Like most numbers, geometric objects such as angles, are abstraction from properties of real objects and quantities. For example, the idea of “two-ness” can be abstracted from real objects such as two apples, two chairs, two goats, etc. It will not take along for a learner to figure out what the idea of two means. Abstracting angles from real objects this way is not as easy as one might think it is.

Look around you and find something that to you looks like an angle. Chances are you would identify corners as forming an angle. That’s easy because you see two sides meeting at a corner. But doesn’t the door also forms an angle when you open it? But where is the other side? How about turning the door knob? Doesn’t it form an angle also? But where are the two sides there? It doesn’t even have a corner!

Mitchelmore and White (2000) of Australian Catholic University conducted a study of 2nd, 4th, 6th and 8th grade students understanding and difficulty about angles. They found that students do not readily incorporate ‘turning’ in their idea of angles. They found that it is the line (or arms) of angle which are the key to students identifying angles in different physical situation. Their study showed the easiest angles for students to learn are the **two-line angles**. These are angles in which both arms are visible such as corners of geometrical figures, corners of rooms, blades of a pair of scissors. The second group of angles are the **one-line angles**. In these angles, only one arm is visible. The other line must be imagined or remembered. Examples are the angles formed by an opening in a door, a hand of a clock and sloping of roofs. The most difficult for the students to identify are the **no-line angles **in which neither arms of the angles are visible. Examples include the turning wheel and spinning ball.

One can be said to have an understanding of the concept of angle if he/she can recognize all these types of angles in physical objects and is able to see that they all share the same property: they all consist of two linear parts (even if they are not visible) and they cross or meet at a point and that the relative inclination of the two parts has some significance – it defines the sharpness of the corner or the their openness.

So what is the implication of these to teaching? The most obvious is the importance of exposing students to as many different physical situation that can be represented by angles. Starting with the definition *an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle *and then drawing the angle figure on the board is certainly the most ineffective strategy the teacher can do to teach students about angles.

Thanks for this post, Erlina!

This is a very clear explanation of the conceptual understanding students need to develop to competently understand what an “angle” is. I think this is another example where lots of talking and experience with objects like doors, wheels and swings is necessary to really “get it”. Textbooks are at best a poor substitute for these discussions and experiences, don’t you think?

True. I believe teaching basic math concepts which are abstractions from properties of real objects should start from where else, the real objects. But what happens if the ‘real objects’ aren’t available? That’s when technology should come in, I guess.