Solving Geometric-Extremal Problems in a Nondifferential Approach: Low Achievers Learning in a Digital Dynamic Environment

• In Event: 41.072 - Potential and Challenges: Research on Students' and Teachers' Interactions With Digital Mathematics Curriculum Materials

Abstract

Objective
A common practice in daily life is choosing the best possible solution. Numerus such problems exist in economics, technology and more, inducing practical meaning into calculus study. Solving extremal problems in mathematics go back to twenty-five centuries ago. Many methods were developed as part of the theory of extremal problems. A dominant method based on Lagrange principle, turned to be commonly used. This differential approach is solely used in today's high-school math studies. However, although calculus provides systematic and powerful technique, this rigorous instrument might hinder students’ ability to understand the behavior of the objective function and its constraints. It may lead to an instrumental learning as opposed to relational understanding.
The proliferation of digital environments allowed us to adopt a different approach, combining geometry analysis with the use of the inequality of arithmetic and geometric means. The advantages are enhanced when integrated in a dynamic geometry (DG) environment. Hence, the study objectives were twofold: (1) to explore students' construction of knowledge in DG environments while working in pairs; (2) to compare the learnt knowledge of the students with their peers who learnt in classical calculus approach.

Theoretical Position
The study adopts the Abstraction in Context framework to trace knowledge construction processes in the context of solving extremum problems in GeoGebra environment, using the non-differential approach. We believe that the use of DG with purposefully designed tasks provides students with the opportunity to be actively engaged in exploring mathematical situations and foster students understanding.

Methods of Inquiry
Students' knowledge construction was followed in a micro-level by the RBC+C methodology (Dreyfus et al., 2015). Students' achievements were assessed by a test and a comparison of changes in their relative positions in the class.

Data Sources
The learning of a focus group comprised of five pairs of high-track, yet low-achievers 17 years-old students was closely followed and video-recorded for several lessons. Students’ achievements and understanding was further assessed in comparison to their 20 peers who learnt the classical calculus approach, by relating to the individual students grades prior to the intervention and in a following test.

Results
Our findings allowed us to pinpoint the contribution (and pitfalls) of the digital environment to students learning at the micro-level. We observed enthusiasm and motivation by the students while using and exploring the extremal geometric problems with GeoGebra In addition, 60% of the students improved their achievements as compared to their peers.

Significance of Work
The study pinpoints an area where students may leverage digital-based learning methodology over traditional curriculum materials. The study focuses on a somewhat less studied students' population – the low achievers among the high track students. The fact that these students, following a relatively concise yet purposefully designed intervention, were able to understand the meaning of extremum points on a set of problems, may lead to a promising path towards a more meaningful, e-based learning.

Authors

• Michal Tabach, Tel Aviv University

• Assaf Dvir, Levinsky College, Tel-Aviv