Learning by doing.

I hated school, it felt like prison. I never learned a thing in class. I passed the exams thanks to self-study mostly. What is the point in repeating what is in the textbooks? Students can read, can't they? Teachers can and should do better. To my surprise I read this article about a ' revolution ' in mathematics teaching. A revolution?!

Revolution in mathematics teaching

Chris Finlay's Blog

Chris Finlay has more degrees than blogs but here is his blog: http://chrisfmathsphysicsmusic.blogspot.com/

P.S.
I owe Chris a Thank You. He knows why.

A beautiful ( Norwegian ) theorem

#mathematics# #norway#

Theorem:

Every abelian group is the direct product of its Sylow subgroups.

Perhaps it is not the theorem in itself I like so much but what this theorem illustrates about the nature of mathematics. Most laymen think of mathematics as the scribbles of physicists they see in science documentaries, i.e. partial differential equations, stuff they call 'formulas'. So in that sense the theorem above may not even be recognized as mathematics, let alone beautiful mathematics.

Mathematics starts with a very precise, razor blade sharp, use of the tool that differentiates us humans from the rest of nature: language. Einstein once said “If you can't explain it to a six year old, you don't understand it yourself.” (*). He must have meant the "root of your knowledge tree", I suppose. Because the beauty of the theorem lies in what it represents: a large graph of concepts with - ( abelian ) group, direct product and ( Sylow ) subgroup - in the center. To anyone 'owning' these concepts the particular relation between an abelian group and its Sylow subgroups can be described in one sentence with no room whatsoever for misinterpretation. The construction of all that knowledge is the collective work of thousands and thousands of mathematicians before us.

P.S.
(*) The simplest way to explain a group is ( as far as I know ) "A collection of movements with no visible effects ( = symmetries )".

Sylow

Both Abel and Sylow were Norwegians. So was Lie, another giant, a special branch in group theory is named after him: Lie Group Theory. It is amazing that a small country like Norway ( measured in population ) can have such an impact.

355 / 113

355 / 113 = 3.141592.....

How would you -prove- that this is the best approximation of pi using only integers less than 1000 ? We can use a 'By Cases' / Brute Force approach and utilize a computer to go through all the possible quotients. Even going through a million of them is a piece of cake nowadays. I wonder if there is a computer-free approach. - Even with a computer its an interesting problem. I.e. what's the shortest program to prove it. The fastest? Go visit Project Euler and you'll be amazed what clever programmers can do in their favorite language. Even if, or especially if, you are a programmer yourself. Amaze yourself. Or take the challenge...

jenn3d

I came across a free tool for visualizing Coxeter polytopes, jenn3d. I suppose this program visualizes the Coxeter groups of polytopes. Polytopes are geometric objects in the n-th dimension with flat sides.



Link: jenn3d.org

Higgs Boson Buzz

It's buzzing in the media, on Twitter, everywhere. It will be announced tomorrow. The God Particle. It exists after all...

Here is a great animation about how the Higgs Boson particle works.

Gephi

I just discovered a new ( free, open source ) tool for graphs and networks. What Photoshop is for pictures, Gephi ( supposedly ) is for graphs. Software is definitely a requirement for Graph Theory. I haven't looked at Gephi in-depth yet, I might. It is exciting though. There is so much going on. When you turn away from a certain area for just a brief period things might have changed quite a lot when you get back. I 'worked with' ( read: used to study graph theory ) Mathematica with the Combinatorica package which is really a gem by itself.

Link: Gephi

Fujitsu Cracks Next-Gen Cryptography Standard

Fujitsu and its Japanese partners have taken 148 days to carry out a cryptanalysis that had been estimated to take hundreds of thousands of years.

Source: http://www.techweekeurope.co.uk/news/fujitsu-cryptography-standard-83185

What surprises me here is that they had the estimates =completely= wrong. Development in all sectors is getting faster and faster. Maybe it is going so fast that predictions have become useless, became impossible.

Educational tools for symmetry

#graphica# #m336#

This website from Prof. Dr. Loren Williams is worth exploring. Don't forget to visit the Crystallography Tool section.

P.S.
Notice the man in the space suit

Fixing study problems(2) - "I skip the proofs."

#openuniversity#

( Continued from yesterday. )

Lack of mass.

Take trigonometry for example. I remember myself saying things like "Why do I have to study this? Is this really important? Will I ever be able to apply this?" Readers of this blog know that trig is very important in mathematics. And in the physical sciences, engineering, game development and even in financial engineering, trig skills are essential. Still, at this very moment, millions of children are struggling with trig because trig is =difficult= (*). As a result however they start asking questions like I used to ask 'why do I have to learn this?'.

Do you ever speak about the topic you have to study at the moment ( besides that you don't get it )? If you do, then it has mass for you, and lack of mass is not the problem. So if you have a study issues at all, they occur further down the line. But if you think your issues are related to the topic itself then the following things may be of interest.

Looking at your topic from a different perspective might help, read books written for the general public, watch documentaries. You need to find a 'click' with your subject. So that finding out things about it comes natural and not as a struggle just to collect some tma points.

Imagine a student fairly good in math but with little affection to abstract algebra. A typical reaction might be, that he/she likes 'applied math', or likes to 'calculate stuff', or the notorious:

'I skip the proofs.'.

Most scientists see that numbers are everywhere, but it takes mathematical maturity to see that numbers are merely properties of mathematical objects at a ( much ) deeper level. My point being that before the student can get over that reluctance to study abstract algebra he has to know that it is literally everywhere.

Talking to people who love your subject helps a lot. And thanks to the Internet you can always find them. You can always decide later that studying your topic is not for you anyway. It is important to recognize that you are not to blame for not 'getting it'. But it is not the subject fault's either. Studying a subject only works if you like it. Is that all there is? I am afraid so. Its like communicating with a person over Internet that you have never met before. Its easy to give up on them.

So:
#1. There is no such thing as 'not getting it'.
#2. Get to know your topic first.
#3. Studying comes natural if you ( start to ) like your topic.

More later.

(*) A more important question is if trigonometry is really difficult or just made difficult.

Fixing study problems (1)

#openuniversity#

Studying, esspecially mathematics, is a consecutive series of a-ha moments, of I-don't-get-its I do get

Studying, especially mathematics, can be hard, very hard at times. There are moments when you -don't get it-. Studying is, as I see it, continuously working your way out situations like that. When you browse through a new math book everything looks totally uncomprehensible but when you put the book away, when the TMA is done or when the exam is done, you -own- the topic. At least that's 'the plan'. Here are a few tips for if the plan doesn't work.

Red flags

The only problem a student can have is -not getting- the subject. Red flags to watch out for are:
- you are putting off study time;
- studying doesn't seem as interesting as it used to be;
- when you study your mind wanders off;l
- concentration periods decrease ( you have to check facebook, e-mail );
- doing assignments is hard or impossible;
- after you read some text you forgot what you were reading;
- the list is endless.

There are four possible causes of why you don't get it which can be handled quite easy providing you are able to determine what is blocking your study progress.
- 1. Distraction. You are NOT working in a distraction free study environment.
- 2. 'Lack of mass'. A technical term for the situation where you have no connection at all with your topic. Your brain literally stopped. It can't process any information about the topic.
- 3. You know exactly what you are trying to learn. You think that you have mastered the topic but when you try to do some of the ( harder ) exercises you are lost.
- 4. The learning went well, exercises went well but the result on the exams were below your expectations or when you confront your understanding of the topic to one of your peers your comm isn't understood or is invalidated by your peer.

Distraction.

We can't switch off internet while we are studying since it has become indispensable for students and knowledge workers so we have to control it. Even self-discipline can be mastered but you have to take the first step. We are all possessed by litle demons that need to be fed with entertainment and pleasure. If you feed them they'll ask for more next time. They are never satisfied.

( ... More later ... )

Unexpected behavior of C2mm symmetry.

#openuniversity# #m336# #symmetry#

I read that a US Senate Commission is worried about the F-35 project because they are building planes while the testing of the F-35 is in full progress. The thing is that the Joint Strike Fighter ( F-35 ) is in fact a flying super-computer running on state-of-the-art software. Sound engineering principles don't apply to these machines. Software is never 'done'. Software is always in development and in testing and ( just ) released at the same time. There will be upgrades for the F-35 until the end of its life. Politicians think ( or say they think ) that when the plane is done, its done.

This thought crossed my mind because planning, by definition, implies uncertainty about the future. Unexpected things can happen, will happen, at a moment when its least expected.

To the point.

Mathematics is unpredictable too. From time to time you'll see unexpected things. Among various other topics I am studying plane symmetries at the moment. I have several books well illustrated with all sorts of patterns that can occur. For me, programming is an effective way to study, so I wrote a program that plots patterns using the symmetries I am studying. One of these symmetries is C2mm which is basically rotating a diamond lattice 90, 180, 270 and 360 degrees. While I was testing the C2mm symmetry in Graphica ( the name of my symmetry program ) I noticed that the patterns are very sensitive to the center of rotation.

I made a video ( of only part of the screen for size and performance reasons ). The second half of the video shows several unexpected patterns while changing the center of rotation. Watch and you may experience the same awe that I felt. All I expected was that the symmetry could generate a diamond lattice from a triangle.