Mathematical Playgrounds for Guided Discovery: From Experimentation to Formalisation with JSXGraph
Dr. Kinga Sipos
Mathematical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
Keywords: JSXGraph, guided discovery, mathematical experimentation, interactive visualization, mathematical intuition, multiple representations, introductory mathematics, economics education.
Abstract
How can interactive visualisations be used not merely to illustrate mathematical concepts, but to introduce them through exploration and experimentation? This contribution presents a collection of JSXGraph applets developed for an introductory university mathematics course for economics and social science students. Conceived as mathematical playgrounds rather than tools primarily intended for practice or assessment, the applets invite students to investigate new notions, develop intuition, and formulate conjectures before or alongside their formal mathematical treatment.
The underlying pedagogical approach combines mathematical experimentation with guided discovery. Students manipulate mathematical objects, vary parameters, and explore deliberately selected examples, non-examples, and limiting cases. By observing which properties change and which remain invariant, they can identify essential features of a concept and recognize the limitations of their initial intuitions. Carefully chosen contrasts and counterexamples encourage students to question apparent patterns, refine their conjectures, and appreciate the need for precise mathematical definitions and conditions.
A further design principle is the coordination of multiple mathematical representations. Dynamic connections between graphical, numerical, algebraic, and contextual representations allow students to investigate the same mathematical relationship from different perspectives. Changes made in one representation are immediately reflected in others, providing opportunities to explore their correspondence and connect concrete observations with abstract mathematical structures. Visualization thus becomes an instrument for mathematical investigation rather than merely an illustration accompanying symbolic expressions.
The effectiveness of this exploratory approach depends not only on the interactive visualizations themselves, but also on how students are guided in using them. Discovery-guiding questions structure the activities by directing students’ attention towards relevant comparisons, encouraging them to articulate and test conjectures, and introducing counterexamples where appropriate. In selected activities, definitions or results can be revealed after students have formulated their own hypotheses, allowing them to compare their observations with formal mathematical statements. The intended progression is from exploration and observation, through intuition and conjecture, towards mathematical formalisation.
This approach has been implemented across a collection of applets covering fundamental concepts in calculus and their applications. Topics include limits and continuity; the definition of the derivative, monotonicity, convexity, and the extreme value theorem; Newton’s method; Riemann sums, definite and improper integrals, and Taylor polynomials. Economic and statistical applications include consumer and producer surplus, linear regression, the mean squared error as an optimization problem, and profit maximisation using linked two- and three-dimensional representations. The activities employ different forms of interaction, including movable points, adjustable parameters, dynamically constructed graphs, and representations that can be progressively revealed.
The presentation will demonstrate selected applets and discuss the pedagogical and technical decisions involved in their design. Particular attention will be given to selecting meaningful interactions and contrasting examples, coordinating multiple representations, and developing guiding questions that encourage exploration without prematurely revealing mathematical conclusions. As a practice-oriented contribution rather than an empirical evaluation of learning outcomes, it illustrates how JSXGraph can be used to create opportunities for guided mathematical discovery and support the transition from visual experimentation and intuitive understanding towards conjecture and formal mathematical reasoning.