So, yes, I took a bit of a blogging hiatus. I have no reason except that during and after Spring Break it has been a challenge to find the time and the drive to keep posting. I am hoping to make a reasonable effort at it again.
Today's post is a bit short but I would like to point out a minor "re-discovery" I made through helping some students with their Differential Equations homework. I sure that to the seasoned mathematicians out there this is perfectly obvious but I saw it as a fresh idea very relevant to our current topic of Second Order Linear Differential Equations with Constant Coefficients:
Thm: Given and then [tex]Re(x)
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Yeah, that's right. It's Pi Day. Recall that , thus it's natural that March 14th is Pi Day. In honor of this day of celebration, check out a site a recently found called Pi Searcher. You can search for any sequence of digits for where they might occur in the decimal expansion of Pi. A truly useless site yet surprisingly interesting.
A Cornell physicist has discovered an algorithm that can solve all sudoku puzzles. He was working with biologists to improve image processing and stumbled across an algorithm he thinks could have many uses. The first of which is ruining sudoku for anyone who learns the algorithm, apparently.
This algorithm, which was extremely effective in image reconstruction microscopy, was extremely general," he said. "If you just express it in the right mathematical language it could be used in all kinds of things."
Read the article here
I'll confess that I am not much of a pedagogical expert when it comes to how I teach mathematics and yet I have made many assumptions about the effectiveness of certain methods as I present daily lectures to "eager" minds. In light of that, I thought I'd pass on a couple of interesting articles from Keith Devlin discussing the issue of understanding versus procedure when it comes to mathematics education.
In each case, Devlin makes the case that full understanding of mathematical concepts comes only after practice of the procedures, and much of it. While I agree that the sort of understanding to which he is referring, such as a full appreciation of the concept of derivative or even what is meaning of dividing fractions, finally arrives when we have mastered the procedures involved. I cannot make an argument based on research in the area but, in reference to my own experience, the mastery of procedures is most often a result of a commitment to the practice of the procedures and for a majority of "non-math enthusiasts" that commitment is lacking. In my experience, some explanation of the underlying concepts is necessary when it comes to making the point that mathematics is USEFUL. Some understanding of the concepts leads to a motivation for learning, for practicing, and thus for mastery.
There are several good points that Devlin makes and if you have the time, I encourage you to read those articles (if such things interest you). If you do by chance read them, please make comments here for myself and others to read. Or, post them to your blog and link back here. Thank you.
I've been working and tweaking a Teaching Philosophy Statement. I would like to document what I have so far so I am posting it here. Feel free to read and comment on it. Make suggestions, if you have them.
My Teaching Philosophy
Many college classrooms have a reputation of being a dry and monotonous necessity in obtaining the necessary qualifications for a future career. It is my belief that this need not be the case and in fact, a dynamic and interactive classroom environment better prepares students for modern careers.
I love teaching mathematics and it is one of my true passions to help students begin to develop a mathematical mindset in which they can rigorously formulate assumptions and parameters to a model and then use that model to make decisions. In my near 8 years of teaching in the college classroom, I have utilized a mix of traditional and non-traditional classroom techniques to help students learn this mathematical mindset. It has clearly been demonstrated in research and in my own experience that self-discovery is an extremely effective method of learning mathematics but due to time constraints and other constraints often inherent to the particular area of study, it is generally not the only to be used. Of course, by “traditional classroom techniques,” I am referring to purely lecture style with some or little interaction between students and the lecturer. By “non-traditional,” I am referring to interactive approaches such as class participation in lecture, group work, projects, student presentation, etc.
My lecture style has always been one of constant interaction, involving students to motivate the next step. Early mathematics is often taught with the basic approach of explanation of method, followed by examples, then a series of mathematical drills through homework. Although I still employ a similar version of this technique, I prioritize the motivation for the techniques that we cover, explaining that there is reason why we do the things we do. Mathematics makes sense.
For higher level courses, there is much more room for discovery and motivating new directions as the students develop skills for proof and verification of claims. In those classes, it has always been a priority that I keep myself as informed as possible on the current abilities of each student. Drawing attention to gaps in knowledge or reasoning helps students to identify their weaknesses and build on their strengths.
All of the approaches I take in class attempt to minimize the intimidation that many students feel toward the subject of mathematics. I often find it necessary to interact with the students one on one during office hours or simply in the classroom; this helps alleviate many of their fears of the subject. While there should always be a clear separation between professor and student, it need not be one of intimidation. In the end, I believe mathematics to be infinitely fascinating as well as applicable, so I constantly try to draw connections between the world around us and the subject at hand.
I've seen this in a video on Google video before but this Java Applet version is well done and worth another look.
I've heard often people wonder at how BIG God must have been to create a such an expanse as the Universe and yet, as this demonstrates, he must be incredibly small. Infinite and infinitesimal, at once.
I came across this at think again! I had fun solving it so I'll pass it along.
Consider the following quadrilateral
The midpoint of opposite sides have been connected, dividing it into 4 parts. We know the areas of three of the regions. The question is what is the area of the fourth?
I had the pleasure of filling in for a colleague in her Discrete Structure class, which is basically our introduction to proof class. Currently they are covering some introductory concepts from Number Theory. One of my favorite results had been covered in the class just before I needed to fill in: The Euclidean Algorithm. Anyways, toward the end of class we introduced the concept of Pythagorean Triples.
Def: An ordered triple, of positive integers such that is called a pythagorean triple.
EX: (3, 4, 5)
EX: (6, 8, 10)
EX: (9, 12, 15)
EX: (5, 12, 13)
Notice that the first three examples are of the same "type." That is, the second and the third are multiples of the first. In fact, we could state the following theorem.
Thm: If is a Pythagorean triple then for any positive integer , is also a Pythagorean triple.
Now, the last of my examples is fundamentally different from the other three. In fact, we could create a new definition.
Def: If the greatest common divisor of the pythagorean triple is , then it is called a primitive pythagorean triple. (In other words, if they have no common factors it is a primitive pythagorean triple)
I left the class with an question and recommended that if they were interested they should do a little research to find the answer.
Question: How many primitive pythagorean triples are there? That is, are there infinitely many or only some finite number of them?