Bezier curve inflection points

Adrian Colomitchi wrote me an email about an article I wrote. It turns out that I was wrong about the inflection point section. I was trying to figure out what Timo meant by “loop tips”, and I figured it could be an inflection point.

So here’s how Adrian describes inflection points:

If you drive or ride a bicycle: inflection points will happen when you switch your direction from left-to-right or vice-versa.
The inflexion points have 0-curvature (and infinite radius)… for an instant you travel straight (because you are switching the direction).
By contrast, the tip points will have maximum curvature.

Here’s how the inflection points should actually look like (screenshots taken from Adrian’s site):

A Bezier curve with no inflection points:
Bezier curve with no inflection points

A Bezier curve with one inflection point:
Bezier curve with one inflection point

A Bezier curve with two inflection points:
Bezier curve with two inflection points

You can find out more about inflection points on a cubic Bezier curve on Adrian’s website.

If you have anything to add about inflection points (or Bezier curves), comment below.

Elevator waiting maths

John Cook wrote an article on where to wait for an elevator. Where do you wait so you walk the minimum distance to the elevator? Read his article for the full explanation and solution. Here’s a pictorial summary of the solution:

Elevator optimum waiting position

The problem was from a paper written by James Handley (and collaborated with others). Basically, you don’t wait at the average position among all the elevators. You wait at the elevator with the median position.

In reality, however, you still have to enter the space where the elevators are. Therefore, the best position with the minimum distance to walk is

to remain at the point of entry, and to not move at all!

That of course, assumes that the elevators had been activated to arrive at your floor. You know, you need to push the button. How do you push that button if you’re 5 metres away? With a long pole? Other people? Telekinesis?

I have a more in-depth discussion on this in the January 2011 issue of Singularity. Go download it and read it. It’s free.

Here’s something to think about. James Handley is a mathematician. What’s he doing in the faculty of medicine? Because he’s helping with epidemiology. If you’re not familiar with the word, “epidemiology” means the study of epidemics. The spread of viruses, rates of infection and the like.

This reminds me of the time when my thesis mentor (back in my university days) suggested I work in the field of epidemiology. Because my thesis dealt with computer virus behaviour. Oh my god, I just looked at my thesis again… oh dear, ordinary differential equations! *closes thesis hurriedly*

Ancient Egyptians and Chinese got math simple

And here I thought bit shifts were already very simple to understand.

I was actually hoping that the video would shed some light on the Udja Eye thing, but it didn’t… oh well.

Fibonacci infographic

Pierce Brown created a beautiful infographic on the Fibonacci sequence. It has rabbits, sea shells and sunflowers. Go take a look.

Maths, context and culture

I was reading this post by Dan Meyer on pseudocontext in maths problems.

If we invite pseudocontext in our classrooms without condition, it becomes harder and harder to tell the difference between the real and the unreal.

Back when I was young, a lot of maths problems made little sense to me. In those days, the maths syllabus up to primary 6 (at 12 years old, or grade 6 if you’re in America) wasn’t particularly hard. At least to me. I’m not bragging, I’m just saying that the education system made things more difficult by introducing word problems. The epitome of conquering a maths exam paper was solving all the word problems at the end.

Word problems were created to introduce another element into elementary math (to make them difficult?). They added language. Suddenly it was something like:

John, Fred and Ken had $5 total. John bought 10 red marbles and Fred bought 12 blue marbles. If 1 red marble costs $0.10, and 1 blue marble costs $0.15, how many blue marbles can Ken buy if they still want to have $1 left?

Your command of the English language became a factor. But it was still ok, because the wording usually formed a pattern. It was marbles, people’s ages, number of apples or oranges in the basket, or some such. In a normal situation, if I really wanted to know your dad’s age, I’d just go ask him. I don’t really need to infer that your dad is 2.5 times your age, and then I figure the answer out (assuming I know your age).

Students here kill each other with A’s

Now if you don’t already know, it’s bloody competitive here in Singapore. Students are afraid of not doing well in school, of heads shaken by their friends, teachers, parents and relatives. Parents send their children to tuition classes (in addition to the normal school classes), regardless of their children’s grades. If the grades are bad, then improve them. If they’re great, great! Now perfect them. Go do your ten year series!

I went to tuition classes till I was 10 years old (primary 4 or 4th grade). I stopped because my dad couldn’t afford to pay for the classes. Being able to eat and pay the bills were more important. It’s a good thing I was disciplined enough to get good enough grades (and imbue enough motivation for all subjects, not just maths).

When I was in university, to supplement the cost of education, I looked into giving tuition. I was surprised that everyone from primary one to university level (?!) were asking for help. Let me just say, I make a lousy tuition teacher. I don’t really know the current syllabus well enough to help the students. Once, I brought up the subject of video games, using the position of battleships to illustrate … something. I can’t remember. I think it was x- and y-coordinate stuff. I was trying to interest the young boy I was teaching. It fell flat. I suck…

The Singapore Math Method

Which brings us to curriculum. It turns out that under the Singapore maths curriculum, Singapore students rank high for maths internationally. It’s so good that America has adopted the method. There’s even a name for it: Singapore Math Method. Let me tell you, I’m simultaneously amused and confused.

I’m even more surprised that Israel adopted the method in 2002, translating the textbooks to Hebrew. I was browsing in the bookstore reading Start-up Nation (Amazon link). It told a story of how Israel, being surrounded by hostile countries, had to innovate hard. Their brightest people are in the universities doing research and are also in the top military ranks. The book told a story of how the “flat” nature of their military translated to their way of doing businesses, in particular start-ups. My friend Christopher told me that per capita, Israelis were the richest in the world. It’s their culture that made them more inclined to creating wealth. I was also told about the Jewish mother syndrome… So I’m a little surprised that this group of people want to know about our (Singaporean) method of teaching maths.

I still believe in solving real world problems. I believe we’re not injecting enough curiosity into our students. That Singapore Math Method seems to have less force-feeding of concepts, and more of coaxing the student to question. The Singapore culture doesn’t seem to require curiosity for the students to do well (have I mentioned the parents are bloody competitive?). Hopefully, that’s changing.

This is going to be a cynical view, but I think most Singaporeans are striving for wealth, and wealth alone. Wealth translates to a better life. There’s nothing wrong with that. Singaporeans strive hard to attain wealth so they can forget about (seemingly) miserable lives. Ok, let me take that back. Apparently, Singapore is one of the happiest places in the world. There’s a “but” though…

Singapore ranks high on evaluated happiness, but not on experienced happiness

Alright, this is starting to depress even lil’ cheerful me…

So. Problems are formulated, and then given to our students to solve. But they have to learn how to formulate problems too, and that comes from asking questions, from being curious, from being disciplined and persistent. And that comes from cultural and societal influences, not from educational systems.