## Equal Understanding

According to an article from Science News titled, Students’ Understanding of the Equal Sign Not Equal, Professor Says, one of the reasons students in the United States are so far behind their foreign peers in mathematics is that they have a lesser understanding of the equal sign. Research conducted at Texas A & M university suggests that roughly 70% of students in middle grades in the U.S. have misconceptions about what the equal sign means. This is a staggeringly large proportion considering that students in some Asian countries have almost no misconceptions about the equal sign.

A large part of the problem is believed to come from students memorizing procedures rather than learning how to interpret the symbols. An example given in the article is 4+3+2=__+2 where students are asked to fill in the blank. Many U.S. middle grade students would solve this by recognizing that 4+3+2=9, and thus filling in the blank with 9. The source of this problem is thought to be the textbooks that students use. In comparing textbooks from China with those from the U.S., researchers noticed that the Chinese texts offered examples explaining things such as the equal sign, while those in the U.S. did not.

## Mandelbrot on Markets

In my last post on Benoit Mandelbrot’s final TED talk, I mentioned that he had begun his career in finance. The article ‘Economist’ Obituary Recalls Mandelbrot’s Scrutiny of Financial Markets goes into more detail about this. The article discusses the obituary written in the Economist for Mandelbrot (he died on October 14, 2010).

Benoit Mandelbrot believed that fractals could be used to model financial markets. The Gaussian (normal) distribution had been being used to model financial markets, and if Mandelbrot was correct, the Gaussian distribution would not be an effective means of modeling. While it has not been proven that fractals can be used to model financial markets, it is now widely accepted that such markets are not Gaussian.

## Math Magic

Today I read the article, Colm Mulchay’s Fibonacci-Inspired Trick. This article described two card tricks that use math. The first trick uses the Fibonacci sequence and good shuffling skills.

The trick is done by having the first six numbers in the Fibonacci sequence [1, 2, 3, 5, 8, 13] (1=Ace and 13=King) at the top of the deck. Then the deck is shuffled thoroughly, ensuring that those same cards are on the top of the deck (even if they are in a different order). You have someone draw the top two cards and tell you their sum. Then you tell them what the two cards are. This is where the Fibonacci sequence comes into play. All of the possible sums of these cards are unique, and thus you know the value of each card in the pair. Tah-dah!

## Mapping the Internet to Hyperbolic Space

The title of the article, Internet Has Been Mapped to Hyperbolic Space, says it all. But what does this mean?

Unlike Euclidean space, hyperbolic space does not require parallel lines to be paired uniquely. That is, given a line A and a point B which is not on A, infinitely many lines can exist that intersect B and are coplanar to A but do not intersect A. If that wasn’t clear enough (I’m not sure I even understand it, someone please correct me if I am wrong), just think of every point as being able to be a hyperbola.

Anyways, with the capability of mapping the internet to hyperbolic space, the article explains that glitches could, in theory, be entirely gotten rid of, while speed would be greatly increased.

I’m not sure what it means to “map the internet”, and I don’t really fully understand what hyperbolic space is, but this sounds like an interesting topic. Given the time and proper resources, I would like to learn more about this.

## Year-Round Schooling: Good for Math?

In the article, Year-Round Schooling Helps Students Recall Sine and Cosine, the benefits of year-round schooling are discussed. It turns out that a long break from subjects in which repetition is an essential part of learning (like math) causes students to forget much of what they learned. Thist means that after the summer, time must be spent re-learning already taught material.

What I found most interesting about this article is that recent research showed that students from a lower-income background were more prone to forgetting things. This is due to less structure during the summer. Without stimulation from trips to museums or spending a week at a summer camp, kids forget more during the summer than students who are involved in activites.