It's Time To Abandon Computational Algorithms
It's time to recognize that, for many students, real mathematical power, on the one hand, and facility with multidigit, pencil-and-paper computational algorithms, on the other, are mutually exclusive. In fact, it's time to acknowledge that continuing to teach these skills to our students is not only unnecessary, but counterproductive and downright dangerous.
My intent in so brashly taking on one of the lingering pillars of basic elementary education is neither to be unduly confrontational nor impishly irresponsible. I am not seeking to wrap myself in the banner of radical reformers, nor to stir up the fundamentalists. Rather, my purpose is to raise an issue that is not going to disappear. It's an issue that must be discussed openly and honestly, and then resolved swiftly and clearly, if we are to realize our aspirations for truly world-class schools.
First, let's clarify exactly what is being proposed. This is not about reducing emphasis on one-digit addition, subtraction, multiplication, and division facts. These facts and the self-confidence that comes with their mastery are more important than ever. Nor is this about abandoning computation, which remains an indispensable part of the mathematics program, so long as it is done mentally, with a calculator, or via estimates. It is about the formal, paper-and-pencil computational algorithms that constitute the core grade-school mathematical experiences of most American youngsters. It's about mindless procedures like "carrying threes into the tens column,'' "six times seven is 42, put down two and carry the four,'' and "eight from two, can't do, cross out the five, make it a four, and borrow 10.'' It's also about memorized rules like "yours is not to reason why, just invert and multiply'' that, for most students, meaninglessly enter one ear and leave the other.
Usually this is all it takes to start an avalanche of dismay. The most common reactions are: "But weren't those rules and procedures good enough for us?'' "Isn't this just the 'new math' all over again?'' "But what if the calculator is lost or the batteries die?'' And "What about the basics?'' So let's make the case. Let's begin with what we know and what most of us can agree upon. Then let's see how acting on these agreements leads naturally and logically to abandoning the teaching of computational algorithms.
We know that less than a generation ago, real people in real situations regularly put pencil to paper, used an algorithm that had been practiced in school to the point of automaticity, and computed a solution to a problem involving numbers. We know that without the knowledge of a systematic procedure, the needed solution was usually unattainable. We know with equal clarity that this is no longer how the real world works. Today, real people in real situations regularly put finger to button and make critical decisions about which buttons to press, not where and how to carry threes into hundreds columns. We understand that this change is on the order of magnitude of the outhouse to indoor plumbing in terms of comfort and convenience, and of the sundial to digital timepieces in terms of accuracy and accessibility.
Shouldn't we be as eager to end our obsessive love affair with pencil-and-paper computation as we were to move on from outhouses and sundials? In short, we know and should agree that the long-division "gazinta'' (goes into, as in four "goes into'' 31 seven times ... ) algorithm and its computational cousins are obsolete in light of everyday societal realities.
We also know that a curriculum dominated by a strict hierarchy of skills and procedures has meshed perfectly with the historically perceived mission that schools serve as society's primary sorting mechanism. What better vehicle for anointing the few and casting out the many than demanding mastery of increasingly complex computational procedures--most often taught and learned in mindless, rote fashion? And how else can we account for the obeisance paid to the norm-referenced standardized test's bell curve of student achievement? But what is increasingly clear is how radically society's needs and expectations for schools have shifted. No longer are schools expected to serve as social and economic sorting machines. Instead, schools must become empowering machines. No longer simply perpetuators of the bell curve, where only some survive and even fewer truly thrive, schools and their mathematics programs must instill understanding and confidence in all. In short, we now understand that teaching formal rules for adding and subtracting decimals, like their "gone and not missed'' cousin, the square-root algorithm, remains a vestige of a sort-'em-out approach that continues to fail both kids and the society that so desperately needs a far more mathematically powerful citizenry.
But none of these larger societal issues is as compelling as what we know about the sense of failure and the pain unnecessarily imposed on hundreds of thousands of students in the name of mastering these obsolete procedures. We know that many students are bored to death and frustrated to tears when faced with "exercises number 1-29 (odd) on page 253.'' Compare the energy and enthusiasm of a class cooperatively learning statistics with bags of M&M's to a class mindlessly and individually inverting and multiplying meaningless fractions to arrive at equally meaningless answers. Compare a class where students are estimating costs for a shopping spree from newspaper fliers prior to using calculators to see who comes closest to $100, and a class tediously finding sums of columns of context-free numbers.
A few short years ago we had few or no alternatives to pencil-and-paper computation. A few short years ago we could even justify the pain and frustration we witnessed in our classes as necessary parts of learning what were then important skills. Today there are alternatives and there is no honest way to justify the psychic toll it takes. We need to admit that drill and practice of computational algorithms devour an incredibly large proportion of instructional time, precluding any real chance for actually applying mathematics and developing the conceptual understanding that underlies mathematical literacy.
We also know what is important and what no longer matters; we know what's the baby and what's the bath water. The "baby'' is having a $10 bill and seeing that Big Macs are $1.59 each. It's formulating questions about change and taxes and it's figuring out how many could be bought. The core of mathematical power is explaining why you think six Big Macs is a reasonable estimate. This "baby'' is explaining--orally or in writing--why division is an appropriate operation in this situation. It's interpreting the 6.2893082 on the calculator display, or it's presenting alternative approaches using repetitive addition or repetitive subtraction or trial and error with the multiplication key. Meanwhile, the "bath water'' is a pencil-and-paper procedure for dividing 10 by 1.59 on which no sane person relies. We know that it's time to change the "bath water.'' We also know that the more we continue to focus on the bath water of the mathematics curriculum, the less chance we have for really saving the baby.
So why do we continue to impose these skills on our students and teachers? Because despite all the powerful reasons for change, schools are equally powerful perpetuators of what they've always done. Ask an educator why long division is still taught and you will hear that it's in the text, on the test, part of the curriculum, and/or has always been taught. Never do you hear that it's needed or that it's important.
One wag has argued that it's easier to move a cemetery than to delete a topic from the school curriculum, so we shouldn't underestimate the enormous obstacles to making these changes. However, what better way for school administrators and boards of education to exhibit leadership than by convening community and parent forums on this issue. Why can't these forums be followed up with clear and direct policy statements about the appropriate focus of mathematics programs? And why can't we expect these policies to result in strong recommendations to simply skip certain chapters or pages in currently used textbooks and eliminate the use of the computation subtest on all standardized tests?
If we are true to our professed goals, the course is clear. It's time to build mathematics programs that engage and empower, unencumbered by the discriminatory shackles of computational algorithms. It's time to banish these vestiges of yesteryear from our schools and from our tests.
Steven Leinwand is a mathematics consultant with the Connecticut
Department of Education.