Posted by tpc at August 17th, 2010
I have no idea who Don Tapscott is and I probably will not read his books, but in the local papers today, he mentioned something on learning in university.
It’s not so much what you know when you graduate that counts as it is your capacity to learn all your life.
Posted by tpc at July 13th, 2010
There are lots of nice videos out there which would be ideal to show to students before lectures or during breaks just to break the monotony. Here’s one about Fibonacci numbers and nature that’s pretty impressive, although I don’t think one will learn much maths from it.
Posted by tpc at May 11th, 2010
Cool article about Pixar and some applications of harmonic function. Would have been useful last year when I taught the maximum principle but didn’t have much else to say about it.
Moving Remy in Harmony: Pixar’s Use of Harmonic Functions
by David Austin.
via: JohnDCook’s twitter.
Posted by tpc at September 24th, 2008
While perusing some blogs, while waiting for the rain to stop, I’ve found more evidence to support learning maths than replying on calculators.
A TRAFFIC warden gave parking fines to innocent motorists – because he did not know how to tell the time.
The bungling parking attendant had to use a calculator to work out the expiry time on tickets displayed in motorists’ windscreens.
But with calculators working in decimals rather than minutes and hours, the ticket-happy warden had his book out before realising his maths was letting him down.
One of his victims was IT manager Dave Alsop, who tried his best to show the warden the errors of his arithmetic ways. But to no avail.
The 29-year-old from Torbay had parked in the Terrace car park close to Torquay harbour and paid £1.20p for 75 minutes. His ticket was issued at 2.49pm and would have covered him until 4.04pm.
But when he returned at 3.41pm, he discovered he had been given a £50 parking fine.
He found the traffic warden nearby and asked him why he had been booked when his ticket clearly showed he had time remaining.
The warden disagreed and tried to prove his point using a calculator.
He tapped in 14.49 and added 0.75 to produce a total of 15.24, which he claimed meant Mr Alsop’s ticket expired at 3.24pm – 17 minutes before he returned to his car.
Mr Alsop said: “I tried to explain to the warden but he didn’t have a clue. He thought he was doing things correctly. He just carried on doing other cars parked there.”
The warden insisted he was right and issued fines to two other unsuspecting motorists.
Mr Alsop, who works for Pavey Insurance brokers in Torquay’s Abbey Road, appealed, had his fine waived and received an apology from Torbay Council.
Via -> Natural Blogarithms -> 360 -> God Plays Dice -> Eric Berlin
Posted by tpc at May 25th, 2008
The sad truth is that many of the students in the university are not here for an education. They are here to socialise, play freesbies in the field, run hall activities and yes get a degree. While some still attend classes, many subscribe to the mantra “studying worked solutions will help them pass exams”. To a certain extent, it becomes a self-fulfilling prophecy because the university do not want to fail too many students, and given that the students did not work hard to learn, the only way to let them pass is to set exams which resembles tutorial questions and provide clear written solutions to tutorial questions.
Now in my view, studying solutions to sample problems is really not a pedagogical sin. Well at least not in mathematics. One very important aspect of learning mathematics is mimicking carefully selected examples. But that is only the start. A student need to move on to the next stage of actually solving problems on their own. Many never do.
Because I do not want to adopt the high handed approach of failing 70% of the class, I intend to assign homework for my courses next semester. Simply put, it is to force students to do work which they are supposed to do on their own but never get round to it. Homework is not a popular activity, although the dept has recently made it mandatory for certain courses - a good sign. It requires a lot of extra effort in assigning problems, writing clear solutions, collecting, collating marks and worst of all, marking. Fortunately, we have graders to help us do the latter.
But collecting homework brings with it a new problem. Plagiarism. Some pretend it doesn’t happen, which is the easy way out since if we catch students we also need to go through the difficult process of discipline. Yet, it becomes a farce if the homeworks are merely copied, which renders everything completely meaningless. I especially pity the graders. (Although the graders might actually like to grade 100 copies of the same assignment.)
The university has licensed a system called turnitin, and I attended a talk by a chemistry lecturer who shared his experience. I really appreciated that. The surprising(?) result is that he had a chart that says almost 50% of the lab reports had been plagiarised. Now, the 50% can be interpreted in many ways, but it does sound high. The twist to the story is that the students knew they had to submit their report to plagiarism checking. So they could in theory, lift a passage from wikipedia, changed the adjectives, rearrange the sentences and escape detection. I do not believe the engine is that smart yet. But it turned out that those reports submitted early was not flagged, but those that were submitted later were flagged. The revelation is that many copied from past year reports from seniors which were not in the database of turnitin but were flagged because their classmates who submitted earlier had the same answers.
But with mathematics, I don’t think turnitin will work. For example, in those chemistry lab reports, an example was that when there were exponents like “x 10-5″ , it got flagged. Plus, if I wanted typed reports, I’d get ms word documents! And most important of all, I wouldn’t be surprised if original solutions to mathematical problems look identical.
I’m still thinking of a good way to implement this. I might try turnitin if I decide to ask the students to write some essays on mathematics history, if that day ever comes.
Posted by tpc at April 30th, 2007
This article in BBC news about a competition from the Royal Society of Chemistry, made its rounds last week. I had wanted to submit my solution for the 500 pound prize, but decided not to when I realized how quickly the news was spreading. A related article has a Professor Shaw claiming that the article was not fair because of curriculum differences.
Of course, everyone is entitled to his or her own opinion. If you want mine, I’m of the view that mathematics training in the UK has been watered down over the past years. Singaporeans usually take the Cambridge GCE O and A levels, and have easy access (thanks to overzealous parents, teachers and publishers) to all the past year exam papers dating all the way back to the 70s. A cursory inspection would reveal that the questions are getting easier. So, the difference in difficulty between the two questions reflect not so much China vs UK standards, but perhaps UK standards of the 70s vs now. It seems that the general curricula is regressing everywhere. Will the same thing happen in China? It is already happening in Singapore. The ministry has just announced the Primary school examinations will incorporate the use of calculators - the nation is well on its way to innumeracy soon.
On a lighter note, some enterprising local produced a tongue in cheek variant of the entrance exam
Now, part (ii) was pretty obvious to me. Since nothing is faster that the speed of light, we can expand the denominator as a series. I cheated for part (i) because I forgot what was the formula for momentum. I gave up on parts (iii) to (v), but I hope everybody knows the answer to part (vi) is 42.
Posted by tpc at April 14th, 2007
I was randomly surfing when the title Weapons of Math Destruction caught my eye. This is a site/blog about “new” math taught to American school children. There are some comics and also this interesting link to a video. In the video, a supposedly well known personality illustrates how two textbooks used in schools are no longer teaching traditional multiplication and long division, but substituting it with a more heuristic approach. It’s pretty interesting if you have 15 mins to spare. At about the 10th minute, the video quotes the following passage:
The authors of Everyday Mathematics do not believe it is worth student’s time and effort to fully develop highly efficient paper-and-pencil algorithms for all possible whole number, fraction, and decimal division problems. Mastery of the intricacies of such algorithms is a huge endeavor, one that experience tells us is doomed to failure for many students. It is simply counter-productive to invest many hours of precious class time on such algorithms. The mathematical payoff is not worth the cost, particularly because quotients can be found quickly and accurately with a calculator.
At the end, there is a plug for Singaporean mathematics textbooks. The irony is that at the pre university level, the Ministry of Education in Singapore is doing exactly the same thing. Pushing for expensive graphing calculators as a substitute for mathematical ability.
Posted by tpc at January 5th, 2007
I quite enjoy reading teaching evaluations. Perhaps, it’s a little egotistical but I regard it as the pat on the back for a job well done.
Anyway, there’s this strange comment from my evaluation under the category “suggestions for improvement”
Erm.. Mr toh, u look fatter in the pic than u are now. Dun be work too hard. And pls smile a bit. U look soo serious in the pic. U look better when u smile
Posted by tpc at December 3rd, 2006
I find it strange that students don’t bother to read instructions. Even on exams!
I have conducted 3 semesters of mini matlab training. At the end, the students need to do a quiz where they use matlab to perform some very routine computations. In order to make them do the computation with the software, as opposed to doing it by hand, the answers are usually ugly decimals. In spite of the written instruction on page 1 of the quiz that they must give their answers in decimals, some still use rational format — matlab will approximate the value with rationals and sometimes the same computation gives fractions that look different. It’s not a major sin to use rationals, just makes marking slightly more difficult, so I just let it go.
But I discovered some other instructors are not as forgiving. In one final exam, the instructions are that students can bring in one A4 sized cheat/help sheet. Some assumed it meant two sides, so they brought in two single-sided sheets and promptly had one of them confiscated. In that same exam, the instructions are that pencils are not to be used at all. That’s fair. But it seems that if you wrote in ink but used pencil to draw some accompanying graphs (like what I’m used to), the graphs will not be accepted.
Posted by tpc at October 20th, 2006
Sometimes, you feel sad when after you put so much effort into teaching your students, they still show that they don’t know anything. But then they don’t seem to mind. I mean I don’t see them seeking me for help when they are obviously not following. A case of having no time? Or simply lacking interest. You really think whether or not you should mark up the standard and fail 50% of the cohort. There are also those who affirm that whatever happens, it’s not your fault. I received an email from one who is supposedly in my lecture group but did not know that I was away for the past week. Surely something must be amiss if you went to lecture and found that there is no one there. (I had cancelled the lectures.) Was there a ghost lecturing on my behalf?
Posted by tpc at November 12th, 2005
If you go to the zoo, the chimpanzee will wave to you. I do not understand their psyche, but I presume they do not understand what they are doing, except when they wave, they get food. I still remember seeing a tourist throw an ice-cream into the enclosure to the adorable chimp.
It’s sad when students are taught monkey tricks, i.e. techniques to solve problems without understanding. One example is at A level when they are required to diagonalize a 3×3 matrix (with distinct eigenvalues.) Some schools actually teach a method to find eigenvectors by using cross products. I know this because a few students asked me whether they can use that method in our course. I’m not sure how it goes, but I guess it’s this:
To find the basis for the nullspace of 
, take two rows and compute the cross product. This works because the rowspace is the orthogonal complement of the nullspace and nullity=1.
The problem with this method is that it only works when A is 3×3 and the eigenspace is one dimensional, but I have a good feeling the students don’t know this. I recall a friend (JC teacher) who tested his students by asking them to diagonalize a 2×2 matrix and some of them who knew how to do it for 3×3 matrices couldn’t do it in this simpler case!
Posted by tpc at November 12th, 2005
Back when I was an undergrad, the maths classes I attended could be broadly classified into two types - general and small classes. The latter are mainly algebra and topology classes which attract an enrolment of less than 15. The former are large classes, usually essential modules or courses that are thought to be easy. I never had much problem with those large classes. The reason being I was among the top end of the cohort and the lecturer needed to “dumb down” in order to cater to everybody. Even Lebesgue integration was never too intimidating. On the other hand, I had a lot of trouble with the group theory and topology and I often wondered during those days why I signed up.
In those big classes, there was one particular lecturer who was very well liked in general and always won awards. I didn’t like him because I felt he was “playing” to the audience, dumbing down too much, oversimplifying and very hand-holding.
Fast forward to the present. As a lecturer, I have this dilemma. The class is huge and the ability of the students diverse. As much as I want to teach rigourously, I find myself guilty of “dumbing down”, to the extent that one particular student commented that I should be replaced. Retribution, I guess. I’m still not sure how to find the right balance, considering that only 10% of the cohort may eventually end up graduating as math majors. How do you teach rigour without at the same time putting off the rest of the 90%, or worse still, extending the bad impression that they may have about mathematics.
Posted by tpc at November 2nd, 2005
By Denis Guedj. I read it (the English version of course) years ago, and picked it up again recently when I ran out of light reading. It has a wonderful plot although the ending fizzled out. Read Simon Singh’s review here.
The book can really be thought of as a walk through the history of mathematics, and despite knowing most of the stories already, it’s still fun to re-read them again. Actually, if it was up to me to decide, I really wouldn’t mind using this book to teach a course on the history of mathematics at my faculty. Murder, mayhem, pistol duels, crazy genius, there are enough stuff to whet the appetites of undergraduates, no matter what their mathematical inclination. I can even throw in Fibonacci and the Da Vinci connection.
A more scholarly textbook could be The Adventure of Numbers by Gilles Godefroy. It’s interesting that both books are all originally French.
Posted by tpc at April 24th, 2005
The most difficult part of teaching trigonometry to kids (more specifically, pre-university and below) is proving trigonometric identities. Typically, the stronger students would relish the challenge, while the weaker would despair. Some of them would even question what is the point of all these. Another potential problem is that students are told to accept certain identities and use them. One chief culprit would be the addition formula:
.
In all my years of formal mathematical education, I do not recall ever seeing that identity being proved. I sort of figured it out for myself, while learning complex analysis, which to me is the best proof. One version is given here. Ah, the beauty of complex analysis!
The shortest path between two truths in the real domain passes through the complex domain.
-Jacques Hadamard (1865-1963)
More info on the above quote given here.
Of course, a purely geometric proof exists and involves drawing one triangle on top of another. A more interesting second proof uses Ptolemy’s theorem.
Posted by tpc at March 25th, 2005
I had been putting off writing about this until I found this post. I have yet to read it, and will do so after finishing my post, to compare the differences.
The Ministry of Education in Singapore will be introducing graphing calculators into the new A level syllabus. This is bad. Let me stress that I’m not a stick-in-the-mud techno-phobe. In fact, I make heavy use of Maple for my work, and believe in using computers for computations. But I certaintly feel that unleashing high powered calculators to students do more harm than good. I have seen students who can’t add two-digit numbers without calculators. Giving them calculators that can graph and do algebra would be akin to giving a chain-saw to kids who still can’t hold a knife. Simmons share similar views and wrote in the preface of his book of seeing students who instead of factoring 
, or some other similar quadratic, use calculators to graph and solve the equation.
The biggest benefactors are not the students but Texas Instruments. Do you know that they fly teachers and no doubt MOE officials, to conferences in Australia and other places, so that they can see and hear other people rave about TI calculators? The seed money has paid off handsomely, since TI can now ship 20,000 units of calculators every year to Singapore.
Posted by tpc at January 26th, 2005
This week, I have to teach predicate calculus to first year computer science students, in addition to the vector calculus for engineering students. I find that I really have nothing to value-add for logic. Perhaps it is because I’ve never really fancied all that logic stuff, but seriously, how do you make things interesting when all that is required is the mundane task of filling up truth-tables?
