Thursday, August 31, 2000: The handout on Wednesday for doing group
work programming their calculators to do long division was a disaster. Other
years, this project has worked very well. The problem here is that almost no
student has had experience with any technology in a mathematics class. This
makes me worry about the projects too. However, I expect that if the students
understand that it isn't that hard for them to learn how to use technology to
work with the material, they will mostly get over their difficulties. To that
end, I sent out a class e-mail giving them some (extremely basic) instruction,
as well as some pedagogical reasoning behind doing technology in the classroom.
We shall see if what I sent out will improve the situation or make it worse.
Truthfully, there is one student Jill in particular in the class, that seemed
almost outraged that I was asking them to do this as part of group work. I have
a very bad feeling about how things are going to work with her on this particular
homework. Also, I am suddenly aware how much the students fear doing any math
on their own. For them, writing this program is NOT mathematics, and many of
them will not try it on their own. What they would do in a classroom when confronted
with a need to use technology, I cannot say. My guess is they would take the
attitude of their instructors here that told them "Thou shalt not use technology."
(The words of one student in the class.)
In my office today, I met with Mary and Ellen about the program, and I think
I really got through to them both on the reason for the project, and on the
power of the technology in their hands. Mary was particularly excited about
learning how to use her graphing calculator. Meanwhile, Ellen seemed to be very
interested in using an Excel spreadsheet for the project. I think a lot of student
attitudes are similar to theirs. Fear of doing something new and different.
I hope that once they see how really easy what I wanted them to do is, that
they won't be so fearful the next time.
I am also trying to figure out what other assumptions I am making about the class/students here are wrong and will come back to haunt me. Along the same lines, in my memo to the class, I mentioned different levels of understanding long division, and suggested that the highest level was to be able to use the algorithm for other things. Then I pointed out that two ways I could think of two assess this understanding would be
Moreover, I am hoping that doing the first will also improve their understanding of the algorithm so that the second will be easier. I guess I shall see tomorrow whether this was a productive way to present the material or not.
I have now collected all of the responses to the first questionnaires. I will
try tomorrow morning to put together a smorgasbord of excerpts from them for
the whole class to discuss. My feelings having read them all. Most students
believe that we teach math for problem solving skills in addition for utility.
Few seem to be able to mention any utility beyond basic consumer math, however.
I would like to raise the issue of what this reasoning says about teaching higher
level algebra skills (factoring, rational functions, etc.). I am hoping to get
them to see that how we teach these skills is as important or more important
as the skills themselves. Thus, if utility is the issue, then we need to prepare
students to be able to utilize their training. If problem solving is the issue,
then we need to prepare students to see how to generalize their problem solving
skills, although for me, this last is not so much to generalize the skills outside
of mathematics, as to see what tools they use so that they can generalize them
outside of mathematics. Some of them do recognize, however, the connection between
mathematics and writing! Interestingly, their view of what mathematics is about,
comes down to explaining the world. This strikes me as the "mathematics
is the language of the universe" approach. Personally, I don't like this
definition because most mathematics is and was done for no clear practical purpose.
Rather the mathematics gains purpose after scientists get done with it. I don't
consider myself a pure Platonist in the sense of Hardy, however. Rather, I feel
that if we put forward that the reason to learn math is its utility, then we
will set ourselves up for trouble. Also, their belief then seems to become that
it is OK not to understand higher level math because it is chaotic since the
world is. This view is anathema to me as I think of higher level math as significantly
more orderly than lower level math. It is simply more abstract because we are
delving deeper into the patterns. That said, I think only two of the 14 students
mentioned the importance of abstraction.
Notes for future classes:
September 8, 2000: (12:08 AM) On Thursday, I met with many students over their first homework assignment. The assignment is extremely difficult for them, which is what I expected. The students that are seeing me are probably going to do pretty well on the assignment. I am more concerned about those that haven't seen me yet. In class on Monday we went slowly through the proof that the definition of addition is well-defined for the rational numbers. I discussed with the class how a misunderstanding of what is behind this leads to difficulties in students abilities to add fractions. In class on Wednesday, we went through the proofs of the irrationality of 2 in great detail. In particular, I showed the students where two shows up in the even-odd argument. Curiously, when the students have seen this argument before, they report to only generalizing even-odd arguments, not generic prime factors. So far, the students that talk to me (about half the class has come to my office so far), tell me that they are beginning to realize why the square root of two is irrational, as opposed to being able to mimic the proof. In conversations, this appears to be true. I believe that in today's class, I will wrap up the proofs of irrationality of square roots and move on towards the numbers e and pi. My goal is to finish them shortly and begin to discuss numbers as length, although I think I will need to take a day to discuss modular arithmetic. It seems that modular arithmetic is discussed in less detail at MSU than it is at BGSU. My guess is that BGSU students benefit from the Discrete Mathematics bridge class, in that they see modular arithmetic twice before this stage, whereas the MSU students have only seen it the one time. I am trying to come up with a group work project on modular arithmetic, but I think I will instead do some applications and magic tricks with modular arithmetic so that they can see how they might teach it at the high school level instead.
I would like to get the class doing some group work soon so that I can see how the students are progressing on my process goals, but the next natural place to do this is when we discuss multiplying lengths of segments and trying to recreate Descartes's description of how to multiply magnitudes so that you get a magnitude out.
Positive comments from students. Mary says that she feels like she really understands
the arguments we are doing now, and she also reports that every time she leaves
my office, she truly feels as if she has learned something. I think, though,
that it might be beneficial for Emily to see me some without Mary. Ellen lets
Mary do most of the talking, and I suspect it might be helpful for her to try
and get more details down. Jim also told me that so far, he is really seeing
how one can relate this mathematics to teaching. As usual, this is the point
in the class where it is relatively easy to do that. I don't know how students
will feel later on in the term.