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.
