Concept image and concept definition¶
A distinction between a learner's associated understanding of a mathematical concept and the words used to define it.
Core Idea¶
In mathematics education, a concept image is all the mental pictures, properties, processes and other ideas a learner associates with a mathematical concept.[1] A concept definition is the words used to specify it.[1] The learner's personal definition may differ from the formal definition accepted by mathematicians, and a learner may not be able to give a personal definition at all.[1]
A particular question may bring only part of the concept image to mind.[1] One response is therefore evidence about what the task evoked, not a complete record of what the learner knows.[1] Image and definition may agree, differ, or contain ideas that could conflict when evoked together; conflict is not required.[1]
Scope of Application¶
This distinction helps analyze understanding of mathematical concepts. Tall and Vinner used it to examine limits and continuity, while Ayvaz and colleagues applied it to prospective teachers' understanding of polygon diagonals.[1][2] These studies do not establish how common any response pattern is across all learners or whether a particular teaching intervention works.
Clarity¶
Separate four things: the mathematical concept, the question asked, the ideas visible in the answer, and any definition the learner gives. For a diagonal, the formal criterion is a line segment joining nonadjacent polygon vertices.[2] Some of the seven prospective teachers in the geometry study said “line” or omitted the nonadjacent-vertex condition in their definitions.[2] Their other geometric judgments are additional observations, not proof that an investigator has seen their whole concept image.
Manages Complexity¶
A learner can have many associations with one concept, formed over time and brought forward by different prompts.[1] The image/definition distinction keeps a short verbal definition from standing in for all of that associated content.[1] It also prevents one awkward example response from being treated as a complete measure of the learner's understanding.[1]
Abstract Reasoning¶
First identify the mathematical concept and a formal criterion for it. Then record the task and the learner's actual response. If the learner also states a definition, compare the wording with the response and formal criterion. Describe agreement, divergence or potential conflict only as far as the observations allow. A potentially conflicting idea need not have caused actual cognitive conflict.[1]
Knowledge Transfer¶
The same questions can be asked of different mathematical concepts. In one study, students explained a function limit and attempted a definition; in another, prospective teachers defined a diagonal and judged geometric properties.[1][2] The tasks and mathematical standards differ, while the separation between elicited associated ideas and definition wording remains useful. The evidence supports this transfer within mathematics education, without establishing a cross-domain Prime.
Example¶
Tall and Vinner asked 70 students entering university mathematics to explain a particular function limit and to define a function limit if they could.[1] Four gave a correct formal definition.[1] Fifty-four used a dynamic approach to the example.[1] These are separate counts; they do not show that all 54 lacked a correct formal definition. The example shows why the researcher examines both a worked explanation and definition wording rather than assuming they are the same kind of understanding.
Ayvaz and colleagues' seven-participant study provides a different geometric setting: the prospective teachers' diagonal definitions and judgments about quadrilaterals supply distinct responses for bounded comparison with the formal diagonal criterion.[2] The paper does not establish a participant-by-participant definition/property cross-tabulation.
Neighborhood in Abstraction Space¶
Concept image and concept definition sits in a moderately populated region (56th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Language, Mind & Meaning-Making (69 abstractions)
Nearest neighbors
- Dual-Character Concept — 0.85
- Lesson Study — 0.85
- Concept Map — 0.85
- Basic Category — 0.85
- Latent Learning — 0.85
Computed from structural-signature embeddings · 2026-10-25
Not to Be Confused With¶
- A picture alone. A concept image includes associated properties and processes too.[1]
- A personal definition automatically matching the formal one. The two may differ.[1]
- A diagnosis from one answer. One prompt evokes only part of an image.[1]
- A proven teaching method. These sources describe and analyze responses; they do not establish a universal repair outcome.
References¶
[1] Tall, David, and Shlomo Vinner. “Concept Image and Concept Definition in Mathematics with Particular Reference to Limits and Continuity”. Educational Studies in Mathematics 12, no. 2 (1981): 151–169. DOI: 10.1007/BF00305619. Full author manuscript inspected; evidence locators use manuscript pages, not verified publisher pagination. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r
[2] Ayvaz, Ülkü, Nazan Gündüz, and Figen Bozkuş. “Understanding of Prospective Mathematics Teachers of the Concept of Diagonal”. Journal on Mathematics Education 8, no. 2 (2017): 165–184. DOI: 10.22342/jme.8.2.4102.165-184. Full published qualitative study inspected. registry ↩a ↩b ↩c ↩d ↩e