Slashdot Mirror
Science and Math For Adults?
Peter Trepan writes "Like most Americans, I made it through high-school and college without a thorough understanding of major scientific and mathematical concepts. I'm trying to remedy this situation both for personal betterment and so I can supplement my *own* kids' education. The problem is, most textbooks are not designed to convey an understanding of the subject, but to squeeze in all the 'facts' required by state law. I'm looking for books that don't just tell me an equation or a concept works, but also explain *why*. Would you please list books that have helped you gain a greater understanding of the basic concepts of algebra, chemistry, calculus, physics, and other core areas of science?" This is similar to an earlier question, but with a broader focus.
Try enrolling in some night classes at your local Community College if you have the time. It's pretty cheap, and you may be able to get your employer to pay for it.
The solution that most math texts take then is to give you *lots* of problems/drills so that the mechanics get ingrained, allowing the insight to come later.
When I screwed up my second year calculus course *really* badly (like 6% on the midterm...) I used a Schaum's Outline to get back on track (and eventually ace the final). It's main benefit is *heaps* of problems to work through. That made me a convert to the problems approach to math teaching.
The key is to do all the problems, in order.
That said, I can't really recommend one math text over another, just so long as there are lots of problems, and hopefully a solution key in the back for at least half the excercises.
The problem is, most textbooks are designed to be companion references, with all the 'facts' squeezed in so the teacher can spend time helping everyone understand the concepts etc. The two work together.
Simple answer is, you need to take adult education classes. I left college barely half-way through, and ended up taking night classes- intro to calculus was one; another was an intensive Economics class. I found them worthwhile; I probably would have enjoyed the class more if I wasn't young enough to be most of the other student's kid(you would fit in FAR better, from the sounds of it.)
Without the classes, you don't get the benefits of peer learning, in-class interaction("Did everybody get that?" [blank stares] "Heh, ok, let me explain it a different way...") the discipline that testing creates, nor the resource of having a Really Smart Person(professor) to go to when you need help. There are also other benefits- making friends(you're probably all in similar 'boats' so to speak, so people socialize pretty readily), and networking. My old boss decided to do part-time classes for an MBA, and got a lot of networking out of it(granted, those were business classes, more prone to networking activities, but you get the idea).
Please help metamoderate.
When it comes down to it we don't know everything.
The base upon which most science is built, save for math which is pretty much a mental exercise, has a bit of uncertainty.
We observe the real world. Then we try to describe it, preferably mathematically. It says nothing about why it works, nor does it ensure that our current description is correct. Just that as far we know that's how the world is.
And strangely, I still find more comfort in this than god.
...and very little from the books.
I suppose it depends on the type of learner you are, but frankly, I imagine seeing and using the information being delivered to me. Rather than simply "knowing" the things I learned, I understood them and used what I learned to add more peices to the puzzle I call "reality."
In more simple terms, everything you (should have) learned should be assimilated into the way you operate within your environment. Ever heard "you use it or you lose it"? There's a lot of truth to that.
Rather than try to get what you missed from books, perhaps it's time to make a much more grand display by going back to school. It doesn't have to be thought of as "remedial" but rather as a "brush-up" or simply continuing education. If you show your children that learning only ends when you die, their minds will be open for life with the expectation that they can grow and improve themselves at any point in their lives... not just during the beginning phases. By the time they reach it, "middle aged" will be 50-something anyway.
Best advice? Go back to school and pay attention this time.
A Tour of the Calculus is a particularly comendable book. It only covers the more basic tenants and theorems of Calculus, but gives you an immense sense of the power behind such theorems and of the near-glacial process which has formed them and the calculus as a whole. Reading it gave me a much deeper understanding of the particular topics it covered, as well as the Calculus and math in general.
~metal_llama out.
---
move every sig!
In each book, there is a bibliography of the sources that it used, in case you want to do additional research on the subject.
As an added bonus, each book is less than $15, and they can be picked up at any Barnes & Noble. So its worth picking up to see if you are interested in a certain subject.
Hope it helps, I've enjoyed them.
010_digital_100
"Keep in mind that during the 80s-90s (I think), there was a revolution of sorts in the way calculus was taught in colleges. Professors supporting this reform movement wanted students to understand the concepts instead of memorizing the formulas."
The concept of "new math", and the resultant ill effect on thousands of mathematics students, was a corruption of some really good ideas. There's no doubt that some bureaucracy was at fault in this madness. They took the idea that mathematics students should not only think about the "how", but also the "why", and corrupted it into the notion that students really don't need to learn their multiplication tables or memorize trigonometric identities. It was tried before in the 1800's, the 1900's, and recently in the 80's and early 90's. Every single time the message of learning "why" got corrupted.
Given that you, yourself, are not very math/physics savvy, text books alone may not be enough. You might easily end up in a situation of the blind leading the blind when trying to help your kids. Understanding math/physics will often go beyond what any textbook can tell you. You might do a lot better from a person you can interact with who can see how well you are grasping a concept.
:)
If you literally want to go to the trouble of hiring a tutor, then you'd get him/her for your kids obviously, but I don't know what to recommend for adult education. Given the current economy I'm sure the tutor might be willing to help you out as well in a package deal.
For what you do, it might be useless, but for people in Engineering and other fields, calculus is a VERY important subject. As a current CS major, I agree with what you say about descrete math and linear algebra, but I think you are discounting the need for Calculus.
RonB
It is human nature to take shortcuts in thinking.
What is the point of avoiding difficult but important classes simply to preserve your GPA? Are you in school to get an education or to simply achieve some arbitrary GPA? I've been in the position of hiring people for technical positions and I've always been far more impressed by a mediocre GPA in a substantial curriculum then a high GPA in an easy curriculum.
I recently returned to school myself, so I do have sympathy with amount of work required to do really well in a course, and I do understand that those planning to continue to a four year school or go on to graduate school need to match minimum requirements, but in my opinion you'll be better served by reducing the number of classes you take in a given term then by trying to ditch the challenging courses.
On the topic of calculus, don't learn anything past calculus I (well, bits of calculus II are useful). The rest is completely useless and you'll forget about it all in a couple of years anyway because of its uselessness. If you want something that's useful go for discrete math and/or the good bits of linear algebra. Your comment is completely offbase. Actually, Linear Algebra is about as important as Calculus in many scientific/engineering disciplines.
More importantly, you claim that anything more advanced will be forgotten, but the later courses often serve to reinforce earlier material. For example a course on Fourrier theory reinforces both Linear Algebra and Calculus.
Most math departments have a course somewhere after the introductory sequence which teaches basic proof techniques often by studying the definition of numerical systems from logical axioms.
These basic proof techniques are the very basis of mathematics. The reason so many people get through high school with little understanding of math is that they are never forced to do any proofs outside of Geometry.
In short, if you cannot prove anything, you know practically nothing about mathematics.
http://yetanotherpoliticalrant.blogspot.com
I misspoke. Being lazy with my words. He certainly understnad the stuff well, but I didn't find it particularly useful for learning the stuff from. Found other textbooks much clearer. I think they're good to go back and look at once you've learnt a subject and have some understanding of it, but in my experience, they just don't cut it when it comes to learning for the first time.
While I'm sure that the people recommending GEB and Hawking have your best interests at heart, they're answering the wrong question. If you want to learn math, you're going to have to start at the beginning and work your way up. "Popular" math and science books won't help you with the basics.
What you'll want to do instead is what they do in school. Start with some basic number theory(nothing fancy, maybe just enough to know the difference between integer/real/rational/etc). After that, assuming you understand how to add, subtract, multiply, and divide, you're going to want to get into some basic algebra, then calculus, then geometry or whatever else you want. Unfortunately, I learned algebra way back in middle school so I don't have a textbook to name, but I do have some advice that applies at all levels:
* Do the problems in the book. Then do some more. Then do even more, just for good measure. Some of the other posters have complained about doing problems. Ignore them. Nothing will give you a better feel for how algebra and calculus work than actualy *doing* them.
* Understand each piece of information before you move on and how it relates to the whole. Any decent textbook should offer problems that use both new and previously gained knowledge. Make sure your textbook of choice has lots of examples and that those examples are worked out well. Never underestimate the value of a fully worked out problem. It may be worth it to get multiple textbooks, look them over, and then return the ones you don't want.
* Be persistant. Children learn math by doing it every(other) day for years. You're an adult. You can learn faster and better, but that doesn't mean you get to be lazy. Do a bit every day, even if it's just working one or two problems. Daily practice will ingrain concepts in your brain and also make it easier to pick up a book and start on something new.
* Don't get too formal. Wanting to know "why" is great, but "why" must often take a backseat to what is being learned. Often, the reason for doing something may not be obvious until you already know how to do it.
* Have I mentioned doing problems?
Now I do have one actual book to name, and that's:
Calculus by Larson, Hostetler, and Edwards
This book has tons of examples and illustrations, as well as excellent problems. It even features a two chapter algebra/pre-calc review!
Some people have mentioned the calc book by Stewart. We use that book at my college, and given the number of people who seem to have problems with it I cannot recommend it for self-teaching.
Good luck!
Visit the
I could never do that. I need the explanation of why and always have.
I doubt that. Ever learn to eat? Or walk? =)
I'll acknowledge that you are much more motivated to learn the WHAT if you've a notion that a WHY will follow, but I'd suggest that you CAN'T learn the why without first learning the what. For example...in 1776, the United States declared its independence from England. Why, you ask? It's impossible to explain WHY without first explaining WHAT occurred in the years leading up to 1776. I'm not saying you must have recall knowledge of those events (ie have them memorized and know them cold) but you must have at least recognition knowledge (as in "oh, yeah, those taxes.") To give a more mathematical example:
Solve -3x+8=20. Solve for x. Why does x have that value? Your answer may involve arithmetic. If so, why does the arithmetic work? Your answer may involve properties of the real number field. If so, why do those properties exist? Your answer may (if you've done serious undergraduate work in math) involve Peano's postulates. Why do those postulates work? Now you're beyond me. Yet to have an satisfying intuitive understanding of why x=4 in the above equation, you needn't be too concerned with anything beyond the arithmetic. You've mastered the WHAT (as in what to do when faced with an equation like that) without having a deep understanding of WHY.
Real math involves proofs.
This is true, but I doubt the fellow with the question is interested in real math. Quite frankly, the proofs are a hindrance to understanding the mathematics. Proofs are often the result of hundreds of years of mathematical development. Consider calculus and the limit theorems involved in the proof of derivatives. I can explain to a ten-year-old why the derivative of X^2 is 2x, and I can utterly convince him that it can't be anything else, but I can't prove it. Why? Because proofs involving infinitesimal quantities require a fair bit of knowledge of limit theory*.
Practical math should be more than "Here's 50 problems of progressive difficulty," but it needn't involve proof. An intuitive demonstration suffices for most people; those who demand proof are generally capable of producing it given the clues in the intuitive development. Otherwise, there'd be no progress in mathematics. Mathematicians begin with an intuitive notion of how the mathematics should work and go on to proof. Intuition yields conjecture yields proof.
*Not entirely true, actually; google for hyperreal numbers - they formalize the notion of infinitesimal numbers and make Leibniz's dx/dy approach mathematically valid.
Calculus is INCREDIBLY important, and from a philosopical point of view it might even be dangerous. :)
Imagine a field of mathematics that explicitly has at it's underpinnings the hypothesis that as you break up a line into smaller segments, eventually if you make each segment have no length, they still all add up to a lenght.
Philosopy aside, it's an INCREDIBLE tool for particular applications. Need the area of a sphere, no problem. A cone, still no problem. An oddly shaped object that looks like a art-deco running shoe? BIG problem, that is unless you use calculus.
I just wanted to reply concerning the cost issue. If you find something you think will work, and can learn easily from it, it's worth the price. You'd be surprised what a good foundation of scientific principles can do for you, at work and at home.
It's not only the facts you know about things; those give you the ability to carry on a discussion with a specialist in any given field. It's also the process of discovery and fact-checking. Every time you work a problem, or follow the progression of a historical great discovery, you teach yourself how to apply your natural curiosity in a productive way. Invaluable.
...
"little" gem?
I shudder to think what would qualify as big.... There's a tradition of starting GEB and never quite managing to finish it.
This reminds me of some stuff my mom has talked about a lot (she's an elementary school teacher). Ever heard of "Bloom's Taxonomy" -- a general theory that there's different levels of knowing. There's knowledge, which you need to be able to build comprehension. Can't understand without examples to guide you there, in other words. Can't apply without understanding, at least in a _real_ sense -- in math you can use formulae as a crutch and replace comprehension with knowledge, and get by for a while. But in a situation like the one here, you're only able to analyze if you have knowledge of the events leading up to it, comprehend the subject and the pressures of those events and their causes, and can apply that understanding of the pieces to the whole.
I remember there used to be a sign in just about every classroom at my elementary school with that on it. But nobody ever explained what it meant there. Guess that was a break in the whole "knowledge" base
Feynman's books are a little too advanced for ANY casual reader--in fact they WERE textbooks for the CalTech freshman physics class in 1964 (I think this was the year at any rate) but were quickly scrapped. Feynman adds outstanding insight to those who already "know" physics (those who took the course and did reasonably well), but otherwise I think he befuddles and confuses. On the other hand, 6 Easy Pieces is good, as is "The Character Of Physical Law." Many mathematical physics books would be great to help you pickup both math and physics, but I would recommend starting at the high school level and then moving up (the high school books are written on a college level anyways--after all, how many high school students are interested in mathematical physics!). GEORGE POLYA wrote a book you should check out called "Mathematical Methods in Science" and it is an excellent start.
I don't mean to discourage you here, but the only way to really understand things like math and physics, especially math, is to sit down with one of those text books with all the "facts" and go through it till you understand everything. If you don't understand the basic stuff in that text, you need to get a text from a previous topic in the same subject, like if you don't understand trig, you need to go back and learn algebra better. It's not that the texts are flawed, it's that your understanding of basic concepts is flawed and you need to review.
This is all assuming you really want to understand the subjects. If all you want to do is fool the average person who has no idea what you are talking about anyway then by all means, get a "physics for dummies" book, but there really is no shortcut to math and physics besides putting in the time.
It's not like there is a new and different way to learn math and science once you are an adult. It's the same way you learn it when you are younger, study really hard.
"The problem is, most textbooks are not
designed to convey an understanding of the
subject, but to squeeze in all the 'facts'
required by state law. "
If text books are no good, then where do all the college students around the world get all their understanding? Is there some secret they know that you don't?
I wouldn't have had a problem if the post had just asked for a source of information to help their children with homework and such, but everyone trying to find a shortcut to understanding math and physics are just insulting those who put in the time to study it.
If you find and enjoy Hardy's A Mathematician's Apology go on to Robert Kanigel's The Man Who Knew Infinity, A Life of the Genius Ramanujan next.
I also recommend Timothy Ferris' books, like Coming of Age in the Milky Way and for history of technology, James Burke's The Day the Universe Changed and Connections. I have heard him speak in person and if you appreciate humor, he is your author. Another winner is James Gleick's Chaos, Making a New Science.
Don't forget periodicals, like Scientific American which convey the excitement of discovery that drives most practicing researchers while still being accessible.