by Robert Craigen and Barry Garelick
It is obvious to many parents that something is off in their children’s math classes: Instead of learning math facts and standard methods, their kids must use cumbersome procedures, find multiple ways to do simple tasks, and explain in writing what they have done. In general, they need more help than their parents did when they were in school.
For two decades now parents and children have been collateral damage in a struggle that has come to be known as the “math wars”. Opinion is sharply divided on how best to teach K-12 math. The tension is between conventional, or traditional, instruction versus what is known by various names including “reform math” (to proponents), or “fuzzy math” (to critics).
Reform math differs from the conventional approach in many ways. To help parents ascertain whether their children are being exposed to reform math-borne illnesses, we have set out a brief guide to key symptoms of a reform math approach.
“In the past students were taught by rote; we teach understanding.” First, ‘rote’ literally means ‘repetition’ — and this is a good idea, not a bad one. Second, it is simply false that teaching was without understanding — by design, in any case — in the past. There have always been teachers who taught math poorly or neglected to include a conceptual context. This does not mean that conventional math was/is never taught well.
Under reform math, students are required to use inefficient procedures for several years before they are exposed to and allowed to use the standard method (or “algorithm”) — if they are at all. This is done in the belief that the alternative approaches confer understanding to the standard algorithm. To teach the standard algorithm first would, in reformers’ minds, be rote learning. But this out-loud articulation of “meaning” in every stage is the arithmetic equivalent of forcing a reader to keep a finger on the page, sounding out every word, every time, with no progression of reading skill. Alternatives become the main course instead of a side dish and students can become confused — often profoundly so.
“Drill and kill is bad.” Reformers believe that making students do repetitious ‘rote’ exercises will deaden students’ souls and impede true mathematical understanding. Actually the reverse is true: 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 you take away the child’s primary scaffold for understanding. As for killing fun, that all depends on the spirit of the exercise. Drills are boring only if they are made boring.
“The guide on the side, not the sage on the stage.” “Guide on the side” is also known as “student-centered learning”. Sounds wonderful until you realize it means that it reduces the teacher to a mere facilitator of holistic “inquiry-based” or discovery learning experiences. Students teach themselves. Providing information directly is regarded as “rote learning” – and a bad thing. Recent meta-studies in cognitive science by Sweller et al, and Mayer, have shown that “minimal guidance instruction”—the corresponding term in that field — is a very poor way to teach novices, though it has some merit for teaching experts. To be clear, “novices” would include elementary school children when learning arithmetic and 7th and 8th grade students learning algebra.
“‘Just in time’ learning.” This approach prescribes giving students an assignment or problem which forces them to learn what they need to know in order to complete the task. The tools that students need to master are dictated by the problem itself. For example, students might first encounter long division in a lesson, late in their education, about repeating decimals, where it is an essential ingredient. Many reformers consider long division too tedious and unproductive to teach until it is absolutely needed. The question of how repeating decimals work supposedly motivates students to overcome this barrier. This is like teaching someone to swim by throwing them in the deep end of a pool and telling them to swim to the other side. The teacher shouts the instruction to the students, who are expected to swallow the method whole along with mouthfuls of pool water, in one go. The students who by some miracle make it to the other side are apt to say, “I don’t know how I got here, but I sure don’t want to do that again!”
“Ambiguity is a great way to learn.” Another aspect of discovery learning. It reveals an underlying pattern that dominates Reform/Fuzzy math: it has no bottom-up structure, lacks coherence, and uses deliberately confusing elements to force a child to decide for themselves how to do this or that, and what, if anything, constitutes a correct answer. While children are psychologically unsuited for lack of structure and ill-defined expectations, reformers hold that “struggle is good”. For experts, struggle is suitable; e.g., an expert swimmer may struggle to perfect a swim stroke whereas a novice may struggle to keep from drowning—a struggle that doesn’t teach them how to swim.
“Flip the classroom!” Flipped classrooms can be implemented in a number of ways, but a trend emerging in poorly implemented reform math programs is the class becoming a homework-like learning-lab environment. The student is expected to learn at home by watching videos on the internet — videos consisting of direct instruction on mathematical procedures. The direct instruction of the classroom is often replaced with “stimulating and engaging activities”. This puts the onus on children to (1) have access; (2) be in a good home environment; and (3) self-motivate to pick up the lessons. But if a student does not understand something in the video, the rest of the lesson is not going to make sense. How much time does the teacher have the next day — in a lesson packed with inquiry-based activities — to backfill what students didn’t understand from the video?
And another inconvenient question: Isn’t the education community’s avalanche-like acceptance of the flipped classrooms a tacit admission that learning procedures is important?
“We’re making students think like mathematicians.” Professional mathematicians are often puzzled at what is meant by this. Mathematicians know that students need both to master procedures and to have a basic understanding of their conceptual underpinnings. Reformers make the mistake of not distinguishing between how novices learn and how experts think. Reformers are often heard to exclaim, “I wish I had understood it this way when I was learning it”. But children do not have that adult’s many years of experience. Denying them the foundational mastery to acquire mathematical expertise deprives students of essential formative experiences.
“Group learning.” Working in groups is not limited to just math classes. It has been a trend over the past two decades that shows no sign of letting up. Group work can be a healthy supplement to teacher-driven lessons or for highly social kids. But it is an inefficient way to get through a lesson in which new technical skills are to be learned. Here are four groups for which this approach is a particularly bad idea: (1) very poor performers—who shrink from participating and can panic at exposure among peers; (2) very high performers—who resent that others in the group look to them to carry the burden, (3) students with social handicaps—for obvious reasons; and (4) students with communication deficits—such as, but not limited to, having a different native tongue as classmates.
Finding a cure for a system that refuses to recognize its ills has proven futile. Parents confronting school administrators are patronized and placated. School officials will agree and say something like, “Yes, students should learn math facts and procedures (and we do this!). Yes, teachers ought to actually teach, (and we do this!). And yes, students should do drills (and we do this!)” This is all followed with: “We use a balanced approach,” which is often followed with: “We’re saying the same things; we’re in agreement”
The purpose of these bromides is twofold: 1) to make everyone feel good, and 2) to make parents go away. Pressed to define what “balance” means, the reform camp will say, “Show why things work first to gain understanding; then use the understanding to teach traditional mathematical operations!”
Such statements reveal internal biases about priorities — priorities that intrinsically lack balance. Whether understanding or procedure comes first ought to be driven by subject matter and student need — not by educational ideology.
And in answer to the statement that we’re all saying the same thing: No. We’re not saying the same thing at all.
Why don’t those arguing for better math education (and who insist they are using a balanced approach) look at what those students are doing who are succeeding in pursuing majors in science, engineering or math? If they did, they would see students learning standard algorithms and practicing many drills and problems (deemed dull, tedious and “mind numbing”) and other techniques that they believe do not result in true, deep, and authentic understanding.
But such an outcome based investigation is not occurring. Some parents whose children are not doing well in math believe what they hear from school administrators that, “Maybe your child just isn’t good at math.” Parents who recognize the inferior math programs in K-6 for what they are get their children the help they need. Unfortunately, parents who lack the means have fewer options.
Robert Craigen is associate math professor at University of Manitoba and co-founder of Western Initiative for Strengthening Education in Math (WISE Math).
Barry Garelick has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California. He is co-founder of the U.S. Coalition for World Class Math.