2,846 research outputs found
Beyond the Instrumental Use of AI to foster Mathematics students’ Metacognitive Awareness and Critical Thinking
No full textThe contemporary challenge of integrating Artificial Intelligence (AI) in education [1--5] highlights the need for a pedagogical approach that goes beyond mere instrumental use, focusing instead on fostering metacognitive and critical thinking skills [6]. Ennis's framework [7] and Flavell's theory [8] explain how learners develop self-regulation and critical evaluation of their own problem-solving strategies. In mathematics education, Schoenfeld's seminal work [9] further emphasizes the importance of metacognitive awareness in mathematical problem solving, highlighting the necessity for students to monitor and regulate their mathematical thinking processes. More recently, in mathematics education research, Contel and Cusi [10] and Miranda [11] have explored how ChatGPT supports metacognitive processes and contributes to shaping mathematical identity during university-level problem solving. The presence of AI as a collaborative "participant," sharing its solution after students complete their work, stimulates critical thinking, and aligns with Vygotsky's [12] view that interaction with external tools fosters knowledge construction and internalization of higher cognitive processes. This pilot study, as part of a broader interdisciplinary project investigating the metacognitive potential of AI in education, examines how structured AI integration in mathematical problem solving can foster: (1) metacognitive competencies in university mathematics students; (2) critical evaluation capabilities of problem-solving strategies; and (3) deeper conceptual understanding through comparison of human- and AI-generated solutions
Undergraduate mathematics student-generated videos as an inside-outside resource for meaningful learning
No full textBack and forth between Euclidean geometry and Taxicab geometry To foster students' theoretical thinking in digital contexts
No full textHigh school students are expected to develop skills to understand definitions of concepts in Euclidean geometry, to handle conjectures and proofs, and to use them later to prove other propositions of the theory. Students do not generally find this shift easy. One of the reasons would seem to be related to the understanding of the definition of a concept (Moore, 1994). In geometry, properties of figures derive from definitions within an axiomatic system: a figure is “controlled by its definition” (Fischbein, 1993). Research shows that by exploring the properties of figures and making conjectures, especially through dynamic geometry systems (Baccaglini-Frank & Mariotti, 2010), students can better develop their understanding. Furthermore, comparing different geometric worlds aids helps the advancement of Euclidean knowledge. In our ongoing experience, both high school and undergraduate students are involved in exploratory cooperative learning activities within taxicab geometry (Krause, 1975) using digital tools, and comparing them with Euclidean geometry. They are required to investigate what concepts can be defined in terms of the taxicab distance by analogy with those existents in the Euclidean one (Berger, 2015) as well as whether a theorem or an open problem in one of the geometries is valid or open within the other. We face the issue of promoting students’ conceptual understanding and theoretical thinking by transferring their knowledge of concepts or the validity of statements back and forth between Euclidean and taxicab geometry, with the goal of providing evidence that productive learning processes are activated when students engage in these activities
On sequentially-k-spaces
No full textA topological space X is called k-space if X is a Hausdorff space and X is
an image of a locally compact space under a quotient mapping. A natural
question arises: when a k-space satisfies that its product with every k-spaces
is also a k-space? Michael showed that a k-space has this property iff it
is a locally compact space. A similar question, related to the class of quasi-k-
spaces, Hausdorff images of locally countably compact spaces under quotient
mappings, was answered by M. Sanchis.
The study of k-spaces and quasi-k-spaces suggests to define, in a natural
way, the class of sequentially-k-spaces, namely, the class of Hausdorff images of
locally sequentially compact spaces under quotient mappings.
In this paper the class of sequentially-k-spaces is introduced and investigated
A note on a Category in Point-free Geometry
No full textThe notion of interval semimetric space leads
to introduce a geometry whose primitive notions are
regions and interval-valued maps, following the
A.N. Whitehead's ideas. A Point-free method to complete metric
spaces by abstraction processes is proposed. The
relations between categories of interval semimetric spaces and
metric spaces are investigated
EXPLORING THE ROLE OF 3D PRINTING TECHNOLOGY IN SUPPORTING UNDERGRADUATE STUDENTS' TOPOLOGICAL CONCEPTUAL KNOWLEDGE
No full textWe explore the educational potential of using a 3D printer in a university mathematics setting. We investigate how 3D printing technology might help undergraduate mathematics students improve their conceptual knowledge and theoretical reasoning. Our approach intends to engage students in both the design of topological objects and their 3D printing manufacture as well as the study of their properties through the manipulation of physical examples. Extending the "example space" and moving from one semiotic representation to another are fundamental learning opportunities. The former is crucial in reifying a mathematical concept by including even a material example, and the latter is critical in activating cognitive processes to acquire conceptual understanding. Preliminary findings highlight that the 3D printing technology effectively integrated students’ learning experiences
Extending example spaces in topology to aid undergraduate students’ transition to generalization and abstraction
No full textStimulating students to construct knowledge by themselves helps them to experience a more successful transition to the next educational step. Research in university mathematics education suggests that it might be beneficial to introduce undergraduate students to theoretical objects by engaging them in the autonomous construction of example spaces. This approach is generally fruitful, but students sometimes have difficulty in providing counterexamples to false conjectures. Inspired by the “boundary examples” construct, we investigate its potential to overcome these obstacles, through “bridge page” tasks. We present the first findings on the role of bridging examples in reaching generalization and abstraction in topology
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