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The article aims at articulating the potential impact of Radford’s theory of objectification on technical research on mathematics teaching. First, instead of Radford’s metaphor of mathematical activities as an orchestra, we proposed an alternative metaphor—mathematical activities as an ad lib music session. With that in mind, we focused on laughter as affective expression in class and conducted phonetic analysis of the laughter in a Japanese high-school mathematics classroom. The analysis revealed that a fun atmosphere that included laughter transformed students’ treatment of a mathematical model in the lesson and that the proposed metaphor is more suitable than the original. We, thus, conclude that affective factors might determine the quality of mathematics learning in an ad lib manner., This work was supported by Grants-in-Aid for Scientific Research (KAKENHI) from the Japan Society for the Promotion of Science, Grant Numbers 18K13162, 18K13175, 19K14219, and 20K14013., The 44th Conference of the International Group for the Psychology of Mathematics Education (Virtual conference, Hosted by Khon Kaen University, Thailand) Date: July 19-22, 2021

The article reveals how the processes of subjectification and objectification proceed in a mathematical activity at a Japanese preschool and how the roles of finger gestures change for children during these processes. The theoretical construct used was joint labor as proposed by Luis Radford throughout his work. We relied in part on his methodology and in part on a microgenetic approach to analyze a scene of addition involving children and a teacher in a Japanese preschool. The analysis captured children obtaining help from the teacher to reconstruct the meaning of fingers as tools for solving quizzes rather than preserving the meaning through unprompted practice. The analysis also showed that the role of finger gestures was reconstructed in class to solve a conflict between children’s differing solutions to an addition problem., This research was supported by Grant-in-Aid for Young Scientists (18K13162, 18K13175,19K14219), and Grant-in-Aid for Young Scientists (Start-up) (19K20947) in Japan., The 44th Conference of the International Group for the Psychology of Mathematics Education (Virtual conference, Hosted by Khon Kaen University, Thailand) Date: July 19-22, 2021

International audience; This study assesses preschool children’s understanding of length and area measurement through the activities of a mathematics program in Japan. Japanese preschool children do not formally learn mathematics; thus, we designed mathematical measurement activities for young, less-experienced preschool teachers to implement for nine five- to six-year-old children. We conducted structured clinical interviews with the children individually, both before and after the measurement activities, and qualitatively analyzed the results by comparing their answers and connecting them to the activity contents. Results indicate that children learn to understand direct comparison through activities; however, it is more difficult to establish understanding of measurement by non-universal units.