tag:blogger.com,1999:blog-20497910668657613582024-03-13T16:35:14.428-04:00Learning from Teaching: MathematicsFormally: Math for Profs. My thoughts on improved college math instruction.Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.comBlogger21125tag:blogger.com,1999:blog-2049791066865761358.post-81879085579518645332010-12-22T09:00:00.000-05:002010-12-22T09:00:32.600-05:00What math is and is notRobert Lewis has a great essay on math, as the most misunderstood subject. I couldn't agree more!
In addition to an analogy with sports training similar to one I have here, I especially liked his parable of the hostile party goer. He is confronted by a man complaining that he was forced to memorize the quadratic formula, and yet has never had to use it. Lewis compares Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-32798045170537195552010-12-14T12:11:00.000-05:002010-12-14T12:11:22.985-05:00Attendance IssuesAt about the midpoint of this last semester, students here at Coastal had a day off (a fall "student holiday"). Not surprisingly, many student took the day before off as well. In my Math 139 classes, I had 15/40 and 11/37 attendance, respectively. It got me thinking about my policy on taking attendance.
I have long held that whether a student wants to come to class is entirely Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-56809745388198112662010-10-10T09:38:00.000-04:002010-10-10T09:38:54.309-04:00Uniformly badA little over half of the courses I have taught over the years have been "uniform" courses. These lower level math courses have multiple sections each semester, so the department has decided to appoint a course coordinator to oversee all the instructors. While there is some variation on how coordinated these courses are, usually it goes as far as common exams and group grading, Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-31133197993059170542010-09-22T10:25:00.000-04:002010-09-22T10:25:03.587-04:00Separating proofs in low level classesThe semester has gotten off to a busy start, thus the lack of postings. Anyway, this morning, I was thinking about a student who came to my office hours for some review on derivative rules. He had forgotten how to take the derivative of $3^x$. As he flipped through his notes, and eventually found the relevant page, he asked, "Oh right, is that where you sue the frog rule?" Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-24420218641594740172010-08-27T10:57:00.000-04:002010-08-27T10:57:10.321-04:00Something is workingThis fall I'm teaching Calculus I for the third time here at Coastal. Perhaps because I just taught it at the end of the summer, I am finding that it is working particularly well. I'm not entirely sure why. Student interaction is good, and I'm sure that is helping. But beyond that, the lectures seem to be flowing in a way they have not previously.
One possibility is the Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-50640576526519842822010-08-13T11:22:00.000-04:002010-08-13T11:22:39.198-04:00The problem = equal signsNew research out of Texas A&M suggests that 70% of middle school students don't understand what the "=" sign means. The article can be found here. The key point, as pointed out in the article:
The problem is students memorize procedures without fully understanding the mathematics.It is interesting that the problem runs this deep. My calculus students have trouble solving Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-52553347651693140442010-08-02T08:46:00.000-04:002010-08-02T08:46:04.491-04:00Thinking vs DoingWord problems are hard for students. I've never really understood why before. You read the problem, figure out what it is asking, figure out how to answer the question, and do it. Usually the mathematics part is not that hard. So why to students have so much trouble?
One of the reasons might be that many students have never been shown how to think about the problem. Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-70744494984011701182010-07-31T08:54:00.002-04:002010-07-31T08:55:31.109-04:00Implicit DifferentiationI recently taught my summer Calculus 1 class about implicit differentiation. This is usually a difficult subject to teach, as students have trouble understanding why they much include $\frac{dy}{dx}$ whenever they encounter a $y$ term. I think I have stumbled upon a good way to explain this.
Before saying anything about how to differentiate, I take a few minutes to explain what an Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-38633805767067574902010-07-25T16:48:00.000-04:002010-07-25T16:48:38.203-04:00Derivative of SineI recently taught my class the derivative rule for sine and cosine. These are particularly easy rules to apply, and make for nice examples when doing the product, quotient and chain rules, so I like to introduce them early. The trouble is that the proofs of the rules are rather complex. We would never ask students to find the derivative of sin(x) using the limit definition Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-42228512803657797622010-07-23T10:42:00.000-04:002010-07-23T10:42:25.769-04:00Product rule vs. trig functionsWhat is the best first example of the product rue? Often, books like to use $f(x) = x e^x$. But this is a horrible first example. Look at the derivative: $\frac{df}{dx} = e^x + xe^x$. Can you see the format of the product rule there? Not in the least. It would help to use $x^2e^x$, but the real problem is that you do not see the difference between $e^x$ and itsOscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-38166331601615437762010-07-20T09:33:00.001-04:002010-07-20T09:34:41.120-04:00The place for proofsI have been thinking quite a bit recently about the place for proofs in introductory math courses such as trigonometry, calculus, and really anything prior to the "proofs" course. As a mathematician, I realize the importance of establishing results rigorously. My students, however, do not. With the rare exception of the dedicated math major, most students would rather I just Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-19099935597220272092010-07-07T16:08:00.000-04:002010-07-07T16:08:18.915-04:00The "Zeno's Paradoxes and Calculus" ParadoxOne of my fondest memories of taking freshman calculus was the brief discussion of Zeno's paradoxes. For anyone unfamiliar, the particular one I remember is the Dichotomy paradox:
That which is in locomotion must arrive at the half-way stage before it arrives at the goal.
--Aristotle, Physics VI:9, 239b10 That is, if you are walking towards the wall, first you must travel half way there. Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-10511036722994457082010-07-05T10:42:00.000-04:002010-07-05T10:42:47.504-04:00Motivating students motivated by their future careerI stumbled upon a now six month old piece from the New York Times today: Making College 'Relevant'. It discusses the trend, well known to everyone in academia, of colleges and universities catering more and more to students' desires to use college as job training. This is a much graver issue for the liberal arts majors than for the sciences, as a science heavy major is seen as one Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-5930020792652631282010-07-04T14:10:00.001-04:002010-07-04T14:12:48.790-04:00Mnemonics and Acronyms are BAD (Best to Avoid Discussing)Please Excuse My Dear Aunt Sally, SOCATOA, All Students Take Calculus, FOIL, ...
It seems math education is riddled with acronyms and mnemonics. Students love them because they afford an easy way to remember what otherwise might be a challengingly complex concept. What is a little more surprising is that many teachers are also very found of these tricks. Last semester I had a Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-13868013621896857882010-06-30T14:24:00.000-04:002010-06-30T14:24:55.119-04:00Mathematics as weightliftingEvery math teacher has heard it: "Why do I need to learn this." Like nails on a chalkboard! I find the question especially difficult to answer as a pure mathematician. I enjoy mathematics particularly because it is not something that can be used -- the abstractness is exciting. But students don't want to hear this. Increasingly, students go to college for the purposeOscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-1602217267375858182010-06-29T09:15:00.000-04:002010-06-29T09:15:39.767-04:00Probability is hardThere is a nice article on the ScienceNews.org site about the Tuesday birthday problem:
I have two children, one of whom is a boy born on a Tuesday. What is the probability that I have two sons?
The article does a very good job of discussing the solution, as well as why it is difficult (and why "Tuesday" has anything to do with the solution). Commenters have pointed out that there Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-87658648324461855182010-04-19T22:34:00.003-04:002010-07-20T09:37:08.374-04:00Random examplesSome examples are better than others - most of the time. For example, when first teaching the product rule, it is not a good idea to use $xe^x$: since the derivative of $e^x$ is $e^x$, students don't see the form of the product rule explicitly. That said, there are times when the technique being taught are so general, that they would work equally well with any example. In cases Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-45865663007051524802010-04-16T11:59:00.004-04:002010-04-16T12:14:41.436-04:00Latex on BloggerApparently, I have just enabled latex on Blogger. I did so following the instructions found here. If this is working, then $e^x$ will appear instead of $!$e^x$!$.Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-59587954855553116312010-04-16T09:15:00.000-04:002010-04-16T09:15:14.204-04:00Mathematics through puzzlesI love mathematical puzzles. I still remember the first one I ever heard. It was the nine weights puzzle, where you have to find the heavy weight by using a balance scale only two times. I was in forth grade. I remember thinking how clever it was; how simple; how elegant. I can't be sure, but I suspect that puzzle got me on my way to being a mathematician.
Students Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-64884300420934607702010-04-14T22:40:00.000-04:002010-04-14T22:40:43.637-04:00Grading is greatThis semester I happened to teach three section of Trigonometry, and had an undergraduate grader. He was supposed to grade 15 hours a week (5 hours per class) which meant he could easily grade both homework and quizzes for me. This left me with only exams to grade. Sounds great right?
Turns out, not so much. The grader has done a fine job (although having to get through Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0tag:blogger.com,1999:blog-2049791066865761358.post-2172607006865504302010-01-13T09:45:00.000-05:002010-01-13T09:45:38.181-05:00About Math for ProfsWelcome to my new blog. Since entering graduate school in Fall 2004, I have taught a variety of math courses at the undergraduate level. These five and a half years of teaching by no means make me an expert on the subject of teaching mathematics, but then that is part of the reason I wanted to start this blog. I really really like teaching. It is the reason I went to grad Oscar Levinhttp://www.blogger.com/profile/08992520344610665242noreply@blogger.com0