As a novice setting up my new home, my most precious possession was a little notebook with a handwritten set of recipes. Usually jotted down in the kitchen, the ink was blurry and the pages were often smeared with turmeric, but all alone in my new kitchen, these recipes served me well and ensured that though mishaps occurred, we did not starve. As time went by, I was able to improvise and add to them and the little notes in the margins became my personal twists to the dish.
Why did this notebook serve me better than all the beautifully crafted recipe books written by cookery experts on every sort of exotic cuisine? The answer is obvious and became clearer when I observed a geometry class in which the teacher was teaching constructions. I was immediately reminded of my school days, when my teacher gave us concise, efficient steps to create angles of different magnitudes, perpendicular bisectors and different varieties of polygons and circles. By following her instructions carefully, I too was able to serve up all these creations and I observed that the recipes that this teacher was giving his students also enabled them to ‘complete’ the topic of constructions.
But why are constructions taught in high school? Is it a topic that is taught for the product or for the process? In order for students to devise a plan for a construction, they have to be very familiar with the properties of the object they are constructing as well as the relationships that exist between objects. Only then can they exploit these relations and create a personalized construction. Constructions are rendered more difficult because of the constraint that only rulers and compasses can be used but it is precisely this constraint that allows students to focus on the problem solving aspect of a construction.
Before the topic is taught it is assumed that students know how to construct a line segment of a given length using a ruler. They should also be proficient in using a compass to construct a circle of a given radius.
They should understand the symmetry of a circle and realize that when two circles of the same radii are drawn from two different points then the points of intersection (if they exist) are equidistant from the two centres. To truly understand and eventually devise the construction steps, students should also know the properties of triangles and the conditions for congruency.
The author has been a teacher of mathematics for the last 20 years. She works in Azim Premji University and is the Associate Editor of At Right Angles: a resource for school mathematics. She can be reached at firstname.lastname@example.org.