I didn’t lose my cool
Okay, so I almost lost it today. All of my students can attest to the fact that I am generally a very laid back guy. In most cases, my students can give me a hard time in class, try to distract me by encouraging me to chase rabbits in class, they can even jest about how complicated the material is. I generally play along, all in good fun, and even use their comments to springboard into some great illustrations of application.
But every once in a while, I get pushed too far. I don't yell. I don't scream. I don't turn red-faced. I really don't even get angry, nor do I let it distract from the lecture at hand. But I do respond. Today, I responded by making good on an off-hand threat I made last week: "If they continue to deride this subject and make attempts to get me to dismiss class early, I will begin adding problems to their homework." In other classes, I have simply invited the students to leave and drop the course if they are so disinterested in making an effort.
Now, for the confession. I honestly feel bad about "punishing" the entire class for the offense of a couple of students. I put the word in quotes because, honestly, more homework helps them better understand the topic, but I know that in the eyes of the students, it was a punishment. One rationale for increasing the homework load based on the behavior of a couple of students is that it will curb that sort of behavior in the future, both from them and anyone else. But at the same time, there are not very clear boundaries. I allow the regular discussions of "Why is this difficult?" or "How can we use this?". I often play along with their joking about dismissing class early, even though they all know how improbable it is. Nevertheless, after every theorem and every proof, is too often.
Despite by doubts, the current assignment will stand as it is now, after the added problems. If there are future such occurrences, I will recognize the fact it failed to meet the goals of stemming this behavior for the entire class and I will likely direct the "punishment" toward any offending individuals. In the end, I hold no grudges or ill will. Any students involved in today's class have clearly demonstrated their commitment to the mathematics program as a whole so I am not upset with anyone. The primary goal is that the students learn a lesson: Even though there is a casual atmosphere in the classroom (or workplace), we maintain a certain level of respect for the course (or job) and for the professor (or employer).
Linear Algebra: Block Matrix Multiplication
In class today, we finished the section on partitioning matrices. We verified that that block matrices obey the same rules for matrix algebra. In particular, block matrix multiplication works as scalar multiplication as long as the dimensions are appropriate for the sub-matrices to be multiplied together.
We looked at a couple of examples how we can evaluate the structure of larger matrices using block matrices. For example, we prove that the given matrix:
![A= \left[ \begin{array}{cc}A_{11} & O \\ A_{21} & A_{22} \end{array}\right]](/latexrender/pictures/1b62a68571c9e17e7fed4ad84aad5609.png "A= \left[ \begin{array}{cc}A_{11} & O \\ A_{21} & A_{22} \end{array}\right]")
is nonsingular iff 
and 
are nonsingular. We derived that:
![A^{-1} = \left[ \begin{array}{cc}A_{11}^{-1} & O \\[2ex] -A_{22}^{-1}A_{21}A_{11}^{-1} & A_{22}^{-1} \end{array}\right]](/latexrender/pictures/0124ace7f8a4bf56a611ee2e3963ec78.png "A^{-1} = \left[ \begin{array}{cc}A_{11}^{-1} & O \\[2ex] -A_{22}^{-1}A_{21}A_{11}^{-1} & A_{22}^{-1} \end{array}\right]")
I then used a rather manufactured example to demonstrate how we might make use of this fact. Find the inverse of the following matrix:
![A=\left[\begin{array}{cc|ccc} 1 & 2 & 0 & 0 & 0 \\ 2 & 3 & 0 & 0 & 0\\ \hline 4 & 3 & 1 & 0 & 0 \\ 1 & 2 & -2 & 1 & 0 \\ 9 & 2 & 0 & 0 & 1 \end{array}\right]](/latexrender/pictures/86f24539b1429418643907aff7157368.png "A=\left[\begin{array}{cc|ccc} 1 & 2 & 0 & 0 & 0 \\ 2 & 3 & 0 & 0 & 0\\ \hline 4 & 3 & 1 & 0 & 0 \\ 1 & 2 & -2 & 1 & 0 \\ 9 & 2 & 0 & 0 & 1 \end{array}\right]")
I made this matrix up so that 
and 
are easily calculable and thus, so is 
.
I just started to touch on outer product expansions and so I'll finish that up next time. The first exam has been scheduled for Monday, February 19. I handed out a review and will post an answer key to the review in Blackboard.
Faith and Science: The Scientific Cycle
In today's class, we were fortunate to have one of my colleagues from our division direct the class. This is the third time this class has been offered at Wayland. The first two times, this faculty member and I co-taught the class. I'll be honest and tell you that being the sole teacher this semester has been an enormous challenge, much greater than the last two. It was extremely helpful to have someone in class bouncing ideas off of.
In the current scenario, I appear much more as the "expert" and less of the facilitator in the discussion. In an applied math course, that would be fine, but in this course, I am reluctant to accept that role. Today's class was like the "good ole days".
The two main topics we covered in class today were the scientific method and the way in which scientific knowledge develops. In discussing the scientific method, we went back over some of the original discussions of the class, centering on the issue of just "how" faith can affect or inform science.
The class proposed that faith can serve much like a feedback loop to weigh the conclusions of science against. Most of them were uncomfortable thinking of faith as a filter through which we choose to accept or deny the claims of science. Instead, it was determined that our faith and/or worldview helps us to evaluate the conclusions reached by science. In most cases, excepting bad experimental design, we don't throw out the results (or data) but we can choose to re-evaluate the conclusions as long as they are rational and justifiable.
This is the true challenge in letting one's faith affect scientific endeavor. The real difficulty is how you draw the line between allowing for, say, supernatural explanations (if at all) and allowing for paranormal explanations. In a classroom full of Christians with a strong conservative backgrounds, most, if not all, of us are comfortable in accepting the supernatural's involvement in our world. However, far fewer of us would be willing to accept the paranormal.
Thus, in evaluating scientific conclusion we must answer the question of what provides the "best" explanation. Some answers to that question have been posed such as Occam's Razor or the fact that the natural trumps the supernatural for all cases. We did not arrive at an answer that satisfied everyone.
We discussed the scientific method in greater detail. We followed that by a short discussion on the nature of scientific development/revolution as proposed by Thomas Kuhn in his book, The Structure of Scientific Revolutions
Next time, we will begin to cover some specific case studies of the interaction of faith and science throughout the history of science.
We will also discuss some of the claims made in Eric Snow's paper: "Christianity: A Cause for Modern Science". In this article, he gives a summary of a couple of papers in which the authors contend that either Christianity helped to create modern science through its worldview, or at the very least, aided in its development. They also contend that other cultures' worldviews stifled the development of modern science, giving examples from China, India, Islam and others.