When I was a child, I always found math easy. Fractions, calculus, formulas—they were always visual. Fractions were pizzas; equations were deformations of the plane; integration was tracing out a shape using a slope.
So I'm sympathetic to the author's desire for better visualization and teaching tools.
But when I reached college, I became frustrated with math. It just wasn't easy anymore, the way programming was: I could pick up a programming book, read it in a weekend, and understand it. But when I tried to read an advanced math text, I became lost after 10 pages.
Eventually, I figured out what had happened: The information density of college-level math texts is insane. Even if you're bright and talented, it may take you a day to understand a single page. And there's no substitute for working carefully, finding concrete examples, and slowly building a deep understanding.
Here's an example that involves programming. Once upon a time, I needed to understand monads, in hope of finding a better way to represent Bayesian probabilities.
I started with the monad laws, a handful of equations relating unit, map, and join. I read countless monad tutorials, and dozens of papers. I read every silly example of how monads are like containers, space suits, C++ templates, and who knows what else.
I wrote little libraries. I learned category theory. I wrote a monad tutorial. I eventually wrote a paper explaining a whole family of probability monads:
And then one day, I thought about the monad laws again. I realized, "Hey, that's it. That's all. Just unit, map, join, and a handful of equations. Anything which quacks like a monad, is a monad. How did I ever think this was complicated?"
But when I look at the monad laws today, there's this huge structure of connections in my head. All that work, just to grasp something so simple, and so easy.
So I'm all for building better visualizations, and for helping people to understand math intuitively. That's an important step along the path. But math doesn't stop at an intuitive understanding. When you really understand it, the equations will suddenly be easy, and everything will fit together.
And then you'll encounter the miracle of math: Your deep understanding will become the raw material for the next level. Counting prepares you for addition, addition prepares you for multiplication, basic arithmetic for algebra, algebra for calculus, and so on. And someday, I hope that my rudimentary understanding of category theory will prepare me to understand why adjoint functors are interesting.
I had a different problem, I found that word density increases 10 fold, but symbol density and proofs go down as you go through it.
My favorite example to use is this: Complex Variables and Applications by Brown and Churchill. This book has been in print for 70 years or something, and it's somewhere around 400 pages in the current edition I believe. My professor I did research for had a early 80s edition, and it had almost 100 less pages than the current edition. There wasn't really anything new added between the versions (chapters are only about 5-10 pages, so there's something like 65 of them) I ended up using mine for the problems and his for reading because I have ADHD, and the wordiness absolutely kills me. Symbols and relationships are much more meaningful to me than words describing them. The real nightmare with the ADHD sets in because of the break in context when you have to switch between two or three pages to find the next theorem, formula or proof.
I retook that class twice.
On the other hand, I utterly and completely rocked my Advanced Electrodynamics course, outscoring even the graduate students, in a course which even made use of the stuff we were learning in Complex Variables (as well as PDEs and all that fun stuff) Why? I had a crazy russian professor who hated all the current textbooks (I'm looking at you, Griffiths) for the same reason that I hated textbooks, too much words and not enough symbols. So he wrote his own notes to every lesson and made his own homework. He said he originally wrote those notes when he first came here, and his english was worse, so there's little or no explanation, just proofs -- math and symbols. A few of these would span two pages, and very rarely three, but there wasn't the context break you get in many college level books, just beautiful math and lots of intermediate steps. The intermediate steps, almost never provided in most textbook proofs, really help the visual learners like me and provide stepping stones for the inevitable manipulation you will perform with those equations in your homework and on tests.
I still have all his notes, I want to bind them up some day when I get a chance.
Any chance of contacting said professor and obtaining permission to share said notes online?
Then... y'kno. Doing so :)
(I realize that scanning tons of a pages and ensuring quality isn't exactly a small undertaking, but I'm sure the HN community would greatly appreciate your efforts if such a thing were possible)
I really liked Griffiths. I thought it had struck a great balance between explanations, mathematics, and examples. That book had great examples and homework questions, which I find the most useful anyway.
Griffiths' E+M book (actually, also his QM book) is a shining star to me, a perfect example of the way introductory textbooks should be written. It ends up being criticized a lot once you get to the advanced level ("Who'd you learn quantum mechanics from, Griffithslol?"), but that's fine. It's perfect, even - if you really need all the gory stuff, sure, go get a copy of Jackson's book, or some 800 page tome on perturbation theory and the rigorous mathematical underpinnings of quantum theory, if you're going into the field you should absolutely know more than you can get from an intro text. But an introductory textbook should be exactly that. Rigor is awesome once you have the intuition down, but it's often very difficult to go the other way (mainly because a rigorous understanding is difficult to achieve without having the intuition first).
I think Griffiths EM/QM texts are fine if your not studying physics. However if you are they just skim the surface of material and this is were I think a lot of the bashing comes in. Physics curriculum (presumably like all) are a constant stepping stone and when your senior level professors expect you to know Clebsch Gordon coefficients in detail and Griffiths QM didn't then some questioning about his texts start to rise. In my opinion I thought Sakurai's QM was excellent in delivering a solid foundation in quantum mechanics
I don't mind it, this professor hated it, although it was the "recommended" book for the course. He recommended the following book, but it was out of print so he couldn't teach the course with it: http://www.amazon.com/Electromagnetic-Fields-Waves-Paul-Lorr...
It's similar to Griffiths, but I feel like it has more examples and is a little more concise in those examples than Griffiths (although I did borrow Griffiths occasionally) I grabbed one on ebay for $5 or something, and it was well worth it.
I have read both of them and I strongly prefer Griffiths to Corson. Lorraine and Corson are extremely rigorous but IMO their explanation of EM is upside down :-) They first explain special relativity and derive the EM equations using it. But historically, we had the EM equations first which forced Einstein to abandon Galilean transformations and discover special relativity.
But Lorraine and Corson just start out with special relativity, without really providing any impetus for why it is necessary, which can be found in the very first paragraph of Einstein's paper on relativity. Historically, physics has been a mystery that keeps unraveling with time. However LnC destroy the mystery by telling us who the murderer was in the opening chapter :-) We don't really appreciate how relativity was discovered. I think teaching the mystery is a very important and easily overlooked part of Physics education which Griffiths seems to appreciate but LnC don't.
Sounds like a great application for digital textbooks on some future color e-ink reader. For students who prefer verbal explanation, some of the intermediate steps can be set to display: none, and the text reflows accordingly. For symbolic learners, hide unnecessary <span>s of text.
So I'm sympathetic to the author's desire for better visualization and teaching tools.
But when I reached college, I became frustrated with math. It just wasn't easy anymore, the way programming was: I could pick up a programming book, read it in a weekend, and understand it. But when I tried to read an advanced math text, I became lost after 10 pages.
Eventually, I figured out what had happened: The information density of college-level math texts is insane. Even if you're bright and talented, it may take you a day to understand a single page. And there's no substitute for working carefully, finding concrete examples, and slowly building a deep understanding.
Here's an example that involves programming. Once upon a time, I needed to understand monads, in hope of finding a better way to represent Bayesian probabilities.
I started with the monad laws, a handful of equations relating unit, map, and join. I read countless monad tutorials, and dozens of papers. I read every silly example of how monads are like containers, space suits, C++ templates, and who knows what else.
I wrote little libraries. I learned category theory. I wrote a monad tutorial. I eventually wrote a paper explaining a whole family of probability monads:
http://www.randomhacks.net/darcs/probability-monads/probabil...
And then one day, I thought about the monad laws again. I realized, "Hey, that's it. That's all. Just unit, map, join, and a handful of equations. Anything which quacks like a monad, is a monad. How did I ever think this was complicated?"
But when I look at the monad laws today, there's this huge structure of connections in my head. All that work, just to grasp something so simple, and so easy.
So I'm all for building better visualizations, and for helping people to understand math intuitively. That's an important step along the path. But math doesn't stop at an intuitive understanding. When you really understand it, the equations will suddenly be easy, and everything will fit together.
And then you'll encounter the miracle of math: Your deep understanding will become the raw material for the next level. Counting prepares you for addition, addition prepares you for multiplication, basic arithmetic for algebra, algebra for calculus, and so on. And someday, I hope that my rudimentary understanding of category theory will prepare me to understand why adjoint functors are interesting.