The present paper examines a patterning activity that was organised within a teaching experiment in order to analyse the different uses of variables by secondary school students. The activity presented in the paper can be categorised as a pictorial/geometric linear pattern. We adopted a student-oriented perspective for our analysis, in order to grasp how students perceive their own generalising actions. The analysis of our data led us to two broad categories for variable use, according to whether the variable is viewed as a generalised number or not. Our results also show that students sometimes treat the variable as closely linked to a referred object, as a superfluous entity or as a constant. Finally, the notion of equivalence, which is an important step towards understanding variables, proved difficult for our students to grasp.

In this paper, we present the results of a study investigating the reasoning processes of grade 3 students working collaboratively to solve a combinatorial task. The task, which involved colouring and identifying colour combinations, was designed to be playful, engaging, and at the same time meaningful for students of that age. Our analysis drew on six mathematical reasoning actions, adapted from existing frameworks: noticing similarities or relations, developing conjectures, generalising to identify commonalities, justifying based on already‑understood ideas, verifying that a common property holds for each member of a group, and generalising properties. We also employed an established framework for categorising solution strategies in combinatorial tasks, comprising four strategies: random selection of items without rejection of inappropriate items; trial-and-error involving random selection with rejection of inappropriate items; emerging patterns in item selection with rejection of inappropriate items; and consistent, complete cyclical patterns in item selection with rejection of inappropriate items. Our results show that the young students demonstrated a wide range of reasoning actions during their interactions, some of which attained the level of generalisation. All nine groups displayed at least three reasoning actions, while four groups demonstrated all six actions, thereby reaching the most advanced level of generalising properties. With respect to solution strategies, all groups progressed systematically from random selection to consistent and complete cyclical patterns. The reasoning actions developed cooperatively through student interaction, and all groups successfully solved the task by identifying all six possible colour combinations.

In this paper, we discuss the images of mathematics and its teaching, as expressed by mathematics teachers, in reaction to educational programs broadcasted by Polish public television channels during the first lockdown in 2020. For our analysis, we deployed a theoretical framework mostly drawn from the work of Paul Ernest. Our data came from a variety of sources such as educational fora, social networks, websites and email exchanges. Our results, based on an interpretative analysis, revealed four overarching images of mathematics and its teaching which, although related to the specificities of television, resembled images found by other relevant studies. The first image of mathematics views it as an unambiguous discipline in which there is no room for more than one point of view. The second image refers to the teacher as a presenter who is expected to possess particular verbal and non-verbal skills, especially given the fact that the television programs did not involve any live audiences. The third image refers to the unappealing nature of the broadcasted lessons, while the fourth image refers to the underlying approach of crude memorization associated with ‘traditional’ mathematics teaching. Our study’s contribution is twofold: in the improvement of televised educational programs (and possibly online courses) and in the improvement of the public’s image of mathematics.

EMS Magazine Article from: Bożena Maj-Tatsis, Esther Levenson