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Supply, Anne-Sophie; Wijns, Nore; Van Dooren, Wim; Onghena, Patrick
doi: 10.1007/s10649-022-10187-9pmid: 36277373
The many studies with coin-tossing tasks in literature show that the concept of randomness is challenging for adults as well as children. Systematic errors observed in coin-tossing tasks are often related to the representativeness heuristic, which refers to a mental shortcut that is used to judge randomness by evaluating how well a set of random events represents the typical example for random events we hold in our mind. Representative thinking is explained by our tendency to seek for patterns in our surroundings. In the present study, predictions of coin-tosses of 302 third-graders were explored. Findings suggest that in third grade of elementary school, children make correct as well as different types of erroneous predictions and individual differences exist. Moreover, erroneous predictions that were in line with representative thinking were positively associated with an early spontaneous focus on regularities, which was assessed when they were in second year of preschool. We concluded that previous studies might have underestimated children’s reasoning about randomness in coin-tossing contexts and that representative thinking is indeed associated with pattern-based thinking tendencies.
Bofferding, Laura; Aqazade, Mahtob
doi: 10.1007/s10649-022-10166-0pmid: N/A
Thirty-seven third graders and thirty-two first graders engaged in solving a tangram puzzle in the shape of a fox. They had five minutes to solve the puzzle, and after this time, they received guidance on the particular piece they had difficulties with. Through the lenses of navigating flexible abstraction, reinterpretation, combinations, and borrowing structure to expand upon the existing 2D shape composition and decomposition learning trajectory, we examined ways in which students’ puzzle-solving processes and their challenges related to the fox puzzle’s features. Students’ initial shape placements suggest that parts of the fox puzzle primed the use of particular pieces, which reduced the abstraction of the puzzle. The most challenging part of the puzzle for students was navigating reinterpretation to place the square and two small triangles on the fox’s head in nonstandard orientations. Even though students faced challenges at different steps, they overcame them similarly by trying new combinations and by borrowing structure. Some students did not complete the puzzle even though they used flips and turns (reinterpretation) strategically. The results suggest potential modifications of the current learning trajectory to account for differences between tangram and pattern block puzzles and differences due to tangram puzzles’ features. Because the puzzle’s features played a role in students’ challenges, future work needs to focus on the interaction between students’ puzzle-solving processes and puzzles’ features for a variety of tangram puzzles.
Christiansen, Iben Maj; Corriveau, Claudia; Pettersson, Kerstin
doi: 10.1007/s10649-022-10167-zpmid: N/A
Within the commognitive perspective, ritual and explorative routines are used in a very particular way to distinguish students’ routines according to whether they are driven by social reward or by generating a substantiated narrative. Explorative routines in this theorisation may refer not to inquiry-based activity but to the result of a student’s routine moving from being process-oriented to becoming outcome-oriented, a deritualisation. Choice of tasks as well as a teacher’s moves offer students different opportunities to engage in rituals, explorative routines and deritualisations. Through nuancing the space spanned by opportunities to engage in rituals and explorative routines respectively, we describe and contrast classroom practices in three lessons from three contexts. The lessons share a commonality in encouraging explorative routines as a starting point, yet being adapted towards ritual activity through decreased openings for student agentivity, fewer invitations for students’ own substantiations or both. We argue that such adaptations are driven by the teachers’ commitment to reach mathematical closure in a lesson, to balance considerations of the classroom community and individual students and to meet curricular requirements. Our model helps interrogate the nature and relevance of hybrids of explorative routines and rituals.
Haghjoo, Saeid; Radmehr, Farzad; Reyhani, Ebrahim
doi: 10.1007/s10649-022-10168-ypmid: N/A
Mathematical objects are the outcomes of human discourse and come to life through the process of objectification. Primary and concrete discursive objects (d-objects) play an important role in this process, forming different realizations through the objectification process. In the present study, we analyze the written discourses about the derivative at a point in fourteen Year 11 and Year 12 Iranian calculus textbooks over 42 years (1979–2020) in terms of primary objects, concrete d-objects, and the realizations used in them, here based on Sfard’s commognitive theory. The research method is a qualitative content analysis in which the analysis units were texts, examples, activities, and exercises in the textbooks. The results show that out of the 14 textbooks, only five included primary and concrete d-objects. The other textbooks only used abstract objects in their written discourse about the derivative. The pictures (visual mediators) that were used as primary objects gradually changed from schematic to real colored photos, and locally developed technologies and cultural elements were included in the new textbook editions. Additionally, the number of concrete d-objects increased in the new editions. In terms of the realizations, the number of roots also increased over time; however, none of the textbooks had all five main realizations (i.e., symbolic, graphical, verbal, numerical, and physical) for the derivative at a point. The realization tree developed as part of this study can be used as an analytical framework to analyze calculus textbooks in other contexts and can be used by teachers and lecturers to help students have explorative participation in discourse on the derivative.
Huang, Xingfeng; Huang, Rongjin; Trouche, Luc
doi: 10.1007/s10649-022-10172-2pmid: 35937038
Due to the COVID-19 pandemic in Shanghai, China, all school classes were delivered through an online environment from February 24 to May 22, 2020. To support this transition, the Shanghai Education Commission led expert teachers and specialists to develop a series of online video lessons based on the Shanghai unified curriculum, and suggested students watch the online video lessons individually from home, followed by an online synchronous lesson supported by class teachers. This study investigated what primary mathematics teachers learned from addressing these challenges through a case study. By following two purposefully selected teachers over 2 weeks during the transition, multiple data sets including online video lessons, online synchronous lessons, daily reflections, and post-online teacher interviews were collected. A fine-grained analysis of the data from the lens of the documentational approach to didactics found that teachers adaptively used online video lessons as important resources for their online synchronous lessons and virtual Teaching Research Groups as a teachers’ collaboration mechanism supported them to develop online video lessons and address various technological constraints. Finally, implications of this case study for mathematics education globally are discussed.
Weber, Keith; Fukawa-Connelly, Timothy
doi: 10.1007/s10649-022-10177-xpmid: N/A
Mathematicians frequently attend their peers’ lectures to learn new mathematical content. The goal of this paper is to investigate what mathematicians learned from the lectures. Our research took place at a 2-week workshop on inner model theory, a topic of set theory, which was largely comprised of a series of lectures. We asked the six workshop organizers and seven conference attendees what could be learned from the lectures in the workshop, and from mathematics lectures in general. A key finding was that participants felt the motivation and road maps that were provided by the lecturers could facilitate the attendees’ future individual studying of the material. We conclude by discussing how our findings inform the development of theory on how individuals can learn from lectures and suggest interesting directions for future research.
Bini, Giulia; Bikner-Ahsbahs, Angelika; Robutti, Ornella
doi: 10.1007/s10649-022-10173-1pmid: N/A
Mathematical Internet memes are examples of how the creative thrust characterising the Web 2.0 environment reaches the field of mathematics, translating mathematical statements into a new digital form endowed with an epistemic potential that is capable of initiating a process of mathematical argumentation. The research presented in this paper aims to shed light on the creative process of mathematical memes, contributing to building a body of knowledge on mathematical memes that, prospectively, could enable educators to profit from these objects in their teaching. Theoretically, this is based on a widened concept of creativity that focuses on the connection linking digital culture with mathematics, and on distinguishing and merging three perspectives to disclose the meanings of mathematical memes. Methodologically, the process of mathematical memes’ creation is investigated through a reverse engineering approach on a dataset of about 2100 items collected in a 3-year-long ethnographic observation within online communities. The result is a heuristic action model of the creation process, that is validated by creating two new mathematical Internet memes that are shared online within the observed communities to explore if they retain the mathematical and epistemic characteristics of Web-found ones.
doi: 10.1007/s10649-022-10188-8pmid: N/A
Our study aims to determine how Habermas’ construct of rationality can serve to identify and interpret the difficulties experienced by university students in the mathematical problem-solving process. To this end, a problem which required modelling and solving a differential equation was used. The problem-solving processes of university students were analysed based on rationality components. The findings demonstrated that the problems in epistemic rationality such as predominance of the figure over the definition and/or theorems, use of dogmatic knowledge, intuitive generalizations, lack of prior knowledge, incorrect recognition of the differential equation prevented the choosing and using of an appropriate problem-solving method, leading to problems in teleological rationality. It was determined that the student performance in communicative rationality was negatively affected by problems in epistemic rationality such as using of knowledge acquired by rote and predominance of figure prototype on the definitions and/or theorems. Throughout the analysis, it is required to define two new sub-components, named “geometric” and “algebraic” representation under the modelling requirements of epistemic rationality. It is advised to use the extended version of Habermas’ construct of rationality to examine the performance of students in mathematical activities to get more detailed and accurate results.
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