Viewing Category: Learning
I've been slugging it out with C# and Managed DirectX for the past couple of months, and haven't made as much progress as I'd like. My conclusion: I need people to jam with to develop momentum. I am pretty much building my development process up from scratch, as it's been years since I've done this full time, and I miss having other people to talk to about technology, game design, and interactive ass-kicking.
There possibly is a group already meeting that does this; I should check out the long-running Boston Post Mortem again, our local area game developer social group to see what's going on. I figure there's got to be a few people in Southern New Hampshire / Metro Boston North--and it only really takes a few--who are motivated and skilled enough to tackle some game development topics and develop best practices. The ideal candidate already knows one of the following so they can bring some skill to the table:
- C# / Microsoft Programming Environments
- Game Development, especially tool chains and workflow
- Object Oriented Programming
- DirectX 9 / Win32 Programming
I used to work in the game industry, so I know something of the second topic though it's pretty out of date. I'm a decent-though-workmanlike programmer, something of a architectural purist that's learned to make concessions to just getting things done. I love documenting APIs for some reason. And I dig algorithms, version control tools, debugging, and Joel on Software.
If you're interested in forming a local study group, let me know and let's get together. If long distance is workable through online collaboration tools, then we can try that too. This is related to my main project for 2008 to develop a large-scale museum exhibit, so I will be working on this full time. I'm not out to create the next cutting-edge graphics demo; I just want to have a decent architecture built on some good tools.
Drop me an email via my contact form or leave a comment here. To maintain high signal-to-noise, I will classify interested parties as either a contributor or a subscriber:
Contributors are developers with experience in one of those three areas I mentioned, and are genuinely interested in understanding the technology to apply it to their active projects. We expect results from each other, in other words, though we are all working on our own projects.
Subscribers are people who are interested in what we're doing, but may not have the requisite time or experience to contribute knowledge and code. Nevertheless, they want to know what's possible, and maybe learn something about how they could start doing what we're doing. 3D and 2D Artists, interactive developers who use Flash/scripted environments, museum exhibit designers, advertising technologists, and experience designers would probably fall into this category. The input from this group of people is a necessary part of developing anything that has life in it, and communicating how this stuff is technically achieved will provide valuable insight. Or so I hope.
Of course, this presumes that people are actually reading this and care :-) At least I tried :-)
One of my colleagues out in San Francisco recently told me me about the One Laptop Per Child organization's program Give 1, Get 1, in which North Americans can donate $399 to the cause. This donation will pay for one XO Laptop--what used to be known as the $100 laptop--for a child in a developing country. Additionally, you will also receive your own XO. $200 of your donation is tax-deductible. You have until November 26 to participate!
I saw the laptop at SXSW earlier this year, and at the time I thought it was cool, but didn't know all that much about it. I'm not sure if this is a production model or not (I saw it at the "Is The Book Dead?" panel). The official specs look awesome from the design and software philosophy side of things. It's small. It's rugged. It's cheap. The software is open source, and it's designed for mesh-networking in both the technological and social sense. It's a fresh piece of design, one that really appeals to me in that it puts computing in the hands of people outside of the office. It reminds me also of one of my favorite books ever, The Diamond Age, a story about a disadvantaged girl that uses a "Primer" (essentially, a very powerful interactive computer in the form of a book) to become Awesome Individuals. The OLPC is, in my world view, one of the first steps toward creating a hardware platform that can accomplish something along these lines. Oh, the machine is "underpowered" by even 5-year old benchmarks, but what it lacks in raw CPU is portability, durability, extremely low power, and outdoor usability even in bright sunlight. That's where the money went, which is eminently more useful give its intended application. This is a machine that, I hope, that can be used more like tool in the hand. By contrast, most of our creative computer-based tasking is related to correlating, compiling, and assembling bits of data into a finished product; we are not really using the computer as a tool as much as the computer is using us to steer the creative process. We shall see what happens. I would like to try using one of these for a few months to see how it changes my working relationship with data and communication.
You can read more about the OLPC and XO on the official website. Check it out!
As technology gets newer and I get older, learning new things becomes frustrating. For example, I want to learn how to work with MySQL for web development, program 3D games, and play the guitar, but my lack of ability in these areas prevents me from achieving my overall life goals. There's also day-to-day stuff that, as a 40-year old American male, I feel I should know: balance my household finances, invest in the markets, ride horses, and flirt with women without throwing up. These are tasks that are, from my perspective, hard to learn for several reasons: a lack of good mentors, reference materials, and classes. And that's even without mentioning the magnetic properties of my ass with respect to my couch.
When I overcome these obstacles, I still hit the proverbial brick wall; for whatever reason, my brain can't quite deal with the important task of learning before getting bored or sleepy, and I end up going to get a sandwich instead or watching Age of Love on TV.
It's easy to presume, as I join the ranks of the Newly Old, that my mind is becoming less flexible. This is the common wisdom; for example, people say that it's tough to learn languages when you're older, and that we should have done it when our minds were most facile: around the age of 6, I think. Although I don't have any studies to back me up, I'm pretty sure that other factors are greater contributors:
We're self-conscious about not being competent in front of other adults, so we iterate less and thus learn more slowly.
We're not particularly motivated, given that mass media tells us it's supposed to be easy. When it's not, we give up.
Since we're all grown up and have our own money, we expect to be able to buy knowledge and expertise readily. It's really amazing just what you can buy, and we've grown to expect the easy access or we get real mad. It doesn't help that our advertising, at least here in the USA, tends to emphasize the quick and easy fix. We expect instant gratification, and thus we've forgotten how hard it was to learn our first lessons, and we've also perhaps forgotten how to learn for ourselves. I wonder if kids these days even know what it's like to have to wait for anything.
A LESSON FROM THE PAST
Yesterday morning I was doing my morning coffee thing, glumly looking at all the things I wanted to do that I was unable to follow through with due to a lack of understanding. One of the main ones has been transitioning my blog to Expression Engine, which I think will allow me to more easily expand the content offerings on my website while improving overall service. I had met up with Mark J. Reeves recently for lunch recently to catch up, and asked him about the possibility of writing a web service that would save data from my Flash Apps and integrate with the Expression Engine user management system. Mark, who's a competent execution-oriented web developer, told me exactly what I needed to do: write some SQL queries to access the pertinent database tables, maybe even repurpose the underlying blog engine to store data for me in custom fields. The problem: I don't know SQL, or what tools to use, or even how to talk to the database. I am paralyzed by not knowing what the best practices are, haunted by issues of scalability and security, and most importantly of all: I was not looking forward to learning all that stuff. I could not readily apprehend the structure of the material, and therefore I could not approach it logically.
Reflecting on this experience, I found myself reminiscing about my youth, when I first started learning about computers. Today, computers don't scare me at all, and it's because I have experienced them nearly to the transistor level of operation. As a result, I can look at a computer system and "read its aura" to figure out what's really going on. That is now, but when I was in the 7th grade, computers were as mysterious to me as, um, MySQL is to me right now. I vowed that I would master the computer and learn all its secrets. Somehow.
And so I started my notebook:
Some historical notes first: this actually isn't the original notebook: it's a manually-transcribed copy. The original copy went to a kid named Derek Bumpas, which I handed to him just before I graduated; he had a good attitude and was eager to learn even though he was in the 8th or 9th grade. He recently contacted me, some 20 years later, to say that getting that notebook had meant a great deal to him, and had helped put him on his path in computer science. That was really nice to hear.
When I first started taking these notes---recall that there was no Internet, hard disks, or multi-window multi-tasking operating systems---paper was the only way to simultaneously take notes and learn. Every nugget of wisdom gleaned from hours of tinkering was transcribed, as cleanly as I could, into this notebook so I could share information with friends at school. There were all kinds of things in the book, all of them interesting to me. It was, in essence, my spellbook. Here's some of the entries:
INCANTATION
An Applesoft BASIC routine to reformat a long string so it would display nicely on a 40-character wide text display; in today's desktop publishing terms, it does "ragged-right justification" for a monospaced font. This was a common task that I had to solve in my various text-based programs, as I was pretty obsessed back then with things looking right. I finally wrote it down...my first incantation.
TRANSMUTATION
As a high school kid without any money, we often "had" to copy software. The difference was that software back then came on 5.25" floppy disks that were copy-protected using peculiar algorithms; it was a fun challenge to try to figure out exactly how to elegantly disable them by rewriting the program code. This was my real education in computer software debugging. The code listing shown here is written in 6502 assembly language, revealing the method behind the protection. By understanding the principle behind the interaction with the disk hardware's imperfections and the software code that exploited them, a copy-protected disk could be transmuted into one that was easily-copied with everyday copy utilities.
SORCERY
As I started to understand assembly language, I learned how mapping the interface between code and hardware (the "input/output", or I/O routines) allowed one to zero-in on the game logic itself. For example, say I wanted to be able to change a shooting game so I had "unlimited bullets". By looking for the specific code that read the joystick button state (e.g.: is it pressed?) I could easily find the code that was responsible for checking how much ammunition was left. And once you decode one piece of code, you can infer the purpose of surrounding code. I was able to modify the game Beyond Castle Wolfenstein (the original Apple II one from 1984) to give me a 30-round submachine gun with burst capability, and rewrote the opening story to explain why you had one in the first place. This changed the nature of the game quite dramatically. By documenting the logic behind the software and noting the location of critical routines, the granting of unnatural abilities within the game world became possible.
ENCHANTING
I attended an American high school in Taiwan, and software was difficult to find for teaching purposes. Taiwan being a rather gray area in terms of copyright, my science professors would sometimes enlist my help to help them make backups of the US-sourced software; the humidity and mold in Taiwan tended to eat disks very quickly in the sub-tropical environment. Most of the time, educational software was protected with fairly straight-forward techniques using off-the-shelf protection systems. Because they were rather generic, the same general process could be used to remove the locks with just a few steps. In this photo, I wrote down the process of disenchanting this particularly piece of software, which required the use of a specialized instrument called "Advanced DeMuffin".
THE PHILOSOPHER'S STONE
And then there were the great unsolved challenges against which I beat my head. Electronic Arts had a very advanced protection routine that was designed specifically to defeat the casual copy breaker; you needed special hardware installed in your computer to even get at the code, and then you needed to understand it. I spend many hours trying to understand just how Electronic Arts' incredibly fast boot system worked, and once I understood that I tried to trace how they were doing the copy protection. Smarter crackers in the U.S. had already done it, but it was beyond my abilities and knowledge to follow. Here's a fragment of my research on the area...to me, to be able to understand this code would be like transmuting lead into gold. I should search online to see if any back issues of Computist describe this. I still kind of want to know how it worked.
THE MODERN SPELL BOOK
It's been years since I've kept any kind of notebook like this, with the exception of my patentable ideas journal. There's so much material out there now that the task of learning is equated with finding resources: the right teacher, book, or online tutorial is perceived as the "magic bullet" that will get things done. However, what I have forgotten is that the process of distilling these ideas into a form that I can invoke at will is necessary as well. It's my missing link.
I went out and bought some of the larger Quadrille-ruled Moleskine Cahiers ($14.95 for 3), and pasted a paper label on the front of it (the picture is at the top of this post). The idea is to start recording the same kind of notes that I used to do in the 7th through 12th grade; looking back, it was a highly productive period of time for me, though I didn't recognize it then. I'm thinking of just writing down really basic things that are currently mystifying, by hand, for reference in this book. I know there are plenty of reference books and online sources that purport to do this already, but do you think any aspiring wizard would buy their spellbook off-the-shelf? NO WAY! They would be told by their cantankerous mentors to go find a sturdy book and pen, and transcribe their spells themselves by hand. Because that's the way you learn, and that's the way you bind the magic to yourself.
WHAT WAS OLD IS NEW AGAIN
Ok, you may have figured out that this whole "spellbook" thing is just an amusing way for me to start learning again. The main takeaway is this: by assembling my own book of "recipes" that actually make something, I'm much more likely to maintain some kind of focus on learning. In the past, what I've done is just read everything and picked out the main principles as they've revealed themselves. What I have forgotten is that transcribing the nuggets is just as useful. I think I probably forgot this because it's so easy to just let the actual implementation replace documentation: Photoshop files, HTML, javascript libraries, etc. I don't think this is a good foundation, because you can cut 'n paste your way very quickly into the structural equivalent to spaghetti code.
Packaging the information into nuggets as I learn, which I used to do when I was younger, may be the way for me to approach the new technologies that are making my head hurt now. As an adult I had expected things to get easier, but really they are just as hard. Fortunately, I now remember how I worked through the challenge.
We'll see how it goes. In the meantime, I'm just pleased with the way my new SQL notebook looks :-)
A few days ago I wrote about reacquainting myself with mathematics. What started out as a mild recollection of past educational experience ended up raising negative feelings that flapped around my consciousness like angry ghosts. A buddy pointed this out, and after some reflection I laid them back to rest; there's no sense in remaining angry about the past.
While Math had been a source of frustration for me, it wasn't due to the math itself; it was the clouds of confusion surrounding the practice of Mathematics that so confounded me. This was exacerbated by my insistence on understanding something before doing it. Now that I know that there's something good to be said for "learning to recognize and reproduce patterns" before understanding something, I am a lot more interested to tackling mathematics once again.
So here is how I'm seeing the challenge:
- I like the mysterious qualities of Math, and want to be more fluent with the ideas and concepts behind that.
- To do that, I'll need to reconstruct the knowledge for myself, so I can have a clear picture of the relevant facts, principles, and standards of practice.
- However, there are barriers to reconstruction inherent in the presentation of the material: spotty instruction, missing/misunderstood key insights, ambiguous writing in textbooks, and misleading illustrations and diagrams. I groused about this at some length in my earlier post.
Turning the Tables
When I was a kid, I hated it when the facts weren't clearly presented. Today, I have much greater insight into how I learn, and am excited about trying to fix this perceived deficiency in my educational background. In fact, I could make this into a role playing game by thinking of education as a crime scene, where terrible wrongs have been committed against learning! The crime scene is a confusing, muddled puzzle: ambigious wording in textbooks obscure the true relationship between the facts, if they are indeed correct and unbiased to begin with. Key steps are mysteriously withheld, or buried somewhere in the back of the book. Upon cross-examination, key witnesses who spoke with confidence on the scene crumple in the face of sterner questioning.
Yeah, I'm totally going to be a Math Scene Investigator! I already am thinking of the cool notebook I'll get for it!
The Math Scene Investigation Process
Concidentally, I had stumbled upon the Taylor Series Expansion entry on Wikipedia a few days ago. I remember the Taylor Series as being a particularly odious waste of my time in high school, embodying all the qualities I didn't like about math:
- It expanded simple functions into VERY LARGE ONES...seemed like a net loss to me at the time.
- It introduced notation that was tedious to write out when showing your work, though I actually did like drawing those ziggy E things a lot.
- I had no idea really what it was good for, and because at the time I thought math was supposed to be about understanding rather than replicating process, I was very frustrated. The beauty of math was unknown to me.
The Taylor series is a good candidate for the Math Scene Investigative Process, which is proposed below!
Divine the modus operandi of the Mathematics
This is right out of the crime scene reconstruction article I was reading. I pulled up the Wikipedia entry and let it flap its gums for 8 paragraphs of irrelevant detail; they were descriptive facts, but not helpful at all in understanding WHY the Taylor Series even had a place in Mathematics to begin with; I really wanted the big picture first. If I don't know what something is for, how can I evaluate it? Instead, the article told me "what" it was, in terms of mathematical detail. That doesn't really help.
Around paragraph 9, the article came clean: by breaking down a difficult-to-transform function into simpler components, one can actually transform things more easily. That is pretty cool, just not in the context of high school mathematics; for me, it was just another one of those lame exercises you need to do that cramps up your hand and uses up a lot of paper. It's only NOW that I can understand that there's actually some kinds of interesting analysis buried in there; the harmonic analysis, for example, is what's behind all those cool MP3 Player spectrum analyzer displays and other cool digital signal processing tricks. But I digress...I have a little bit of the story now: The Taylor Series has the power to break down tougher calculations into easier ones, a special kind of mathematical solvent. And there are all kinds of interesting side properties that opened up entire new fields of analysis. Pretty serious stuff. Time to move on.
Reconstruct the Facts and Events of the Mathematics
Apparently, I had stumbled in the middle of a pretty serious piece of mathematics; a kind of universal solvent that has far-reaching implications throughout the field, with fingers in modern computer consumer technologies. I would have to step carefully; this was no two-bit axiom I was dealing with.
The rest of the Wikipedia entry turned out to be pretty dense. If the first 8 paragraphs were hard going, paragraph 10 and beyond were incomprehensible, filled with self-referential assumptions and inside jargon. It was a jigsaw puzzle that I would have to unravel piece by piece, reconstructing more than just the Taylor Series knowledge, but its relation to other fundamental players like Power Series and who knew what else. The material certainly wasn't written to explain, in nice bite-sized chunks, what the heck was going on to a newbie MSI like me. Still, it's my job to wade in and extract the real story. If there was a tome called "A History and Timeline of Mathematical Insight and Philosophy", that would help a LOT. Instead, I would have to do things the hard way to figure out what the math was supposed to be telling me. That meant tracking down and explaining every piece of detail on the page, from the funny notation to the significance of every fact. Some of it, I imagined, would be irrelevant in the big picture, but even the tiniest shred of evidence might shed light on another mathematical principle.
This would take some time. I decided to move on to something else.
Interview the witnesses and get their testimony.
Who are the witnesses? People who actually use or teach the mathematics. By talking to them, I'll gain a better picture of what's happening in the world of the Taylor Series, and probably mathematics in general. That perspective will allow me to put together more pieces of the puzzle and form working hypotheses as I build my case.
It's a known problem that witnesses are notoriously unreliable in their observations, and are subject to biasing influences that make their testimony subject to unintentional falsification. A seemingly-confident witness at the scene may crumble in the face of sterner questioning; I've seen this happen over and over. It's important, though, not to hold that against them: people generally mean well, and their testimony and experience provides important clues in understanding just what happened at the crime scene. If anything, you'll be getting new ideas from those witnesses, so treat them nicely.
Still, I have to remember that their testimony, no matter how well-intentioned, may have some flaws or misinformation in it. It's my job to piece together a story that makes sense to me, is supported by the facts as they have come out, and explain the modus operandi of the mathematics. And...it's all got to convince a real math teacher that I have a strong case.
Another challenge is that it's very hard to explain things, especially to someone who isn't familiar with your work and the context in which you perform it. Teachers will have an advantage, as an "expert witnesses", but even their testimony may be inaccessible or flawed. Like I said, it's tough to keep all those details straight. I've got my job cut out for me.
Put it all together
This is one of the best parts of Monk, when he gets to explain how everything fits together despite his debilitating neurosis. Um, I'm not quite there yet with this Taylor Series thing...the investigation is just beginning!
Wrapping Up for Now
Essentially, I'm thinking of approaching Math without the skepticism I've had in the past. Now that I'm older, I'm confident that there is some logic underneath all the obfuscating jargon and material; I just need to treat it like evidence at a crime scene. In the past, I had assumed that everything printed in a book or came out of teacher's mouth was guaranteed to be 100% accurate and true, and if I could not literally understand it there was probably something wrong with me. I know now that understanding comes in many forms; by taking a more investigative approach to with the assumption that the facts are not out in the open, I may be able to make some additional strides.
There are a couple of important resources I have now that I didn't have then:
The Mathematics Wikipedia Entry, from which I can skim the world of mathematics to build my own "big picture" view of how it's all related.
Access to Mathematic Frameworks for all 50 states, a byproduct of the trend toward National and State Standards in Curriculum. For the first time, I've been able to discovery just what we're supposed to be learning, and why. That's important context to have, representing a "pragmatic view" of math education in this country.
It would be interesting to look at a real college-level mathematics curriculum, for people who major in it. I'd be curious to find out how it's different. I guess I need to track down some real mathematicians. Anyone out there?
I've been thinking a lot about education lately, as part of my general interest in how people learn to empower themselves. Also, my love-hate relationship with Mathematics has recently been rekindled.
I've taken my fair share of math, but nothing very theoretical. I was in the more advanced math courses in high school, took Calculus / Differential Equations / Linear Algebra in college as part of the engineering requirement, and was exposed to numerical methods in grad school. Throughout the entire experience, I felt like I was missing some point and lacked an intuitive feel for the material...it bothered me. On the other hand, my favorite math class was one taught by an actual Math Department professor, who would get up in front of the class and excitedly derive all the steps for solving a particular type of differential equation. I would get caught up in the excitement of watching the hunt unfold before my eyes, but would invariably bomb the exam when I would try to reproduce his steps. As it turned out, what everyone else was writing down was just the end result: "if the equation matches form A, apply pattern A."
Oh.
A Failure to Understand
What I liked about that math class was the passion of the math professor. Looking back, I think that sense of vicarious excitement obscured a critical realization: the rules of communication still apply to technical and scientific education. Communication, which I am defining as a knowledge transfer between two or more individuals, is difficult when the contextual basis for understanding is unknown by one of the parties. For example, when the business guy talks to the web developer guy about improving some aspect of the company web site, confusion reigns if they do not have a common vocabulary. Same if their motivations are different. The business guy needs to learn to speak Webbish and the web guy needs to learn BusinessSpeak if there's any hope of understanding what the other really means and what their needs actually are, with emphasis on understanding.
I was recently working through a discussion of the P vs NP problem in Computer Science. At first glance, it meant absolutely nothing to me, but because a friend was working with it I was more determined than usual to cross my gulf of understanding. The gist of the P vs NP problem is, I think, whether certain computational problems thought to be "solvable" only through brute-force computational methods are actually solveable through some yet-do-be-discovered shortcut; should such a shortcut exist and be proven, it would have tremendous ramifications for the world of mathematics and computer security. I'm fairly confident that I'm not stating it completely because there is a LOT of background material that I am not very familiar or equipped to understand. I'm not confident about making an accurate statement, and this bothers me.
To have confidence in my understanding of P vs NP, I would need to have a deeper foundation in the underlying mathematical concepts of computer science, down at the hoary theoretical level: turing machines, for example, which is are theoretical underpinning of the von Neumann architecture that I studied, briefly, as a bright-eyed young computer engineering student.
My understanding of Turing machines is not very mathematical. I know that turing machines are named for Alan Turing, the "father of Computer Science", so-called because he came up with the idea of "computable numbers". That is, if a machine could write numbers in some fashion and then read them back to perform simple operations with them, that would unlock the entire range of mathematical operations that any person could conceivably do. People had already been engineering machines that could just "read" numbers, fixed into the mechanics of the design (all pre-computer machines are "read-only" in this way). However, my understanding starts to falter because I have not read Turing's original 1936 paper; I just peeked at it and my attention span bounced off it like hot oil on teflon. I just happen to know the story behind it, and that's been enough for me...until now.
Mathematics in Story
One of my favorite childhood books was Alvin's Secret Code, which introduced me to the idea of codes, ciphers and the stories around them. It also taught me how to be aware of codes in everyday use; there's one part in the book where Alvin recognizes that the tags on retail items encode the model number and year of manufacture, and he uses this knowledge to catch a salesperson in an outright lie. Yes! This book probably is what formed the basis of my interest in computer programming, 3-4 years prior to me actually finding out what a computer was, by establishing a framework of assumptions. The assumption might be stated as follows:
You can encode things to mean other things, it's just a matter of knowing what the code is and what the intention is behind it.
It occurs to me that having this imprinted on me in the 4th or 5th grade may have been absolutely critical to my development as a person. I have also just discovered that there was MORE THAN ONE ALVIN BOOK, and that THESE BOOKS ARE NOW WITHIN MY GRASP. When I was a kid, I was overseas and our school only had the one copy. But I'm starting to digress...back to Turing!
Another one of my favorite books is The Diamond Age by Neal Stephenson. Toward the end of the book the protagonist, Nell, learns about Turing machines in the context of an interactive learning simulation that she has bonded emotionally with. The Turing machine is represented as a giant mechanical puzzle castle in which she's trapped, with all the denizens controlled by links of special chains. By learning how to alter the chains, Nell learns how to manipulate them to eventually control the entire castle. In the process of playing through the story, she has learned how to understand Turing machines and computer programming at the most fundamental level, eventually achieving a level of mastery that serves her later in life.
The most recent book I've read has raised the Ghost of Turing yet again: PopCo by Scarlett Thomas. It's about Alice, a 20-something toy designer, who grew up with her mathematician/cryptographer grandparents. This gives her a somewhat unusual perspective on life because she's grown up with mathematics and mind puzzles; her idea of quality time is helping her grandfather factor tables of prime numbers. While this book doesn't talk much about Turing specifically except that Alice's grandmother knew him, it did rekindle my interest in the mathematics from the cryptography/story side of things, much as Alvin's Secret Code did when I was 12 years old. There's a lot of math in PopCo too, and apparently the Math community finds it mostly accurate. As I read, I was struck by the sheer number of interestingly-named concepts in Math, like Godel's Incompletion Theorem and The Travelling Salesman Problem. It had been a while since I'd considered the rich history of thinking in Mathematics, and I found myself wistfully thinking that I had missed out. I had only really experienced the memorize or be punished side of Math, and you know what? That sucks!
Careful readers may have noticed that it tends to take two observations for me to jump to a conclusion. When three form a chain of reflection, that's when I am moved to action. It comes down to this: I don't want to be in the dark about Math anymore. It's time to wade back in and see what I can do about it.
What's in the Way of Learning?
In grad school, there was this one guy who had a very strong mathematics foundation, and he could run rings around the other students in the class when it came to naming theorems. He and the professor would fence mathematically over the theory while the rest of us watched with sullen fascination. This was a class I did particularly poorly in, feeling the familiar sense that I really didn't know what was going on at an intuitive level. I'd characterize the feeling as follows:
I didn't have the sense of connection between the idea being expressed and the desired result. In other words, I don't know why this is important, other than the professor says so. Even when I asked, the answer was expressed in a way that I couldn't connect with.
I didn't have a strong theoretical foundation that draws from a layered understanding of the basic theoretical elements of mathematics. Therefore, my footing was unsure, and the few connections I did have were tenuous at best.
I therefore concluded that I was screwed. :-)
At the same time, this was actually a rather enjoyable class because I learned some interesting ideas about numerical methods and finding solutions computationally. However, I couldn't really navigate the landscape because of my lack of solid foundation. I postulate the following barriers existed for me:
Jargon. There's a lot of it. Special terms mean special things, each backed by fundamental concepts that forms the basis for further argument. Without that basis, it's difficult to understand anything that's going on. In more technically-oriented groups, the jargon itself starts to resemble paper currency in that it's no longer backed by a "gold standard" of conceptual understanding; that is, you can actually get stuff done by just trading patterns back and forth, because they're assumed to work. The tendency is for a technician to think and communicate in terms of the symbols, not the ideas behind the symbols. Or, they can only teach in one way; they're just not used to thinking of other ways to communicate the idea.
The ramification is this: a technician can teach process very well, but not necessarily understanding; if I were to give myself a lot of credit, I may have been a victim of this type of teaching in my earlier years. Being realistic, I would have to admit that I just wasn't very interested either, except in those instances when I could see how the mathematical concept was relevant to something I could touch or see.
Concept. There are a lot of interesting concepts in Mathematics, though to this point I have only been able to appreciate them on the level of story. I like stories where someone has wrested an entirely new way of thinking out of the commonly-accepted view; effecting a change in perspective, so to speak. I also like the stories where someone grasps a critical distinction in an existing process, alchemically transmuting our understanding of a long-held truth into something profound. Seeing beneath the surface is one theme behind both these types of stories.
The challenge is learning all the various concepts. Since concepts build upon each other in a certain way for them to make sense, the concept chain needs to be built rather carefully. Even more important, the chaining insight needs to be provided and made very clear. I've lost count of the number of textbooks that buried that insight between 15 steps of otherwise-useless derivations. Why? Does no one see this except for me?
Acceptance. I have difficulty being interested in things I can not see and verify. Why am I so untrusting? It's because I have been burned numerous times by making face-value assumptions. The first time I remember is when my Dad had given me a disappearing coin box magic trick made out of wood. You could put a coin in it and it would seem to disappear after you closed and opened it. So enthralled with this marvel of deception that I brought it to school. As I was showing it to some classmates, the box became jammed and I worriedly expressed my concern. A fellow 2nd grader proclaimed, "Oh, I can fix it" with such authority and confidence that I unquestioningly handed it to him. He then raised one hand, and slammed the box with as much force as he could muster. It shattered into three pieces. A moment of shock ensued, after which he gathered himself and said, "well, I guess that didn't fix it" before beating a hasty retreat.
Even after this seminal experience, it took me a long time to realize that there is a world of difference between belief and knowledge when evaluating a person's claimed expertise. In terms of my Mathematics education, it meant that I reacted more to the the ability of the teacher to credibily explain the depths of the theory in a manner I found satisfactory, than his/her thorough knowledge in the process. I needed to know the why, otherwise I was stuck. I am just now starting to loosen up in this regard, because I've realized that conditioning and training without so much darn THINKING is incredibly useful.
Relevance. Any group of people will have the tendency to look inwards toward the group, not outside toward the rest of the world. The reason for this, I think, is that our attention is shaped and incentivized by what's going on right in front of us. In other words your attention will be on the group and its relevant issues because you can't get away from them...you're in the issues, and they naturally form the default frame of reference. This allows the group to speak with greater efficiency, but when it comes to passing on information to a dissimilar group, a gulf of understanding forms that is sometimes difficult to breach. This leads to apathy and disinterest.
I didn't see much of the relevance of the math when I was a kid; it was a world with which I could not form a common frame of reference, not with teachers nor with my life. It was just something I was forced to do. The textbooks would try, making feeble attempts to show "Math in the Real World" between chapters. Typically, they would retell the fable of some "famous" mathematician, using a somewhat book-reportish style that didn't appeal to my life as a plugged-in child of the 80s. Or they'd explain how if you knew math, you could be a rocket scientist! Neither approach was particularly compelling or real. Just to be sure of this, I recently went to a used book store and flipped through half a dozen math textbooks of various vintages. The ones from the 80s and 90s were particularly awful, which brings me to my next point.
Presentation. By this, I mean the visual design, the quality of writing, and the typography of our textbooks. When the math teacher failed to reach me through lecture, the textbook was my last hope. They have all been, in my experience, uniformly bad---I just didn't know it at the time. I just knew my head hurt and I was confused, and assumed it was MY FAULT.
It comes down to this: a lot of textbooks are poorly written, even when the author is an expert in their field. It's partly because of the observations I've just made above, and also due to a lack of familiarity with writing and presenting an accessible argument. An OK textbook should, at minimum, transcribe facts accurately in context with the processes that bind them. A GOOD textbook would present the reasoning behind those processes. The hypothetical GREAT TEXTBOOK would have the facts, processes, and reasoning presented in a fashion unencumbered by ambiguous grammar, lousy semiotics and eye-straining typography. The layout will take information design principles and semantics into account to support the logical structure of the material. AAAAND, as if I'm not asking for too much already, the book would be written to facilitate conversation between people. That's a very different focus from anything I've yet seen.
It's not the Material, It's the Presentation
So I've listed a bunch of personal hangups I've had with math-related textbooks, and I haven't even engaged the mathematics yet; all that other stuff is in the way! I suspect I am not alone.
I don't think it's a question of people understanding that MATH IS COOL. MATH IS INACCESSIBLE to the average person who isn't lucky enough to have a grandmother that cracked German and Japanese Enigma transcripts during WWII. METHOD of PRESENTATION that we have access to, on average, has been rather MEDIOCRE.
The same argument can be applied to other fields of course, not just Math.
It's Also the Attitude
Accessibility is an issue that sometimes comes up with Jazz (not that I really know anything about Jazz, but bear with me). Some people will argue that it takes a refined ear and a knowledge of the history of the art to truly appreciate the more esoteric material; I'm fine with that. What bothers me is when it this knowledge is used to divide people, raising the "elite" at the expense of the "non-elite". I don't have a problem, mind you, with being selective; after all, you can't expect to have a meaningful conversation about a specialized field with a non-specialist, or expect a non-specialist to do specialist work. My issue is when the elitist attitude leaks into the educational context, demoralizing students. Therefore, I think it's important for the material to deliberately project an empowering, inclusive vibe via Mathematics as a tool that can be mastered. This is different than just being "automatic encouraging" and showering kids with gold stars...kids know when they're being pandered to. What's far more important, I believe, is for kids to feel that they are being empowered and included by someone else that means something to them. It could just be their own self, driven to understand something that has tickled their fancy in some way. It could be through loving, intelligent parents, or that cool auntie, or the mysterious former spy in the house next door teaching you something about codes.
Extrapolating My Next Step
I've just vented a bunch of things that have been on my mind, with the intention of clarifying the following:
- What's wrong with me and Math?
- What can I do about it?
My immediate intention is to start reviewing Math from some arbitrary starting point, really taking a hard look at what is being said, and why. Ultimately, I would like to be able to understand the aforementioned P vs NP problem and be able to follow the mathematical logic behind it. In terms of impossibility, this is pretty much like John F. Kennedy pointing at the moon in 1961 and saying "Yeah, we're going to put a guy ON THAT, Nikita!"; it's a very deep rabbit hole, and it happens to be one of those Millenium Prize problems. I'd be happy to just know what the heck they're talking about, and it happens to tie in with many childhood interests that I have never been able to engage at a higher level. It would be great to make some progress there, and slay a few demons at the same time.
I've tried this in the past, actually, but this time I have a new angle: Treating the educational experience like a crime scene. I'll write a bit more about that sometime soon.
I've always been curious about how the mind works, and in theories of how things work in general, so when I found a whole book filled organized by cognitive concepts I was tempted to order it right then and there. I became aware of its existence in this article on Automaticity. Why was I looking up Automaticity? Because someone used the word in an online forum, and I didn't have a clue what it meant...
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