(last updated on October 10, 2020)
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.
p>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?