May 2001 -- When I was a boy in school, my father told me, often, that mathematics was the most important subject. "And the thing in mathematics," he always advised, "is to know the formulas. You have to be able to remember the formulas." I didn't agree with him, at least not about the formulas. Most of my classmates, and most of my teachers, memorized and applied formulas and algorithms (fixed, step-by-step procedures for solving mathematical problems) with little idea of their meaning. This seemed ridiculous. The important thing in learning mathematics, I thought (though I would not have been able to phrase it this way at the time) is not the formulas; it is grasping the conceptual structure of the subject.My disagreement with my father contained the essential elements of the current Math Wars, the debate that is going on today over the way that mathematics should be taught.
The reformers, who represent the educational establishment in the Math Wars, take the position that mathematics curricula should stress conceptual thought and should de-emphasize mastery of the traditional algorithms. Their guiding document is the National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics, which was published in 1989. A new version, subtitled Standards 2000, of which I have a draft, was issued last year. The document is clear regarding this point:
Many adults are quick to admit that they are unable to remember much of the mathematics they learned in school. In their schooling, mathematics was presented primarily as a collection of facts and skills to be memorized. The fact that a student was able to provide correct answers shortly after studying a topic was generally taken as evidence of understanding. Students' ability to provide correct answers is not always an indicator of a high level of conceptual understanding (Standards 2000 Discussion Draft, p. 33).
What do the reformers advocate in place of the traditional approach? They advocate introducing students to mathematics through many concrete examples. They expect students, with guidance, to generalize from such examples. In this approach, teachers do not teach students methods for solving math problems; they introduce students to particular problems, then guide them to discovering methods from their experiences with these problems.
The Standards 2000 draft is filled with examples of how this should happen: The teacher proposes a math problem. Either in groups or individually, students thrash out the problem as best they can. The teacher visits each individual or group during the process and nudges them Socratically past sticking points.
For example, on pages 197-99 there is a scenario of an exercise in decimal arithmetic. Groups of third-to-fifth graders are trying to determine the amount of gold possessed by a jeweler who has kept 1.14 grams, .089 grams, and .3 grams of shavings from three ring resizings. One group has added 3, 89, and 114 to obtain the answer 206 grams. The teacher asks, "What happened to the decimal numbers?" The students in the group are unanimous in the opinion that the decimal points were simply an impediment. The teacher steps it up: "Are you saying, then, that if you start out with 1.14 grams of gold and some other little bits that it adds up to two hundred and six grams of gold?"
The role of teachers in the reform approach is elaborated in the many subsections ofStandards 2000 that address the issue explicitly. Here are quotations from a couple of such subsections in the Grades 3-5 chapter, from which I took the example of the jeweler:
Teachers must also routinely provide students with rich problems centered on the important
mathematical ideas in the curriculum so that students are working with situations worthy of
their conversation and thought. . . . Teachers must make on-the-spot decisions about which
points of the mathematical conversation to pick up and which to let go . . . [The teacher working
with the jeweler problem], for example, chose to let one group of students struggle with the
fact that their answer was unreasonable. . . . Teachers must refine their listening, questioning,
and paraphrasing techniques, both to direct the flow of mathematical learning and to provide
models for student dialogue ("What is the teacher's role in stimulating and improving commu-
nication?" p. 199).
Teachers should select tasks that help students explore and develop increasingly sophisticated
mathematical ideas. . . . They should ask questions to encourage and challenge students to build
arguments and develop strategies based on the mathematics they know ("What is the teacher's
role in helping students make mathematical connections?" p. 203).
The reformers are opposed in the Math Wars by a grassroots movement of parents, and others, who fear that children will not learn math if they are taught by the reformers' methods. (H.O.L.D. 1 [Honest Open Logical Debate on math reform] and Mathematically Correct 2 are two of the most active opposition organizations.) These folks think that de-emphasizing correct answers and competence in basic procedures, as the reformers wish, will produce students who simply cannot do the math they need to.
The reformers' opponents advocate teaching the traditional algorithms and drilling students to competency in them. They advocate clear, concrete standards based on actually solving math problems. They press school boards to reject reform curricula and to adopt curricula that stress competence in basic skills. They promote scholarship on the importance of teaching children mathematical fundamentals, and they make the case against reform curricula.
The difference between the reformers and their opponents can be seen clearly in their respective statements of standards. One of the broad standards in NCTM Standards 2000 is that students should "use computational tools and strategies fluently and estimate appropriately" (Standard 1. See page 155). For example, a specific version of this standard for grades 3-5 is that students should
develop and compare whole-number computational algorithms for each operation that are based on understanding number relationships, the base ten number system, and the properties of these
operations, and by the end of grade 5, develop efficiency and accuracy in using these algorithms (P. 157).
Here is an analogous standard, for fourth graders, from the Mathematically Correct Standards 3 :
The student will add and subtract with decimals through thousandths.
There are other issues in the Math Wars that I will not address here. For example, the reformers have restructured the mathematics curriculum so that middle school and high school students are no longer taught the traditional subjects—algebra, plane geometry, trigonometry, solid geometry—all at once; each year, they learn little bits of algebra, geometry, and trigonometry with some truth-table logic and probability interspersed. Their opponents would prefer that these subjects be presented in the traditional manner, as integrated fields. Another source of contention is the leftist/P.C. slant in some of the textbooks that the reformers advocate; certain of their opponents object to that. But the central issue is this: conceptual thinking versus the traditional algorithms. The reformers think that students should struggle with mathematical problems on their own and that, from these struggles, methods of solving the problems will emerge. Having devised these methods themselves, students will understand the abstract conceptual structure of the methods. Their opponents think that unless students are taught the traditional algorithms, they will not be able to do math.
I think that the reformers are wrong and their opponents are right: Students should be taught, and made to master, the traditional algorithms. I think that this is the correct way to teach mathematics by the reformers' own standard: The best way to advance students' conceptual thinking about mathematics is to have them master the traditional algorithms.
Of course, mere rote memorization of mechanical processes as an end in itself is pointless. But a mastered algorithm in the hands of a student is an incomparable tool for laying bare the conceptual structure of the mathematical problems that the algorithm solves. With such tools, and with the guidance of good teachers in their use, a student can grasp and integrate in twelve years a body of mathematics that it has taken hundreds of geniuses thousands of years to devise.
Oddly, in spite of their emphasis on conceptual thinking, the reformers have not, to my knowledge, presented an account of the nature of conceptual thinking and how their approach promotes it. In order to make my case, I am going to begin with a short account of some relevant features of conceptual thinking.
The first thing to say about conceptual thinking is that the adjective "conceptual" is there for emphasis, not for qualification; there is no other type of thinking. We add the emphasis because, while concepts must be used in all thinking, they are often used poorly. We use the term "conceptual thinking" to refer to good thinking, thinking that makes efficient use of the human capacity for abstraction.
A concept is a mental integration that is achieved through abstraction. By identifying similarities and abstracting away from differences among certain particular things—differences that are unimportant in some contexts—we unite these things in thought. Examples are easy to come by, since all words except for proper names correspond to concepts. The concept "ape" refers to any of a wide variety of animals; it abstracts away from the differences between, for example, Koko (the famous signing gorilla) and J. Fred Muggs (the TV star chimpanzee). The concept "seven" refers to any instance of that particular quantity; it abstracts away from the differences between Kurosawa's samurai and Disney's dwarfs.
Concepts are a way of organizing information efficiently. They provide a cognitive economy that allows us to structure into manageable units the massive amount of information we receive through our senses.
As a mathematical example of the property of unit economy, consider the statistical concept "mean." We have to summarize quantitative information about molecules, auto accidents, sales, and, most important, baseball. The concept "mean" allows us to address problems in all of these fields in one fell swoop.
A method is a prescription for achieving a goal. Like concepts, cognitive methods are abstract, and their abstractness is crucial to their power. The syllogism, a logical method for inferring new truths from established truths, applies equally to biology and physics and mathematics. I might use a syllogism in reasoning about the skeletal structure of wrens:
Birds are vertebrates.
Wrens are birds.
Therefore, wrens are vertebrates.
And I might use the same method to establish that there is only one prime-number factorization of 35 (namely, 5x7):
Natural numbers have unique prime-number factorizations.
35 is a natural number.
35 has a unique prime-number factorization.
As a mathematical example, consider the traditional addition algorithm, which is a method for determining the sum of two numbers. It works for any pair of numbers; it lets me add 15 and 7, and it lets me add 73948578 and 13384.
Another feature that is essential to the cognitive power of concepts and methods is that they are automatic. Once we have grasped a concept, once we have identified the similarities among particulars on which the concept is based, and performed the abstraction to condense the information integrated by the concept, the everyday use of the concept is virtually effortless. When, having grasped the concept "dog," I encounter a collie, or a beagle, or a mutt, I classify it automatically as a dog, and I know without detectable mental effort that it must eat and that it cannot fly. Likewise, once we have mastered a method, its application is automatic. Using methods is not as nearly effortless as using concepts; long division is a chore. But it is a comparatively straightforward bookkeeping chore, not a complex cognitive chore; the vastly more difficult cognitive chore was completed once and for all when the method was formulated and grasped. In doing long division, one works hard to ensure that the steps of the rote procedure are performed properly. But one does not have to analyze the relations among the relevant mathematical concepts and organize the sequences of steps from scratch.
The automatic feature of concepts and methods lets us apply the results of cognitive labor without repeating it. When I need to add two numbers in the course of learning to solve a differential equation, I can do so without diverting conscious focus from my difficult new challenge.
Words are the concrete symbols that we use to identify and apply concepts. Assigning a word to a concept completes the process of abstraction. The word is a perceptual tag, a pointer that directs our attention to a concept's referents, and to the information about them that the concept has condensed. Rote procedures play an analogous role for cognitive methods. The formal mechanics of an algorithm, for example, allow us to bring a deep, detailed analysis to bear on a newly encountered problem, and to solve the problem by associating it with a simple superficial pattern.
The advantages of the abstract, automatic nature of conceptual thought come with liabilities. It is easy to lose track of the facts that gave rise to a concept; the things to which the concept refers; the context for which it was intended; the similarities on which the abstraction was based. It is easy to lose sight of the reasons that a method works, of the facts that make it valid and efficient. Because concepts are abstract, because they refer to open-ended classes of things, it is easy to let them float—to treat them as if they did not refer to anything in particular. Because methods are implemented as rote procedures, there is a tendency to lose touch with the meanings of the relevant concepts.
This is why it is important to cultivate the habit of defining concepts, of consciously identifying the facts on which they are based. It is also why the practice of drilling students in rote mathematical procedures is dangerous. We have all known students who mastered algorithms but never grasped the underlying theory; sad to say, we have all known teachers who taught algorithms without understanding the underlying theory.
The reformers' approach to this problem is to have students devise their own methods, rather than have them learn the traditional algorithms. The idea is that the inventor of a method must grasp the facts on which the method is based in order to formulate the method. Of course, this is true. Therefore, a student who has devised his or her own method for achieving some mathematical goal will understand the relevant mathematical concepts. To quote again from the NCTM Standards:
By talking about problems in context, students can develop meaningful computational algorithms (Standards 2000 Discussion Draft, p. 220).
The problem is that this is not true. If by "meaningful computational algorithms" we mean things like the traditional addition algorithm, or long division, then the fact is that the average student will never develop such an algorithm.
Archimedes was a creative genius in physics, engineering, and mathematics. Euclid set the standard for mathematical rigor and clarity by his deductive formulation of geometry. Aristotle defined the laws of logic. They all did arithmetic. None of theminvented the addition algorithm. 4 It was invented long after their deaths, on another continent. The Greeks of the classical period used a notation similar to what we call Roman numerals, a system in which the streamlined simplicity of our addition algorithm is impossible.
Identifying an important new concept, or developing a valuable new method, is hard cognitive work. The vast majority of us are not conceptual innovators on that scale. Most of us never originate a major new concept; we simply use the concepts that we have learned from others. Likewise, very few of us ever devise a significant new cognitive method. It is easy to lose sight of the difficulty of cognitive innovation because of the automatic property of concepts and methods that I discussed above. Having mastered the application of the addition algorithm, say, it is easy for us to regard it as simple and obvious. But it is not. If you are reading this dense article on mathematical pedagogy, odds are that you are educated, intelligent, and thoughtful. How many concepts have you coined? How many cognitive methods have you originated?
The Standards 2000 Draft is full of (barely) plausible examples of children "devising their own strategies" for solving math problems. On page 114, a girl in the second-grade supposedly recounts her strategy for adding 153 and 273:
Well, 2 hundreds and 1 hundred are 3 hundreds, and 5 tens and 5 tens are 10 tens or another
hundred, so that's 4 hundreds. There's still 2 tens left over, and 3 and 3 is 6, so it's 426.
This real or imagined student is one exceptionally bright and articulate second-grader. But we must be careful about what we mean if we are going to say that this solution is the result of a strategy that she devised herself. Her strategy is really a minor variant of a method that someone taught her. She has clearly been taught how to represent numbers in base ten, what such representations mean, and how these representations can be used to simplify the addition of two numbers.
It is good for students to experiment with "devising their own strategies" in this manner. It is a way for them to explore the structure of the methods that they have been taught. But it is not the development of a "meaningful computational algorithm."
Considered as a cognitive method, the traditional addition algorithm for adding two numbers is a masterpiece. You memorize 55 sums: 0+0 through 9+9. You learn the rote procedure. And you can add any two natural numbers. The algorithm is a model of the efficiency and universality that characterize good conceptual thought. It is a tour de force of abstraction, a condensation of a literally incalculable class of cases into 55 sums and a simple procedure—infinity for the price of 55.
This is one reason that we should teach students the addition algorithm, and the other traditional algorithms, if we want to teach them good conceptual thinking. These algorithms are exemplars of conceptual method.
Another reason that students should learn this algorithm is the reason it was invented: so that they can add numbers fast and reliably. Of course, eventually students will go on to learn to use calculators to add numbers fast and reliably. But if encouraging good conceptual thinking is our goal, then we must teach students to compute sums by hand. If the student needs a calculator to determine the answer to a problem, the gap filled by the calculator is a gap in the conceptual chain that should tie the student's ideas to reality.
Yet another reason that we should teach students the addition algorithm, and the other traditional algorithms, is one that I mentioned above and that I will elaborate upon below: The mastered algorithm gives students access to the conceptual structure on which it is based. Before I discuss this, however, I want to give some analogous examples.
Counting. Counting is a cognitive method, a method for determining how many. There is a familiar rote procedure associated with it: You recite the number sequence while attending successively to each of the things in the collection whose quantity you want to determine.
We teach children to determine quantities by teaching them this rote procedure. These days, I happen to be doing exactly that with my year-old daughter. I point to the horses on the carousel and I say "one, two, three. . . . " I do the same for her fingers, for the glow-in-the-dark planets on her bedroom ceiling, for all sorts of other things. At first, my daughter saw no particular similarity between the horses and the fingers. In fact, the first similarity that she saw was that I associated the rote procedure with each. Were it not for my repetition of the rote procedure, and her slow memorization of it, she would probably never identify the similarities in quantity on which the number concepts are based.
Imagine if I tried to get my daughter to learn numbers without using the recited number series; imagine if I tried to get her to devise her own method for determining how many.
The memorized rote procedure provides a mental locus at which my daughter can store and integrate the data that she needs to organize conceptually. Beyond this, the structure of the rote procedure reflects the structure of the underlying facts; it provides my daughter with clues about what she should be attending to.
It is true, of course, that epistemologically the understanding of what quantities are and how they are related is fundamental, whereas the verbal form of the number series and the gestures we use for counting are secondary. But pedagogically, it is these expressions that give children access to the more fundamental facts.
Language. A child's learning of words provides an even starker example. At first, words are meaningless sounds to a child. He remembers some of them after frequent repetition. Then he starts to associate the empty sounds with particular things. He notices the sound "duck" being repeated often and notices that certain similar things are present when he hears it. Eventually, he will focus on similarities of shape, color, motion, and on other similarities among these things. But the thing that calls his attention to these similarities, the most salient unifying feature of his early experiences of them, is that whenever he is around these things, he hears the sound "duck."
Epistemologically, parroting the sound "duck" is a very different thing from understanding the concept "duck." Pedagogically, memorizing that word gives us access to the facts that underlie the concept. The word is a label on the mental file folder that is the concept, and memorizing the word opens that folder.
Imagine trying to get a child to grasp the concept "duck" without using a word for it. This would not be education, it would be charades.
Cooking. A person who is learning to cook—I am speaking here with the voice of experience—follows recipes with a to-the-milliliter precision at which savvy cooks scoff. The recipe is a rote procedure that the novice does not understand but that produces the desired result.
With the recipe followed, as he is making and eating the meal, the novice uses the rote procedure to glean insight into the facts on which the method is based. He sees that the cornstarch thickened the gravy—so he understands its role, and he can adjust it to his needs in the future. He tastes the spice cardamom in the rice, he recalls how much was called for in the recipe, and he grasps how to alter this amount to change the results.
Imagine a perfect novice trying to learn to cook without recipes.
Physics. Even at a fairly sophisticated level, memorization is useful and probably essential. Consider some bright high school students learning physics. Their teacher tells them Newton's first law—a body in motion will continue in a straight line at a constant speed unless acted upon by an outside force—and they memorize it. At first, they do not understand the factual basis of the law. Indeed, the law's claim appears to them to be obviously false; everything eventually slows down, and virtually nothing travels in a straight line.
Here, the students are memorizing a proposition. But, just as memorizing words helps children learn concepts, and memorizing the rote procedure helps them learn to count, so memorizing this proposition helps them to learn mechanics. Their crude understanding of the memorized proposition guides them in accumulating empirical support for the law; they may never have wondered what forces might act on a knuckleball, a planet, or a compass needle, but the claim of the law urges them to do so. As they acquire data, the memorized proposition provides a mental locus at which they can integrate them.
A student who has mastered the addition algorithm can use his mastery to investigate and understand the algorithm, and particularly the base-10 representation of numbers, on which the algorithm is grounded. The idea of representing numbers in this manner is crucial to all of modern arithmetic, 5 and thus to all science and technology.
A child uses the algorithm to add 54 and 78. He adds the 4 and the 8 and gets 12. He puts the 2 in the units column of the answer and carries the 1; thus, the 12 becomes 10 and 2, and the 10 goes to the next column, to the 10s place, where it becomes 1 ten. He adds the carried 1, the 5, and the 7 in the tens column to obtain 13 tens, which becomes 3 tens—he writes the 3 in the tens column of the answer—and 1 hundred, which he carries. There being no hundreds digits in the original numbers, he writes down the carried 1 as the hundreds digit in the answer, and he is done: The sum is 132.
The attentive student—or the student made attentive by a good teacher—finds revealing trends as he repeats this algorithm. The tens column is aptly named; any multiples of ten in his sum of the digits in the units column are brought over to the tens column, and only the remainder is left in the units column. Likewise, the hundreds column, the thousands column, and so on. The mechanics for each column are exactly like those for the preceding column; the hundreds column stands in relation to the tens column just as the tens column stands in relation to the units column. He notices that two two-digit numbers sometimes sum to a two-digit number, sometimes to a three-digit number, never to a four-digit number. Why? He notices that when he adds two numbers, he either carries 0 or 1, never anything larger. Why? However, when he adds 3 numbers, he sometimes carries a 2. How many numbers would he have to add before he might carry a 6? Why start at the units column? He tries starting at the other end of the number (the approach taken by the brilliant second grader in the Standards 2000example); he learns quickly why this is a bad idea, and he grasps more deeply the idea of carrying the overflow from one column to the next higher column. He wonders how this would all work if he represented numbers in terms of powers of a number other than 10; he learns to write numbers in other bases; he sees that the addition algorithm works without alteration (except that he must remember to carry when he gets a multiple of the relevant base), and he grasps more deeply yet the logical structure of the algorithm and the underlying system of representing numbers.
He does all of this as a result of his mastery of the algorithm, and with the guidance of a teacher who is determined to school him in conceptual thinking. He does this because the algorithm reveals patterns that express its terse, elegant, logical structure. A student who could never devise such a method can grasp its workings more easily than its inventor because the algorithm itself makes obvious the logical subtleties on which it is based.
Thus, the lesson for the opponents of the math reformers is this: They have had some success in the important and dreary task of battling the politically entrenched educational establishment. But if they want to succeed fully and finally, they should not yield the pedagogical high ground, the realm of conceptual thinking, to the reformers. By advocating mastery of the traditional algorithms, the reformers' opponents have in fact established themselves as the defenders of conceptual thinking in the Math Wars.
David Ross, Ph.D., is a mathematician at Kodak Research Labs and a frequent speaker at The Atlas Society's summer seminars.This article was originally published in the May 2001 issue of Navigator magazine, The Atlas Society precursor to The New Individualist.
Notes
1 http://www.dehnbase.org/hold/2http://www.mathematicallycorrect.com/
3http://mathematicallycorrect.com/kprea.htm#G34 Georges Ifrah’s recent book The Universal History of Numbers (New York: John Wiley, 2000) provides an entertaining and detailed account of the history of number systems. In particular, it contains a fascinating explanation of the perfection of the modern system of numeration in India 1500 years ago. This is the achievement on which the traditional addition algorithm—and the other traditional arithmetic algorithms—is based.5 What is crucial is not the particular base of 10, but the scheme of representing numbers in terms of powers of a fixed number of moderate size. See Ifrah’s book.