Christopher Tsang
Sophia Bennett
Affiliations:
Christopher Tsang — Chifley College, Australia
Sophia Bennett — Nextus Institute of Science & Technology
Corresponding author: Christopher Tsang — christopher.tsang2@education.nsw.gov.au
Students often approach mathematical tasks with the linguistic resources of everyday English, especially when familiar terms appear in unfamiliar contexts. This paper examines how such interpretations shape student reasoning. Four classroom episodes—calculating an “average”, responding to “find x”, drawing a square “with three lines”, and interpreting “average” as “typical”—show that many non‑standard answers are “80‑cent logical”: coherent within everyday language even when they diverge from formal mathematical meaning. These cases illustrate how linguistic cues influence mathematical reasoning and how hybrid phrases invite interpretations that sit between everyday and formal registers. The analysis argues that responding to such interpretations requires collective structures rather than individual teacher intervention. Moderation processes, course reports, and external examining provide ways to document and synthesise these observations across modules and cohorts. Recognising linguistic ambiguity as a systemic feature of mathematics education shifts attention from isolated errors to the interpretive processes that shape students’ reasoning.
The questions that motivated this paper emerged not from a classroom but from a family living room. Two grandchildren—one a graduate student, the other a first‑year undergraduate—were watching Minecraft videos when the younger one asked, “Is calculus difficult?” His sister replied, “Only as hard as the word calculus sounds,” highlighting how linguistic impressions shape perceived difficulty. Minutes later he returned with a new question: “What is differentiation?” Before an answer could be given, he clarified that he meant standard deviation. Then came the question that started this paper: “Is the mean the same as the average?” His confusion was not mathematical but linguistic. The everyday sense of “mean” as “unkind” and “average” as “typical” collided with the formal statistical definitions. This moment illustrates how students often approach mathematical terminology through everyday English, even before they encounter formal instruction.
Research in mathematics education has long documented the role of language in shaping mathematical understanding. Everyday words such as mean, find, line, table, similar, and square carry multiple meanings across contexts. Students frequently interpret mathematical instructions through their everyday senses rather than through formal definitions, especially when the task wording appears familiar.
Sfard’s notion of “thinking as communicating” emphasises that mathematical reasoning is inseparable from the linguistic forms through which concepts are expressed. Pimm and Morgan similarly argue that mathematical language is not merely a neutral medium but an active participant in meaning‑making. When hybrid phrases appear—such as “draw a square with three lines”—students must negotiate between everyday and disciplinary registers. This negotiation often produces responses that are linguistically coherent but mathematically unexpected.
The arithmetic mean is often introduced procedurally: add the values and divide by the number of values. While operationally correct, this obscures the conceptual structure of the mean as a balancing point.
Consider a set of values represented as weights on a number line. The mean is the unique point at which the total “moment” balances. This interpretation explains why the mean can be a value not present in the dataset (e.g., 1.43 vans). It also clarifies why students object to such results: they interpret the mean as a physically realisable quantity rather than a mathematical equilibrium.
Students object to averages such as 1.43 vans because they interpret the mean as a literal count of objects. Their reasoning is coherent within everyday language: you cannot have “0.43 of a van.” The mathematical meaning of the mean as a balancing point is invisible to them.
When asked to “find x,” some students circle the letter x on the diagram. This response is not careless; it is linguistically logical. In everyday English, “find” means “locate.” The mathematical meaning—“determine the value of”—is a specialised register that must be learned.
Students often draw a three‑sided shape. They interpret “with three lines” as “made of three lines,” not “using three strokes.” The ambiguity arises from the dual meaning of “line” in everyday and mathematical contexts.
Students frequently interpret “average” as “typical,” leading to responses such as “the average student is someone who studies sometimes.” This interpretation is linguistically valid but mathematically misaligned.
Moderation processes, course reports, and external examining provide ways to document recurring interpretations and revise task wording. In distance‑learning contexts, anticipating everyday interpretations becomes particularly important because students cannot immediately clarify ambiguous instructions.
Assessment designers should avoid hybrid phrases that invite everyday interpretations, define specialised meanings when everyday meanings conflict, test task wording with multiple readers, and incorporate linguistic analysis into moderation reports. These practices shift assessment design toward a more student‑centred model.
AI systems can surface multiple plausible interpretations of a task, revealing ambiguity before students encounter it. Large language models can simulate novice reasoning, identify unintended interpretations, and flag hybrid phrases that may confuse students. AI‑supported task design shifts assessment from staff‑centric intuition to student‑centric clarity. Instead of relying on individual lecturers to anticipate ambiguity, institutions can use AI tools to generate alternative readings of task instructions, detect contradictory requirements, evaluate cognitive load, and test clarity across diverse linguistic backgrounds.
The Scientific Literacy Writing Task booklet contains several structural features that contributed directly to Jasmine’s misunderstanding. The assignment requires students to “write a 1000‑word popular‑level article on a scientific topic of your choosing” and to cite “exactly six source articles.” Nowhere in the main description of Part 1 is there any mention of prohibited topics. The prohibition appears only later, on a separate page titled “Topic Examples – NOT TO BE USED,” which states: “You must not choose one of the following topics, or anything closely related.” The checklist reinforces the ambiguity. Students are instructed to “Read the Marking Rubric…,” “Provide exactly six different references,” and ensure “Turnitin similarity score should be 15% or less.” However, the checklist contains no reminder that certain topics are prohibited.
A further contradiction emerges from the similarity requirement. Students must keep similarity below 15%, yet the assignment booklet itself scores 24% when uploaded to Turnitin. This undermines the credibility of the requirement and confuses students about what constitutes acceptable similarity. Compounding this, Turnitin did not flag Jasmine’s topic as disallowed. Her Writing Task was accepted with an 11% similarity score and no warnings. She had no reason to suspect that her topic was forbidden. These structural issues align with the forthcoming monograph “Inward Learners,” which identifies systemic defects such as the “Appropriate Yet Forbidden Semantic Trap” and the “Passive‑Aggressive AI Policy Paradox.” These traps arise when institutional documents contain hidden constraints that students cannot reasonably detect.
AI tools can simulate student interpretations, detect ambiguous phrasing, and identify unintended traps before assessments are released. Administrators should encourage staff to use AI‑assisted instructional design tools to check the readability, clarity, and cognitive load of their notes, assignments, and exam papers.
The episodes presented earlier reveal how linguistic ambiguity shapes mathematical reasoning. Students’ interpretations are often coherent within everyday language even when they diverge from formal mathematical meaning. This suggests that non‑standard responses should not be treated as isolated errors but as evidence of deeper interpretive processes. The implications and AI‑supported approaches discussed in Sections 5 and 6 highlight the need for systemic solutions. Rather than relying on individual teachers to anticipate ambiguity, institutions should adopt moderation practices that document recurring interpretations and revise task wording accordingly. AI tools offer a promising way to simulate student reasoning, identify ambiguous phrasing, and test clarity across diverse linguistic backgrounds.
The Scientific Literacy case demonstrates that linguistic ambiguity is not confined to mathematics. Ambiguous task wording, contradictory requirements, and unenforced constraints can produce failing grades in any discipline. Addressing these issues requires a shift toward learner‑centred assessment design supported by AI‑assisted moderation.
Many non‑standard answers are “80‑cent logical”: linguistically coherent even when mathematically incorrect. Linguistic ambiguity is a structural feature of how students encounter mathematics and other disciplines. The Scientific Literacy case shows how ambiguous task wording, contradictory similarity thresholds, and unenforced constraints can produce failing grades. These issues arise partly because higher‑education staff are hired for disciplinary expertise rather than instructional‑design excellence. AI‑supported assessment design offers a practical way to close this gap. By simulating student interpretations and detecting ambiguous wording, AI shifts assessment design toward a more student‑centric model.