Teaching Mathematics for Conceptual Understanding

There is a common thread with many students I have come in contact with over the last few years, when teaching and discussing mathematics.  Many students are:  1. not remembering math from year to year; 2. not able to easily transfer math skills to other subjects, and 3. not able to problem solve. As a teacher of mathematics and math coach, this is troubling. 

In this microwave era of, “I want it now,” and “what’s taking so long,” many teachers of mathematics have resorted to equipping students with a series of algorithms and procedures to commit to memory.  However, as we are preparing our students to be college and career ready, long gone are the days where mathematics equates to numbers only.  To equip our students to compete locally and globally, our focus must shift to helping students develop insight, versus only procedural skills.  Thus, the much debatable topic of teaching for conceptual understanding and learning is worth delving into. It is not surprising that when polled, many K-12 teachers are not familiar with what teaching conceptually looks like, and how it is carried out in the classroom. Many of us sat in classrooms where we watched a teacher go through solving a problem, wrote down the “steps,” and then proceeded to practice fluency by working another 25 to 50 problems. No wonder many parents have labeled their child’s math homework as “new math!” This is understandable because of the early focus in education to produce industry workers (i.e. assembly line workers, welders, automobile workers). But what this teaching lacked was equipping students to: make sense of problems and persevere in solving them, reason abstractly as well as quantitatively, construct viable arguments and critique the reasoning of others, model with mathematics, use appropriate tools strategically, attend to precision, look for and make use of structure, and look for and express regularity in repeated reasoning.  Sound familiar?  It should, because these are the Standards for Mathematical Practices, the way in which we should deliver math content. Now, our task as teachers is to prepare students for jobs and careers that may not exist yet, and teaching through the lens of the Standards for Mathematical Practice will ensure conceptual understanding. This knowledge takes them far beyond procedures and fluency, arming them with the critical thinking and problem solving skills needed to be successful today and in the days to come.

The Learning Principle from the NCTM Principles and Standards for School Mathematics (2000) is a good resource to gain an understanding of conceptual knowledge.  This principle states: “Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.” This supports the fact that rote memorization is not the key to high achievement in mathematics, especially if our students don’t understand the math.  However, I do not discount procedural fluency in any way.  But, there must be a balance in order to improve student achievement. 

So, how do we correct this lack of conceptual understanding in the mathematics classroom?  Allowing students to model concepts, use manipulatives in class, and express their findings in words will help students gain an understanding of complex ideas.  Give students opportunities to express and defend their thinking, as well as receive constructive feedback from peers and the teacher. This should not be limited to learning at the elementary level.  Take time to incorporate the practice of estimating.  Also, giving students an opportunity to express math concepts in multiple ways leads to conceptual understanding, in essence, understanding the why before practicing procedural fluency (the how).  Dan Meyer says it best in his TED talk, Math Class Needs a Makeover, and offers examples on how to begin changing our delivery to best meet the needs of students of mathematics.

After sharing these ideas with my middle level colleagues, I was pleased to find that several teachers have changed their practice to include the modeling of concepts with an understanding of why/how it works, before delivery of procedural knowledge. Also to my delight, while conferencing with a reflective teacher, she shared a new practice she will incorporate to begin assessing conceptual understanding.  By simply inserting a “think check” component into her daily lesson, she will learn if students are taking away an understanding of why, and not only how.  This will also remind her to plan for conceptual understanding in her delivery, so students will be able to answer such questions each day. 

Here is an image of an example:

Related Resources to aid in teaching mathematics for conceptual understanding:

Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 6-8 (Volume III) (2nd Edition) (New 2013 Curriculum & Instruction Titles) - Illustrates what it means to teach student-centered, problem-based mathematics, provides references for the mathematics content and research-based instructional strategies, and presents a large collection of high quality tasks and activities that can engage students in the mathematics that is important for them to learn.

LearnZillionInstructional Videos -  See visual, conceptual explanations of the Common Core State Standards, along with guided practice and note-taking guide.

3-Acts Math Tasks Inspired by Dan Meyer - Storytelling to provide a framework for certain mathematical tasks that is both prescriptive enough to be useful and flexible enough to be usable. One minute of video or one photo to tell the start of a mathematical story that will engage learners in asking a question.

Mathalicious -Real-world lessons help middle and high school teachers address Standards while challenging their students to think critically about the world.

Conceptual Math.org - Promotes math as a tool for understanding yourself and the world around you.

Statistics Education Web - Improve statistics education at all levels, with relevant, useful, and meaningful applications.

Works Sited:

National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.