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Background

This web page is for people who hated/feared/were-confused-by math in school, but have decided they want to learn more on their own.

I have no particular credentials for writing this page, except:

  1. I did ok until differential equations, and then bombed out. Years later I went back and through self study learned enough math to enjoy getting an EE degree. After that I dabbled in quantum mechanics, quantum field theory, and chaos theory.

  2. My daughter was confused by fractions in 5th grade. Through home study she was ready for calculus by 7th grade.

The Approach

The keys in both cases were:

  1. Find and use the very best tools.

    We'll discuss these below. I typically looked at perhaps a dozen texts or tools for each one selected.

    In addition to specific primary texts, I like the Schaum Outline Series. They are cheap; they give a no-nonsense description of the technique; they provide clearly worked examples; and they provide lots of answered problems.

  2. Find a way to make regular sessions fun -- and to stop each session before it becomes drudgery.

    For my daughter, we set a timer for 30 minutes. Totally focus for the 30 minutes. Stop in mid problem when the timer dings. For myself, I tend do a section at a time. Either way, the idea is to concentrate intensely for a short time, and then get away before you burn out or slow down.

  3. Practice each skill until it is trivial.

    This is a problem for both group classes and self-study. Math is defined and taught with very little redundancy. So if you miss something early on, you are in serious trouble later.

    My rule of thumb is this: Do all the problems available in the text. Repeat any that you can't solve as fast as you can write.

  4. Move on immediately after making the skill trivial. Don't keep pounding on it.

    Again, because math is built up as a house of cards, the lower level skills need to be fresh when needed for higher levels.

    This is mainly a problem in group classes. The best students get bored waiting for the rest of the class to catch up -- and then can't remember the material when it is finally needed.

    For self-study, the implication is that if you do take time off, you probably need to refresh on the more basic material.

  5. Keep your work -- and make it readable.

    Keep a notebook with all the problems and solutions worked out. These represent your brain's approach to the problem domain. Years later you can look at it and say "Hmmm. So that's how I do Laplace transforms." Re-doing a few of the problems can bring you back up to speed.

    Making it readable includes:

    • Date each page
    • For complex problems (e.g., "story problems"), do 1 per page.
    • Clearly define the problem, drawing diagrams and labeling parameters as needed.
    • Describe how you will attack the problem. Convert this to mathematical form (e.g., sets of equations)
    • Solve the math symbolically, then insert parameters and solve for the specific case
    • Convert back to reality as needed.
    • Draw a rectangle around the answer(s).

Making it Fun

There is no way you will study hard enough to learn math if it isn't fun. Part of the fun is mastering a difficult topic. Part is simply knowing the topic exists. I find it helps to mix history/biography with raw math and application problems.

Another possibility is to build the study program around a hobby. The classic story is the boy who learns math to do baseball box scores. That can be carried on to the physics of bat hitting ball, aerodynamics of a curve ball, etc. Or it can be built around fingernail polish or horses, or ...

All that matters is that you are interested, and stay interested for several hundred hours (in 30 minute blocks) over several years.

Getting started

I never found a text appropriate for this phase, so I've written down my own approach. It assumes a teacher/coach (typically a parent) is there every single minute, in an intense one-on-one partnership.

See: K-6

Precalculus

Here we assume you have a gut feel for and casual comfort with:

  • Naturals (0...n)
  • Integers (-m...0...n)
  • Rationals (a/b)
  • Reals, in decimal form (0.125, 0.333333, etc.)
  • Enumeration, identity, associative, commutative, distributive laws
  • Multi-digit addition, subtraction, multiplication, division
  • Rote skill for single digit addition, subtraction, multiplication, and division.
  • Introduction to exponents

Tools

A graphing calculator, or a computer program which does graphs.

Swokowski's Precalculus [swok90] (whatever the current edition happens to be).

Make photocopies of the insides of the covers (various formulas). On the page with trigonometry, add: 

periodic: y = a cos(bx+c) + d
          amplitude= |a|      period = 2pi/|b|    frequecy = 1/period
          shift    = -c/b     offset = d

   d+a|.             .       
      | .          . |  .    
   d  | -.- - - -.- - - - - -
      |   .     .    |    .
   d-a|     . .            
      +--------------+-------
     -c/b      (2pi/b)-c/b

Put these copied pages in plastic page protectors, and put these in your math notebook. These cover pages are the whole story. The text itself just gives you practice in using the formulas. You can use those cover pages for many years after you finish the book.

Exercises

The trick is to really understand each topic as you go. Don't say "Oh, I'll figure it out when I need it later." To really understand, do all the odd numbered problems (i.e., the ones with answers). At least do them up to the story problems.

Story problems are a special case. You don't need them to learn the math technique itself. But if you want to actually do something with math (instead of just play with abstractions), you need practice converting reality into math and math into reality. In a way, you are really learning physics, chemistry, surveying, engineering, etc. instead of math at that point. So do the story problems in each section, but don't panic if you can't see how to get a few of them turned into math.

Also, once you figure out the reality-to-math transformation, it is usually obvious. So if it was a hard one, throw away your work paper and redo the problem on a fresh sheet -- it will flow smoothly and you'll be proud of yourself.

Introductory Calculus

The biggest problem with calculus is that people think it is complex. So they spend a year or so on all the nuances of beginning calculus, get discouraged, and never get any further.

The truth is, you need to do a few difference limits to see what's going on, memorize a few formulas for derivatives, and have access to a table of integrals. In real life, you use computer programs which do symbolic and numerical differentiation and integration.

So how can you learn just the essentials?

Tools

Paper and pencil, plus the graphing calculator.

Hahn et al Hurricane Calculus [hahn95].

If you need more examples, get Ayres's Calculus (Schaum Outline). But do NOT try to do all the problems in it -- they are overwhelming. Just look at examples when you don't undertand a particular topic in Hurricane Calculus.

Exercises

Read the material, then solve the Additional_Problems (no more than 15 per section). If you are stuck, try this:

Go back and write out at least some of the worked examples, trying to understand each step as you go. It is like a beginning novelist typing out classics or a painter copying the masters -- it helps you focus on the mechanics of the process, which in turn helps you see the big picture.

  1. Limits. If you like finding abstract pattens, do all the limit stuff up front. If you just want to use the stuff, skip to section 3.2 "The Easy Way".

  2. Derivatives. The table in 3.2 is the whole story on derivatives. The cover pages for Kreszig (see below) have a nice listing of the formulas. Learn to use the formulas, but don't bother trying to memorize them.

  3. Integrals. The essential point is this: Integration, except in special cases found in calculus texts, is hard or impossible. In real life we use lookup tables, or use a computer program which does symbolic and/or numerical integration.

    DO NOT BOG DOWN HERE. Read Ch 8 (Techniques for Solving Mind-Bending Integrals), be impressed that someone thought of those tricks, and keep moving.

  4. Sequences and Series. As noted above, solving integrals directly is difficult or impossible in many cases. Instead we make approximations of the formula as a series of ever-smaller correction factors. You just keep calculating the corrections until the answer is good enough. In some cases you can get terms to cancel, so the series gives an elegant solution. That's what math texts are for. In most cases you just have to grind it out. That is what computers are for.

    Either way, you are responsible for assuring the series converges (and how quickly), so take this section seriously.

  5. Remaining chapters. Skip them. Kreyszig does a better job.

That's it. We took one of the smallest calculus books round, and chopped out major sections. The key is to keep moving -- don't get stuck in the quicksand.

Math for Engineering

This is the bulk of what scientists and engineers do in terms of math. It assumes fluency in the previous tools, so just jumping here doesn't help. You really do need to know algebra and exponentials and such from Precalculus, plus derivatives and series approximations from Calculus. But this time they are applied to more complex things, such as differential equations, matrices, and vectors.

Tools

Paper and pencil; calculator.

Kreyszig's Advanced Engineering Mathematics [ krey88]. I've had various editions -- just get the latest one available.

Copy the inside cover pages, put them in plastic page protectors, and save them in your notebook.

Exercises

Ok, if you are working through Kreyszig, you have nothing to be ashamed of. You can leave it on your desk top and impress friends, family, and co-workers. Of course, you actually have to do the problems...

In fact, you should plan on doing all the odd-numbered problems. They build nicely from simple cases, to tricky cases, to applications. If you jump right in to the story problems, you will be wasting time even if you solve them. Think of it this way: The fastest way to build a yacht is to build its dinghy first (to learn the process) and then build the yacht. By the time you hit the story problems for a section, the mechanics should be rote.

This could take years. So what? You are doing it for fun and personal development. Is TV really better than being able to say "So that's what Laplace Transforms are about."?

  1. Differential Equations. Plow on through to Ch 3 (systems of differential equations). Many real world problems translate to systems of differential equations. (These are often converted to finite difference equations, which can then be solved by computer.)

    If you like math for its own sake, work through the power series material. You can see all the old algebra skills coming into play at a higher level. But it might be interesting to sci/engr folks only if their subject domains need the formulas.

    The Laplace Transform is a winner. If you can get your problem stated in the Laplace form, you are down to straight algebra. Control systems are generally defined and analyzed this way.

  2. Linear algebra and vectors. You've already covered much of the material in Precalculus, ch 8. So either do some review, or skip directly to vector differential calculus. There, you are sure to bog down in grad, div, and curl. Just take your time and do the problems.

    [I don't have a shortcut. I have trouble with left and right in the first place, much less the "right hand rule". I think Kreyszig does as good a job of explaining and demonstrating as anyone.]

  3. Fourier analysis. After a lot of gyrations, you end up with variations on Fourier Transforms. In other words, translate from waves to spectra and back. With any luck you can fit the problem into a FFT (Fast Fourier Transform), and use existing software to solve it. It's more fun if you can apply it to sound or image processing.

    If you are a glutton for punishment, go look at wavelets. Fourier transforms define events in terms of continuous waves, but that leaves inconvenient side bands. Wavelets use little blips of waves as the basis, allowing much tidier modeling of single events.

  4. Complex analysis. Here you get to do everything you've learned so far all over again, but with complex numbers. To be honest, I didn't complete all the sections. I do remember completing all of Ch 12 in 1 day (and have the solutions in my notebook to prove it.) Unfortunately, I have no idea what it's about right now...

  5. Numerical Methods. Everyone needs to know about this, but in real life you use pre-built libraries (or at least well-known algorithms). So scan the text, then implement your own routines as a learning exercise. E.g., my m3na (source: m3na.tgz ). Then go use serious libraries such as atlas, blas, lapack, gsl, fftw.

  6. Optimization. Generally this is taught as Operations Research in MBA classes. You might want to implement simplex. But use an algorithm text.

  7. Graphs. Again, an important topic. But get it from an algorithm text.

  8. Statistics. Another important topic. Kreyszig does a good job.

    [I also have some notes I'll put on the web sometime which takes a single problem and evolves statistics as the answer becomes more sophisticated.]

Suggested algorithm texts:

  • Algorithms [sedge88]
  • Introduction to Algorithms [cor90]
  • Foundations of Data Structures in C++ [horo95]

Math for Physics

Kreyszig covers most of the needed material except tensors and groups.

There is a text (M. Boas: Mathematical Methods in the Physical Sciences, 2nd Ed) which ought to be appropriate here. But it doesn't have answers to problems, so it is hard to use for self-study. However, you can use it to see another view on topics in Kreyszig. Someimes seeing two different people cover the same ground helps fill in the missing links.

For tensors, I came up with Kay's Tensor Calculus [kay88]. Ok, I didn't really internalize this. I kept thinking: I need a computer program to keep track of all the pieces. But even just doing some of the problems is cool. Heck, just knowing about metric tensors is cool.

I haven't yet found a good text for self-study on groups. The best I have on my shelf is ch 2, of Kaku's Quantum Field Theory [kaku93] There is a Shaum Outline for it you might try.

Math for Math's Sake

More topics

By this point you are choosing your own topics, and finding your own texts and tools. In my case, I'm moderately interested in:

  • Logic. esp. in the context of logic programming and AI
  • Category Theory. esp. in the context of programming "types"

When you burn out....

When you can't handle the material, you can turn to biographies, history, and popularizations. They at least give a flavor of what's going on.

A good start point is Kramer's The Nature and Growth of Modern Mathematics [kram81].

Next check out Davis's Mathematical_Experience [davis99] to understand vicariously how it feels to do math. Then look at Mathematical_People (out of print) for interviews with professional mathematicians.

A book often cited is Peterson's The Mathematical Tourist [peter88].

Bibliography

Cor90
T.H. Cormen, C.E. Leiserman, R.L. Rivest. Introduction to Algorithms MIT Press, 1990. ISBN 0-262-03141-9

Das92
S.K. Das. Deductive Data Bases and Logic Programming Addison-Wesley, 1992. ISBN 0-201-56897-7

Davis99
P.J. Davis, R. Hersh. The Mathematical Experience Houghton Mifflin Co,1999. ISBN: 0-395-92968-7

Hahn95
J.B. Hahn, T.J. Dunlap, S.P. Matyus. Hurricane Calculus Prometheus Enterprises, 1995. ISBN 1-886783-00-4

Horo95
E. Horowitz, S. Sahni, D. Mehta. Foundations of Data Structures in C++ W.H. Freeman, 1995. ISBN 0-7167-8282-8

Kaku93
M. Kaku. Quantum Field Theory Oxford University Press, 1993. ISBN 0-19-507652

Kay88
D.C. Kay. Tensor calculus Schaum Outline Series, 1988. ISBN 0-07-033484-6

Kram81
E.E. Kramer. The Nature and Growth of Modern Mathematics Princeton University Press, 1981. ISBN 0-691-02372-7

Krey88
E. Kreyszig. Advanced Engineering Mathematics John Wiley and sons, 1988. ISBN 0-471-85824-2

Peter88
I. Peterson. The Mathematical Tourist I. Peterson, 1998. ISBN 0-7167-1953-3

Sedge88
R. Sedgewick. Algorithms , 2nd ed. Addison-Wesley, 1988. ISBN 0-201-06673-4

Strogatz94
S. H. Strogatz. Nonlinear Dynamics and Chaos Westview Press, 1994. Paperback 2000. ISBN-13 978-0-7382-0453-6.

The reviews say it is exceptionally clear, and that fits. Assumes upper class or 1st year graduate level understanding of math and physics. Works from 1D linear systems to nonlinear, and then to 2D nonlinear, and then on to chaos. Lots of solved problems, with examples from physics, biology, chemistry, and engineering.

Swok90
E.W. Skowkowski. Precalculus: functions and graphs 6th ed. PWS-Kent Publishing, 1990. ISBN 0-534-92086-1

 
Creator: Harry George
Updated/Created: 2010-01-01