Concrete Taxonomy

The role of concreteness in grasping abstract mathematical concepts has been widely explored in the educational research. However, empirical results send a conflicting message: while some concreteness helps students to reach abstract levels of representation, starting with concrete examples may also prevent students from transferring to abstract symbolisms. We argue that such discrepancies might result from a verbal dispute in our field. Indeed, while the words concrete** and abstract are often used in the field of mathematics education, they are not always used with the same meaning. For example, the word concrete can be used as specific, relatable, visual, tangible, while the word abstract can be used as general, reusable, vague, symbolic. In this work, we offer three contributions: First, we empirically identify the verbal dispute. Second, we offer a data-informed taxonomy to organize this semantic landscape and support research inquiry. And third, we offer templates for the design of future mathematics education interventions and studies.

Ongoing project... More information coming soon.

Final taxonomy

Materials

We hope that this taxonomy supports future work, in three ways, and we provide templates to support these endeavors.

First, with our taxonomy, we focused on providing a complete overview of the meanings of the terms "concrete" and "abstract" in the mathematics education literature. Now, the relationships between these axes should be further investigated: What relationships have been explored, and validated or disproved? What gaps remain to be explored? We offer an initial discussion in this paper, but we believe that pursuing this approach thoroughly and with the wider community can raise critical questions and inquiries in the field, and eventually, through empirical evidence, help refine the tool towards a simplified version that can be widely used in practice. (link to template)

Second, we encourage scholars to use the tool to better situate their studies and designs in this landscape. As illustrated in our paper, we believe that this opens the floor to nuanced comparisons of existing studies, and further conversations about both design and learning processes leveraging concreteness and abstraction. We believe that our comprehensive taxonomy as well as these explorations can further facilitate future meta-analyses and reviews of the field. (link to template)

Third, we believe that this tool should evolve through practice and discussions in the community. Therefore, we would like to encourage researchers and practitioners alike to use the taxonomy to reflect on their own studies and design processes, and share their experiences and suggestions. While our approach is anchored in data, this tool is designed to evolve and improve through continued use. (link to template)

Team

• Julia Chatain - Concept, research, tool implementation, data annotation, analysis, paper writing
• Charlotte H. Müller - Concept, research, data annotation, analysis, paper writing
• Keny Chatain - Concept, research, data annotation, analysis, paper writing
• Leon Calabrese - Data collection, data annotation
• Manu Kapur - Supervision

Publications

Chatain, Julia, Charlotte H. Müller, Keny Chatain, Leon Calabrese, Manu Kapur. "Towards a taxonomy of concreteness and abstraction in mathematics education". In the EARLI Conference. (2025).

Long paper under review.