> Alain Schremmer notices that one problem with the > book's organization > is that it makes him wonder what the "story behind > the mathematics" > is, that is what the point of the development of this > math is, where > it leads, what it allows us to do. He has noted that > I still appear > trapped in the usual mode of presenting math as a > series of disconnected > topics rather than as a unified whole but that he can > also tell that > I'm trying to escape that mindset. That is, I > believe he is thinking that, > if I continue the book with this organization, I may > make much further > progress in escaping this mindset than most do but > that I still won't > escape that entirely. > > I wonder where he gets that idea when he hasn't seen > the text of the book. > But maybe he can see something in that organization > that tells him this > fact whereas I don't see it. > > Here are two of his responses: > > http://mathforum.org/kb/message.jspa?messageID=7208094 > &tstart=0 > > and > > http://mathforum.org/kb/message.jspa?messageID=7208113 > &tstart=0
When I had thought more about Alain's remarks yesterday morning, I had realized what he means when he says that a mathematics course tells a story: The course develops ideas along the way that lead to a final big idea in the end, something like a "punch line." Another way to say it is that the course in one continuous stream of ideas that lead to some ulimate goal rather than multiple little streams of ideas that are not connected and do not really lead anywhere. Many math courses, especially general education math courses or survey math courses, do not lead anywhere but instead have students meandering aimlessly throughout different parts of Math Land. That is, such courses are really just a hodge-podge of different mini-courses all thrown together into one course.
Based on Alain's comments, I really do question if my arithmetic book as it stands would really lead anywhere--that is, if the book really does lead to an ultimate goal at the end that makes it clear to the students what the point of arithmetic really is. That is, I will need to make it clear in the book what big reason arithmetic was developed for and not just simply what various smaller questions arithmetic was developed for.
Of course, we know that arithmetic leads to algebra and other higher mathematics, but how meaningful is that goal to students? Even if that happens to be meaningful to students (I really question that), can an arithmetic book really prove that this goal is accomplished? Another goal is number sense, but how meaningful is that goal to students? Even if we use this as a goal, we still need to motivate the need for number sense. In fact, there are two problems with this goal: (1). Can we really prove to students that such a course leads to this goal? (2). What exactly does this goal mean? What is number sense? I have a good intuitive idea of what that means, but I cannot articulate a clear, precise definition of what number sense is. No wonder many students end up thinking that mathematics is just a formal requirement in schools, that math is just "one of those courses you have to take to get a diploma."