Barry Garelick

Math Teacher, St. Patrick’s Catholic School


Misunderstandings About Understanding in Math

The issue of balance between procedural fluency and conceptual understanding in mathematics has served as a dividing line in education. Some believe that understanding of a procedure or algorithm must precede the procedure/algorithm itself—and if it doesn’t precede it, it should come about quickly. Failure to do this results in students who some call “math zombies”. Others believe that procedural fluency and conceptual understanding is an iterative process where one feeds the other.

The learning process encompasses a spectrum that begins at the novice level and extends to expert levels. Students benefit by seeing worked examples which shows how to think about the problem before actually working it. Students then imitate the procedure, which ultimately becomes imitation of thinking.  As anyone knows who has learned a skill through initial imitation of specific techniques, such as drawing, bowling, swimming, dancing and the like, what seems like it will be easy often is more challenging than it appears. So too with math.

Most of the time, the expert level does not resemble the means of its nurture. Just as footballers and athletes do numerous drills that look nothing like playing a game of football or running a marathon, so the building blocks of final academic or creative performance are small, painstaking and deliberate.  As one goes up the scale from novice to expert,imitation of thinking includes many levels of understanding. Even at the most basic levels, the procedural understanding of novices is the foundation that allows them to reason mathematically, to solve problems and to build upon in developing conceptual understanding.


Barry Garelick teaches 7thand 8thgrade math at a K-8 school in California.  He retired from the federal government about 7 years ago.  While he was working in the DC area, he researched math education during a 6 month assignment to a US Senator in Washington DC, and over the past 10 years, wrote several books, and many articles about math education focusing on the evidence behind effective math instruction.  His articles have appeared in The Atlantic, Education Next, and the Notices of the American Mathematical Society..