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Posts Tagged ‘math’

Happy Pi Day Everyone!

March 14, 2011 Leave a comment

Yes March 14 (3/14) is indeed pi day; a day to celebrate everyone’s favourite irrational number.

As a special treat for you, I’m going to show you a way to calculate pi to a high accuracy for yourself (knowledge of a computer programming language helps).

So the number pi is closely related to the geometry of the circle. The area of a circle is represented by the following formula:



Where r is the radius of the circle. So we can use this relationship to calculate pi, but we can do it without measuring anything! Instead, we are going to use something called a “Monte Carlo” simulation

Monte Carlo simulations are very useful in physics. They use random numbers to simulate a real-world system. Since they are based on random numbers, the technique is named after the famous Monte Carlo casino.

Ok, so imagine you have a square that is 1 meter by 1 meter. Now imagine you place one-quarter of a circle with a radius of 1 meter over top of this square, like so:

Since the radius of the circle is 1 meter, and we are looking at exactly one-fourth of the complete circle, the area of this “slice” of circle is:

The area of the square is

And since it is 1 meter in length and 1 meter in width, the area is simply 1 meter squared:

Since the area of the square is simply 1, the ratio of the area of the square to the area of the slice is:

So we can see that pi is directly related to the ratio of the area of the slice to the area of the square.

Don’t worry, I’m goin’ somewhere with this…

Now comes the Monte Carlo part of the problem. We are going to generate 2 random numbers between 0 and 1. The first random number will be an x-coordinate, and the second will be a y-coordinate. For illustrative purposes, lets say the numbers turned out to be 0.5 and 0.5. So lets put a red circle at (0.5, 0.5):

We can see that this point lies inside the circle. In fact, any x and y values which satisfy this equation:

will lie inside the circle.

So what if we generated a huge number of points and counted the number that landed inside the circle? If we did this, then the ratio of the number of points which landed inside to circle to the total number of points would be:

Which you see is exactly the same as the equation above which related the area of the slice to the area of the square, except for one little difference. The “squiggly line” equal sign in the above equation means “approximately equal to”. It indicates that this method is an approximation because we are only using a finite number of points. If we generated an infinite number of points, that “approximately equal” sign will change to a regular equal sign.

So now all we have to do is run this calculation for a large number of points (N) and we will find that:

So how accurate is this method? Well again it will depend on the number of points. For example, if I run this method using 10 points I get a value for π of about 3.6.

If I use N = 100 points I get π ~ 3.16

And if I use N = 1 000 000, I get π = 3.140916

which is getting pretty close to the actual value of about 3.1415926539…

There you go. I’m sure your life will be much richer and meaningful now that you have this information. Now if you’ll excuse me, I’m going to eat some pie.

Math Can Be Cool. No, Seriously!

January 25, 2011 1 comment

I did physics in University, and I had to take math courses. I hated the math courses.

And I like to make jokes at my friends who did Applied Math or Pure Math as their degrees, because I just found it so boring.

But dammit if this isn’t the coolest thing I’ve seen all day:

Notice it starts with the Fibonacci Sequence, makes a Fibonacci spiral, and from there it just keeps going.

Awesome.

Simple Writing Exercise Boosts Grades of Female Physics Students

November 29, 2010 Leave a comment

Although men and women are equal in their abilities concerning math and physics, women still seem to lag a bit when it comes to grades in the Physics classroom.

Consider the classroom of Akira Miyake from the University of Colorado. Female students got grades, on average, 10% lower than male students. A significant difference. Miyake knew that this had nothing to do with ability, so how does one fix it?

Well he tried a technique that had been shown to increase the exam scores of black students in American high schools. These students were victims of the “stereotype threat“, which is when an unfair stereotype is applied to a group to which you belong (i.e. black students do worse in school than white students). However, after utilizing a technique called “values affirmation”, the scores of the black students dramatically increased. In fact, the increase was greatest amongst the poorest students. What happened?

The technique involves a writing exercise in which the student writes a few sentences about their core values (family, religion, creativity etc.) and write why those values are important to them. The exercise takes no more than 15 minutes. This was done twice at the start of the school year, and the results show a large increase in the students scores.

So could it work for women too? Women also suffer from the unfair stereotype that they are not as good at math/physics as men. Therefore, they don’t have as much confidence and don’t do as well. It is a self-fulfilling prophecy.

But Miyake did an experiment. He divided his introductory Physics class into 2 groups, with equal amounts of males and females in each group.

At the start of the year, he asked Group 1 (the “values affirmation” group) to write about their core values and why they are important to them. He then separately asked Group 2 (the “control” group) to write about the values of someone else and why they might be important to another person. Both groups were told that this was an exercise in writing ability, not physics grades.

He then compared the results of the 2 groups with their final marks in the course, as well as how well they did on the Force and Motion Conceptual Evaluation (FMCE), which is a standardized test about basic physical principles. These were the results:

As you can see, the women’s scores increased, whereas the men’s scores remained largely unchanged.

This is quite a result. With a simple writing exercise designed to instill confidence and reduce the effect of the stereotype threat, exam scores in the class increased by an average of about 5%, and about 13% for the FMCE.

So what we find is that it’s not enough to debunk the myth that men are better than women at physics, we have to get women to believe it as well. To believe in themselves.

Why am I so interested in this issue? I have written about it a couple of times. Mainly because it frustrates me. I tutored during my time in University and many more girls asked for tutoring than guys, because they didn’t do as well. This was mainly because, in my opinion, they simply didn’t have as much self-confidence when it came to the exams.

“I’m no good at Physics, I just want to pass,” was a phrase I heard more than once.

“No, you will not just pass,” I told them, “you will do well.”

But making them believe it was tough, because of all the stereotypes they no doubt encountered through their lives. So I would tell them about these kinds of studies, how girls are just as good as guys at science/math. I even told them how I came in 3rd place in my senior year research project class. 1st and 2nd place went to two girls.

So it’s an uphill battle, but progress is being made. If I ever have a daughter, you can be damn sure she will not grow up thinking that she is not as good as guys at science or math.

Who Is Better At Math, Guys or Girls? Answer Found Within…

October 15, 2010 1 comment

It’s always been an unfair stereotype that girls are not as good as guys at math. A lot of girls feel like they are not supposed to be good at math; that math and science are boys territory.

Science teachers and parents have been trying to spread the word that its OK for girls to be good at math, and to like math. And now, we have the science to prove it.

A recent meta-analysis of the published research in this area was performed. In total over 1.2 million people were studied between 1990 and 2007. Students from grade school through college level were included, as well as the results from several long-term, large-scale studies. The results all show that there is no significant difference between the math skills of men and women.

JS Hyde, one of the authors on the paper, discussed why there is still a stereotype in our society,

There is lots of evidence that what we call ‘stereotype threat’ can hold women back in math. If, before a test, you imply that the women should expect to do a little worse than the men, that hurts performance. It’s a self-fulfilling prophecy.

So having the data to show that women are just as good at math as men is only half the battle. We have to start making a change in our culture and encourage more girls to pursue their interests in science and math.

Women + Science = Awesome

August 11, 2010 1 comment

My undergraduate physics class had about 30 students who got their B.Sc. at the same time as me.

Only 4 of them were women.

But dammit, they were equals every single step of the way. We studied together, we wrote tests together, and we got drunk after we wrote tests together.

I have heard of a time when women studying STEM (Science, Technology, Engineering and Math) was frowned upon. They were ignored by their peers and were not taken seriously.

Boy, how times have changed.

But unfortunately, there is still a large gender gap in the physical sciences. Although women are considered equals, enrollment numbers, particularly in Engineering, are still dominated by males. Recruiting programs have been set up to try and get more women into Engineering (although being the only girl in a class full of engineers probably has its advantages in terms of attention received). And while the number of women studying physics is climbing, it is climbing slowly.

So what is the solution? How do we get more women interested in science and technology?

I think a big part of it will be letting young girls know that its OK to like science. There are probably a huge number of girls who would rather play with chemistry sets than barbie dolls, but feel strange doing so; especially when its only the boys playing with toys like that.

While I have never been a girl myself (though I was called one repeatedly in elementary school), I imagine its not easy for girls interested in science to grow up in a culture of many girls who don’t.  Even though we have come a long way in terms of equality, I think STEM still has a stigma as being for “Boys Only”.

So how do we get the word out that science is for everybody? The internet is a great resource that, up until a decade or so ago, was not available.

For example, a dear friend of mine from University has a blog called Technolochick. This is a site written by women for women who are interested in technology. Its a great idea for a blog and you should really check it out.

Celebrity involvement never hurts either. Amy Poehler, SNL alumnus and star of ‘Parks and Recreation‘, currently helps run a website called ‘Smart Girls at the Party‘  which takes an entertaining approach to encourage girls to follow their passions in STEM.

So ladies, don’t be shy. If you love science, say so! If you want to study it, do so!

And even though I considered all those in my class as equals, if one of the girls batted her eyes at me, I was probably more likely to give her my answers to the homework problems.

So you have that going for you as well.