Do you like this example as much as I do?
I am driving a car with an 8- year-old child inside. I deliberately start naming all green cars we see around as being “red”. Red cars continue to be red ones. The child identifies the problem immediately and has a growing concern about what is happening with me. Soon we approach a street intersection with traffic lights ahead. The red light is on. I slow down and eventually stop, waiting for the green light. When the green is on, I start driving and hear a huge sigh of relief. The conclusion follows: “You are joking!”
The child’s reasoning is based on proof by contradiction: “If you are in trouble with colours, you won’t drive across the intersection when the green light is on”.
From the child’s point of view her reasoning does not have any relation to mathematics. From the teaching point it shows that many conceptual constructions in mathematics can be successfully introduced rather sooner than later. For example, the use of proof by contradiction presented here, as well as many other useful methods and structures, can be seen everywhere through elementary and higher mathematics. The teacher’s task is to keep focus on them all the time—while moving from topic to topic, extending the content knowledge and improving problem-solving skills.