Abstract
This study concerns geometric prediction, a process of anticipation that has been identified as key in mathematical reasoning, and its possible constructive relationship with explorations within a Dynamic Geometry Environment (DGE). We frame this case study within Fischbein’s Theory of Figural Concepts and, to gain insight into a solver’s conceptual control over a geometrical figure, we introduce a set of analytical tools that include: the identification of the solver’s geometric predictions, theoretical and phenomenological evidence that s/he may seek for, and the dragging modalities s/he makes use of in the DGE. We present fine-grained analysis of data collected during a clinical interview as a high school student reasons about a geometrical task, first on paper-and-pencil, and then in a DGE. The results suggest that, indeed, the DGE exploration has the potential of strengthening the solver’s conceptual control, promoting its evolution toward theoretical control.
Highlights
Geometric Prediction and Dynamic Geometry as Resources for Supporting Mathematical ReasoningPrediction is the act of anticipating the outcome of a process, building on information at one’s disposal at that moment
ReIsnultthsisofsethcteioAnnwaleysaensalyze the most significant aspects of Ilaria’s exploration within the paper-and-pencil and dynamic geometry environments as concerned the shifts in her focusI,nththeisunsefoctlidoinngwoefahnearlypzreedthicetimonosptrsoicgensisfeicsa, nant dashpeerctesvoidf eIlnacreiag’senexepralotiroant.ioTnhwesiethaisntfpuhoesceecudtpssa,twhptheeemerr-eautnnioddfaeo-dpnldvteiinafnincegicdleoafbinnahdfseeedrrdeypnnorcaenedmstihicactebigooaenunotapmtlrhyeoettcircesyospselvernosev,crie’arsdoncundomrnheceedenrpetesstvucairasdilbeccenoodncnecitenrgroenSlneeoedcvrteiatohrtinetohns3eh..3ifTf,itghasuenisrndee.hwaeserpects were identified based on the analytic procedure described in Section 3.3, an11dowf 2e7 u4.s1e.dIltahrieam’s ItnotaerdvvieawncdeuirninfegrethnecResesaobluotuiot nthoef tshoelvGeirv’esncoTnasckeptual control over the figure
The study presents the case of a 9th-grade high school student solving a prediction open problem, first within a paperand-pencil environment, and within a Dynamic Geometry Environment (DGE)
Summary
Geometric Prediction and Dynamic Geometry as Resources for Supporting Mathematical ReasoningPrediction is the act of anticipating the outcome of a process, building on information at one’s disposal at that moment. Research in science education highlights the fruitful practice of asking students to make predictions when they are learning about new phenomena (see, for example, [1]). According to Kasmer and Kim [3], mathematics teachers create precious learning opportunities when they ask students for a prediction at the beginning of problem-solving activities. Prediction seems to stimulate processes of visualization, sense-making, and creating connections between previously learned mathematical concepts and new ones that can be developed as the prediction occurs [3]. Making a prediction yields the opportunity to acquire new knowledge; gaining further insight into processes of prediction in mathematics seems to be a promising direction in Mathematics Education, as well
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