In this Newsletter:

1. Current math events
2. Math tips - (a) Math Dictionary; (b) Numerical methods
3. Poll results - Summer math
4. From the math blog
5. Your brain and math

1. Current math events

Recently, a reader asked for information on “current math events”.

I have been running a series of articles in the squareCircleZ blog on recent applications of math:

• The melting Arctic - a disturbing application of math
• Interesting semi-logarithmic graph - YouTube Traffic Rank and Another semi-log graph from Alexa - imeem
• Math Optimizes Kidney Matches
• The maths of music
• Math in computer games and Math in computer game development

For a really extensive list of recent research and developments in math (56,000 articles), check out ScienceDaily.

2. Math Tips

(a) Looking for a Math Dictionary?
Each new topic in math involves many new vocabulary terms. Do you have a brain like a sieve when it comes to math definitions? This free Math Dictionary from MathWords.com may be just the thing. There is an alphabetical list of math words as well as definitions grouped by topic, including numbers & symbols, geometry, algebra, trigonometry, calculus, probability & statistics, and real world applications.

I hope you find it useful.

(b) Numerical Methods
Have you ever wondered how the early mathematicians developed the formulas that we use every day?

Usually there was some important problem that needed solving and the best the mathematicians could do was to approximate the answer using numerical methods.

For example, how would you find the area of a circle if you didn’t know the formula A = πr²?

As far back as the 3rd century BCE (over 2,200 years ago), the Greek mathematician Archimedes worked out a remarkably accurate value for the area of a circle. He was aiming to find a square that had the same area as a given circle. This process is known as “squaring the circle“.

Archimedes approached the problem by drawing a square inside a circle, so that the 4 corners of the square touch the circle. The area of that square is easy to find (actually, the square roots involved gave him some trouble), but its area is clearly smaller than the area of the circle as you can see in this image.

He then figured that if you draw another square outside the circle, so the circle just touches the 4 sides of the square, then the area of the circle must be somewhere between the areas of the smaller and larger squares.

He had a reasonable approximation for the area of the circle by taking the average of the areas of smaller and larger squares.

To improve on this, the next step was to draw a regular pentagon inside the circle. As we can see in the image, the area of the pentagon is closer to the area of the circle, but still too small.

Once again, we draw a larger pentagon outside the circle and find its area.

Our desired area is somewhere in between the areas of the 2 pentagons.

We can keep going with this and each time we do it, we get a better approximation for the area of our circle. Here’s what a dodecahedron (12 sided polygon) looks like inside the circle.

You can see that the area of the dodecahedron is getting quite close to the area of the circle. What Archimedes has done is to use a numerical approach to find the area of the circle.

You can see more on this topic, with calculations included: here and here.

[BTW, the topic of "squaring the circle" is where this math blog gets its name: "squareCircleZ".]

3. Poll results - summer math

The most recent IntMath poll asked readers “How much school-related study will you do during summer?”. Just over 40% indicated that they were going to summer school, which is higher than I expected, but probably not that surprising since if you are doing math at summer school, you’ll need to visit Interactive Mathematics. The rest indicated between zero study and “maybe a week”.

The results for “how much study will you do in summer?”:

43% I’m going to summer school
20% Maybe 1 week
20% None
11% Maybe 1 hour
  7% Maybe 1 day

Total votes: 500

The current poll asks you about how inflation is affecting you personally. You can vote on any page in Interactive Mathematics.

4. From the math blog

1) Friday Math Movie - I Will Derive
This week’s movie is a fun summary of when and how to apply differentiation.

2) EduSim - free 3D multi-user virtual world
EduSim is a 3-D virtual world that comes as a free download. It’s worth checking it out.

3) Google search, math and latent semantic analysis
There is sophisticated math going on behind the scenes when you perform a search on Google. It involves a technique called latent semantic analysis which is based on matrix operations.

4) Friday Math Movie - The Amazing Origami of Robert Lang
An origami specialist talks about how math helps him to create beautiful art.

5) Math in computer game development
Behind every computer game lie many applications of mathematics.

5. Your brain and math

Last newsletter, I linked to a special article on Math problem solving and brain activity.

Thanks to all the readers who wrote giving feedback that they found the article very interesting and useful.

It is quite important that teachers — and students — understand how the brain deals with math and where the stumbling blocks can occur. In this age where there are thousands of Weapons of Math Distraction and we all suffer from attention deficit disorder, brain research helps us to understand how we process problems and what we give our attention to, and why.

If you missed the article last time, here’s the link again: Math problem solving and brain activity.

Till next time.

This is fun but will probably only make sense if you have studied differentiation already.

The out-of-tune piano in the opening and the mournful singing add to the wacky and nerdy atmosphere.

Thanks to Maria at HomeschoolMath for the link.

EduSim is interesting and has a lot of potential.

Yes, EduSim is engaging and clearly a lot of fun for the little kids shown in the movie. Edusim provide some lesson ideas, including one on geometry (whose screenshot didn’t look too exciting to me).

I’m wondering about applications for older kids. Like SecondLife, probably EduSim’s greatest potential is in getting the students to build new 3-D worlds. There is scope for learning about 3-D geometry there.

Edusim is also integrating Wii.

Google has become the dominant search engine because of its relevance and efficiency. Relevance is achieved through its propriety PageRank algorithm, which determines which pages are the most likely to satisfy your search query. Efficiency is achieved by using thousands of PCs rather than big servers to hold all the indexing, document and media information.

I wrote about this a while back in Math that made Google rich.

Now let’s move on to an aspect of matrices that search engines use, called latent semantic analysis.

Here’s what Wikipedia has to say on the subject:

Latent semantic analysis (LSA) is a technique in natural language processing, in particular in vectorial semantics, of analyzing relationships between a set of documents and the terms they contain by producing a set of concepts related to the documents and terms. LSA can use a term-document matrix which describes the occurrences of terms in documents; it is a sparse matrix whose rows correspond to terms and whose columns correspond to documents, typically stemmed words that appear in the documents.

Let’s put this in everyday language. Simply put, latent semantic indexing is something the search engines do when they analyze the content of a Web site in order to figure out what the site is about.

It’s actually what we humans do every day of our lives — try to figure out the meaning in what we see, hear and feel.

That Wikipedia article delves into the matrix operations that are involved in latent semantic analysis.

(If you are a bit rusty, see an Introduction to Matrices.)

Origami is the art of paper folding. In Japanese, ori comes from “oru” which means “fold” and gami comes from “kami” which means “paper”.

Robert Lang is a full-time origami artist. In this movie, he explains how he uses a math to make his excellent creations. As he says,

Math is much broader than what most people think.

[The movie is from wired.com and there's advertising at the beginning.]

You can get a better idea of the math in origami that he is talking about at geometry behind origami (paperfolding.com).

A common cry in mathematics classrooms is, “When am I ever gonna use this stuff?”

I think it’s a perfectly reasonable question and it is evidence of the learner seeking meaning in pages of mindless algebra.

Well, who uses math and how do they use it?

One field where math is alive, kicking and vital is the multi-billion dollar computer games industry.

I came across this Math and Physics forum in gamedev.net. It gives a fascinating insight into the things that game programmers are trying to achieve in their games and the math that they need to achieve it.

Some recent forum topics:

• Collision detection (vectors, coordinate geometry, distance formula)
• Mollweide projection (3-D geometry)
• Checking if an x,y point is within an ellipse (analytical coordinate geometry)
• Poker (probability)
• Angular velocity
• Matrix operations

I’m always impressed with how people freely give of their time and expertise on such forums.

That forum link again: Math and Physics forum

See also Math in computer games for another example of how math is being used every day in the computer games industry.

Check out mathtv.com.

MathTV has lots of short videos covering a large range of math topics from numbers through to simple calculus.

Unfortunately, there is no real-world context given for any of the problems (they are algebra-based explanations only), except for some of the application examples (which are not real-life applications for the most part).

There is limited reflect time given - there were many times where the explanation rushed on before most students would have absorbed the current step.

But hey, it’s free.

A few weeks back we had Hector the Droid teaching us about quadratic formula.

Here’s another effort at tarting up what is normally a very dry topic in school.

In this Newsletter

1. IntMath future development - survey results
2. Math tip - Solving math word problems
3. From the math blog
4. Some brain math

1. IntMath future development - survey results

Thank you to all those who completed the survey on future IntMath developments (which I mentioned in the last IntMath Newsletter).

The survey showed there is strong interest in each of the following:

• A forum for help on homework. Also, several teachers asked for a teachers’ forum.
• Video lessons on various math topics, and also on how to use computer algebra systems.
• e-Books on various math topics, including math anxiety.

There were some great suggestions that you gave in the written part of the survey and I will progressively develop the best ideas over the coming months.

Thanks again for your input!

2. Math tip - Solving math word problems

Most students struggle when they are asked to solve word problems. There is a good reason for this — your brain is super-busy while figuring out such problems.

This tip became a bit long for this Newsletter so I made it into a separate article. Check it out here:

Math Problem Solving and Brain Activity

3. From the math blog

1) Fractal Science Kit
Fractals are beautiful mathematical art objects. This article describes a tool for creating them.

2) Google gadgets - cool extras for your website
Google Gadgets allow you to easily add fun stuff to your Website or blog, including math games.

3) Friday Math Movie - Math Test Anxiety
This strange anime illustrates one student’s nightmares about math tests. Warning - it’s very strange and may be disturbing for some viewers.

4. Some brain math

To finish, I thought I’d tell you some interesting facts about the brain.

• The human brain has 100 billion neurons, each linked to as many as 10,000 neurons
• The brain is 60% fat
• The brain can handle about one billion instructions per second
• The brain weighs 2% of total body weight, it needs 15% of the body’s blood, consumes 20% of total body oxygen, and requires 25% of total body glucose (sugar)
• The brain uses up 1.5 calories per minute during crossword puzzle-solving

[Source]

Summary: You have a remarkable organ between your ears. Look after it well.

Until next time.

I believe that solving math problems is a very important issue. Mathematics is more than just “doing algebra”. If we know how to solve a real-world problems, then we’ll have a powerful and important ability, and best of all we’ll see why we are doing all this math.

Let’s say you have a math problem that is in sentence form. Most people struggle with word problems and one reason is that math word problem solving uses many parts of the brain.

Here are some tips on how to attack math problem solving — and I have included an indication of what goes on in the brain during each step. Here is a diagram of the brain so you can follow along. The front of the brain is on the right of the diagram.

Image source.

a. Read over the whole problem: Understand the whole question first and take special note of the what?, when?, which?, how many? parts of the question, usually at the end.

The areas of the brain that you use for this portion of the math problem are the very back of the brain (your occipital lobe) where you process what you see. You also use the language areas of your brain (a large portion of the left hemisphere, surrounding your left ear).

b. Write down or draw what the question tells you: By listing the information in the question, it helps you to sift through what you already know and it reminds you of the math that might be involved. Once again your left brain is involved in the writing part, in particular Broca’s Area and Wernecke’s Area, which are above your left ear.

If there is any geometry involved (or graphs, or moving objects, or any other visual element) in the question, draw the situation. For drawing, you use the areas towards the rear top of your brain (the parietal lobes), the area at the top of your head (the sensorimotor region) and the back of your brain (vision).

c. What do they want? Is the quesion asking for a speed, or a time, or a length, or a position, or a cost? Many students answer a word problem by giving an answer that is not what the question actually asked. In the real world, will your boss be impressed if you give a time answer when they actually asked for a cost?

At this point, it is good to estimate the answer and to write down that estimate for checking later. Estimation involves non-language areas of the brain (while exact arithmetic involves language areas).

So far in our math problem solving, we have a good idea what the question has told us and we know what we need to find. We also have an approximation for our answer.

d. Identify the math required: Now you need to make a decision about the math. Will it involve algebra? Or maybe trigonometry? Or logarithms? Maybe it will involve differentiation or perhaps integration? This is where you see the need to actually learn the formulas in each section of math that you study, and not rely 100% on formula sheets. The best way to recognize the math that you need to use, is to know that math in the first place.

This step uses the higher-order thinking areas at the front of the brain (the frontal lobes) and the memory areas of the brain (which tend to be all over).

e. Do the math: Now we need to churn through the algebra (or whatever) to get our answer. Assign variables to the known and unknown quantities in the question. In the brain, this step involves the frontal lobes and the area behind and above the ears.

Your answer must include units (if there are units in the question).

f. Check, check and check: Firstly, check if your answer is close to your estimate. If not, it’s back to the drawing board.

Next, read over the question again and check that you have actually found what the question was asking for.

Finally, check all the algebra and arithmetic steps.

Phew! We are done.

Now don’t be scared about all the brain activity involved in math problem solving. It’s like most human abilities — the more you do it, the easier it becomes and the brain can begin to relax.

George Polya contributed greatly to our understanding of how to solve math word problems. You can see a summary of his approach from his 1957 book “How to Solve It” at G. Polya: How to Solve It.