I propose here ideas teachers need to know and pay attention to when teaching mathematical proofs and how to prove.
A. What is a (mathematical) proof?
I define proof as a relational network of claims (propositions and conclusions), substantiation (established knowledge that makes the claim legitimate) and appropriate connectives so sequenced to justify why the conclusion is a logical consequence of the premises. Continue reading “Tips for Teaching Proofs and Proving”
The following problem (or proving activity, if you like to call it that) is a typical textbook geometry problem. It is tough and guaranteed to scare the wits out of any Year 9 student.
When I used the given condition to construct the figure using GeoGebra, the only thing I can move is A or B and what it does is simply to reduce or enlarge the circle. Pretty boring. So I thought of making C dynamic. The way to do this is to construct point C along the circle and then construct a perpendicular line to AB. With C moving along the circle, the once static and close task is now a dynamic, exploratory and task.
Your students will observe that for triangle ABC to be an equilateral triangle, CD must be the perpendicular bisector of AB. You can now ask them the problem: Given that CD is the perpendicular bisector of radius AB, prove that ABC is an equilateral triangle, which is what the textbook is asking them to do.
In presenting the problem the way I’ve shown above, you did not only make the problem more interesting (hopefully) and accessible to the majority of the learners (I’m sure most of them can answer the questions), you have also given learners the chance to explore the problem first and be familiar with the situation.
Note that you will be doing a disservice to your students’ geometry life if you will stop at #5 and not give them the opportunity to prove. Proving is what makes mathematics different from other disciplines. It would be a shame if they will go through life only complaining about x and not of proving as well. I’m joking but you know what I mean. You may want to check some of my favorite post about teaching geometry through problem solving: Unpacking mathematics – a geometry example and Problem Solving Involving Quadrilaterals.