The Development of Students Geometrical Thinking through Transformational Processes and Interaction Techniques in a Dynamic Geometry Environment

Abstract

The paper draws on a didactic experiment conducted in a secondary school mathematics classroom in Greece which aimed to explore a) ways in which students develop problem representations, reasoning and problem-solving, making decisions and receiving feedback about their ideas and strategies in a DGS-supported environment b) ways in which students develop rigourous proof through building linking visual active representations and c) ways to develop students' van Hiele level. The mathematical problem the students engaged with -either in the Geometer's Sketchpad dynamic geometry enviroment (Jackiw, 1988) or in the static environment -generated potentially insightful data on the issues focused on the comparison between the experimental and control groups. Initially, three pairs from the experimental group explored the treasure problem within a dynamic geometry environment. The discussions and results of the discussion were videotaped. The problem was then reformulated by the researcher taking into account the research group's retroaction, and re-explored by both the control and experimental groups in a paperpencil test. The researcher then (semi) pre-designed multiple-page sketches detailing the sequential phases of the solution to the problem using rigorous proof, and in so doing transferring her classroom reaching style into the software design, drawing on the chain questioning method of Socrates, which aim to stimulate interaction. For this reason, she linked all the software functions/actions using the interaction techniques supported /facilitated by the Geometer's Sketchpad v4 (DGS) environment (Jackiw, 1988) to better allow students to discover solution paths and to reason by rigorous proof. This mode of design and the results of the

A Research Synthesis Using Instrumental Learning Trajectories: Knowing How and Knowing Why

“…Μ8: Yes, but there might be a second domain (pointing to the values of Ν) from which point forward the sequence tends to zero. The students consequently had an environment of linking visual active representations (e.g., Patsiomitou, 2008aPatsiomitou, , b, 2010Patsiomitou, , 2012aPatsiomitou, , b, 2019 in which the figure of the fractal had been linked with the table of the measurements via the functional process of iteration, which continuously could be linked with the graphic representation of the sequence.…”

“…Meaning could they use the figures' secondary properties to construct a quadrilateral by reversing the process they have followed until now? These thoughts led me to create a "S.M.A.R.T" tool (i.e., Simple, Multiple use, Active/Alive, Reversing use, Tool) (e.g., Patsiomitou, 2006aPatsiomitou, , 2012aPatsiomitou, , b, 2018. I created the custom tool to help students visualize the meaning of central symmetry in correlation with the meaning of a segment's midpoint.…”

“…Transformation due to the using of hide-show The construction and use of the action-buttons on the diagram modifies the overall diagram or parts of the diagram on screen, which leads the user to focus on particular elements of the diagram. The gradual appearance of additional elements on the diagram, which is connected to the sequence of the construction, may lead the user to a sequential understanding of the structure due to the reflective visual reaction (RVR) (Patsiomitou, 2010(Patsiomitou, , 2012a. 5.5.2.…”

Eξ αποστάσεως Εκπαίδευση & Αναπαραστασιακά Συστήματα στα Μαθηματικά στην Εποχή του covid-19

“…Απώτερος σκοπός είναι οι μαθητές/τριες να έχουντη δυνατότητα να κατανοήσουν τις μαθηματικές έννοιες, δίνοντας νόημα σε αυτές, το οποίο σχετίζεται με τα ενδιαφέροντα τους και το μαθησιακό τους στυλ (Πλατσίδου & Ζαγόρα, 2006).Υπάρχουν Μαθηματικά εκπαιδευτικά λογισμικά τα οποία -υπό κατάλληλες παιδαγωγικές συνθήκες -είναι δυνατό να συμβάλλουν στην επεξεργασία της πληροφορίας, καθώς επίσης, στον πειραματισμό και τον έλεγχο υποθέσεων από τους μαθητές/τριες. Για παράδειγμα, ως γνωστόν τα δυναμικά γεωμετρικά λογισμικά (Geogerba, Geometer's Sketchpad, Gabri) δίνουν τη δυνατότητα στους μαθητές/τριες να πειραματιστούν με τα δεδομένα και τα ζητούμενα, για να κατασκευάσουν προσωποποιημένα την μαθηματική τους γνώση (Patsiomitou, 2008). Επίσης, υπάρχουν και άλλα λογισμικά, όπως το autodesk sketchbook, που μπορεί να προσφέρει έναν διαφορετικό τρόπο προσέγγισης της γνώσης με παιγνιώδη τρόπο, καθώς είναι σχετικά εύκολο στη χρήση για τη δημιουργία εικόνων κόμικ, όπως και με την εξοικείωση με τη χρήση γραφίδας στον υπολογιστή.…”

İlköğretim 8. Sınıf Öğrencilerinin Geometrik ve Cebirsel Düşünme Düzeyleri Arasındaki İlişkinin İncelenmesi

“…Matematiksel düşünme, değişik alanlarında kullanılan matematiksel tekniklerin doğasına bağlı olarak kendi içinde "geometrik düşünme", "cebirsel düşünme" ve "olasılıklı düşünme" gibi farklı biçimler almaktadır (Dindyal, 2003). Özellikle geometrik düşünme (Halat, 2006;Kılıç, 2003;Patsiomitou, 2008;Usiskin, 1982) ve cebirsel düşünme ile ilgili araştırmalar (Çağdaşer, 2008;Çelik, 2007;Dikkartın ve Uyangör, 2007;Kaş, 2010; Öner Sünkür, İlhan ve Kılıç, 2012) alanyazında dikkat çekmektedir.…”