Conceptual knowledge maps for mathematics

About working and studying, the main topic of this blog: well, it is no longer an option. It is mandatory, society needs people who are committed to their jobs and invest in their lifetime education. Everything changes at an ever faster pace and it is thus our responsibility to stay of value in all of our dynamics.  It is an issue we must align regularly.

Studying in a particular field will gradually align your life to your new interests. For example, I have recently, more or less -landed- in a job where some of my colleagues are mathematicians. In fact, a lot of the work I do involves mathematics. More about that in posts to come.

Everything changes. So do study skills. And a vast amount of tools for computer aided learning is now available. From MIT lectures via YouTube to software for flash cards to sophisticated knowledge databases for personal use. I would like to mention two items I have found recently.

InfoRapid Knowledge Builder is a graphical tool to document conceptual knowledge maps, free for personal use at http://www.buildyourmap.com/. ( Before this I used Mindjet and TheBrain. ) I am working on a map with topics in Analytical Number Theory.

Another item I found is a book. It is called On Course, Strategies for success in College and Life. ( And if you work and study read the title as Strategies for successful combining work and study.) The book is packed with information. It is not a book you should read once but it is a companion you can use, consult, for years. I have the Study Skills Plus edition.

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

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 ... )

Study Tip - 4 ( Plan, plan and plan )

It has been a while since my last study tip.

Learning style

I am not a type that can sit still for hours to read, think deep mathematics and make an occasional note in my virtual notepad in my memory. Although I would like to be like that it is not going to happen. They say Euler was like that. That's why he had no problem to continue with his mathematics after he became blind. Stephen Hawking also has this ability to do theoretical physics in his mind. In the Netherlands we have a former checkers World Champion, Ton Sijbrands, who can play and win or draw close to 30 games blind. It proves what the human brain is capable of, in principle. - I can't do it though because I have to be active, I must do something, better: create something. Otherwise I can literally doze off. A friend once said that I have a kinematic learning style. Different people have different learning styles. I can't sit in a lecture and listen to a lecturer either. It literally goes in one ear, leaves the other, without any storage or processing in between. I get extremely bored in lectures ( or courses ), days seem ages because I want to go home to -do- something.

Check-sheet

Over time I organized my style of studying into a series of to-do items. The first to do is always 'Make a detailed plan.' For example if I plan studying a booklet ( Open University study materials are delivered in booklets which often take two weeks to study, or something between 16 and 32 hours ) I create a check-sheet for it. ( If you have ever done a course in scientology, you know the concept. ) If you study from a check-sheet you won't have those blockages. They really work. The most important thing is that you create a plan that works for you. A plan helps you go through the materials and you know when to do what and no more. That way you won't fall in the 'I never have time-off trap'.

Statistic

Maybe you study to make a step on the career ladder, expect to make more money or maybe you simply want to be able to do your current job better. Or you study to get a(nother) job, or you finally want that degree. In any case you have a deep interest in the subject you study. Final study results come in after many years of study. They are hard to measure on a daily basis, even success on a module which usually takes a year is not a good measurement. TMA results? Yes, somewhat. But what if you got that 'good' result by cramming the TMA out in 48 hours of almost continuously cramming? You will have forgotten the materials on the exam. You have to set a study statistic you can measure on a daily basis. I make ( = do ) flash cards with the open source program Anki. For example when I study a definition, of a graph for example. I type in Anki Q: What is a graph. A. A graph G is a tuple [V,E] where V is a set of vertices. etc. Anki makes sure that I will never forget the definition by monitoring my recall on the question in a very efficient manner. But most of all, I am doing something while styding because I am alert and awake as to which questions I have to put in Anki. Naturally I try to formulate the answers in my own words. Anki then provides me a with a range of graphs and statistics about my progress.

Feedback

Have you ever made a plan which worked well for onze day but you threw away the next day because all the time unexpected things happen? I often made plans to study on times when I was not physically up to studying. I felt tired, spent too much time in traffic, got hold and so forth. That is very valuable information you can use when you make your next plan! And if you record the time spent on each topic your time-estimates will improve and improve.

That is what I mean by "plan, plan and plan".

Oh, I no longer think in time, my unit of doable work is the Pomodoro. More about pomos another time.

Heatwave : day off study.

Took a day off study today. Have been putting a lot of time in studying lately. Felt like work instead of fun. We have a mini-heatwave in The Netherlands. For the 1st of October it was the hottest day ever ( since recorded weather anyway ). I don't like summers, especially when they turn up in my favorite season autumn. I mean, I think everybody has been off-schedule today. - Tomorrow, I take a day off as well: I really missed working with Mathematica. I got really interested in the foundations of computer science lately. Will read about formal languages, grammars and parsers tomorrow. And of course will have a look at the Mathematica built Lisp interpreter. - I am working on a program myself, SceneGraphica, I need to spend time on that as well. I think I will start all over. Nothing will be lost though. I wouldn't have had the ideas I have now without the effort put in the early versions.

Study Tip - 3 ( mathematics does not have to be difficult )

Being a programmer who studies math (, a math studying programmer, whatever ) gives me access to the literature of the foundations of computer science. Now and then I browse the libraries. I thought to have a found a book that fits my level, so I read the preface to check if I was right. The author Alan Parkes made the following interesting statement.

For Students : I wrote this book partly because when I studied this material as part of my own Computing degree, I had to work really hard to understand the material, a situation which arose not because the material is too difficult, but because it was not well presented ...

from Preface of A Concise Introduction to Languages and Machines by Alan P. Parkes, Springer 2008.

I have seen video of a lecture on MSRI where the researcher / professor held 'a talk' ( there is a subtle difference between 'a talk', a presentation and a lecture, 'a talk' is a euphemism for a not well prepared presentation ) about a subject from scribbled notes. Most of the lecture he stood with his back to the audience mumbling while scribbling math. End of lecture. Goodbye. It was expected of the audience to write up notes from his mumblings and scribbles. - Folk like that, probably based on a brilliant thought they had decades ago, get the opportunity to write books too. Once in the hands of ( then ) students like Alan Parkes they make the learning experience troublesome to say the least.

Well written, illustrated mathematics followed by examples, more examples, examples of exercises, exercises with written solutions and so forth make studying mathematics easy. ( Think of an Open University booklet, a good one ). Open University booklets are written by course teams for students and not by professors who write to impress... who actually?

Tip 4: Change books

If you have difficulty with a ( mathematics ) subject then find another book about the subject. Check how the other author explains the topic. Try a third, fourth if necessary. This method is guaranteed to work. When you are close to revision for an exam it stimulates to read through the material written by someone else. Different exercises, and so on.

Finally: authors who write for students are read, respected and remembered.

Study Tip - 2

Do you know any professional musicians, dancers perhaps? History shows that art and mathematics thrive in the same places. I, sadly, don't. Although yesterday I witnessed a pianist's daily practice routine. It started with loosening up the muscles. Then, slowly, player and instrument become one sound generating machine. To me this explained why musicians can have such deep relationships with their instruments. The musicians we see on stage ( all of them, not just the 'stars' ) have practiced at least ten-thousand hours to reach that level. I don't think it's such a bad guess to say that, any mathematician who is at the forefront and is creating new mathematics, carries at least the same weight of practice hours on his belt as a professional in the arts or an athlete.

Tip 2: Exercise daily

Work on a difficult exercise every day. Find a booklet with Olympiad level exercises or any book with exercises that are a challenge for -you-. Exercises you can do won't make you better. Hard exercises do.

Creating new mathematics ( ... )

Suppose your assignment was to teach a group of friendly aliens, just arrived from the Pleiadians, the rules of the game of chess. A student has completed the course with success if he is able to play a game according to the rules. That's not too difficult you think considering the number of six year olds able to play a descent game of chess. Your study materials are: lots of chalk and a blackboard. No chess pieces, no boards are available in class. Your teaching assistant will type your lecture as you speak, therefore it is not allowed to use drawings or symbols that can't be typed instantly.

This may seem difficult ( it is ), but compare it with the creation of new mathematics (...)

Mathematics is a parallel universe which we can enter with our mind only. Although bodiless, exterior, we are free to travel in this spectacular universe. When we come back however we lack the words to describe our observations, to communicate what we have seen with our mental ( mathematical ) eyes. Each and every observation must be recorded and analyzed before we can attempt to describe it. Definition by definition we try to create a consistent picture of what we have 'seen'. In this notion of mathematics, for example the number e always existed, it just took an Euler to describe it properly. Obviously the mathematical world is not some parallel library where we can go to and lookup the answers to the current unsolved problems. - In a BBC documentary about Andrew Wiles and Fermat's last theorem, Wiles describes his research as entering some space, then by touching things by hand in the dark he had to form a mental picture of what's in that space, and so on.

Describe what you observe (...)

Study tip - 1

( For a while I have been thinking about writing -the- great (...) post listing zillions of study tips. Since it is not hard to imagine such a post will never be written I just start with tip 1 ( in random order ), then see how far I will get and maybe, one day, compile them into one post or page. )

Tip-1: The next item on the (study-)list

If you want to start studying immediately in the time you have allocated for study make sure you know -exactly- what you are going to do when you start. Don't lose time on deciding if you are going to read, revise, do exercises, work on assignments or whatever it is that you do for studying. The best time to plan a session is at the end of each study session. This already structures a session into study / plan next session. This plan can be as short as 'Do TMA questions 2 and 3'. Or 'Read pages 12-28'. Very quickly go through it and write your plan down in your agenda ( whatever system you use ). Visualize yourself starting the next session and starting with these tasks. - The trick is that your subconscious already starts working on it. Programs, prepares you for the task. Next session, starting the task will be easy and enjoyable. It works. Make a habit of it.

The Dunning-Kruger effect

We learn and learn, become more knowledgeable every day. Time passes, your skills improve. Then one day you'll have to face ( much ) younger colleages at work. For example one who thinks older people are trash no matter what.

The Dunning–Kruger effect is a cognitive bias in which unskilled people make poor decisions and reach erroneous conclusions, but their incompetence denies them the metacognitive ability to appreciate their mistakes. The unskilled therefore suffer from illusory superiority, rating their ability as above average, much higher than it actually is, while the highly skilled underrate their own abilities, suffering from illusory inferiority. Actual competence may weaken self-confidence, as competent individuals may falsely assume that others have an equivalent understanding. As Kruger and Dunning conclude, "the miscalibration of the incompetent stems from an error about the self, whereas the miscalibration of the highly competent stems from an error about others"

Years ago I thought ( had the illusion ) that I knew quite a lot about computer science, programming, even mathematics ( ... ). Today I seriously have the feeling I know almost nothing. I attributed that process to a sort of more = less principle. It seems to be a psychological law. Part of growing older, maturing, becoming more proficient in your field. - A side-effect is that it creates a division between younger and older workers. - Have you ever worked with a say 10 ( or more ) years younger colleague who thought he was a G_d in his field? ( They tend to call everything that is not coded by their golden genius fingers -trash- ). The tensions that a relation like that can create can be explained by the D-K effect, I suppose.

See this question at /* Programmers */ ( although closed as off-topic ).

Tips for Study Tech

I read two articles by Sato on the QED Insight blog.

How To Read A Mathematics Textbook
Posted on April 5, 2011 by Santo

and

“Students Don’t Read Textbooks”
Posted on April 8, 2011 by Santo


I have summarized the articles and added info where needed with the typical OU mathematics student in mind:

About reading:
* Find alternative books, which approach the subject from a different perspective than the course textbook.

* Read with pencil and paper by your side. By the end of each reading session you should have written out a nice list of questions to work on; If you can't answer the questions by yourself ask your tutor or visit a mathematics site like Planet Math, or The Math Forum

About exercises:
* Work through a large number of exercises. Top students when pressed for time, fall back on just doing the assigned exercises.
Some specialized books that you might consult for training in problem-solving are:
- The Art and Craft of Problem Solving, by Paul Zeitz;
- Problem-Solving and Selected Topics in Number Theory, by Michael Th. Rassias
- How to Solve Problems, by Wayne Wickelgren
- How to Solve Problems New Methods and Ideas, by Spyros Kalomitsines

About Study Tech:
* Daily work is key. Do your best to budget your time so that you devote some time every day to each of your courses.
* Be systematic. Keep a notebook of questions (might be electronic), and note your answers, too. This will become a wonderful record of the evolution of your understanding. Also keep solutions to your exercises and problems in a neat notebook. Review the notebooks often, ideally daily.

Always remember that a desperation for marks is counterproductive to learning.

Memorizing digits of PI - revisited

A quality shared by all great mathematicians of the past is that they did not try to make an impression with their knowledge but instead shared their knowledge to the benefit of all. Someone who keeps his knowledge to himself demotes knowledge to a set of tricks and himself to a mere magician.

I don't know the author of the comment below but in the tradition of the best, he wrote the following comment to the post 'How to remember 1000 digits of Pi' which I received earlier this week but only read today. Not often do I receive a comment that I promote to a full post. As you will see it deserves to be.

Remembering the first 1,000 digits of Pi is a very simple task. It sounds daunting at first, but I assure you it is much easier than you think. The method you are using only complicates the process. You need to use the Dominic Method with the Journey Method. The Dominic Method uses 100 characters, each representing a number and letter code.

For instance: 11 on my list is Andre Agassi; his action is playing tennis, naturally. To remember 4 digits at one time, you pair 1 character with the 2nd characters action. So for the first 4 digits of Pi, we have: 3.1415 which corresponds to: Andy Dick writing on a blackboard (1415). AD=14, AE=15. AE is Albert Einstein and his action is writing on a blackboard. So pairing the first character and giving him the second character's action gives you a sequence of 4 digits.

The NEXT step, after memorizing your 100 person list (which gives you 10,000 memory storage locations) is to put them on a Journey. Take a familiar Journey around your house, neighborhood, etc. For instance, start in your bathroom: You have Andy Dick writing on a blackboard IN YOUR BATHTUB. Then you have Norman Bates (92) playing with toys (65)IN YOUR TOILET...then at your sink....you get the idea.

If you want clarification on this or help with your 100 person list, let me know. People have used this technique to memorize 75,000 digits of Pi. I have a journey with 1,000. A simple 50 mile round-trip stretch of highway with landmarks to "Put" these characters on will suffice for 250 storage locations of 4 digits (1 person with 2nd action) at each, which equals 1,000 digits. Email me at: 1fastbmw328@gmail.com if you want more info/help.

The author clearly knows what he is talking about as he refers to the Dominic method and the Journey method, both seem well known methods in the realm of memorizing. ( I must admit that I was not aware of either of them. Another reason why I enjoy learning new stuff. ) It seems that the key is Memorizing the 100 Person List. That is do-able, I suppose.

Clearly, I will start learning more about the methods and start experimenting with them. More soon.

How are your basic math skills?

I don't know how the situation is elsewhere but in The Netherlands there has been a lot of commotion about the basic math skills of teachers in primary schools. Since then students have to do an extra exam in basic math skills. The root cause was that basic math skills were only taught at the primary school level. In secondary school they got dependent on their calculator.

Link: Basic Math Skills Quiz

Google Logic

I found an excellent summary of the Google Search Logic and tons of other study tips at the site of McGraw-Hill.

Link: Power Google

17. Be Competent.
17-1 Look,
17-2 Learn,
17-3 Practice.

The Way to Happiness, LRH