What Does It Mean To Know Math?

I just received a new book by Van Walle.  Van Walle works are based on developmental thinking and learning theories.  I love his work because it discusses both sides of the "math coin".  Many people believe math is about memorizing facts, rules, and procedures.  Although this is one part, or one side, of the math coin, it does not take into account the other side "mathematical thinking and reasoning".  Many people do not even realize that there are two sides of the coin.  Why?  Think back to how we all learned math in school.  The phrase, "We don't know what we don't know.", is so true.  If a person has never been exposed to thinking about math as thinking, reasoning, and problem solving, our lenses, which we view math is very narrow. 

In my job I have spent many years working on the learning theory in mathematics.  In order to be an effective math teacher there are two broad components: 

1.  Teacher needs an understanding of how children learn math.

2. Teachers need to understand math.  There is a difference between knowing a procedure to solve a problem vs. knowing why the procedure works.

 The rest of this blog will be focused on what does it mean to know math?  In the early 1980's our country began to reform mathematics.  The focus began to include problem solving as an important strand in the math curriculum.  In math we have content standards (Number Sense, Geometry, Algebra, Statistics, and Measurement) but NCTM also included process standards (Communication, Problem Solving, Representations, Reasoning, and Connections).  Unfortunately many teachers and parents aren't completely aware of these process standards and how it affects how mathematics is taught.

So where does a teacher begin?  Remember, we don't know what we don't know.  Many of us are doing the best job we can with the knowledge we have.  Part of a teacher's professional duty is to be willing to learn.  The necessary traits of a math teachers includes persistence, positive attitude, readiness for change, and self-reflection.  

Many elementary teachers will admit that math is not their strong point.  Well, writing is not my strong point, but I still practice, by writing this blog, and I don't just say "oh well, I just can't write".  Math is not our strong point because of the way we were taught math.  I'm not saying it's our teacher's fault neither.  It's just we can only teach in ways we are aware of.  We all are doing the best job we can.  Just like we want our students to be persistent, we need to be persistent.  We must realize that struggling is a part of the learning process.  Just embrace it!!

The second trait is positive attitude.  Our attitude, without ever saying a word, becomes the culture of our classroom.  Students can tell if their teachers don't like math.  In fact, the amount of time spent in math is directly correlated to the amount of time spent learning math in the classroom.  Hmmmm....

The third trait is to be ready for change.  Math reform is reflected in many of our state standards and curriculum resources.  Once again, continue to be a lifelong learner, practice what we preach.

The fourth trait is self-reflection.  Metacognition is important.  The thinking about our thinking.  We must take time to analyze what are our strengths and what areas of mathematics, or any other area, do we need to improve.  Then focus our time to learn.

A person who knows math has a profound, flexible, and adaptable knowledge of math content.  This person can:

  • think of real-world examples that fit the circumstance,
  • are able to use alternative strategies to solve the same problems,
  • can estimate a reasonable answer,
  • is able to draw a diagram to model how a process works, 
  • are capable to see if their strategy will lead to a reasonable solution.  

WOW!!  How's your math understanding of fraction?  Let's give it a try:  17 1/2  divided by  3/9 is about what?

  • Can you give a reasonable estimate?
  • Can you think of a real-world problem to go with this process?
  • Can you solve this in more than one way?
  • Can you draw a picture to model what this means?

Learning and knowing math is a complex system.  It is important we understand the difference in knowing a procedure (algorithm) or understanding the concept.  We need both.  My goal is to help as many teachers and students to really know mathematics.

Please respond if you have any questions, thoughts, or another perspective.