Learning Science · Concepts and changing explanations
Examples, nonexamples and boundaries
Use carefully varied examples and near misses to discover what makes membership in a category matter.
Use the examples to connect this lesson with the earlier topics. No specialist equipment is needed.
Back to the chapterAfter reading, you can
- Separate defining features from incidental features.
- Choose contrasting examples that test a proposed boundary.
What changed and what stayed?
Suppose every triangle in a lesson is blue, points upward and has two equal sides. A learner might unknowingly adopt blue or upright as part of the definition. Showing one red triangle helps rule out colour. Rotating another helps rule out orientation. A nonexample with curved sides tests a different condition. Example selection influences which features are available to compare.
An example belongs to the target category; a nonexample does not. A useful nonexample is often a near miss that changes a relevant feature while preserving much else. A square is a clearer contrast for triangle than a photograph of a cloud because both are closed straight-sided plane figures. The difference in side count becomes easier to isolate.
State and test a candidate rule
Ask the learner to suggest a rule, classify a new case, and give the reason. Then choose the next case to test the rule. If the proposed rule is three corners, show an open path with four straight segments and three bends: it has three corner-like bends but is not closed, so it tests whether closure has been understood. If all chosen cases are extremely similar, success can reflect a narrow visual habit rather than the intended concept.
Do not make every nonexample obvious. Once the basic idea is established, include cases near the boundary and examples with unfamiliar surfaces. Explain why each belongs or fails. The goal is accurate discrimination, meaning noticing the difference that matters, rather than memorising which pictures received a tick.
| Case | Classification | Reason |
|---|---|---|
| 0 | Even integer | 0 = 2 × 0 |
| −2 | Even integer | −2 = 2 × (−1) |
| 3 | Nonexample | Not twice an integer |
| 2.5 | Nonexample | Not an integer |
Some boundaries need a stated convention
Not every category has a single simple rule. Everyday categories may admit disputed cases; scientific classifications can depend on an explicit framework. A whale is a mammal even though its body shape resembles some fish. Relevant biological relationships outweigh a superficial feature. Learners need the framework and its evidence, not a demand to guess the teacher’s preferred label.
Ask whether a difficult case is genuinely outside the category, ambiguous because information is missing, or classified differently by competing conventions. For a precise mathematical definition, the boundary follows that definition. For a judgment such as reliable evidence, explain the criteria and uncertainty. The same example strategy should not pretend these two kinds of decision are identical.
Worked example
Teaching even integers
A learner sees 2, 4 and 8 and proposes that even numbers are positive whole numbers ending in 2, 4 or 8. Which further cases test that rule?
- Add 6 and 10. Both are twice an integer, but their final digits were excluded by the proposed list. These cases reveal that the listed endings were an accidental restriction.
- Add 0 and −2. Zero equals 2 × 0 and −2 equals 2 × (−1), so positivity is not required. State the definition: an even integer equals twice another integer.
- Contrast 3 and 2.5. Three is an integer but is not twice an integer; 2.5 fails the integer condition. Ask for the different exclusion reasons rather than only a tick or cross.
The examples revise incidental restrictions while keeping the defining relationship. Zero and negative even integers belong; three and 2.5 fail for different reasons.
Try it yourself
Think through each question before opening the answer.
Which helps test an upright-triangle misconception: a larger upright triangle or a rotated triangle?
Show the answer and reasoning
The rotated triangle changes orientation while preserving the defining geometry. Size variation tests a different possible restriction.
Rotation preserves straight sides, closure and side count while changing the surface feature in question. A larger upright triangle leaves the suspected orientation restriction untested.
Why include 0 when teaching even integers?
Show the answer and reasoning
Zero equals twice the integer zero, so it satisfies the definition. It challenges the incidental belief that even integers must be positive.
The defining relationship applies to zero: it is twice the integer zero. Examples restricted to positive cases could hide that boundary from the learner.
A learner classifies correctly but says "it looks right". What next?
Show the answer and reasoning
Ask for the relevant condition and present a near miss. Correct labels without reasons leave the basis of classification uncertain.
A correct classification can be based on an irrelevant visual cue or guess. A near miss and an explicit reason provide more information about the feature controlling the choice.
What to carry forward
- Vary incidental features and contrast defining ones.
- Use boundary cases to test rules rather than to trick learners.
References
Sources supporting the explanations and learning checks in this lesson.