In the short answer section of the survey, there were four questions. Below we present each question and Neal's answers to them.
Question 42: Describe the most interesting mathematics problem you have ever worked on.
Before: To be honest, I can't remember - probably very largely due to the fact that my success in mathematics has not been spectacular in the past two years - ones that I thought were interesting I always did wrong.
After: TEN CARDS! This problem brought together many things like 31, 33, 34, 40, +especially 41. The research (and card tricks) was fun! Putting it all together gave me the greatest sense of accomplishment.
Comments: The numbers refer to questions on the numerical survey. The Ten Cards project was his group project in the class.
Question 43: What is the average length of time you spend on a mathematics problem before going to someone else for help?
Before: Probably half hour and as much as an hour or so.
After: 2 hours or so.
Question 44. What is mathematics?
Before: A natural science that studies different relationships involving certain numbers and/or certain order/structure.
After: The ability to think critically/logically in terms of solving problems. It is an area of natural science to which many truths of the world can be shown through the order we create in mathematics.
Question 45. Describe what features you think make a mathematical problem interesting.
Before: forms a bridge between an ordinary math concept(s) to a higher level of concept(s) in which you must use them to reach something entirely new.
After: In order for any problem (math or other) to be interesting, there must be an inner motivation sparked inside the individual …
Myself - something that builds off of another - creating a system of linked ideas
- patterns/order, where you have to be creative.
- the problem is that once I got into theory, until THIS level, I could not afford to be creative, I no longer found mathematics interesting - there was no inner motivation…
Neal's responses show pretty thoroughly how disaffected with mathematics he was prior to taking this class. In particular, as his answer to question 42 showed, he had been unsuccessful since calculus, and as his answer to question 45 showed he found himself unable to be creative in his mathematics classes at that level. The question from this is then: what effect did this have on his understanding of mathematics? Looking at his answer to question 44, we see that critical thinking and logic were not part of his original definition of what mathematics is, but were added at the end of the class. This perspective is very much in line with his disaffection, since his main experience in college mathematics appears to be struggling to do problems but only over short periods of time before going elsewhere for help. Tied into this, we see that the idea of inner motivation becomes critical to his end response to question 45. Again, the notion that such was possible was not something he did not appear to recognize on the first survey.
Question 45 was repeated in the interview I did with Neal four months after the class was over. Neal's response at this point was:
One that starts thought, mathematical thought, and interests to the individual student. Whereas, back to the project again, it was good to be able to have a choice, a list of projects, not just projects, but a list of problems, and being able to pick something that interests you. Going along with that, there need to be motivation involved, because if there is no internal or external motivation for mathematical problem, it is pretty hard to explore, and to explore the mathematics I guess.
Again, we see the importance that interest and motivation play in his answer
to this question. In fact, if anything he has moved further from the idea that
a good/interesting problem is one that is externally motivated than he was in
the survey response that he gave at the end of the course.