What Am I Teaching?

When we teach math, what are we teaching?  I think of it in layers.
 
On the surface, students learn to solve problems of increasing complexity and sophistication, using mathematical tools of increasing sophistication.  They learn how to multiply, add fractions, solve equations, factor quadratics, etc.  But the goal of math education has to be more than that.  In fact, if that is all we have achieved, we have literally failed.
 
So what is below the surface?  Mathematical thinking.
 
There is a reason that we have fields of study that have endured for thousands of years.  Literature, music, science, philosophy, mathematics all contribute a useful and unique way to view and process the world.  Mathematical thinking is a way to deal with quantities and patterns that can be valuable in many contexts.
 
Broadly, I think there are three areas of mathematical thinking we deal with in primary and secondary education.
 

• First of all we teach numbers.  What is 3, 4/5, -7, or sin(2π)?  Starting with whole numbers, we help students understand what these quantities represent, and what their existence implies.  There are relationships within and between number families that students need to explore and investigate on an ongoing basis.
  

• Secondly we teach operations.  We have to remember that the symbols on the page are code for something else.  Think about what is written here: 4/5.  There is a 4, a line, and a 5.  The symbols don’t tell you anything about what 4/5 means.  When we teach operations like addition or multiplication, we can teach those operations in a purely symbolic way: what to do with the various numbers in different positions on the page.  What does it mean to divide 4 by 5?  What does it look like?  When we multiply two fractions, what does it mean to take 2/3 of 4/5?
  

 
I want to be clear: I am not saying we need to teach students why the various algorithms work (perhaps with the purpose of having them remember them better).  I am saying we want students to understand what the operations we use in mathematics represent.  When we divide 4 by 5, what are we actually doing to the 4, and what role does the 5 play in the operation?  Mathematics describes and uses certain operations on numbers (and algebraic expressions) that our students should explore and understand.

• Finally, we teach generalization.  Sometimes people refer to this as abstraction.  The process of noticing patterns and making up “rules” or routines that apply in general occurs on so many levels – all students have an opportunity to do it on a daily basis in math class.  The process of observing something, finding patterns, and making generalizations, is a process called inductive reasoning.  The concept is powerful but not perfect - the further we get from pure numbers the more carefully it should be applied.  But this is the crown jewel that mathematics and we math teachers have to offer the world.  I would say that there is no point to teaching math if we do not teach this.  And we have important choices to make about our teaching which will determine whether it is present or absent from our classrooms.  If we train students in manipulating symbols on a page, they have no opportunity to experience the process of generalizing, let alone learn to do it themselves.  But if we talk to students about the process and ask them on a daily basis what they see, what they notice, and help them grow in the ability to make conclusions and develop algorithms, we can grow their skills in generalization.
  

 
What is most important?  I think we have to have a zen-like duality in our minds, keeping a close eye on both computational competence and mathematical thinking.  Both measure our progress, but either one alone is not enough.  And they are certainly not mutually exclusive.  Does it hurt students to teach them an algorithm?  Not if you also build a classroom culture in which students understand that doing mathematics means exploring and reasoning and building generalizations for themselves.