In CRAM Math, algebra is viewed as a language. When students don't get this idea, algebra is something foreign and not related to real life, making algebra that much harder to study as an academic subject.
- When algebra is viewed as a language for expressing ideas, it becomes much more useful, like using single letters to represent words in texting.
- This entire page explains why your algebra teacher generally can't help you to understand algebra in one semester, or in one academic year if you have to repeat the course. If you do understand it at the end of a year, consider yourself lucky or gifted.
- Sections 4 and 5 are not absolutely required to understand CRAM Math, but they may help you become more clueful about your algebra teacher's place in the world.
The main complaint
One of the most common questions students ask instructors about algebra is: "What does this have to do about real life?", and the instructor most often just gives the asker a condescending look, like he or she is thinking, "This student will never cut it in graduate school..." without offering an answer to the student's question except to say "Read chapter one again, darnit!"
A formal language like algebra is not a human language like English, Spanish, or Mandarin Chinese. Many people refer to human languages as natural languages.
- Human languages have grammar rules, but people thought, spoke, and wrote in human languages before the rules were known and realized. Human language leads to rules about the language which are its grammar.
- When somebody makes up a new rule about a human language and says everyone should follow it, it is time to measure that person's ego to see if it has gotten too big for that person's importance to the world.
- Formal languages only exist because somebody had to create the rules for the language to be used. Formal grammar expressed as language rules lead to the existence of formal language, and not the other way around like in human or natural languages.
Algebra is a proper subset of human language
Amazingly, everything in algebra can be said out loud or understood inside your brain as a part of whatever human language you happen to speak ordinarily!
- This is why computerized word processing grammar checkers are a joke. If you study algebra or any computer language thoroughly, you can understand everything about that formal language because it is a formal language, but your computer will never understand everything you might happen to think, say, or write because for the computer, human languages do not compute!
The avoidance of thrashing
The concept algebra is only a part of any human language is the key to avoid thrashing (wasting time by guessing) until you give up on a particular homework problem or walk out disgusted in the middle of a quiz.
Because algebra is a formal language, moving through a problem from its original statement to its final solution is a path. Thrashing means that you wandered off the path into Wonderland in the world of Daydreams and Other Distractions.
- Formally-trained mathematicians refer to a path to a reasonable solution as a proof, especially when the proof also shows annotations (notes) for the reasoning behind each step along the path to a solution. In mathematics, she who has the proof is queen for all eternity, or at least until someone comes up with a shorter, more elegant proof and chops off her head. Every homework, quiz or final exam problem that is successfuly solved can be considerd to be a proof.
In algebra, there can be many paths to a solution, BUT THEY ALL LEAD TO A SOLUTION THAT CAN HELP YOU PASS THE CLASS!
- Corollary: There is usually more than one way to skin a cat.
- Translation: There is often more than one "right way" to solve the problem. Just because your teacher doesn't have the time, energy, or inclination to mention this fact does not mean those other ways do not exist!
Today's need to avoid thrashing in algebra
In post-industrial Information Age economies, there is a vital need for mathematicians and scientists as a raw human resource for building future industries; where this raw capital is not available, there is a natural barrier to economic growth that acts as an impediment to market dominance.
- When members of a society's general population are of the opinion that math is too hard to learn and has no relation to their lives in a modern economy, then it is easy to predict the fall from or absence of market dominance of that society's economic component in the global economic system.
Formal language resource
A book that explains this idea of all formal languages are proper subsets of any human language is J. Glenn Brookshear's Theory of Computation: Formal Languages, Automata, and Complexity, which can be found using this tailored WorldCat link.
Comment on training new algebra teachers in college
Unfortunately, Brookshear's Formal Languages, Automata, and Complexity is almost never required in college for teaching future algebra teachers about teaching algebra; I only had to read it because I was a computer science major.
- Since future algebra teachers generally are not required to make the connection between algebra and useful language in college, most algebra teachers can't show you how to make that connection, either.