My Research and Technology

Over the past year I have been doing math research with a professor on campus and a research partner. Over this time I have been slowly adjusting while I transition from math student to math researcher. It is one thing to be able to learn mathematical rigor and constructs, but it's another thing to just start with nothing and work your way up. My professor understands this and is doing his best to make this transition easy. He directs me to mathematical research websites like arXiv, a website run by Cornell that lets users upload research papers for others to work from and review, so I can see how research papers are structured and to orient me in my research. Likewise, I browse websites looking for answers and similar results. Websites like  The On-Line Encyclopedia of Integer Sequences make it very easy to plug in a few numbers and see every series that share numbers with your input. Websites like these make it very easy to quickly see connections and share information. But I am finding that this has problems as well. With publication being as close as a click away, false results are sometimes uploaded and referenced in multiple research papers whereas in the past, results had to be carefully peer reviewed before being published. And while this was slower and thus led to a "stunted" growth in the mathematical field, it can also be said that this led to more rigorous results.

As a future teacher, I need to consider these things. Is it more important for my students to get answers? Or is it more important for my students to get the concepts of mathematical rigor? The obvious answer is the second one but it is the sad truth that most schools orient themselves more towards the first. With answers so easily available online and with, what seems like, no answer unasked, should I only be checking for correct answers, correct work, or should I even assign written homework at all? Should I instead assign students oral exams and grade them on their thought process? This is my problem with Scantrons. Scantrons may be easier and quicker but they also invoke a binary system of right and wrong that I don't want to subject my future students to. I want them to be able to try things and to see their thought process. A blank Scantron bubble is wrong, but there could be so much thought and work behind it that is important to consider.