Conceptual Understanding vs. Procedural Fluency

What does it mean to teach students mathematics for conceptual understanding and procedural fluency? The National Academies (Adding it Up) highlight five strands that supports students to become good math learners!

Students must develop conceptual understanding of mathematics as well as procedural fluency. They also point out that students also need to develop productive dispositions, the ability to reason logically and be able to formulate, represent and solve mathematical problems. Basically, these ideas are highlighted in the Common Core Standards of Mathematical Practice document.  The Inside Mathematics website provides further explanations and video examples of each practice.

What does this mean for lesson planning and whole class discussions?  Let’s explore!

What is conceptual understanding? You can read more about conceptual understanding by clicking on the link.

Basically, it means that students need to have a flexible understanding of mathematics to use their knowledge as a tool to solve problems.  The Common Core document outlines what students should understand and be able to do. You can also look at the progressions documents provided in this blog.

An example:

Let’s take a look the third grade Common Core standards for on Operations and Algebraic thinking.  The standards indicate that third grade students must be able to multiply and divide within 100. However, it does more than that! The standard also points out  what students should also understand conceptually.  For example,representing and solving problems, and understanding properties of multiplication is essential for developing conceptual understanding. Therefore, simply having students memorize multiplication and division facts is not enough. Understanding the math concepts helps student solve problems other related problems. This means knowledge becomes a tool to support learning.If the students forget how much 5x 6 is, then they can use their knowledge of 5×5 and their understanding of multiplications to figure out the answer.

What does this mean for you as a teacher? The Whole Class Discussion book has a chapter on Lesson Planning (Chapter 4).

It provides you with tools to find the “big mathematical ideas,” and think about planning lessons  to support conceptual understanding and procedural fluency.  This book describes three levels of lesson planning.  Long term planning, short-term planning and even planning and adjusting while teaching. This builds on the idea that supporting student learning of math involves understanding the big picture of what students should learn and adapting your teaching to build on student understanding.  This way, your teaching becomes a lot more efficient and effective!


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