by Paul Dunham
Education is a field fraught with fads. Trends swing back and forth like pendulums, and those involved tend to cluster on one extreme or the other. A decade ago the “Reading Wars” raged while opposing groups pulled the “whole language/phonics” pendulum to one extreme or the other. Finally it came to rest somewhere in the center. The math education pendulum swings between the extremes of content and context. It’s high time for the math pendulum to settle in the center as well.
Content can be described as the basic building blocks behind math. For example, what is division and how does one divide two numbers without a calculator? What are the rules, symbols, and techniques of Algebra and how can they be used to solve an equation? What kinds of numbers exist, what are their characteristics, and what is necessary to perform operations on them? Mathematics content is a highly structured and very well understood topic. Content traditionally taught through high schools in the US has been around for several hundred years, without substantial change. Learning mathematics content is a process of study that necessarily starts with a foundation of basic principles and builds upon them in a strictly ordered manner.
As an engineer, I’ve been trained to see the entire world as an enormous word problem. In other words, I’m comfortable mapping concepts of mathematics to real world contexts. This is only possible because I learned the content so well and have applied it so long it has become second nature. Much of the success I have had in my career as an engineer I owe to a few excellent math teachers I had when math was still being taught the way it had been taught to generations of this county’s youth. Looking back on the accomplishments of these generations, and my own, we didn’t do too badly. Just what was the problem that needed to be fixed? What needs to be “reformed” about math ed in the US, anyway?
What the proponents of modern “reform” mathematics dislike about the way math content has been taught is that it can turn into a very dry exercise. I had a few good teachers, but they weren’t all good, and many students aren’t lucky enough to have even one. Students presented with abstract problems to solve may become so focused on the mechanical procedures behind the process that they fail to learn why mathematics behaves the way it does or how it maps to actual situations in our environment. Many reform efforts over the last 15-20 years have focused on teaching concepts of why and where math can be applied, which is not in itself a bad thing. Unfortunately, they have taken the pendulum too far. Reform curricula tend to neglect the teaching of content in favor of the more fashionable emphasis on context. Their authors promise that by doing this, students will be more engaged in learning and will learn things that are more useful for their future lives. From what I have seen in high school classrooms, student engagement in the act of learning is, if anything, worsening as a result of curricula such as Core Plus. But student engagement, attitudes, and discipline are all topics well beyond the scope of what I’m writing about here. Suffice to say that these problems won’t be made to vanish with the adoption of a new math curriculum. And if readiness for college is considered a desirable outcome of a high school education, the boom in private math tutoring and math remediation in local colleges shows that reform-approved Core Plus isn’t serving them well in that regard either.
I have had many students in my community college classes that typify, I think, what proponents of reform math are afraid of. They have come to see learning math as a process of memorizing procedures, and they tend to overlook the principles behind their application. They’re reluctant to embrace the idea that math might actually be useful in their lives or in their jobs, and wish only to be able to replicate a set of procedures for an exam so that they can move on and forget the entire experience. Their skills at carrying out those procedures may be sound, but they have trouble identifying which procedures to choose for a particular problem. For example, they may remember the Pythagorean Theorem, and the quadratic formula, but can’t remember which of the two is used for what kind of problem. It’s as if, when presented with the need to cut a piece of wood, they can’t decide whether they need a hammer or a saw. You can hammer a nail with a saw, but it won’t work very well. You can also cut a piece of wood with a hammer, but that won’t work very well either.
Some will argue that many in our society can get along fine just knowing the mechanics of simple mathematical operations, and I suppose that this is true to a degree. Does a carpenter or surveyor need to know just why the Pythagorean equation works? Perhaps not, so long as they can keep track of where it is valid and where it is not through other mnemonic means. But that’s pretty limiting. An understanding of the basis of such a concept is necessary before one can recognize new applications for it. This is the difference between a technician and an artisan, an assembler and an inventor, a proofreader and a novelist. We need both assemblers and inventors in our society, and it is not the role of a teacher, an administrator or a textbook publisher to decide who will be what. Their roles should be to make both paths available, and to keep open the gates that allow students to find the path that serves them best.
My experiences teaching math in a community college have made it plain to me that lack of skills with content is the main roadblock that students face when attempting to apply math to a particular context. It is very difficult for me to fathom just how the reform math folks can think that learning nothing but concepts and contexts, in lieu of content, will prepare people to do the kind of things that I do.
Every analogy has its limitations, but in many ways I see Math as a tool box (This is not to be confused with the Core-Plus concept, which for most students winds up being little more than a crib-sheet). A carpenter (or mechanic) needs more than just a box of tools. They need to develop skills to use the tools appropriately and efficiently. Here’s where the tool-box analogy needs to be expanded a bit. The skills needed for the use of hand tools have two components. One is intellectual, context based (why is the saw the appropriate tool to cut the wood, and what attributes does the saw have that make it so) and the other is muscle memory (the strength and hand-eye coordination needed to cut a straight line). It’s the intellectual skill part that proponents of reform mathematics emphasize, and I believe that it’s an important part of what math students need to learn. Without it, content is superficial and unlikely to be remembered long enough to come in handy. But the deeper, more visceral component of skill, necessary for true mastery, comes only through practice. Proponents of reform math have generally discounted practice as base and demeaning, and often malign it as “drill and kill”. Practice is often not popular with students either, and yet ironically, it is the students who like it the least that need it the most. Practice is needed to attain the level of skill that college level work in technical fields requires.
People who merely endured a series of math classes long ago and moved on, never to do math again, may not have developed this level of mastery. As a result, they are unlikely to recognize its value. Those who find themselves in college without these skills can always enter a program in an Education school, of course, and some find themselves teaching math in our schools. Some of those leave the classroom and go on to positions of authority as Math Education Professionals. Having no first hand basis for an appreciation of the level of mastery needed for real applications, these Math Education Professionals will dwell instead on the intellectual component of skill and dictate to teachers, departments, and districts how math should be taught. It’s a peculiar and unfortunate fact that those in charge of training our youth for technical careers have little if any first hand knowledge of what those careers require. I believe that this warped setup is principally responsible math education pendulum’s current extreme position.
Math Education Professionals have developed their own community independent of both the Mathematics (content intensive) and the Technical (context intensive) communities. This community is a theocracy rooted in theories that members must adhere to in order to gain and maintain professional stature. The result is the widespread use of curricula such as Core Plus, which pull the pendulum so far to the context side that it puts students in a position of fumbling with tools that they aren’t given a chance to become familiar with. Content is cast aside in favor of context, and practice for personal mastery is cast aside in favor of group interaction. The teacher’s role is as “guide on the side” to loosely direct what happens in class: “Here’s a piece of wood. Let’s talk in groups about whether we would use a hammer or a saw to cut it with. Try it each way and tell each other how it makes you feel. OK, now let’s move on”. This is little more than Math Appreciation training. A few attentive and diligent students might glean a glancing appreciation of the intrinsic nature of the tools and why they are used for particular applications, but aren’t given an opportunity to take it to the next level. Most of the students, however, are just enduring the ordeal and won’t retain anything of value in the way of either content or context. They’d be better served if they were given a pile of wood, a saw, and assigned to develop the skill to make clean, straight cuts.
Students and their parents must demand that the pendulum be allowed to settle on a sensible balance of content and context. Alternate tracks (yes, a very un-PC term) must be re-established to allow college bound students to gain the skills they’ll need without private tutoring, and those not headed for college to establish some core competency with basic content. Core Plus is an abomination that serves few, if any, students well. The education community is fond of saying that all students are different; it’s time they stopped insisting that all be served from the same trough of faddish pedagogical gruel.
