Fluency.

 

The fourth overall topic for teaching for mastery is fluency.  Fluency is defined as being able to use what they learn effectively and easily.  The foundation of fluency is conceptual understanding, strategic reasoning, and problem solving to do math automatically.  In other words, students connect conceptual understanding with strategies and methods so the methods make sense to them.

Unfortunately, fluency is often associated with computing basic facts quickly.  Fluency is this and so much more.  It is more than memorization, accuracy, and speed.

Fluency allows students to solve deeper problems that are more meaningful and are likely to encounter in real life. It also helps students do better in the math class and in life.  Fluency is considered to include how efficiently a student solves a problem using the fewest number of steps.  They do not get bogged down trying to use too many different steps to solve the problem, and they are able to immediately use the results.

They also need to be accurate in their record keeping (how they show their work), knowledge of number facts, and other relationships.  They are also double checking any results.  Finally, they need to know more than one way to solve a problem. They need to have a flexibility which allows them to choose the most appropriate method to solve the problem and a different method to check their work.

It has been shown they need to be fluent both in procedure and in concepts.  Conceptual fluency helps them understand why.  When they don't understand why, they often forget or remember incorrectly.  Furthermore, when they memorize everything, they have difficulty bringing meaning to their work later on. 

Students need to have procedural fluency and conceptual fluency to be considered fluent.  It is important to include activities which help the student develop an understanding of concepts and relate them to procedure.  Manipulatives are a good place to start.  Let me know what you think, I'd love to hear.  Have a great day.