BOTR,
Hi. This is a fascinating and upbeat topic. Like you, hope it doesn't end soon.
For one thing, mathematics has so many different perspectives. An analogy: consider buying a laptop computer and loading it up with software. The software is mathematical, but it could have so many different applications. Same with people.
About the time I came on line with this board, I was looking at the operation of the Kepler extra-solar planet transit observatory and the history of transits. So the name was much on my mind. A second candidate might have been (Pierre) Gassendi. But when I stop to reflect, for much of my career I had been occupied with the outgrowths of the 17th and 18th century mathematics which had been nudged along by needs of describing Newtonian physics.
That historical perspective wasn't exactly what attracted me to the subject when I was a kid. It was: How do you get a satellite into orbit or how do you get a spacecraft to Mars? And the latter pre-occupation was the result of reading science fiction stories by people like Robert Heinlein or his 1940s and 50s contemporaries. Heinlein, an Annapolis graduate and engineer who pioneered aircraft carrier design and operation in the early 1930s, was probably a better children's writer than he was when he was writing to supposed adults. There was an insistence that someday sailors would sign on to ships that would sail oceans wider than the ones of centuries ago - that the ships' building blocks were all there and the sailors would learn to navigate by studying books "like this one here". I worked after school to pay high school tuition at a school run by teaching brothers for one reason among several: they promised to teach calculus, something that Robert Heinlein had highly recommended from his writer's retreat somewhere off in Colorado.
Have had only brief experiences on the teaching side of the classroom - and, just by chance, in the midst of another one with a local STEM (science, technology, engineering and mathematics) program in the school district which involves a summer camp for rocketry and robotics. So far I have only completed the preparatory workshop of assembling and programming toy robots, but it is an opportunity to reflect on what Quendi and others are talking about: What are good teaching practices? What are valuable aspects of mathematics? What is it all about?
Not to recount all my career, but I did not immediately attend college after high school. In four years of military service I did some correspondence and overseas campus courses and then enrolled in a midwest university where I had intended to go to in the first place to study engineering (insufficient funds for sure). And later another school for graduate study that including being a teaching assistant in astronomy - my longest previous teaching experience. As observed, there was a distinct difference between the mathematics department approach to mathematics and that of the physics and engineering departments. At Michigan, the applied mathematicians remained in the math department, but at Washington, the applied mathematicoians were in my home department: aeronautics and astronautics. In either case, my hands were full dealing with their assignments; and I survived rather than excelled. But along the way I did learn some of the things I had wanted to find out about when I was a teenager. And, of course, encountered new problems.
In the meantime main frame computer dinosaurs evolved into the notebooks such as the one on which I am writing this post- and this notebook will host many of the analysis programs developed on mainframes and more.
In the STEM program now underway, there was some introductory remarks about rocketry which included the description of a sounding rocket's path. Much like an arrow or an artillery shell, idealized without atmospheric drag into a parabola that arced up from the ground. It was observed in passing that the angle of elevation for maximum range over a flat earth was 45 degrees. I had seen the solution somewhere, I had thought, and then I got pre-occupied as the lecturer explained about assemblying LEGO robots with trying to figure it out.
It was vexing, because the angle I was looking for kept getting canceled out of the equations. Then there were two parabolas that we were dealing with: the y and x of vertical and horizontal distances - and then the y and t of vertical distance and time. The horizontal distance relation vs. time was the necessary link to setting up a calculus solution. The distance between launch and "impact" on the x axis could be found when the quadratic equation solution for time of flight was substituted into the horizontal distance relation based on an initial and constant horizontal velocity. Seeing a derivative ( =0) of this distance with respect to an angle of launch would provide a maximum or minimum. This turned out to be a an expression in cosines squared minus sines squared multiplied by a constant. cosine A x cosine A - sine A x sine A = cos 2A, I believe. If cosine 2A =0, then A=45 degrees would work.
This took longer to figure out than I had expected and there could have been several false alarms about solution. I make typos in math just like in posts - and that had always been a problem. The main difference is is that with math you believe your own mis-statements. As a grader once, I encountered a similar interesting problem about light sources from stars and three students figured it out ("Use the vector triple product", one of them said later)- I didn't. And so when I got done with that homework set, I wondered if it would be a good excuse to drop a bottle of booze on my head - or just accept it - that there would be days...
In the recent exercise, I think only one or two people appreciated having an answer to the problem: a math teacher and the lead of the workshop. But even in saying that, I could see and appreciate that the other people in the program had competencies and skills that were much different than mine. Some people simply aced the workshop material. Did that mean that in 2012 certain skills have been identified that make 18th century mathematics skills obsolete? Probably not as simple as that. Number series, set theory, encoding, statistics, ... numbers are handled quite differently in these fields than in the exercise just performed - but not necessarily in entire isolation. In the above trajectory example, I think it was an illustration of several tools in trigonometry, algebra and calculus having to be used together in a puzzle. In other areas of study in support of a science, technology or social studies, puzzle solving proceeds in a similar manner. It always helps to be lucky, not error prone and patient as well as enthusiastic.
