Developing a Deep Understanding of Mathematics
February 2022 

Developing a deep understanding of mathematics begins in early childhood, continues through elementary school, and then builds further in middle school, high school, and beyond. A deep understanding of mathematics goes beyond algorithms, procedures, and knowledge—although all are important—to making conceptual connections and understanding underlying mathematical structures. One of my favorite characterizations to differentiate knowledge and understanding is from Wiske (1998), who describes knowledge as “information on tap” (p. 39) and understanding as “the ability to think and act flexibly with what one knows” (p. 40). Wiske goes on to say that students need to be able to use this learning both inside and outside of the classroom (p. 13).

Sound familiar? This parallels what is in NCTM’s Catalyzing Change series. The first recommendation, Broaden the Purposes of Learning Mathematics, states that each and every student should develop deep mathematical understanding; and the fourth recommendation, Develop a Deep Mathematical Understanding, goes further by specifically describing what this means by grade band (webinar handout).

• Early childhood settings and elementary schools should build a strong foundation of deep mathematical understanding, emphasize reasoning and sense making, and ensure the highest-quality mathematics education for each and every child.
• Middle schools should offer a common shared pathway grounded in the use of mathematical practices and processes to coherently develop deep mathematical understanding, ensuring the highest-quality mathematics education for each and every student.
• High schools should offer continuous four-year mathematics pathways with all students studying mathematics each year, including two to three years of mathematics in a common shared pathway focusing on the Essential Concepts, to ensure the highest-quality mathematics education for all students.

When you look at the three grade-band descriptions, what stands out to you? Consider it, write it down. Reflect on why you made those selections. A few key phrases that stood out for me were foundation, quality, practices and processes, coherent, and each and every. I sum it up this way: Developing a deep understanding of mathematics results from a solid, strong foundation of concepts supported through high-quality, equitable mathematics education for each and every learner. Building and developing mathematical processes and practices through a coherent, cohesive approach to mathematical concepts and procedures that values students’ positive mathematical identity is essential to developing this deep understanding. That is a great deal to unpack but powerful!

Now what does that mean for me as a teacher of mathematics? It can mean many things but here are two for us to consider. First it means I need to deepen my own understanding of mathematics. In 2010 Wohlhuter, Breyfogle, and McDuffie called us as teachers to continue our own learning because “developing deep knowledge and understanding of mathematics is a lifelong process, and building the foundation for teachers’ development must begin in preservice preparation and continue throughout one’s professional life” (p. 178).

Second it means we need to seriously consider what it means to teach mathematics for understanding. For a number of years, I have taught a course entitled Teaching for Understanding at my university. In that course, the goal is for the students, who are usually practicing teachers, to develop their own framework for teaching for understanding based on relevant research and effective practices that reflect goals for learning in their discipline. They are often from a variety of disciplines and have found this beneficial as teachers learn from one another across disciplines. They read about relevant theory, research, and practice to examine key elements that should be part of a framework that would support their role as a teacher in teaching for understanding and support their students in learning for understanding. As a group, they explore ideas of what understanding is, what it looks like, what the characteristics of teaching for understanding are, what barriers or challenges to teaching for understanding may exist, and even explore whether the goal is to teach for understanding. Thought provoking, rich discussions to be sure! We spend time reflecting on such questions as What topics are worth understanding? What about them must students understand? How can we foster understanding? How can we determine what students understand?

Maybe such an exercise would be useful for you to engage in as an individual teacher and collectively as grade-level, departmental, or mathematics course teachers (i.e., geometry, algebra, statistics, etc.). Maybe it could be a topic of discussion in a PLC (professional learning community), departmental meeting, a coaching session, or other professional development opportunity. What elements are essential to you to teach for understanding so that students learn for understanding? They might include aspects of the eight Effective Teaching Practices (NCTM 2014, p. 10) I discussed last month or might focus on areas such as assessment, collaboration, task choice, classroom environment, equity, use of technology, the role of reasoning and sense making, or others you might identify.

Whether the topic is number sense, fractions, functions, equivalence, proportional reasoning, probability, area, rate of change, or area under a curve, students need to develop a deep understanding of key mathematical concepts. This deep understanding is empowering and transformative. It builds a sense of agency and positive mathematical identity; it develops students’ confidence as knowers and doers of mathematics; and it opens up opportunities for all students as they pursue their future.

What does it mean to you to develop a deep understanding of mathematics and to teach for understanding? It’s a powerful question to consider!

Trena Wilkerson
NCTM President
@TrenaWilkerson

References

National Council of Teachers of Mathematics (NCTM). 2014. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM.

Wiske, Martha Stone, ed. 1998. Teaching for Understanding: Linking Research and Practice. San Francisco: Jossey-Bass.

Wohlhuter, Kay A., M. Lynn Breyfogle, and Amy Roth McDuffie. 2010. “Strengthen Your Mathematical Muscles.” Teaching Children Mathematics 17, no. 3 (October): 178–83.