Barry Garelick, a veteran math teacher in California and respected observer of math instruction, recently reached out after seeing my Q&A with ST Math’s Andrew Coulson on using visualization to teach math. Garelick is a cogent thinker, clear writer, and author of books including Out on Good Behavior: Teaching math while looking over your shoulder and Math Education in the U.S.: Still Crazy After All These Years. Given all that, I thought his reflections well worth sharing—see what you think.
Rick, I thought your recent interview with Andrew Coulson of ST Math was a fascinating look at how educational products—particularly those that address math—are promoted. In the interview, Coulson states that the “innate ability of visualizing math was not being leveraged to solve a serious education problem: a lack of deep conceptual understanding of mathematics.”
As someone who has been teaching math for the past 10 years and written several books on key issues in math education, this struck a chord for me. I’ve seen the three-decade-long obsession with “deeper understanding” cause more problems than it solves—including overlooking other factors contributing to problems in math education, such as the disdain for memorization, the difference between understanding and procedure, and the issue with trying to teach problem solving solely by teaching generic skills. Undoing these would be a long-overdue step in the right direction to reverse the trends we are seeing in math education.
For starters, many math reformers seem to disdain memorization in favor of cultivating “deeper understanding.” The prevailing belief in current math-reform circles is that drilling kills the soul and makes students hate math and that memorizing the facts obscures understanding. Memorization of multiplication facts and the drills to get there, for example, are thought to obscure the meaning of what multiplication is. Instead of memorizing, students are encouraged to reason their way to “fluently derive” answers. For example, students who do not know that 8×7 is 56 may find the answer by reasoning that if 8×6 is 48, then 8×7 is eight more than 48, or 56. (Ironically, the same people who believe no student should be made to memorize have no problem with students using calculators for multiplication facts.)
Unfortunately, this approach ignores the fact that there are some things in math that need to be memorized and drilled, such as addition and multiplication facts. Repetitive practice lies at the heart of mastery of almost every discipline, and mathematics is no exception. No sensible person would suggest eliminating drills from sports, music, or dance. De-emphasize skill and memorization and you take away the child’s primary scaffold for understanding.
Teaching procedures and standard algorithms is similarly shunned as “rote memorization” that gets in the way of “deeper understanding” in math. But educators who believe this fail to see that using procedures to solve problems actually requires reasoning with such methods—which in itself is a form of understanding. Indeed, iterative practice is key to attaining procedural fluency and conceptual understanding. Understanding, critical thinking, and problem solving come when students can draw on a strong foundation of relevant domain content, which is built through the “rote memorization” of procedure. Whether understanding or procedure is taught first ought to be driven by subject matter and student need—not educational ideology. In short, of course we should teach for understanding. But don’t sacrifice the proficiency gained by learning procedures in the name of understanding by obsessing over it and holding students up when they are ready to move forward.
Finally, although it’s been shownthat solving math problems cannot be taught by teaching generic problem-solving skills, math reformers believe that such skills can be taught independent of specific problems. Traditional word problems such as “Two trains traveling toward each other at different speeds. When will they meet?” are held to be inauthentic and not relevant to students’ lives.
Instead, the reformers advocate an approach that presents students “challenging open-ended problems” (sometimes called “rich problems”) for which little or no prior instruction is given and which do not develop any identifiable or transferable skills. For example, “How many boxes would be needed to pack and ship 1 million books collected in a school-based book drive?” In this problem, the size of the books is unknown and varied and the size of the boxes is not stated. While some teachers consider the open-ended nature of the problem to be deep, rich, and unique, students will generally lack the skills required to solve such a problem, such as knowledge of proper experimental approaches, systematic and random errors, organizational skills, and validation and verification. Students are given generic problem-solving techniques (e.g., look for a simpler but similar problem), in the belief that they will develop a “problem-solving habit of mind.” But in the case of the above problem, such techniques simply will not work, leaving students frustrated, confused, and feeling as if they are not good at math.
Instead of having students struggle with little or no prior knowledge of how to approach a problem, students need to be given explicit instruction on solving various types of problems, via worked examples and initial practice problems. After that, they should be given problems that vary in difficulty, forcing students to stretch beyond the examples. Students build up a repertoire of problem-solving techniques as they progress from novice to expert. In my experience, students who are left to struggle with minimal guidance tend to ask, “Why do I need to know this?,” whereas students given proper instruction do not—nor do they care whether the problems are “relevant” to their everyday lives.
At the end of the day, finding a cure for a system that refuses to recognize its ills has proved futile. Parents confronting school administrators are patronized and placated or told that they don’t like the way math is taught because it’s not how they were taught.
Change will not come about by battling school administrations. There must be a recognition that the above approaches to teaching math are not working, as is currently happening with reading, thanks to the efforts of people like Emily Hanford, Natalie Wexler, and others, who have shown that teaching reading via phonics is effective, whereas memorizing words by sight or guessing the word by the context or a picture is not. Until then, only people with the means and access to tutors, learning centers, and private schools will be able to ensure that their students learn the math they need. The rest will be left to the “equitable solutions” of the last three decades that have proved disastrous.
Barry Garelick is a 7th and 8th grade math teacher and author of several books on math education, including his most recent, Out on Good Behavior: Teaching math while looking over your shoulder. Garelick, who worked in environmental protection for the federal government before entering the classroom, has also written articles on math education for publications including The Atlantic, Education Next, Nonpartisan Education Review, and Education News.