A better initial question is: What is a symbol, and what are the types of symbols used in mathematics? A *symbol* is an object, a mark, a sound, etc. that represents an idea, a process, or a physical entity. Simple examples include stop signs, clock faces, company logos, the music used for the olympics games, badges, red roses for love, etc.

Symbols in mathematics are usually marks on paper, or special characters or pictures on computers. One of the most important set of symbols in mathematics are *numerals*, which are symbols for numbers. For example, the numeral “3” stands for and names the idea of “threeness” (for mathematicians: “threeness” is the equivalence class of all sets that have the same order as the set {*,**,***}). You could also use three tick marks, | | |, to denote that number, but denoting the number 3,431 using tick marks would be very unproductive!

There are three important types of symbols used in mathematics: constant symbols, placeholder symbols, and operator symbols.

*Constant symbols. *These symbols denote numbers, as in

0, 1, 2, 3, 4, 5, -12, 3.14, π, .

The key for understanding constant symbols is that they specify a unique number or object. For example, if we say, “Let stand for the number 3,” then is being used as a constant symbol—not as a variable or placeholder symbol.

*Placeholder or variable symbols.* Letters such as , , and are often used as placeholders for where we might expect to see or place a number. (Note that ONLY 1 number is expected—a variable, contrary to popular belief, does not “vary.”)

A variable symbol is* “*waiting” to be supplied with a unique number. Context is everything in distinguishing between variables and constants. By analogy, the word *he* in, “He is the President of the United States,” is like a variable. With no context or clarifying sentences surrounding this sentence, we cannot determine who “he” is, and so we cannot determine if the sentence is true or false (determining when a sentence is true or false becomes very important in later posts on variables). However, a little context and the word *he* becomes a constant symbol, “Look son, there goes Barack Obama. He is the President of the United States.” The word *h**e *is no longer a variable.

*Operator symbols.* * *These symbols include the addition symbol “+” and the multiplication symbol “×”. There are usually one or two blank spaces associated to operator symbols in mathematics. For example, the addition symbol in a mathematical sentence is really “___ + ___” with the two blank spaces to be filled in with constants or variables. Here are some other familiar operator symbols:

__ – __ |__| -(__).

Maybe surprisingly, the division symbol is not considered an operator symbol in *formal * mathematical logic because not ALL numbers can be placed in the blank spaces associated to the division symbol. For example, placing a 0 in the blank of 3÷___ is not allowed. However, the division symbol is defined in terms of the multiplication symbol, i.e., when , *a*÷*b*=*c* if and only if *a*=*b*×*c*. In school mathematics, it is customary to include the division symbol in the list of operators with the following caveat: the set of numbers that can be substituted must now be specified (like, 3 ÷ *x *for all *x ≠ *0).

Operator symbols are also called function symbols—they are, in fact, shorthand notation for functions (recall that a function is a rule that assigns a unique output of the range set for each input or inputs from a domain set).

Next Up: Part II: Expressions.

CHANNEL: *Engineering School Mathematics*

## Eradicating Textbook School Mathematics (TSM)

It is apropos to begin this website with a short article on Textbook School Mathematics (or TSM), which math education engineers are dedicated to eradicating from U.S. Schools Systems. Coined by Hung-Hsi Wu, Professor of Mathematics, UC Berkeley, it roughly means “something resembling mathematics as defined by standard K-12 mathematics textbooks.” Hung-Hsi Wu talks about it in more detail in his Fall 2011 American Educator article,

Phoenix Rising,TSM to a student is like eating a cold can of Spam for dinner. Sure, it is a meat product that was once made out of real ham, it functionally fills the stomach, it is vaguely nutritious, but somehow it is worse than going to bed hungry. TSM currently

isthe curriculum for both traditional and reform styles of teaching in the U.S. Both traditions get the actual mathematics incorrect (see Wu’s article for examples).Professional teachers have for a long time sensed that there was something wrong with the mathematics they taught—why didn’t it make sense? They often rebelled against TSM, but having grown-up with TSM themselves, they did not have the mathematical background to be able to express why TSM was wrong or how to address its failure.

The goal is to help teachers make math make sense again to their students, to engineer a new mathematics curriculum that is mathematically correct and ready for consumption by school students. The Eureka curriculum writers thought of themselves as

engineering school mathematicsin the sense that we are customizing abstract, university-level mathematics in a way that can be easily digested by school students. Let’s end with another quote from Hung-Hsi Wu article, which states,We only add that “correctly taught and learned” in the quote above is the work of curriculum writers and dedicated teachers, who through years of experience, know how to engineer lessons that bring real mathematics to life in ways that TSM doesn’t.

CHANNEL:

Engineering School Mathematics

© 2015 Scott Baldridge