What is Mathematics?

That doggoned Cathy Carroll…I have still not stopped thinking about her colloquium and podcast which pushed us to redefine “fluency”.  So let’s take that even further!  What is “mathematics” anyway?

If I were to ask a primary teacher, “What is mathematics?”  I would expect them to say something about learning to add and subtract.  Intermediate teachers might include multiplication and division.  Typically, an elementary teacher’s response is something involving arithmetic and I would expect the parent of an elementary student to say the same.  So is mathematics defined as arithmetic or is arithmetic just a slice of that which we call mathematics?

This is not a new concept. In 1998, the National Council of Teachers of Mathematics released its Principles and Standards for School Mathematics.

In it, they defined mathematics content as having six main areas of focus: Number & Operations, Algebra, Geometry, Measurement, and Data Analysis & Probability.  In addition, NCTM defined five process standards: Problem Solving, Reasoning & Proof, Communication, Connections, and Representation. All of these content and process standards were expected to be incorporated into every mathematics classroom, K – 12.

Fast forward to 2010…  the Common Core Standards actually built on these revolutionary NCTM Principles and Standards.  The content is now delineated into 11 different domains (many of the same as NCTM) and there are now eight Standards for Mathematical Practice.

Since 1998, mathematics has been defined much broader than arithmetic, so why are we still confused?  Why has our elementary curriculum put such an emphasis on arithmetic these past 20 years?  (After all, a free 10-key calculator can do all of our arithmetic for us!)  What is mathematics???

According to Wikipedia, “Mathematics has no generally accepted definition.”  Well, that’s not very helpful…

Dictionary.com defines mathematics as, “The study of numbers, equations, functions, and geometric shapes and their relationships.”  They then add in some branches and major subdivisions…

Merriam-Webster defines it as, “the science of numbers and their operations, interrelations, combinations, generalizations and abstractions, and of space configurations and their structure, measurement, transformations, and generalizations.”

So maybe Wikipedia is right but going back to my last blog post about fluency, the ability to express oneself easily and articulately, I think I know what mathematics is.  Mathematics is the ability to make sense of our world.  Mathematics is our way of representing our world – of expressing what we find in our world easily and accurately.  Mathematics is the ability to solve problems – real problems – in our world.  So, yes… arithmetic is an important aspect of mathematics, but it is not the definition of mathematics.  We have to redefine mathematics for our colleagues, our administrators, our parents, and our communities.  Mathematical skills are not enough if we cannot use them in our world.

I’m a big fan of Donald Duck.  In 1959, Disney released “Donald in Mathmagic Land”.  I showed it to my high school students every year.  (I still own it in VHS format.) I’ll close with the definition put forth in this classic movie.  

In the words of Galileo, “Mathematics is the alphabet with which God has written the universe.”

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