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What is the difference between Arithmetic and Mathematics?

My favorite quick answer is…
“Arithmetic is to mathematics as spelling is to writing.”

The dictionary definitions of these two bodies of learning are:

a·rith·me·tic
(1) the branch of mathematics that deals with addition, subtraction, multiplication, and division,
(2) the use of numbers in calculations

math·e·mat·ics
(1) the study of the relationships among numbers, shapes, and quantities,
(2) it uses signs, symbols, and proofs and includes arithmetic, algebra, calculus, geometry, and trigonometry.

The most obvious difference is that arithmetic is all about numbers and mathematics is all about theory. In college, I have a vivid memory of Linus Pauling¹ delivering a guest lecture and after scrawling theoretical mathematics all over three blackboards, a student raised his hand and pointed out that 7 times 8 had been multiplied wrong in one of the earlier steps. Pauling’s answer was, “Oh, that… numbers are just placeholders for the concept.” And, he just waved away the fact that the numerical conclusion was obviously not accurate. Now, that was in the sixties before the plentiful access to calculators and computers, so his point is even more valid today. Learn the theory in mathematics and the calculators and computers will keep you accurate. That said, it is very important to stress that calculators have their place in our children’s education but not to the exclusion of their understanding the material with their own brain.

I have a friend who was a math major at Northwestern University… a real whiz at math with future plans in theoretical math…. Until, one summer when he discovered business and how well he could think on his feet. He could perform complex arithmetic in his head faster than anyone else and with his advanced problem solving abilities he had unique ways of thinking. He now owns 21 stores, over 400 employees, and travels around the world doing business in multiple languages with translators and making deals with his extraordinary ability to manipulate numbers accurately and quickly in his head. His non-dependence on calculators makes him the successful businessman that he is.

To be sure, both arithmetic and mathematics are abstract. There is a passage in “Zen and the Art of Motorcycle Maintenance”² where a father and his 9-year old son are traveling cross-country on a motorcycle and as they pass through badlands country, the father is talking about ghosts to his son. His son then asks his father if he, the father, believes in ghosts. The father answers abruptly and quickly with “Of course, not!” Then, he thinks about it and he explains to his son that maybe he DOES believe in ghosts because he believes in the number system and it is a ghost. A ghost is non-concrete, can’t be touched nor felt, no weight, no mass. What are numbers? They are symbols with meaning attached to them… and, for some, connecting the symbols with the actual counting process is very abstract. When we look at ancient Egyptian numbers, they are meaningless symbols to us unless we have taken the time to study and connect the symbol with its intended meaning. (For a good history of math website, visit http://www-history.mcs.st-and.ac.uk/~history/Indexes/HistoryTopics.html)

And, then, there’s my own experience with arithmetic, which I could do during elementary school, not very fast, but I could always do it. I didn’t “perk up” until algebra – for me, THAT was interesting and became more and more so as my education proceeded. But, arithmetic always “haunted” me in both my personal and professional life. In my personal life, friends were always giving me the check at restaurants to add and divide evenly among us – ugh, that was tedious and they just didn’t get that numbers were not my thing. Professionally, I have stood in front of the class and made some terrible arithmetic mistakes while doing complex math equations, but thank goodness for Linus Pauling, I didn’t take those mistakes too seriously. Its difficult for people to understand that you’re a math teacher but you really don’t care too much for numbers. It’s the problem solving and theories of math that I find fascinating.

Having spent most of my life teaching high school math, it was disheartening to hear my uncle say that what I am teaching is not “real math” – his world was teaching the math of particle³ physics to advanced graduate students at Stanford University. Only a handful of people in the world understood the papers he wrote. His definition of arithmetic is that it is structured and that math is not – in his mind, counting through calculus is arithmetic. The theoretical math in his papers was gibberish to me but symbolic prose to him – the “marriage” of math and science. From his point of view, until you get to advanced physics, the math is not “real” math. Perspective is everything.

In conclusion, arithmetic uses numbers and mathematics uses variables – each discipline has its own complexities and thought processes.

¹Nobel Prize Winner in Chemistry
²The author wrote autobiographically, wrestling with philosophical questions regarding the comparison of a romantic education and a classical education – feelings/emotions versus technology/rational thinking.
³Components of the nucleus of an atom

Note: The bridge between arithmetic and mathematics is basic algebra. See Below...

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