Algebra as language

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 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.

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!"

Formal languages
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 spoke and wrote in the language before the rules were known and realized. Human languages lead to rules about the language.


 * Formal languages only exist because somebody had to create the rules for the language to be used. Formal grammar expressed as 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 fo the computer, human languages do not compute!

Formal language resource
A book that explains this idea of all formal languages as 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 book 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 don't make that connection in college, most algebra teachers can't show you hot to make the connection, either.

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