• Journal of Educational Research in Mathematics
• /
• v.24 no.1
• /
• pp.1-27
• /
• 2014

In the present study, we analyze the four participating 9th grade students' mathematical reasoning processes in their dragging activities while solving open-ended geometry problems in terms of abduction, induction and deduction. The results of the analysis are as follows. First, the students utilized 'abduction' to adopt their hypotheses, 'induction' to generalize them by examining various cases and 'deduction' to provide warrants for the hypotheses. Secondly, in the abduction process, 'wandering dragging' and 'guided dragging' seemed to help the students formulate their hypotheses, and in the induction process, 'dragging test' was mainly used to confirm the hypotheses. Despite of the emerging mathematical reasoning via their dragging activities, several difficulties were identified in their solving processes such as misunderstanding shapes as fixed figures, not easily recognizing the concept of dependency or path, not smoothly proceeding from probabilistic reasoning to deduction, and trapping into circular logic.

• Journal of Educational Research in Mathematics
• /
• v.24 no.4
• /
• pp.515-536
• /
• 2014

In the present study, we analyze four seventh grade students' recognition of geometric properties and the following justification processes while their adopting different construction approaches in GSP(Geometer's Sketchpad). As the students recognized dependency and level-1 invariants by dragging activities, they determined their own construction approaches. Two students, who preferred robust construction, immediately recognized the path of a draggable point and provided step-1 justification. The other students attempted soft construction followed by their recognition of level-2 invariants and the path, and came to step-2 justification.

• The Mathematical Education
• /
• v.60 no.4
• /
• pp.543-554
• /
• 2021

The dynamic geometric environment plays a positive role in solving students' geometric problems. Students can infer invariance in change through dragging, and help solve geometric problems through the analysis method. In this study, the continuous spectrum of the dynamic geometric environment can be used to solve problems of students. The continuous spectrum can be used in the 'Understand the problem' of Polya(1957)'s problem solving stage. Visually representation using continuous spectrum allows students to immediately understand the problem. The continuous spectrum can be used in the 'Devise a plan' stage. Students can define a function and explore changes visually in function values in a continuous range through continuous spectrum. Students can guess the solution of the optimization problem based on the results of their visual exploration, guess common properties through exploration activities on solutions optimized in dynamic geometries, and establish problem solving strategies based on this hypothesis. The continuous spectrum can be used in the 'Review/Extend' stage. Students can check whether their solution is equal to the solution in question through a continuous spectrum. Through this, students can look back on their thinking process. In addition, the continuous spectrum can help students guess and justify the generalized nature of a given problem. Continuous spectrum are likely to help students problem solving, so it is necessary to apply and analysis of educational effects using continuous spectrum in students' geometric learning.

• Journal of the Korean School Mathematics Society
• /
• v.20 no.3
• /
• pp.325-344
• /
• 2017

Since PISA2006, the computer based assessment in mathematics(CBAM) was introduced for the first times and at last PISA2015 used all items in CBAM for problem solving. In this study, we focused on which important properties were considered in constructing geometric 'fence items' used in PISA 2015 to find the future direction over our teacher education, especially for constructing 'computer based assessment items.' For the purpose of the study, we analyzed the fence items on three components such as dependency, invariant, and path found in dragging activities, within a computer environment using the dynamic Geometry Software, GSP. Also, for the future, we provided an open-ended problem related to the fence items, which we could use as the merit of computer-based environment.

• The Mathematical Education
• /
• v.60 no.1
• /
• pp.1-19
• /
• 2021

This study investigates students' conceptions of fractions from a measurement approach while providing a technological environment designed to support students' understanding of the relationships between quantities and adjustable units. 13 third-graders participated in this study and they were involved in a series of measurement tasks through task-based interviews. The tasks were devised to investigate the relationship between units and quantity through manipulations. Screencasting videos were collected including verbal explanations and manipulations. Drawing upon the theory of semiotic mediation, students' constructed concepts during interviews were coded as mathematical words and visual mediators to identify conceptual profiles using a fine-grained analysis. Two students changed their strategies to solve the tasks were selected as a representative case of the two profiles: from guessing to recursive partitioning; from using random units to making a relation to the given unit. Dragging mathematical objects plays a critical role to mediate and formulate fraction understandings such as unitizing and partitioning. In addition, static and dynamic representations influence the development of unit concepts in measurement situations. The findings will contribute to the field's understanding of how students come to understand the concept of fraction as measure and the role of technology, which result in a theory-driven, empirically-tested set of tasks that can be used to introduce fractions as an alternative way.