Some of the major universities such as Yale, MIT, etc. have been filming many of their courses and posting the lectures online. Dozens of these courses have been viewed and the segments where math is used has been catalogued. On this site the following is given: the math being used in the video, the link to the video clip, the time segment within the lecture that contains that mathematics application, the university where the course was taught, the course title, the name of the professor, and a brief description of the content and how it might be used in the relevant mathematics course.
The site is indexed by mathematics application with four main categories: Statistics, Algebra and Pre-Calculus, First Year Calculus, and Advanced Math. Advanced Math means multiple integration, vector valued functions, linear algebra and differential equations. Beginning, Intermediate and College Algebra were all lumped together due to the fact that there is so much overlap. Since there is no exact standard on what content goes into each course, an instructor may have luck searching through the topics of another one of the courses. For example, the hyperbola is listed under Algebra and Pre-Calculus, but some colleges include it in their Trigonometry course. Also, some topics, such as series, are split up with the advanced video clips put into calculus and the less advanced topics put into algebra.
The main use for this site is for an instructor to show the video clip to the class either at the beginning of lecture as a hook or right after introducing the relevant topic to the class. The video clips are not intended to replace a math lecture. They should spark the interest of the students in the math class. Many of the video clips show the professor demonstrating a topic in their science or social science class but the math that is shown is done without the steps provided. This is a great opportunity to have the students see if they can work out the steps either individually or in groups. The time frame for the clip was a best guess on what is needed for the students to get value. It is recommended that the instructor view a few minutes before and after the time frame so that the instructor can properly introduce the clip to the students to provide them with proper context.
The science content of many of these videos will be well over the students heads, but it has been found that even if the students do not fully grasp the science, they are appreciative of the fact that they have at least a hint of how the math that they are learning is used. If the instructor lets them know that they will not be tested on the science, then the students will enjoy the clips and feel more relaxed watching them. Often they will want to talk about the science after the clip, but an instructor can get carried away and lose valuable time to do the mathematics needed in class. Break time or time after class is a great time to chat about the science.
This site uses the word "professor" to refer to the person who is lecturing on the video clip and "instructor" to refer to the person who will be presenting the video clip in a math class. "Student" is used for both the students in the video clip and the students in the math class who will be watching the video clip. It should be clear from the context which "students" are being referred to in the Teaching Ideas piece.
Most of what is taught in the standard mathematics classes is used in courses outside of the mathematics department. By far the most frequently used statistics topic is regression analysis. The scatterplot and regression line are shown especially in social science classes. Only lower division social science class lectures were available online and they often hinted at confidence intervals and hypothesis testing, but rarely explained the statistics in any detail. The most frequently used algebra topics are: Exponents, Logarithms, Rational Expressions, Linear Equations and Their Graphs, Geometric Series, FOIL, Solving for a Variable and Proportions. The main algebra topic that was not found was factoring when the polynomial is not a difference of squares. Algebra courses spend so much time on factoring, but if applications are the main motivation for learning algebra maybe factoring should be deemphasized and the freed up time could be spent on the more useful topics. Just about every topic covered in a standard first year calculus course was used. The second year calculus content was not found as frequently, but that is most likely due to the fact that most of the science courses looked at were lower division courses that did not have second year calculus as a prerequisite. One surprise was that the generalized binomial theorem, the MacLaurin series expansion of (1 + f(x))a was used over and over again in physics. This is a very small topic in a second semester or third quarter calculus class, but maybe it deserves its own full section. Also 2 occurred as a base of the logarithm more often than 10, especially in computer science courses. The standard notation for log2x is lg(x). This is not emphasized in intermediate or college algebra, but maybe it should be. With computer science gaining its foothold as a standard science course on par with chemistry, we should rethink what bases other than e needs to be focused on. Finally Euler's equation: eix = cos x + i sin x was used over and over again in physics, computer science and engineering courses. This is barely touched on in the calculus courses, but it might deserve its own section and could be introduced in detail in the standard trigonometry course.
As expected, the physics sequence used mathematics the most. Much of physics is a complete duplication of what is done in calculus. Chemistry, economics, and computer science also rely on mathematics but not quite as much as physics. Social science makes heavy use of statistics but does not use as many of the algebra or calculus concepts. The courses looked at from outside the United States did not differ at all from US courses with respect to how mathematics was incorporated.
This collection was completely created by a single individual, Larry Green. It is intended for use as a resource for instructors and not as a scientific study of mathematics applications. The selection of the courses was completely based on the interest of the author and did not use any scientific process. There are over 550 applications catalogued and if printed the collection would need over 200 pages to print it all out. With such a large collection compiled by one individual, there are bound to be many typos and other errors. If you find an error please contact the author by email at: DrLarryGreen@gmail.com.
Dr. Larry Green is a full time mathematics and statistics instructor at Lake Tahoe Community College and has been teaching there full time since 1996. Before teaching at Lake Tahoe Community College, he taught at Sonoma State University. Dr. Green received his bachelor's in mathematics at UC San Diego and earned his Ph.D. specializing in algebraic geometry at UCLA. He is the past president of the California Mathematics Council Community Colleges (CMC3) and is the current chair of the mathematics editorial board of MERLOT (Multimedia Educational Resource for Learning and Online Teaching). He is an elected school board member of Lake Tahoe Unified School District and serves as the vice president of the El Dorado County School Boards Association. Dr. Green begins every day watching a lecture while doing his daily workout, multi-tasking the mind and the body, and recommends this mental and physical workout for everyone.