Scaffolding Fifth Graders’ Learning with a Digital Multi-Representation Applet: Design Research on Focusing Multiplicative Structures with Dynamic Dot Arrays

Abstract Digital multi-representation applets offer promising opportunities for promoting students’ conceptual understanding by dynamically converting graphical and symbolic representations and for promoting discoveries and justifications of mathematical properties. However, learning effects of open-ended multi-representation applets seem to depend on targeted scaffolds for students’ processes of conversions, treatments, and conversions of treatments. While these design challenges have come to be well-understood for dynamic geometry and functions, more design research is needed for multi-representation applets for multiplicative reasoning. We report on a design research study in Grade 5 in which the multi-representation applet of a dynamic dot array was integrated into a digital teaching–learning environment with sequenced tasks and more targeted scaffolds. The qualitative analysis of 30 students’ processes revealed multiple potentials of the dynamic array to (a) connect static representations by articulating the underlying multiplicative unit structures (conversions), (b) describe systematic variations (coordinated treatments), (c) explain systematic variations in representations by articulating unit structures (conversions of variations), (d) discover multiplicative commutativity (treatments), and (e) go towards justifying multiplicative commutativity by restructuring the array and its unit structures (conversions of treatments). Design experiment Cycle 1 revealed that all potentials were partially addressed by some students. To enable students to exploit more of these multiple potentials, attention scaffolds and language scaffolds were integrated. In Cycle 2, more potentials were exploited while still leaving room for improvements. In total, the study shows how design research can contribute to embed multi-representation applets into teaching–learning environments with well-sequenced tasks and targeted scaffolds.