Logical reasoning is of great societal importance and, as stressed by the twenty-first century skills framework, also seen as a key aspect for the development of critical thinking. This study aims at exploring secondary school students' logical reasoning strategies in formal reasoning and everyday reasoning tasks. With task-based interviews among 4 16-and 17-year-old pre-university students, we explored their reasoning strategies and the reasoning difficulties they encounter. In this article, we present results from linear ordering tasks, tasks with invalid syllogisms and a task with implicit reasoning in a newspaper article. The linear ordering tasks and the tasks with invalid syllogisms are presented formally (with symbols) and non-formally in ordinary language (without symbols). In tasks that were familiar to our students, they used rulebased reasoning strategies and provided correct answers although their initial interpretation differed. In tasks that were unfamiliar to our students, they almost always used informal interpretations and their answers were influenced by their own knowledge. When working on the newspaper article task, the students did not use strong formal schemes, which could have provided a clear overview. At the end of the article, we present a scheme showing which reasoning strategies are used by students in different types of tasks. This scheme might increase teachers' awareness of the variety in reasoning strategies and can guide classroom discourse during courses on logical reasoning. We suggest that using suitable formalisations and visualisations might structure and improve students' reasoning as well.
Due to growing interest in twenty-first-century skills, and critical thinking as a key element, logical reasoning is gaining increasing attention in mathematics curricula in secondary education. In this study, we report on an analysis of video recordings of student discussions in one class of seven students who were taught with a specially designed course in logical reasoning for non-science students (12th graders). During the course of 10 lessons, students worked on a diversity of logical reasoning tasks: both closed tasks where all premises were provided and everyday reasoning tasks with implicit premises. The structure of the course focused on linking different modes of representation (enactive, iconic, and symbolic), based on the model of concreteness fading (Fyfe et al., 2014). Results show that students easily link concrete situations to certain iconic referents, such as formal (letter) symbols, but need more practice for others, such as Venn and Euler diagrams. We also show that the link with the symbolic mode, i.e. an interpretation with more general and abstract models, is not that strong. This might be due to the limited time spent on further practice. However, in the transition from concrete to symbolic via the iconic mode, students may take a step back to a visual representation, which shows that working on such links is useful for all students. Overall, we conclude that the model of concreteness fading can support education in logical reasoning. One recommendation is to devote sufficient time to establishing links between different types of referents and representations.
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