An International Journal of the IMA, 2022
CERME 10, Feb 1, 2017
We show how a combination of two theories, Abstraction in Context and Proceptual Thinking, served... more We show how a combination of two theories, Abstraction in Context and Proceptual Thinking, served as basis for design decisions in the framework of a research study about learning the integral concept in high school via constructing knowledge about accumulation.
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Papers by Anatoli Kouropatov