Often enough, elementary school children will identify these figures as “triangles” even though they can say that triangles have three sides. Why? Partly because children (and, for the most part, adults, too) learn words more from context than from definitions and explanations. Especially when we have no other word for this shape, we choose a best-fit category. These shapes look more like the concept image of triangles that children build up from examples like than they look like any other shape for which the child has a name, so that is the category the child lumps them into.
By contrast, it is common enough for children not to recognize as a triangle, despite its fit with the definition, because it is visually so different from the common examples . Similarly, people often enough refer to as an “upside down triangle” (even though nothing about the definition of a triangle specifies what way it must sit) because the common image of a triangle sits on its base.
Examples help people “get” an idea in the first place, or extend or clarify an idea. But examples, as you have seen, can also create misunderstanding. This article shows ways in which examples are essential, risks of poorly chosen examples, ways to make the best of examples, and some limitations of examples.
Clear examples illustrate the essential elements of an idea without distracting the learner with irrelevant patterns. The picture limits the learner’s image-of-triangle with three irrelevant patterns: all figures sit comfortably on a horizontal side at their bottom (triangles don’t have to sit that way); all figures have symmetry or right angles or both (triangles are more varied); and all figures can be packaged in a box that is roughly as long as it is wide (triangles can be “extreme”). Including and helps the learner see the essential idea (three sides) and not inadvertantly include irrelevant ideas (orientation on the page, symmetry, or extreme skinniness).
Although those extra examples can correct wrong first impressions after they’ve occurred, presenting those examples first helps avoid students jumping to wrong conclusions. It takes more work to correct misunderstandings than to avoid them. Order matters. The first examples are especially influential when you are teaching without talking (“silent teaching”), and the examples you use are your only way of communicating. Those first three or four examples must contain enough information to help students not jump to wrong conclusions.
An unfortunate choice of examples in a silent teaching activity: The teacher is playing “Guess my rule” as a way of introducing the idea of square numbers. He chooses these first two example (or might present them this way , as in the Difference of squares activity). These, unfortunately, can lead students to patterns other than the one the teacher wants. (Any set of examples can mislead — see the limitations of examples below — but well-chosen examples are less ambiguous.) One possible consequence is , in which both the units digit and the tens digit increase by one in each successive case. The induced pattern is logical — an example of good thinking — but it is not about square numbers and so the example failed at its purpose; it contained two patterns, not just the one we wanted students to find.
A single example is never enough information. Any single example, by itself, suggests too many possibilities, but some first examples are more ambiguous than others. 7 → 49 calls fewer things quickly to mind than 2 → 4. The latter is a poor example of squaring because it is also an example of doubling or adding two. Of course, from a single example alone (without words to highlight the intended features), we can’t ever be sure what was intended. 7 → 49 is also an example of multiplying by 7 or adding 42 (or infinitely many other patterns like “add 3, multiply by 5, and then subtract 1”), but these other possibilities are far less distracting, less insistent on standing out, than the possibilities for 2 → 4.
Even with as examples, a learner does not have enough information to know what is not a triangle. Selected non-examples, like , help focus attention on details that might otherwise be missed. The “three sides” must be straight, not curved; there can be no extra frills or bows or hanging-over bits of line (line segments must intersect only at their endpoints); the “points” can’t be “cut off” (the shape is bounded by only three segments); the figure must be closed (all endpoints must be joined).
These non-examples were selected to be “near-misses,” very close to the image people have of triangles. When children give verbal descriptions of triangles, they often mention “three lines” or “three corners,” but omit the details that eliminate even fairly distant misses, like , which may sometimes be useful non-examples to help children improve their verbal descriptions.
In everyday conversation, definitions are of little real help. Try to think, for example, how you would define “chair” to include all the different kinds of objects, wooden, plastic, stuffed, formal, etc., that are “chairs,” and how to exclude superficially similar objects that are not chairs. Or think how to define “cat.” Alternatively, imagine that you did not know these words; then look in a dictionary to see how much you must already know in order to understand the definition! Finally, think how little that definition really contains of the “cat” in your head. Definitions are not easy routes to meaning, even for adults, until one already has a fair idea what the word means from use in context—that is, from examples! It’s not uncommon for adults to notice, when asked (perhaps by a child) the meaning of a word that they’ve long understood and used, that they don’t really know, and have to look it up. For casual use, context and experience are enough to give us “the general idea” of a word, and make it useful even if we can not give a definition.
This is as true for children as it is for adults. Young children acquire vocabulary at an astonishing rate, a full 50% of their expected adult vocabulary by the age of five! They do that entirely from use in context, and generally without any specific attention to didactics. New words are acquired extremely rarely from a definition and never solely from one. And, in fact, when children do look up definitions before they have the general idea—a kind of vocabulary-learning that occurs only in schools—the results are generally strange. They look up “extinguish,” see that it means “put out,” and write “before I go to bed each night, I extinguish the cat.”
In a way, examples are bits of context — ways to give information other than “saying what the word means” — allowing children to acquire vocabulary in school a bit more the way they do out of school, at which they are so adept. Examples allow teachers to use a word communicatively until students are able to use it as well. Teachers can use the word rather than explaining it because the example provides the context and carries the meaning. Only then, when the students already have a rough meaning from communicative use in context can one effectively clarify the meaning formally with other words, through discussion and/or definition.
But examples are not enough. Even after a rough idea is acquired from examples, extreme examples, and non-examples, it is still the case that definition is needed. It is easiest to show why by example, using a whimsical category called “smanglings.” Imagine developing this category inductively, through examples.
Each of these is a smangling:
So far, it would seem that smanglings are squares of any size or orientation. When we learn that these are also smanglings, we realize that the category is broader, and might decide that smanglings are probably four-sided shapes of any kind. But even these are smanglings!
So, now what do smanglings appear to be?!
For pedagogical purposes — maybe for any purposes other than sheer perversity — the order in which the examples were presented was terrible. In order for examples not to be misleading (as we deliberately were in this case), it is important that they be as varied as the category permits (so, not just squares, not even just four-sided figures, if in fact, more variety is permissible). As stated above, it is often (but not always) the case that the variation should come quite early, so that the first impressions are not misleading, as they were here.
We cheated. This, too, is a smangling:
Now what do smanglings appear to be?
It is tempting, now, to conclude that smanglings can be any closed two dimensional shape. But it is not logical to draw that conclusion, just as we could not (logically) conclude, before the circle was included, that smanglings could be any polygon. The correct answer to “What is a smangling?” is “We can’t tell, because we haven’t seen anything yet that isn’t a smangling.”
So, here are a few non-smanglings:
Without these non-examples, how could we have guessed that color mattered? In fact, it would also have been illogical to make such a guess, as there was no evidence one way or the other.
Here is another non-smangling: Apparently the border must be visible!
At this point we do have a lot of information. Let’s see how far it gets us.
|Testing the concept:
Which of these is a smangling?
We can be fairly sure that a and d are smanglings; we can be equally sure that g is not. But what about the rest? Case c is the right color and has a black border, but we don’t know whether decorations on the inside are allowed. No non-example rules them out, but no example shows them. This exercise (because it is purely inductive, without definitions) is more like science than mathematics: we can hypothesize pending further testing, but we cannot decide. It seems likely that case e should be ruled out as it has no black border, but case f is harder to rule out. It has the black line, and though it is not the right color on the “inside,” it can’t be because it doesn’t have an inside! We have no evidence about this case. It certainly doesn’t fit the examples, but it isn’t ruled out by the non-examples, either. We have the same problem with case b. It is the right color on the inside, and the border is visible, but we’ve learned that color matters. Maybe the color of the border matters, too, and we simply don’t know.
In fact, no matter how numerous and varied our examples and non-examples are, unless they are exhaustive (i.e., the set of smanglings is finite, and we have encountered every one of them as an example), examples alone are insufficient to allow us to decide all cases, because they provide no way of knowing whether or not some perverse exception lurks among the cases that have not been seen. But the examples — and especially the task of trying to choose among the unknowns and then defend that choice — make it much easier to understand a definition now than it might have been at the start. One advantage for students of encountering this meta-mathematical idea is that it helps motivate what otherwise often seems like bizarre over-particularity in the wording of definitions. There is a lot we must say to define a smangling in a way that allows us to decide, definitively and without question, which of the unknowns is and isn’t a smangling.