It's an understandable requirement. Our B-school curriculum has a very strong quantitative component. Microeconomics, accounting, and data analysis are math-heavy subjects, and even the instructor for the core marketing class has illustrated theory with mathematical equations. To give you an idea of the types of questions I'm dealing with, here's an example from the practice microeconomics exam:
You have a patent on a drug that has been approved for sale in the U.S. The U.S. demand for this product, for which you are the monopoly producer, isThat was actually one of the easier questions using a standard economics equation with price, marginal cost, and elasticity variables. The more difficult ones involved transfer pricing and monopoly pricing, which use calculus. Moreover, I'm in the room with bankers, scientists, engineers, and others who have used math in their day-to-day careers. Even if you don't have this type of professional background, everyone has to be able to keep up to get the most out of the curriculum, and maintain an intelligent and productive level of discussion with classmates and faculty.
ln Q = 3.4 - 1.5ln P + 0.5 ln A
where Q is millions of tablets sold, P is the price per tablet, A is expenditure on advertising, and ln denotes a natural logarithm.
If you are maximizing profits, and if the marginal cost of a tablet is $0.90, what price should you charge?
So why not stick with the concepts covered in the GMAT, which is intended for people heading to business school? While GMAT test preparation concentrates on algebra and geometry basics, it's basically 7th-grade math. Precalculus goes beyond typical GMAT review topics, and in my opinion, the advanced math review is more appropriate for business school. For instance, in my microeconomics class, we had to deal with logarithms and graphing supply/demand curves -- two areas which were *not* part of the Barron's or Kaplan GMAT review, but were covered extensively in precalculus. The online precalc class I took was not through MIT, but rather through a well-regarded public university's continuing education division. The class has been in existence since at least early 2006, and during that time has apparently been taught by the same instructor. On the class bulletin board, I noticed that there were other students from UPenn/Wharton, the University of Chicago, and NYU who were taking the same class before starting their respective MBA or masters programs. The precalc class was a for-credit course, but the credit will not be transferred to MIT. The main purpose of taking the class was to ensure that we come prepared for some of the concepts and exercises that are now being thrown at us, as opposed to checking off a math credit.
So, how was the class?
The curriculum was standard. It started with a basic algebra review, went on to quadratic equations and graphing, spent a few chapters on functions, and ended with three or four chapters on trigonometry. There was one chapter on logarithms, too.
The convenience was great. We could go at our own pace, and start at any time during the year. Beginning in April, after putting the kids to bed, I would go downstairs to the living room to spend about 2-3 hours on readings and homework, and about every week, the chapter tests. I never had to deal with driving. Homework and tests were completed online. I was able to make very steady progress, with the aim of completing the course by the time my full-time MBA program started in Cambridge in early June. I was able to finish the required chapters in about two months and take the proctored final exam in downtown Boston the day after Memorial Day, just a few days before heading to MIT.
I worked very hard at the online precalculus class, and did very well in the homework, tests, and final. Most importantly, I feel that I learned a great deal and came to business school well-prepared. The value of taking precalculus was very apparent in the on-campus microeconomics class, which had a substantial math component. The online math preparation allowed me to focus my attention on economics theory, instead of getting hung up on the calculations in the problem sets.
But there were drawbacks, too. For years, I've heard criticisms from other students, faculty, and even supporters of Web-based distance education, relating to a lack of interaction between students and faculty. I can verify that this was indeed the case in the online class that I took. Here's what observed:
- There was no shared sense of community, or any efforts by the school (the state university that offered the online course) to create one, beyond setting up an online message board. Many of the students used this to introduce themselves at the start of class, but by chapter 1 or 2 in the book practically all shared dialogues had stopped using the official message board.
- What few questions that were placed on the message board -- either relating to the course content, tests or the online software -- were never answered by the instructor. One sad example: "Can you please provide more information on the final? Will it be similar to the Practice Final? How many questions will there be? Please let me know when you have a chance." I am sure others had the same question, too, but it was never answered on the board -- although the teacher did answer this particular question in a private email, when I asked her.
- Because old comments from previous students were never removed from the board, it gave the appearance of an abandoned ghost town -- the MySpace of math.
- The only comments that I saw from the teacher were the first comment at the top of each thread ("please feel free to email me, etc.") which dated from March 2006. Three times, she responded to students who introduced themselves, but by mid-2007 even these responses had stopped. There was no shared response by the instructor to any question about tests, math problems, or software issues. It is uncertain if the students who asked them gave up, or attempted to contact her by email afterward.
- The lack of an easy mechanism to ask complex questions was very frustrating. For instance, the trigonometry chapter covered difficult concepts and methods relating to trigonometric functions and equations. In a classroom setting, the instructor would be using the board to work out equations and would be referring to the unit circle while students asked questions. In an online setting, I could use the textbook, online exercises, and pen and paper, but I still had a ton of "why" questions that could not be easily described or diagrammed via email.
- On the other hand, the teacher was very responsive to those questions that were asked by email. I sent more than 10 specific queries over the course of the semester, most relating to grading errors with the MyMathLab software we used to complete assignments and take tests, or questions relating to the final. She responded to every one within 12 hours. This was impressive, considering many university professors I've had contact with in classroom settings sometimes takes days or even more than a week to respond to email from students.
- Even though she was responsive with email, I did not observe any spontaneous communication from the instructor in the way of asking about problems, or even "keep up the good work" encouragement.
- The $170 precalc textbook
contained two extra books which were never needed for the class, as well as a login key for the MyMathLab section. Interestingly, the book was reproduced entirely online, reducing the need for even buying a physical text.
- The instructor prepared "lectures", which were actually explanatory essays with diagrams. The quality of these documents was generally quite good -- I'd say they were much clearer than the Sullivan precalc textbook
- However, the text "lectures" did not encourage a shared dialogue, and seldom/never change from year to year. I found this out when a link in one of them directed me to an external website, which generated the following error: "Web Hosting from beeb.net closed on 30th June 2008." In other words, the link had been added at least two years before and was never revised.
- MyMathLab contained video clips of various concepts and exercises produced by Pearson employees, but there was no classroom video of the instructor. I watched one of the MyMathLab videos, but found they were much less engaging than the free videos produced by Khan Academy.
- Grading was easy. As long as you studied, understood the concepts and questions likely to appear on the homework and exams (which were driven entirely by textbook content), it was nearly impossible to do poorly. For the online homework on MyMathLab, the system allowed unlimited attempts on each question and even gave step-by-step instructions on how to solve tricky problems. This does not mirror the homework or testing scenarios typically found in physical classrooms, in which you get one chance to get it right, and in the case of tests, cannot have open browser windows or the ability to communicate with other people at the same time.
- The tests followed the textbook lessons very closely, and you were allowed to take practice tests as many times as you liked. The questions that appeared on the practice and real tests were practically identical. Rarely was there a "trick" question.
- Only the final was proctored (I did it at the New England College of Finance in downtown Boston). There was no monitoring on any other graded content. This, combined with an almost complete lack of student/teacher interaction, makes it very easy to cheat on homework and chapter tests.
- Because the homework and tests corresponded to the textbook lessons, it was more efficient for me time-wise to take notes and practice problems from the textbook and do the homework and tests without even reviewing the redundant (but better-written) "lectures".
- The textbook had a fair number of word problems, but these almost never appeared on the homework or tests. I wish they had -- the practical applications of mathematics is where a lot of people struggle, but is the best way of illustrating abstract concepts.
Further, struggling students tend to suffer in an environment when teachers aren't there to help, or even notice there's a problem. I wonder how many dropped out of my class, after attempting to make contact on the online message board, or getting hung up on the software? In the absence of any monitoring system for the exams and homework, how many have turned to cheating?
On the other hand, the convenience of taking a class at home was addictive. It was very easy to incorporate these classes into my home life, without dealing with wasting time or money on commuting to class. And, most importantly, I learned what I set out to learn.
Would I take an online class again? Maybe, if the topic lends itself to rote memorization and hands-on problem solving that does not require interaction with other students or faculty.
But for most college- or university-level subjects, online education is a poor substitute. In my opinion, the most effective learning takes place in the classroom, where you can easily raise your hand, engage in spontaneous discussions with classmates and faculty, turn to the person next to you to ask for clarification, or approach the professor after class or during office hours to ask questions or exchange viewpoints in a way that practically guarantees an instant response and is not constrained by typing, software interfaces, or waiting for a response.
To give you an example, in my on-campus microeconomics class, I suspect that about 3/4 of us were partially or fully baffled during our professor's first explanation of concepts like two-part tariffs and double-marginalization in certain transfer pricing scenarios. The only way we were able to "get it" were by some students raising their hands and asking the professor to explain a particular element, other students sharing their own experiences from their careers (with responses from other students or the professor), and the problem sets being explained in person by the TA, with more questions from us. Not everyone raised their hands or participated in the debates, but everyone in that classroom heard them, and learned something from them. I doubt 10% of this interaction would have been possible online, even using technologies that allow instant feedback from remote students -- it's too easy for people to multitask, read email, or browse the Web while attending class, and un! less sophisticated two-way video systems are involved (such as telepresence), it will be difficult for faculty to get important visual feedback cues from the students they are teaching.
A decade from now, there may be better technologies that truly bring the shared classroom experience to people's homes, but the asynchronous, Web-based technologies that seem to dominate the online education sector don't come close to the real thing.
Other education-related blog posts by Ian Lamont:
- Online education: A teacher speaks
- MBA math review
- For-profit schools take a hit from Frontline
- My Sharp scientific calculator
- A giant step for ALM thesis research
- Review: The Gates Unbarred: A History of University Extension at Harvard, 1910-2009
math word problems online
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