(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**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*modus operandi*of the Mathematics**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?

## 18 Comments

You’re missing the very comprehensive mathworld.com website on your list. That’s my first step when trying to find something math-related. It’s not always very clear, but if you read Wikipedia, then Mathworld, then maybe one of the books referenced… you’re usually able to get a relatively good grasp of what they’re trying to say.

I also like the first few chapters of Knuth’s “The Art of Computer Programming”, where he reviews most of the maths relevant to programming. Not everyone likes it, but it’s well-written and there’s lots of historical background in it.

——-

Hi Dave, very, very fun post. I like the crime solving analogy!

Have a question for you. I like and agree with 1,2, and 4: 1) modus operandi, 2) reconstruct facts, and 4) put it all together. I don’t quite understand what you mean by 3) interview the witnesses. That sounds intriguing, and I’d like to know more. Especially when it’s all written on paper, who can be the witnesses?

On the other hand, a professor I had, super guy!, Chris Peterson, has done a study where he went to the literature and read the biographies of all the winners of the Purple Heart – he was drawing some conclusions about their strengths from the descriptions in their bios. So, it is possible to decipher and learn about witnesses even from the written word. How do you plan to go about “interviewing the witnesses”? Thanks!

(p.s. also really liked the links in this post – about eye-witnesses, biased judgments – v. cool!)

Kena: Thanks for the link to mathworld!!! I wasn’t aware of that one. Knuth’s

The Art of Computer Programmerhas been on my wishlist for a long time…sigh, so many things to buy!Senia: Thanks for asking about #3! I think I lost the critical sentence during a cat-induced data loss episode yesterday as I was typing this up. The witnesses in my case are talking to actual mathematicians, teachers, people who use the math every day! So I would ask them using non-leading questions about the mathematics with my little notebook: “So, Mr. Math Guy, what is your relationship to the Taylor Series Expansion? Uh huh, uh huh, how do you spell that? I see. What are the steps in performing the expansion? Why use this at all?” I’ll go clarify that paragraph…thanks!

That’s very interesting about your prof who studied the literature and biographies of purple heart winners. I think you CAN tell a lot from the text, but it might be tougher in the case of textbooks since they may not be sourced by a single author. Do you have a citation for that study by Chris Peterson? Would love to read it!

Here is a real college-level mathematics curriculum and course materials:

http://ocw.mit.edu/OcwWeb/Mathematics/index.htm

You seem to need a helping hand in your investigation. Math is very easy because all we know how to do is add. Math is also very complicated because all we really know how to is add. Subtraction is the addition of a negative number. Multiplication is the addition of the same number several times. The ‘Taylor series’ takes something called a ‘transcendental function’ (which is essentially something that isn’t adding) and makes it an addition problem so that the answer can be computed. I’m not an expert, just a college student, so there are probably exceptions I haven’t heard of. You can blame the teachers but most people I’ve met with math phobias are their own worst impediment.

Lochlan: Heh, thanks for the assistance :-) That’s a great statement you wrote: “Math is easy because all we know how to do is add, and it’s also complicated because all we really know how to do is to add.” Awesome! It sounds like you’ve learned to filter out a lot of the surrounding detail and cut right to the chase…if I’d had teachers that explained the math like you just did, maybe I wouldn’t be so mad :-)

It’s not so much the teachers I blame as it is a difficulty to communicate the fundamental issues clearly and with useful supporting material. Since I have a graphic design and communications background now, I can see much more clearly that the problem is not a lack of expertise on either part, but a lack of awareness when it comes to writing a clear sentence in a useful context. This is actually very difficult, because clarity and useful context are often pretty different between any two people. Hence, the “math investigator” approach, in which one FOCUSES on creating one’s own clarity and context.

It might be interesting for you as a “visualization master”.

Some enthusiasts in Russia are conserned about declining interest to science in general and the math in particular from the side of school people. So they chose 3D visualisation of different math phenomena, trivia and simply amusing things as a way of attraction and to provide motivation to students.

Here at the top (sorry, the page is in Russian) you can find links to movie clip with different resolutions. The clip describes Reuleaux triangle. It does not have any sound or text (Russian =), and tries to demonstrate all relationships in motion.

If you find it interesting, in top left menu the first item leads to a small collection of such etudes.

Regards,

Pavel

PS Welcome to CS :^)

Hi Dave, thank for that explanation – that’s cool about interviewing people about their opinions about the math, and their take on it! I like that. Don’t have that Peterson reference – if I find it, I’ll let you know.

David,

your problems and skepticism with mathematics are quite natural. From an early age most people are taught most of their mathematics by people who not only lack understanding but also have a deep phobia of it. This no doubt filters down to students from day one. The other problem is that most of what is learned before second year university is at most the grammar of mathematics and not really what I would call “math”.

I am often amazed that anyone can do mathematics at all!

I really like your prposal—it is essentially the scientifc method as it pertains to much applied mathematics research.

Pavel: Thanks for that reference…checking them out! On a side note, I almost bought one of those reference sheets on russian grammar; I was thinking I wanted to do a gross comparison the grammatical structures of italian, spanish, french, german, english, and russian. Buying all of them, for a project that was likely to just go nowhere, was more money than I wanted to spend, though I was tempted to get the russian one because it was TWICE AS LONG as the others, and I was really curious to know what was taking up all that extra space.

Senia: Awesome! If you find it, maybe I’ll read about it also on your site…great going on that, btw!

Jif: That’s interesting, that applied mathematics research methodology is actually the same thing as what I was thinking from the “student investigator” end of things. Maybe there’s an educational process that could work here…I was thinking that it would be cool to make “Math Investigator” kits, maybe learning kits in general. OMG…it would be so cool. I will have to design that!

Incidentally, the investigative process as crime scene reconstruction is actually an adaptation of the scientific method. That link is really awesome; it happens to also incorporate a nice summary of logical fallacies, which I think should be taught more rigorously. I’d love to make some flashcards for that. Or maybe a card game.

Dave,

Russian is quite hard to master as almost all words change forms (conjugation and declension).

I speak French and have read about grammar specifics of German. Among all English seems to be the easiest one.

Mathematic is precise or it is nothing. The wikipedia entries are good: tight, concise, to the point.

The problem lies in the way mathematics is perceived by people without experience in mathematics. Most people are intensely frustrated by the fact that they cannot pick up a mathematic books, and simply “read.” No one can do that with mathemetics they have not already learned.

A good rule of thumb that I have found is that to read and understand 1 page, that is, a SINGLE page of mathematics at the frontier of my knowledge takes anywhere from 1 hour to 1 week of thoughtful deliberation of the material. If the material is too far out of bounds, there may be no hope for understanding without the prerequisite graduate level education.

It’s pointless to attempt to read and understand mathematics without an accompanying pen and paper for taking notes and exploring the behavior of the objects under study.

The Taylor series provides formal basis for probably 1/3 to 1/2 of computational mathematics, basically anything involving computation of a physical quantity is likely to a have one or more Taylor series methods derived for solution. It’s a critical piece of elementary mathematics.

Good luck.

Thanks for the insight, Dave! The question I imagine, then, is finding figuring out where in the frontier one happens to be, and which way to go when one finds that they are in over their heads. My impression, admittedly not based on a large sample of math books, is that the material is written to affirm what someone already knows, rather than to guide or to teach. This is just a hunch, though…I will need to acquire some math books and instruction material and start deconstructing it for myself to see if there’s some way I can make the material more accessible (and by that, I mean just better designed).

Me again.

More really cool stuff. I was stopped cold in graduate school by statistics. I needed the crime scene investigator approach but didn’t have the time (or insight) during a 14 week semester. Dropped out, thinking that I really needed several grad level courses, especially one in number theory to understand all this stuff about expectation and [insert technical name of those test types], etc. And what it was really trying to accomplish.

Math makes me mad too – all the really cool stuff on the other side of it (engineering in general, operations research, complex options pricing models) that I haven’t be able to get to for lack of it.

I’m going to have to add your feed to my read to keep up with your evolving writing. You’re doing too many cool things.

Hello David. I found this very interesting post of yours by searching (After coming to a very simple realization of relationships across the many planes of life, ie.. words in relation to math in relation to electricity in relation to engineering and so on.) using the Question, “Is Math the Ultimate Solvent?”. In addition, I allowed my mind, by “keeping it simple” or “reducing back to the simplest known truth”, to drift on the journey backwards towards the purest truths. I realized that certain words are mathmatical in meaning. For example; plus, also, and, additional, more, above, up, and positive, all stood out as the (+) sign in math. Wheras words like; less, take, minus, subtract, down, negative, all stood out as the (-) sign in math. Then I asked myself a very simple question, “What is Zero?” It is perhaps, “The ultimate even”, “The ultimate Solvent”, “The ultimate equalizer”, and so on. I totally followed everything you so elequently put into words. I do like the analogy of the “Crime Scene Investigator” type exploration. You are the “Detective”. Clues are “evidence”. Witnesses are those who can tell you certain things about certain pieces of evidence. You then are the “Zero” or obtainer of all the “evidence” and “witness testimonies” that can see it, as a puzzle slowly being put together piece by piece and therefore slowly start to see the “whole picture” forming before your eyes. Great Stuff and Hope you have fun in your Journey and hopefully you are putting every single step of your journey down on paper(wait that’s old school) or saving it in a file on your hard drive, hehe. Someday, hopefully, your final draft could be a book similar to “Understanding Math and all it’s involements around us for Dummies”. O.K. Maybe a shorter Title but you get my drift. I am going to have my 15 yr old daughter read this post that you wrote. She wants to be an “Electrical Engineer”. When I was talking to her about how all things are related to one another, she said, “Dad quit confusing me”. Dad then went on a rant about the simplicity of the “Engineering” of anything. She eventually said, “Goodnight Dad”. I laughed and just sighed, hoping theres a tomorrow to continue the conversation all the time realizing that all of tomorrow’s confusion will probably just cloud the memory of the initial conversation that reached today’s set of conclusions, so hopefully when I have her read this page, it will remind one of us to continue on where we left off. Thanks again for writing what you did. Later, Craig.

Sorry about this addition to my earlier response but I just thought of it and thought it to be important “evidence” for your “puzzle”. The matter of fact truths of math: (+-x/=0123456789) the + only performs one action, the – only performs one action, and so on. The = sign screams out to me as a “person who is trying to get two seemingly different set of circumstances on the same page by pointing out how like they really are when it’s all said and done”. Later, Craig.

Hi,

I´m not surprised you did not like maths. Try books that are more conceptual. Hans Magnus Enzenberger wrote a very funny one for 9 to 99. It is written for children, and I bought one for my children. I loved the book myself too. I once came across a site with the presentation of a conceptual method of teaching maths. What you’re describing is a typical top-down learner’s complaint.