Math needs a revolution too

Math Needs a Revolution, Too, writes Barry Garelick in response to The Atlantic story, The Writing Revolution. He first encountered reform math when his daughter was in second grade.

. . . understanding takes precedence over procedure and process trumps content. In this world, memorization is looked down upon as “rote learning” and thus addition and subtraction facts are not drilled in the classroom–it’s something for students to learn at home. Inefficient methods for adding, subtracting, multiplying, and dividing are taught in the belief that such methods expose the conceptual underpinning of what is happening during these operations. The standard (and efficient) methods for these operations are delayed sometimes until 4th and 5th grades, when students are deemed ready to learn procedural fluency.

Students are expected to “think like mathematicians” before acquiring the analytic tools necessary to do so, Garelick writes. Procedural skills are taught on a “just in time” basis.

Such a process may eliminate what the education establishment views as tedious “drill and kill” exercises, but it results in poor learning and lack of mastery. Students generally work in groups with teachers who “facilitate” rather than providing direct instruction.

As reform math has become the norm in K-6 classrooms, high school math teachers are trying to teach algebra to students who “do not know how to do simple mathematical procedures,” he writes.

In math, as in writing, learning the fundamentals may not be fun or engaging. It may require practice. But students who skip the basics rarely develop the ability to “think like mathematicians” or write like “authors.” They’re confused. And bored.

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  1. Neither good writers nor good mathematicians achieved that status without years of studying and mastering the fundamentals. The need for repeated and directed practice in order to master essential skills is completely accepted – and demanded – in athletics and performing arts,but completely disdained in academics. Of course, no one obsesses about those kids who don’t make the varsity basketball or swim team or the orchestra and they don’t obsess over the color/flavor composition of such groups. In academics, it seems to be much more important to enable the pretense of equal outcomes across such groups. The fact that such outcomes are unconnected to actual knowledge and skills is not mentioned.

  2. Planning a curriculum is like writing a novel. When do you introduce the butler or the concept of a function? Where in the sequence to put Geometry or Combinatorics or the notation of Logic? Why suppose that a curriculum that works for a college-bound engineer has the same appeal to an aspiring cook or poet?

    Curriculum planners are control addicts. Markets and federalism institutionalize humility on the part of government actors. If a policy dispute involves a matter of taste, numerous local policy regimes and competitive markets in goods and services permit the expression of varied tastes, while the contest for control of a State-monopoly provider of a good or service (such as a school district) must create unhappy losers. If a policy dispute turns on a matter of fact, where “What works?” is an empirical question, numerous local policy regimes and competitive markets in goods and services will generate more information than will a State-monopoly enterprise. A State-monopoly enterprise is like an experiment with one treatment and no controls, a retarded experimental design.

    The content, pace, and sequence of a Math curriculum involves a mix of value judgments and empirical assumptions. The attempt to march all students through a uniform curriculum at a uniform pace imposes enormous costs on students, parents, teachers, and taxpayers.

  3. Endless drills of math facts in addition, subtraction, multiplication, and division is what I remember from my public school days in grades 1-6. Now, why doesn’t it work today is a better question, is it due to the fact that many students who attend elementary school never learned the basics mentioned above at home?

  4. Everyone likes to talk about balance. I went to a parent/teacher meeting about Everyday Math when my son was in fifth grade and that’s what everyone talked about. But it wasn’t getting done, even for some of the simplest skill parts of the balance equation. The spiral wasn’t working and it wasn’t an ability issue. Bright kids didn’t know the times table. This wasn’t an argument over whether kids needed to be experts at long division by hand. This wasn’t bad implementation. Teachers are told to keep moving through the material and to “trust the spiral”. It doesn’t work. It’s a fundamental flaw. Talk of motivation and engagement puts the onus on the student. Conceptual pie chart understandings of fractions don’t translate into the ability to manipulate rational expressions. Talk of understanding and problem solving hides the fact that simple skills are not being mastered.

    Reform math educators like to unlink understanding from mastery of the skills required to do math. While learning the times table might seem like learning the list of presidents, the same can’t be said for proper mastery of fractions. While rote learning might seem possible for the basics, it’s impossible to do by the time you get to fractions. One’s knowledge might be inflexible, but the solution is not more concepts, but hard work on problem sets that explore many different variations. You won’t get that by spending a lot of class time on a very limited set of real world problems, and whatever motivation those problems provide won’t translate into the motivation required to understand and master all of the variations one will get for homework. That’s how real mathematical understanding is developed.

    Spiraling, real world problem solving, and flipping the classroom put the onus on the student. These techniques assume that learning is some sort of natural process, and if they don’t work, then many just point to IQ issues. They can’t separate the variables and can’t calibrate the connection.

    Reform math curricula pump kids along until it’s easier to believe that the problem is one of IQ, parents, peers, society, and many other things. Even kids will believe it. They will tell you that they are not a “math brain” even though they just have gaps in very simple material. It’s too easy to overlook the variable of not ensurng mastery of the basics in the lower grades, especially when you don’t value mastery and when you see some kids do well. Where is the research that examines that variable? How difficult would that research be? How difficult is it to ask parents exactly what they do at home? Do you want to see my flash cards? Do you want to see my math workbooks? Obviously, schools must know something is going on at home. They send home notes telling parents to work on math facts.

    What is the test that determines whether kids get on the fast or slow math track in 7th grade? What, exactly, are those questions? From pre-algebra on, math consists of a steady diet of homework problem sets. In-class, real world, mixed ability group learning does not provide the understanding or motivation to get that job done.

  5. An important part of the math renaissance includes “answers before questions,” “solution keys” and a reinvention of homework. Same rules as the reading renaissance. Thanks.

  6. The problem with the “think like a mathematician” idea is that the people who create the curriculums are obviously not mathematicians. My kids’ school uses “Investigations in Numbers.” Frequently my kids have problems with their homework and it is almost always due to errors in their worksheets. I have a math degree and my wife has a math minor and often we have to spend several minutes figuring out what the problems are supposed to be about.

    For example, one problem said “Draw three lines of length 21 that start and stop on the same point. How long is the red line? How long is the blue line? How long is the green line?” The point of the problem was that perimeter and area are different. The questions were supposed to be “What is the area within the red line?” but the problem writers and reviewers did not realize this. Basic math incompetence like this is standard within homework assignments. The English skills are not much better. For example, “Cut a ribbon into pieces that are length 7, 8 and 9. How long will the ribbon be?” The ribbon WILL not be any length because you have cut it. It WAS length 24 before it was cut. This would be nitpicky when answering a question but if you are teaching math, you must make it so the question is obvious not misleading.