The Trouble with Triangles

In a large study in India, a group of fourth-graders were asked to choose which of the four figures was a triangle. Figure D was an upside down irregular triangle – no side was equal in length to any other side.  The other three figures were not triangles at all, but were a four-sided figure, like an arrow head, a cone, and an open spiral made of straight lines. Many children could state that a triangle has three straight sides and three corners, but only 30% of the children would accept a triangle that had its horizontal line on the top or did not have three equal sides.  How to best describe their problem?

One might be tempted to say that the children were not listening or that the teachers had not given the children enough examples.  But the children had listened and the teacher had given them numerous examples.  A better explanation can result by considering how the young children “construct” the presented information.  What the children were doing can be called “prototyping.”  As they listened to the teachers and viewed the examples, the children selected one particular triangle, usually an equilateral triangle pointing upward, and treated this specific case as a prototype that they could visualize as an object.  What they were not doing was thinking of a rule or a procedure that would generate a triangle if drawn according to that rule.  Forward, turn right, forward and turn right until facing the spot you started and step forward to that spot.  This rule will always generate a triangle, even strange-looking triangles.

Do children use visual imagery to prototype because they are young or because we teach them to do so? Teachers try to define a concept, like a class of geometric objects, using a list of features and relations the object has, but they do not treat these features as the result of an action rule for producing that class of objects. A rule cannot be remembered as a visual image of an object; an equilateral triangle printed in a textbook can.  Given this method of teaching children look for what an object has and do not think about what features it could have if drawn according to a rule.  They look for something similar to a standard case that they remember seeing.  Consequently they have no way to accept the unusual example of an upside down triangle with unequal sides.

Teachers or video games that try to teach concepts by asking the children to remember a list of features rather than a procedure, the class will often be surprised at how children over generalize a specific case of the information presented.  They over generalize the necessity of a feature that the prototype has even though other triangles do not have that feature.  They have no strategy, other than a mental comparison to the prototype that helps them calculate which features are necessary and which features are not, such as equal-length lines or a corner pointing upward.

Why don’t children simply remember the correct list of features? Three straight sides of any length that form a closed figure.  Perhaps it is because the phrase “of any length” is another way of saying “eliminate the length of the sides as relevant.”   But here again “of any length” does not have a mental image and is itself a rule – a rule of exclusion.  And if the teachers are not orienting the children to think about rules of production or rules of exclusion, this “of any length” will not be easily remembered.  The children continue to think about how triangles look rather than the rules that direct how triangles are made. Teachers need to use the verbs of geometry as much as the nouns.

Reference

How Do Our Children Think (2006). Ahmedabad, India: Educational Initiatives.  www.ei-india.com.

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