People keep whining about why more Americans aren't pursuing math as a college major. I've done some of this whining myself.
But there is at least one good reason why this should be the case. I'm not talking about the economy, outsourcing, insufficient math preparation in high school, or any of the other usual suspects.
The truth is that the Math Department, almost everywhere, has a serious marketing problem.
Think about it. For every other science, the first college course is a survey course. In it, students are exposed at a rudimentary level to all the specializations for that science. Further courses develop each subtopic in depth - e.g., Organic Chemistry (for Chemistry), General Microbiology (for Biology), or Electricity & Magnetism (for Physics). All these topics, plus the many more that make up the degree requirements, are introduced in the freshman survey courses.
But math is different. There is no math freshman survey course. In fact, you can make it all the way to grad school without actually having a good idea about what it is exactly that mathematicians do, or what constitutes a solid undergraduate math education.
Again, science majors all have to take the survey courses for all the sciences. For math, they simply take algebra, precalculus, or calculus. None of these is a survey course designed to entice young students into the major. A math student taking a required biology survey course might decide he loves biology. But what are the odds that a kid considering a science major and forced to take calculus - which is really just a prerequisite for his science courses - will decide to fling his science career to the winds to pursue the siren call of math?
So, the other sciences have two chances of attracting a math major (since all math majors have to take at least two science survey courses), while the math department has close to zero chances of attracting a science major.
This means that Americans who consider a math major are usually simply kids who loved their high-school math courses to distraction. Needless to say, this is a small and ever-shrinking pool.
On top of this, math majors themselves have an unclear (to say the least) view of what their future studies entail, or how this could all possibly make a coherent whole. It's entirely possible to "scattershoot" your way through a B.S. in Mathematics, taking courses of the particular level required, but with no overall plan that pulls them together. For example, you can graduate topheavy in topology or stuffed with statistics, but anorexic in analysis and stumped by set theory - or vice versa.
In their turn, potential employers and issuers of research grants are also clueless about the areas of mathematics. They never had to take a math survey course, either. The result? Comparatively few research dollars go to mathematicians, and their salaries are lower than than those of other scientists.
To give future math majors a fighting chance, I am going to attempt to explain to all PHS readers exactly what "higher" math (the kind Ph.D.'s and Master's candidates study) is all about.
According to Margie Hale in her book, Essentials of Mathematics (Mathematics Association of America, 2003), "You can see that the three main branches of mathematics - algebra, geometry, and analysis - grow out of three basic areas. Logic comprises the rules by which mathematicians operate, the 'grammar' of the language. Set theory provides the vocabulary. And the number systems comprise the most basic content from which the various branches grow."
She includes a diagram of a tree with these branches:
- Linear algebra
- Graph theory
- Topology
- Complex analysis
- Real analysis
- Game theory
- Cryptology
- Number theory
- Knot theory
- Calculus
- Operations research
- Differential equations
- Statistics
- Computational math
- Modeling
- Probability
I would simplify the scheme to the following major branches.
- Geometry/Topology - the study of objects in a plane and in three dimensions. This includes Euclidean geometry, trigonometry, special geometries such as geometry on a sphere or elliptical geometry, topology, and metric spaces.
- Analysis - study of our number systems and their functions. This includes the algebra/pre-calculus sequence, calculus I, II, and III, differential equations, analysis, functions of real variables, and functions of complex variables.
- Abstract Algebra - the study of sets and operations on them.
- Statistics - often a separate degree option of its own, so statisticians won't have to take those pesky topology courses - the study of probability and how it can be applied to the real world
- Number theory - the wonderful world of whole numbers; historically a favorite for amateur mathematicians.
- Logic - the foundation of mathematics. Logic is usually taught as part of a discrete structures course. It includes symbolic logic, set theory, and (fanfare) theory of proof. Proving things using valid logical argument is what mathematics is all about. If you can't prove something, it isn't true (yet).
- History of math. This one is self-evident
To simply know what all these fields are about is to begin to have a grip on what math is all about. Any of these fields then requires you to learn dozens of terms with their definitions, and hundreds of theorems based on those definitions with their proofs.
So What Do Mathematicians Do?
To see what mathematicians do and how they do it, first see what mathematicians of the past have done. The best way to start is to read a good history of mathematics. This will show how various thinkers managed to come up with what we have today, as well as acquainting you with many of the names and terms you will need if you decide to become a math major.
Some good choices for books to read are those used by University of Missouri-Kansas City in their 4000-level History of Mathematics course:
- A History of Mathematics by Victor Katz
- Journey Through Genius: The Great Theorems of Mathematics by William Dunham
- The Historical Development of Calculus by C. H. Edwards
- Euclid's Elements, volumes 1, 2, and 3 by Thomas L. Heath
- A Source Book in Mathematics by David Eugene Smith
Or check the book list online for any other college's History of Mathematics course. Read the books, write a paper or two, and take two semesters' worth of high-school credit for "Survey of Higher Mathematics."
There is no AP course in History of Mathematics, and no required freshman survey course in Fields of Mathematics.
But there should be.