Gaining meaning for expressions with Grid Algebra: developing the CAPS framework

Through examples from the early stages of a study with Grade 7 low-attaining learners from Malta using Grid Algebra, we argue for the significance of actions, pictures and symbols in learners developing concepts for formal expressions. The interrelationship between these forms the basis of the CAPS (Concept, Action, Picture and Symbol) framework as an analytical tool, which is presented in this paper

A short empirical note on perfectionism and flourishing

Flourishing describes an optimal state of mental health characterized by emotional, psychological, and social well-being. In a recent publication, Flett and Hewitt (2015) suggested that perfectionism prevents people from flourishing. Perfectionism, however, is a multidimensional personality characteristic, and its various dimensions show different relationships with indicators of subjective well-being. In the first empirical study of perfectionism and flourishing, we examined the relationships of multidimensional perfectionism (self-oriented, other-oriented, and socially prescribed perfectionism) and self-reported flourishing in the past two weeks. Results from the sample of 388 university students revealed that only socially prescribed perfectionism showed a negative relationship with flourishing, whereas self-oriented perfectionism showed a positive relationship. These results were unchanged when positive and negative affect were controlled statistically. Our findings indicate that not all dimensions of perfectionism undermine flourishing and that it is important to differentiate perfectionistic strivings and concerns when regarding the perfectionism–flourishing relationship

Negotiating between learner and mathematics: a conceptual framework to analyze teacher sensitivity toward constructivism in a mathematics classroom

Context: Constructivist teachers who find themselves working within an educational system that adopts a realist epistemology, may find themselves at odds with their own beliefs when they catch themselves paying closer attention to the knowledge authorities intend them to teach rather than the knowledge being constructed by their learners. Method: In the preliminary analysis of the mathematical learning of six low-performing Year 7 boys in a Maltese secondary school, whom one of us taught during the scholastic year 2014-15, we constructed a conceptual framework which would help us analyze the extent to which he managed to be sensitive to constructivism in a typical classroom setting. We describe the development of the framework M-N-L (Mathematics-Negotiation-Learner) as a viable analytical tool to search for significant moments in the lessons in which the teacher appeared to engage in what we define as “constructivist teaching” (CT) during mathematics lessons. The development of M-N-L is part of a research program investigating the way low-performing students make mathematical sense of new notation with the help of the software Grid Algebra. Results: M-N-L was found to be an effective instrument which helped to determine the extent to which the teacher was sensitive to his own constructivist beliefs while trying to negotiate a balance between the mathematical concepts he was expected to teach and the conceptual constructions of his students. Implications: One major implication is that it is indeed possible for mathematics teachers to be sensitive to the individual constructions of their learners without losing sight of the concepts that society, represented by curriculum planners, deems necessary for students to learn. The other is that researchers in the field of education may find M-N-L a helpful tool to analyze CT during typical didactical situations established in classroom settings

Authors' Response: The M-N-L framework: Bringing radical constructivist theories to daily teaching practices

Upshot: We seek to address several questions and statements made in the commentaries by elaborating on the four main aspects of the M-N-L framework. Before doing so, we discuss the issue of constructivist teaching in the context of schools. We conclude by hypothesizing on what would be lost in the M-N-L framework by taking constructivism out of the picture