Abstraction and concreteness are crucial in math education. But what exactly do we mean by "abstract" and "concrete"? We've studied the literature and extracted 11 definitions. Five definitions are tied to how the learner perceives the materials ("Interaction" part, top), and six definitions are tied to how the materials are designed to represent a mathematical concept ("Mapping" part, bottom). With this tool, you can reflect about the concrete or abstract nature of a representation, and move the sliders to anchor your thoughts. To learn more about this taxonomy, visit the project page.

Note: To export your results in a clean format, print this page as PDF. This tool is optimized for desktop use.

Learner representation

Learner

Interaction (subjective)
Familiarity

Many aspects of the representation are unfamiliar to the learner.

The learner can relate to many aspects of the representation based on their past experiences.

Interest

The representation does not align with the learner's interests.

The representation aligns with the learner's interests.

Simplicity

Several aspects of the representation can be perceived as difficult by the learner.

Several aspects of the representation can be perceived as simple by the learner.

Relevance

Several aspects of the representation can be perceived as irrelevant by the learner.

Several aspects of the representation can be perceived as relevant by the learner.

Sensoriality

The representation is purely visual.

The representation can be perceived by several senses of the learner.

Representation object

Representation

Mapping (objective)
Situatedness

Several aspects of the representation are idealized and detached from the real world.

Several aspects of the representation exist in the real world.

Experientiality

Several aspects of the representation are symbolic with respect to the concept, i.e. someone needs to explain the meaning of the symbols for the learner to understand the representation.

The representation is designed as an experiential mapping of the concept, i.e. the learning can make sense of the representation by observing or interacting with it.

Specificity

The representation supports generality and universality.

The representation supports specificity and examplarity.

Formality

The representation is formal, i.e. in accordance with convention.

The representation is informal, i.e. in disagreement with convention.

Approximation

Several aspects of the representation are rigorously aligned with the underlying concept.

Several aspects of the representation are only loosely based on the concept.

Physicality

The representation is implemented using technology.

The representation is implemented as a purely physical object.

Concept cloud

Concept

Reference

Chatain, J., Müller, C., Chatain, K., Calabrese, L., & Kapur, M. (2025). Concreteness and abstraction in mathematics education: A taxonomy of the semantic landscape. Educational Psychology Review, 37(4), 92.