In my Fall course of Math Models, I have three groups working on projects to finish up the semester. One of the groups have an assignment to explore a model of the spread of a forest fire. The assumptions are that the trees are on a rectangular grid, or a lattice. The time is a discrete variable and at each time step the probability that the fire spreads from one point in the lattice to an adjacent point (up, down, left or right) is given by p. For simplicity, the event that the fire spreads to each point is assumed to be independent of any other point.
Part of their project is to implement a numerical simulation of their forest fire. I couldn't let them have all the fun, so below is an example of my version of the simulation in MATLAB. I have to hold off on posting the code until after they have handed in their project.
In the graphical representation of my simulation, green represents an unburnt tree, black is burnt and red is currently on fire. The fire lasts for exactly one time step. I also implemented a 3-D version, where a height of 1 is unburnt, 2 is on fire, and 0 is burnt. I'll confess to having way too much fun with this.
I have used a 200x200 lattice with p = 0.5.
In my math models course this term, my students have begun working on their projects. They will complete 3 – 4 modeling projects throughout the semester. We just completed work on a Highway Design Problem with some basic curve fitting techniques to join two or three different grades (slopes) of road sections using parabolic curves.
In the current project, three teams are developing a formula to calculate the number of hours of daylight that a given point (of latitude) receives for each day of the year. I’ve introduced the problem and laid out some notation and assumptions to guide them along. Here are some snapshots from the board on Monday. The students asked that I document these notes since their copies did not do justice the concepts they were supposed to represent.
I have the most excellent privilege of teaching a course at Wayland in Mathematical Modeling. The course is designed as a projects course where the majority of the semester is spent working on modeling projects. The typical problem will take groups of 3 - 4 students anywhere from 2 to 4 weeks to solve. They often need to develop and learn new mathematical skills but mostly they will rely upon the mathematics courses they've covered up to this point. This gives them the opportunity to see their mathematics in action; they get to see what an applied mathematician actually does.
I'm always looking around for additional websites that provide real-world problems and not just problems designed for a particular application in a particular course. The problem with many problems that are included with typical textbooks is that they have been shaped and manipulated so that the techniques being covered in the course fit neatly within the problem. In practice, that rarely happens. The problem, not the technique, comes first and a mathematician must develop a reasonable model based on the desired outcome. Then appropriate techniques are used, learned or developed as the case demands.
This post marks the first of many I hope to follow which indexes a number of interesting sites that aid in promoting mathematical modeling and problem solving in the true spirit of serendipitous, constructive learning.
eBourbaki is a mathematical problem-solving company whose mission is to solve the world's mathematical problems using contests to inspire innovation and creativity.
eBourbaki’s mission is to solve the world’s mathematical problems. Our primary role is to host prize competitions focused around pertinent problems on behalf of sponsor organizations. The competition is global and open to everyone through the internet. The only pre-requisite for winning is providing the most innovative practicable solution. Our ultimate and unique agenda is to improve mathematical engagement, education and innovation worldwide.
The site is slick. I like the idea of competing to develop the best solution and I have seen a few different competitions along this line. The first contest was held in 2007. The ultimate goal was to develop a plan to shade downtown Phoenix, AZ during the summer. Below is part of the statement of the problem:
Your task is to devise the most cost-efficient way to distribute trees and structures throughout the downtown area so that the sidewalks and public spaces are shaded for the duration of the working day (8-5).
This year's contest runs from May 5 - 12 and the winning team will receive a cash prize. Here's the teaser (they won't give the full statement of the problem until the contest begins):
London faces serious transportation challenges today. With congestion charges on the rise and increased awareness of the environmental impact of many forms of commuting, cities are turning to bicycle stations to ease traffic, reduce pollution, improve parking, and enhance a green-friendly image. Last summer, Paris joined the ranks, instituting a city-wide network of high-tech low-cost rental bicycle stations.
We ask the question: if London were to embrace this concept, how would it best go about doing so? Where should the bike stations go? How many bikes at each station?
I'm setting a goal of participating in some sort of contest along these lines with a group of our students next year.
Somehow, I thought developing an online math course would be easier. Having taught college level math courses for 9 years now, I've begun to understand what it takes and doesn't take for student to succeed in a college level mathematics course. Here are the things that I think will be the biggest hindrances in the online version of the college algebra course.
- Limited interaction with the instructor during the explanation process. I love being able to spur students on in their learning process. For example, I propose certain ideas that are clearly false and allow them to correct these concepts of algebraic operations as a way of helping them learn. When they go awry, I straighten their course, so to speak. Also, in explaining, I leave the floor open to any questions along the way. If I misstep or jump over a step, the students can interact with me.
In the online version of the course, the students will be watching lecture videos and filling in a lecture notebook. Interaction is lost until they email me with questions over the lecture or homework.
- Students are not forced to write out their homework in a systematic way. The homework assignments will be assigned and submitted through an online web application provided by the textbook publisher. One nice feature that is available now is that they do enter in their answer through some sort of equation editor. This at least forces them to grasp notation which is leaps and bounds better than multiple choice questions. Nevertheless, I will see very little of their handwritten work.
As part of the course, I decided to at least check this sort of work twice during the semester. They are required to take two pencil and paper, proctored exams. This gives us a check on the fact that they are the ones learning the material and not someone else. It also gives us a couple verifications that they have grasped the notation and systematic process of doing these types of problems.
- I'm worried about the ability of the students to communicate fluently with the instructor through the online medium. There are equation editors in the messaging center in Blackboard but their use is so tedious that fluent communication is difficult. It is much better than no such tool being available, but I'll just have to wait and see how well it works for the course.
We'll probably have a bit of a rocky start with this class the first time it is offered but who doesn't. The disclaimer I will put on all my documents at the beginning of the semester is below
"It is a myth to think that an online version of a course is easier than a traditional class. College Algebra is the worst case of all. If you struggled in a traditional math class, you can expect to find this as challenging, if not more. Student be warned!!"
. . Too harsh or does it need even more "teeth"?
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In responding to a comment on the last entry, I answered the questions of what equipment and software I am using to create our online course materials:
For our College Algebra course, we cover the last six chapters from an Intermediate Algebra textbook. There will be a lecture for each section of each of the chapters. It looks like every full section lecture will be about 20 - 30 minutes in length. I am shooting a screen (or slide) one at a time which comes to about 10 slides or so per section. This post was one of those slides. Each of those slides ranges from 2 - 5 minutes in length. I think that comes to a total of about 27 lectures about 30 minutes each. I will probably have the students order a DVD with hi-res versions of the video, but post compressed versions on the web, maybe even as podcasts available through an RSS feed.
In terms of software, I am doing everything the hard way since I am on a PC. I hear this would probably be easier with a Mac. Nevertheless, I am mostly using free software. For the screen capture, I use CamStudio. I piece all the slides together using Windows Movie Maker. The software I use for writing on the screen is the software that was included with my tablet. My first choice would have been to use a tablet PC but the cost was prohibitive. Instead, I went with the WACOM Intuos3 tablet. We purchased the 9x12 tablet which works well, but is probably larger than needed. The Corel Painter Essentials software was included with the tablet and it worked quite well for the screen casting.
I hope this answers your questions. I got most of my ideas from a podcast I watched in iTunes by the name of "Is All About Math".
I am currently developing an online course in College Algebra for Wayland. The hope is to have a series of video lectures, accompanied by a lecture notebook where the students will fill in the book as they watch the lectures. Below is a sample video from on the series.
A GUIDE TO GRADING EXAMS
by Daniel J. Solove
Associate Professor of Law, The George Washington
University Law School
December 14, 2006
It's that time of year again. Students have taken their
finals, and now it is time to grade them. It is something
professors have been looking forward to all semester.
Exactness in grading is a well-honed skill, taking
considerable expertise and years of practice to master. The
purpose of this post is to serve as a guide to young
professors about how to perfect their grading skills and as
a way for students to learn the mysterious science of how
their grades are determined.
Grading begins with the stack of exams, shown in Figure 1
The next step is to use the most precise grading method
possible. There never is 100% accuracy in grading essay
exams, as subjective elements can never be eradicated from
the process. Numerous methods have been proposed throughout
history, but there is one method that has clearly been
proven superior to the others. See Figure 2 below.
The key to this method is a good toss. Without a
good toss, it is difficult to get a good spread for
the grading curve. It is also important to get the
toss correct on the first try. Exams can get
crumpled if tossed too much. They begin to look as
though the professor actually read them, and this is
definitely to be avoided. Additional tosses are also
inefficient and expend needless time and energy.
Note the toss in Figure 3 below. This is an example
of a toss of considerable skill -- obviously the
result of years of practice.
Note in Figure 3 above that the exams are evenly
spread out, enabling application of the curve. Here,
however, is where the experts diverge. Some contend
that the curve ought to be applied as in Figure 4
below, with the exams at the bottom of the staircase
to receive a lower grade than the ones higher up on
According to this theory, quality is understood
as a function of being toward the top, and thus the
best exams clearly are to be found in this position.
Others, however, propose an alternative theory
(Figure 5 below).
They contend that that the exams at the bottom
deserve higher grades than the ones at the top.
While many professors still practice the
top-higher-grade approach, the leading authorities
subscribe to the bottom-higher-grade theory, despite
its counterintuitive appearance. The rationale for
this view is that the exams that fall lower on the
staircase have more heft and have traveled farther.
The greater distance traveled indicates greater
knowledge of the subject matter. The bottom
higher-grade approach is clearly the most logical
and best-justified approach.
Even with the grade curve lines established,
grading is far from completed. Several exams teeter
between levels. The key is to measure the extent of
what is referred to as "exam protrusion." Exams that
have small portions extending below the grade line
should receive a minus; exams with protrusions above
the grade lines receive a plus.
But what about exams that are right in the middle
of a line. In Figure 6 below, this exam teeters
between the A and B line. Should it receive and A-
or a B+?
This is a difficult question, but I believe it is
clearly an A-. The exam is already bending toward
the next stair, and in the bottom-higher-grade
approach, it is leaning toward the A-. Therefore,
this student deserves the A- since momentum is
clearly in that direction.
Finally, there are some finer points about
grading that only true masters have understood.
Consider the exam in Figure 7 below. Although it
appears on the C stair and seems to be protruding
onto the B stair, at first glance, one would think
it should receive a grade of C+. But not so. A
careful examination reveals that the exam is
crumpled. Clearly this is an indication of a sloppy
exam performance, and the grade must reflect this
fact. The appropriate grade is C-.
One final example, consider in Figure 8 below the
circled exam that is is very far away from the
others at the bottom of the staircase. Is this an
Novices would think so, as the exam has separated
itself a considerable distance from the rest of the
pack. However, the correct grade for this exam is a
B. The exam has traveled too far away from the pack,
and will lead to extra effort on the part of the
grader to retrieve the exam. Therefore, the exam
must be penalized for this obvious flaw.
As you can see, grading takes considerable time
and effort. But students can be assured that modern
grading techniques will produce the most precise and
accurate grading possible, assuming professors have
achieved mastery of the necessary grading skills.
DISCLAIMER FOR THE GULLIBLE:
This post is a joke. I do not grade like this.
Instead, I use an even more advanced method -- an
eBay grade auctioning system.
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In my Faith and Science course, when we covered the scientific models for origins of the universe, we studied the "mainstream" model of the Big Bang. I remember making the comment that for many Christians who are scientists, the idea of a Big Bang at the beginning of the universe is completely consistent with the idea of God as Creator. "God spoke and BANG, it was". I also commented that we are unable to determine what happened, if anything, prior to the Big Bang. In fact, the current version of the model only goes back to when the age of the Universe was about 10^-43 seconds old. Admittedly this is very close to the beginning, but it was not possible to know what happened before that or, at least, it was impossible to know what happened before the Big Bang.
Well,scientists at Penn State have begun to try to look beyond the birth of the universe. Using quantum tools with general relativity, Abhay Ashtekar and two of his post-doctoral researchers have developed a model that looks before the Big Bang to see a shrinking universe with much the same physics as ours. "Using quantum modifications of Einstein's cosmological equations, we have shown that in place of a classical Big Bang there is in fact a quantum Bounce," says Asktekar.
Since I can't claim to have the expertise to critique their science, I am curious to watch how this model is accepted in the scientific community.
For more, you can read the article in Scientific Daily:
Penn State Researchers Look Beyond The Birth Of The Universe
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Class Date: Friday, February 16, 2007
Dr. Boyd was in charge for another day. This time he covered topics from the history of science as an illustration of how science progresses. Using specific case studies from the history of scientific endeavor, we learn some important principles that undergird how we understand science as it is today.
Below are some of the topics that we touched on in this class:
- The UV Catastrophe: This serves as a good case study to see how scientific revolutions occur as well as a lesson about the dangers of extrapolation
- Causality: the difficulty of assessing the cause from the effect. Wearing skirts causes an increased likelihood of breast cancer
- Did Science arise in a Christian World? Did the Christian World help to create modern science? (see Eric Snow's Paper)
- Positivism vs. Realism
- Michael Faraday
- Isaac Newton
We finished the concept of directional derivatives, introducing the notation for the gradient of a function of several variables. We proved the formula of the maximum value of the directional derivative, as well as the direction for which it is maximized.
Next time we'll begin basic optimization theory for functions of several variables.