Circumstance : Learning to look around us and contemplate and understand our surroundings is something that comes with maturity. While a toddler may be completely nonchalant by the road traffic, it isn’t because he’s brave, but because he is unaware. Naivete is not bravery. As our toddler starts to pay attention to the world outside of himself he starts to understand circumstance. Circumstance forces us to think outside of ourselves in life and it forces us to look at what is happening around the periphery of a problem in math. In Geometric proofs we are forced to use this skill of circumstance.
Truthfully, some students really struggle with the steep learning curve of completing a proof. They see the multitude of steps and wonder how anyone even came up with them all. The big picture can be so overwhelming that they fail to step back and see that each proof is completed one step at a time. In a proof we start with a given truth (“given triangle ABC on line B-C-D”) and we are asked to show why the conclusion is necessarily true( “prove: angle A+B is equal to angle DCA.”)
To progress through a proof we are constantly asking the same questions: What do I see? What do I know is true? What can I prove is also true from what I know?
These questions are incredibly large and vague. Most students really struggle to narrow them down, so we ask them to think outside of the proof and identify information around them. The most obvious circumstance question is ‘what is going on inside triangle ABC?’, but there’s also the equally important question, ‘what is going on outside of triangle ABC?’
These circumstance questions help us see other truths about the situation that may or may not help us discover the new truth of the conclusion of the proof. In our example above, “what is going on inside triangle ABC?” Leads us to remember the qualities of the triangle. It has three sides, three angles, the sum of the angles is 180 degrees. So angle A + angle B + angle C = 180. This may help us, so we venture forth into the world outside of the triangle. It is big and bright and full of truth and beauty!
