What does it mean to do, learn, and understand mathematics?

John A. Van de Walle, a mathematics teacher, consultant and author, puts it this way, "Doing mathematics means generating stratgies for solving problems, applying those approaches, seeing if they lead to solutions, and checking to see if your answers make sense.  The best learning opportunities are those that engage learners in using their own knowledge and experience to solve problems through social interactions and reflection.  The question "Does she know it?' must be replaced with "How does she understand it? and "What ideas does she connect with it?"  As teachers plan and design instruction, they should constantly reflect on how to elicit prior knowledge by designing tasks that reflect the social and cultural backgrounds of students, to challenge students to think critically and creatively, and to include a comprehensive treatment of mathematics."  from Elementary and Middle School Mathematics, Teaching Developmentally

In order to have a balanced math program, there must be three essential components: Conceptual Understanding, Procedural Fluency, and Application.

Conceptual Understanding: According to the National Council of Teachers of Mathematics (NCTM), "Conceptual understanding must both precede and accompany instruction on procedures." Learning is most effective when students see clear, explicit connections between concepts and procedures, reinforcing their understanding through repetition and iteration. A strong conceptual foundation allows students to develop reasoning strategies that deepen their understanding, rather than relying on rote memorization of algorithms. When students use procedures without truly understanding them, they are more likely to make errors and may struggle to recognize when their answers are incorrect.

Procedural Fluency: Procedural is the ability to accurately, efficiently, and flexibly apply mathematical procedures to solve problems, including the knowledge of when and how to use different strategies depending on the situation, not just memorizing a single method.  The key thing to note is that procedural fluency is best achieved when built upon a strong foundation of conceptual understanding. 

Application: Application of procedural fluency" in Common Core math refers to the ability to not only perform mathematical procedures accurately and efficiently, but also to flexibly apply those procedures to solve real-world problems, choosing the most appropriate strategy based on the context, and demonstrating understanding by explaining the steps taken to reach a solution. 

Teaching math seems simple if your only focus is developing student's skills in completing a procedure.  But as you can see, mathematics is much more than knowing how to complete algorithms.  Be encouraged.  Try new things.  Ask your students what they think about numbers and it's guaranteed that you will be surprised by what they think and what they can do.  Help students to apply their skills and ideas to specific contexts. Develop the BIG ideas in math so they are carried through a student's lifetime.  That's what we remember and that's what it means to learn.  And most importantly, have FUN!