Dr. Jones and the physics class recently ran into such a situation while studying the mechanics of dynamic motion. The context was an object being pushed across a flat surface in such a way that the object moves with a constant velocity. Noting that constant velocity implies an acceleration of zero and applying Newton's Second Law of Motion, we reach the conclusion that the net force on the object in this direction of constant velocity is zero. For many of us, however, this is counterintuitive. How, our rational minds insist, can the net force be zero when the object is moving? Once we take the aforementioned leap of trusting the math, this fact becomes a more palatable. Indeed, after sitting with the idea for a while, it becomes obvious. The frictional force between the object and the surface is exactly counterbalanced by the force pulling or pushing the object. The net force is zero and the object is not slowing down or speeding up.

Another example that lies somewhat comfortably in the zone of proximal development of high school mathematics students is the Monty Hall Problem. In his game show, Let's Make a Deal, Monty had three doors. Behind one of the doors was a desirable prize, behind each of the two others, a goat. The contestant would choose a door. Monty would then have one of the doors with a goat behind it opened. The contestant was then asked whether they wanted to stick with their original choice or switch to the other unopened door. The door of the contestant's choosing was opened revealing their choice. The question is what is the best strategy. Should one stay with the original choice or switch? Does it even matter? The answer is counterintuitive to many. Indeed, the mathematician Paul Erdos refused to accept the answer as did thousands of viewers. Erdos was ultimately convinced when showed a computer simulation of the game. Gardiner's Python Programmers are investigating this question. We started with writing programs to simulate the game. Then, we will write additional programs to investigate the strategies of sticking with the original choice versus choosing. Once we have the answer, we have the opportunity to align our intuition with our findings.

In the fast paced world of Secondary Education we need to take advantage of the opportunities to wrestle for prolonged periods with questions that are not easily answered. When the counterintuitive becomes the intuitive, we have truly shown how flexible and inventive our minds can be. We are taught to really examine and consider carefully the evidence and the argument. There is a healthy place for doubt and speculation, even in the confines of our own certainty. It is not so bad, we discover, to admit that we are wrong about something. To learn to marvel at your own growth is a worthwhile endeavor.