Yesterday evening, I decided to work on a 'fixed angle animation' created by John Mason (available here). Before reading this post, it would be useful to watch the animation.
Here is a screenshot (which it must be noted does not capture the dynamism of the animation):
As I watched the animation, I experienced a feeling of pleasure and surprise as the parabola appeared, followed by bewilderment. This mixture of emotions was enough to make me want to work on the question: How does this construction form a parabola?
I watched the animation a few more times, observing how how each line was formed. I sketched a few cases of the construction, trying to simplify the complexity of the situation. Through the fog came vague images and ideas; similar triangles, parallel lines, equal angles and equal distances, but no clear strategy for solving the problem I had posed as yet.
I spent some time refreshing my knowledge of parabolas, and while interesting, I suspect that thoughts about the focus and directrix were distracting me from tackling the problem more directly.
... and then using geogebra, in order to create lots of examples, and have more control over movement:
Working on geogebra allows me to consider what must be, and what only seems to be, the case. I briefly explored some special cases, such as the one below, but they did not tell me anything useful about the general case:
I was in a familiar situation of trying to determine angles that were not easily determinable, and knew that proceeding down this 'angle-chasing' route was going to become complicated. So I went to bed, and had a reasonably sleepless night trying hard to not think about the problem.
Had I considered this further, I think it might have been useful, but as it involved trigonometry, and seemed a bit complicated, I avoided it. The fact that I am inflexible in my problem solving approaches, favouring certain approaches (geometry) over others (trigonometry) is something I could consider. [This inflexibility is also something I have noticed in my approaches to teaching.]
However, I think that considering the parabola as a projectile led me to focus on the tangent at the origin, or at what could be considered a root, which then led me to then consider something John and I had discussed in the past. He had pointed out that the discriminant of a quadratic could be considered both as the "square of the inter-rootal distance once the leading coefficient is 1, but also the product of the slopes at the roots," something that, at the time, I had not previously been aware of.
In short, there is a relationship between the gradient of the tangent at the root (and then also all of the 'fixed-angle lines') and the distance between the roots (the points where the parabola crosses the horizontal 'axis'). This led to me working algebraically to determine the relative lengths of various segments, which coupled with use of similar triangles, led to my verifying the construction as a parabola.
- My initial surprise and enjoyment as the parabola formed gave me the energy to want to work on the problem (which I had formulated). Without the sense of mystery created by the animation, I would not have worked on the problem for as long as I did.
- I spent a long period of time in a state of bewilderment, which was eased by drawing lots of diagrams. These examples allowed me to identify possible relationships. Generating the images on Geogebra allowed me to create more examples more accurately, allowing me to make and break conjectures.
- It took me a long time to shift to other methods for tackling the problem, due to my preference for working geometrically. I have observed the students I teach getting stuck in similar 'ruts', both while working on a particular problem, but also over time. [I have also observed myself getting stuck in 'pedagogical ruts'.]
- The time spent considering various pieces of knowledge may have been a necessary part of the process, but there was also a sense in which it was obstructing me from working on the problem more directly. That said, I was quick to discard an approach if it was seemed over-complicated.
And finally: working on this problem once again highlighted the importance, as a teacher of mathematics, of working on problems, in order to understand how learners might also work on mathematics.