In the November 15 issue, we discussed student misconceptions in the understanding of triangles in "Too thin to be a Triangle" and the basics on misconceptions. We highlight here, the student responses and identification of shapes. The following article discusses these ideas in detail.

The teacher had presented Rahul, Tanu and Ali with a question below (Fig. 1)

Tanu and Rahul chose B and C respectively and Ali chose D. The teacher decided to probe them on why they chose their particular answers and why they did not choose option D as their answer.

Teacher: Tanu, why did you choose B as your answer?
Tanu: In A, the bottom line is not straight. In C, the lines are not joined and D is too thin. B looks like a triangle, and if we turn it upside down, it will become a triangle. So my answer is B.

Teacher: Rahul, how about you?
Rahul: In A, the base is not flat. B is definitely a cone. D will become a triangle only if we turn it upside down. But C looks like a triangle, so my answer is C.

Teacher: Ali, why do you say D is a triangle?
Ali: Actually, I think even D is not a triangle because it is very thin. It should be fat like A to be a triangle. But I am sure A, B, C are not triangles, so I chose D.

Rahul: I agree with Ali. Also one line in D is straight and the other one slanting. A triangle should have 2 slanting lines as in A here and a sleeping line  (See Fig 2).

Just then Piya approached the group. The teacher presented Piya the question too.
Piya: The answer is D. A has 4 angles. B is a cone. C has many lines.. A triangle is a closed figure with 3 sides and 3 angles, so D is a triangle. Thick or thin it does not matter.

Ali, Tanu and Rahul were a bit reluctant to accept Piya's answer as the correct one.

Their teacher then asked them, "What is the definition of a triangle as you have learnt it?"

Rahul replied immediately: A closed figure with 3 sides is a triangle.
And others readily agreed with him.

The teacher then presented the question again to all of them. Now Tanu and Rahul realised that A, B, C are not triangles. But Rahul still thought D is not a triangle.

Teacher (showing them Fig 3): "Is this a closed figure with 3 sides?"

Children (unanimously): YES.

Teacher: "Is this a triangle?"
Children (unanimously): YES.

Now the teacher rotated the shape (Fig 3.1) and asked the children, "Has the shape changed?"

Rahul, Tanu, Ali (unanimously): YES

The teacher then asked a crucial question.

Teacher: "Is this a closed figure with 3 sides?"
Rahul, Tanu, Ali (after thinking): YES

Teacher: "So, is this a triangle?"
Rahul, Tanu and Ali said in a tone of understanding, "YES. It is still a triangle because it is closed, and it has 3 sides that form 3 angles! Oh! Just like Piya said."

Children are familiar with triangles as shown in Fig.2, and they “prototype” this image as a triangle in their minds. When presented with an image different from their “prototype”, (E.g. Fig 3.1) they are not able to identify it as a triangle.

As geometric thought develops in children, model building or “prototyping” takes place first. It allows them to relate to a particular shape, for e.g. a triangle. It is followed by the ability to recognise identical shapes, a particular shape in different orientations, from a collection of various shapes. Lastly, children learn to describe the properties of the shape.

Identifying a shape merely by its appearance as Tanu and Rahul did is a crude strategy. It is important that children are exposed to shapes in different orientations and sizes and are encouraged to identify shapes based on the properties of these shapes. Identifying a shape merely by its appearance as Tanu and Rahul did is a faulty strategy. The underlying understanding should be that a shape remains unchanged as long as its basic properties remain unmodified.

It is extremely important to bring out misconceptions from a child’s mind during the formative years, so that they don't pile up and hinder the understanding of greater concepts to ensure unhindered development of more sophisticated concepts.

(Educational Initiatives is an Ahmedabad-based organisation working at the forefront of driving change in education through research, large-scale assessment and the development of new learning technologies. Student responses have been included are taken from a series of student interviews conducted by EI.

We also welcome your ideas on how to prevent misconceptions. Please write to us at: misconceptions.dh@ei-india.com)


Educational Initiatives, Ahmedabad