Saturday, December 7, 2013

Why Math Teachers Need Real Analysis and Abstract Algebra

A common complaint heard by anyone that hangs out with future math teachers is, "When are we ever going to use this?"

Only it's not their students that are saying this, it's the teachers themselves complaining about the classes of Real Analysis and Abstract Algebra. 

And yes, I believe the irony is lost on them.

For those of you that don't know, Real Analysis and Abstract Algebra are commonly viewed as the first "real" math classes (and in the case of the former, that's a common pun). Basically, from learning to count and basic arithmetic all the way through high school algebra, trigonometry, functions, three courses in calculus and the miscellaneous other math classes offered to underclassmen and underclasswomen, these have all been the "ABCs" of mathematics.

Depending on the institution, there may be a transitional course between calculus 3 and real & abstract. This course may have a title such as "transitional math" or "foundations of math" or "discrete math", but it's basically just a getting-your-feet-wet course and not the intense introduction to actual mathematics that real and abstract are.

And yet these future enthusiasts of mathematics, these people supposed to guide our children to be the next generation of mathematicians and scientists, are repelled by these courses like Dracula is by a crucifix.

To describe these courses a little more without delving too deep into jargon or a technical discussion, real analysis is basically the course that explains how everything in calculus 1 and 2 work, while abstract algebra is basically the course that explains how algebra works.

Like, ACTUALLY explains it, not just handing it down from on high or saying, "the proof of this is beyond the scope of this course." Those are the courses whose scope includes the proofs that were once-upon-a-time beyond the scope of the courses you took as underclassmen. 

And yet future teachers of algebra and calculus balk at the mere thought of these courses worse than the rookiest of rookie pitchers.

There are many reasons why math teachers NEED Real Analysis and Abstract Algebra, and in no particular order, here are 10:
  1. A math teacher needs to understand actual mathematics.
  2. Teachers should know at least one level beyond what they teach their students (if you're teaching high school, you should know college mathematics: i.e. Real Analysis and Abstract Algebra). 
  3. Math teachers serve as advisors to students and need to be able to advise those that might actually enjoy mathematics. 
  4. Real Analysis explains how calculus works. 
  5. Abstract Algebra explains how algebra works. 
  6. Mathematics before Real & Abstract (from now on, I'll just say R&A) can be thought of as "cookbook" mathematics that requires no actual thinking. Follow the recipe and get the answer without ever understanding what you're doing. 
  7. Curriculum writers without understanding of R&A struggle and fail at writing and re-writing the curriculum of high school level classes. 
  8. Number Theory requires R&A, which is the course that explains arithmetic (so you should take that too).
  9. Teachers should not be a cliche: "Those that can, do; those that can't, teach."
  10. A math teacher should love mathematics, and saying you love math without loving R&A is like saying you're in love with a girl you saw at a bus stop once. Don't be so superficial.
What are your thoughts? I plan to update this list with more reasons why as I work through future sections of Zero Angel's Mathematics (since yes, I do use and incorporate Real Analysis, Abstract Algebra and Number Theory into those books).