Published in Educational Studies in Mathematics 12 (1981) 421-434.
ABSTRACT. Recent advances in the way that adults perform computation in our society require reconsideration of the assumptions underlying current elementary mathematics instruction. The widespread use of calculators and computers for situations requiring precise calculation removes much of the motivation for teaching the current addition, subtraction, multiplication, and division algorithms. Yet precisely this use of computing technology now puts a premium on the exercise of estimation techniques for verifying the reasonableness of computations. These techniques, especially those that can be used "mentally" (without the use of any external tools), have been used informally for years, but never formalized for instruction. This paper discusses a range of estimation techniques, and presents in detail a series of mental estimation procedures based on the concepts of measurement and real numbers rather than on counting and integers. A set of techniques for teaching these procedures is described. These estimation techniques are evaluated against the multiple functions that elementary mathematics instruction needs to serve.
We are in the midst of a revolution in the ways we deal with computation in our society, brought on by the advent of cheap microelectronics. As the pocket calculator has become commonplace, there are fewer and fewer occasions that require adults to perform the long, laborious, and error prone techniques for doing complex calculations with numbers written on paper by pencil. A bank clerk who insisted on adding up long columns of numbers using the techniques learned in school would soon be seeking new employment.
This development has implications for the instruction of mathematics in our schools. Teaching the pencil and paper computational techniques has been the cornerstone of mathematics instruction. Much of this instruction is disrupted when students bring calculators into the classroom. And even when calculators are not present, the motivation of students to learn (and teachers to teach) is problematic if the skill is seen as useless in everyday life.
Let us examine the rationale for the current paper and pencil techniques. There are three main functions for learning to compute: (1) for use in professional employment, (2) for use in non-professional everyday life, and (3) for use as a basis for understanding mathematics.
This is the challenge to current mathematics instruction: to find new ways that aid people in carrying out accurate computation using calculators and computers, new ways to help people perform the mental computations they do in everyday non-formal settings, and that form a basis for further instruction in mathematics. Is this an impossible challenge? Let us attack it by considering what is actually needed to achieve the first two functions.
Other techniques for computation, once considered essential to a proper mathematics training, have fallen out of use, under the impact of improved computing machinery. The use of logarithmic tables ("...without which many of the numerical calculations which have constantly to be made would be practically impossible. . ." (Ball, 1908, p. 195)) to perform high precision multiplication and division of large numbers has almost disappeared. The last major slide rule company recently closed its doors. No computational technique should be considered sacred, but instead examined in light of the functions that it serves.
In a few situations, exact accuracy is required. In these cases, the same "checking" technique can be carried over from paper and pencil computation to calculator use. The computation is repeated, sometimes in an alternate or reciprocal form to see if the identical result occurs.
In most situations, only approximate accuracy is needed. In dividing up a restaurant bill, determining the amount to tip, figuring cost and tax, computing miles per gallon of gas, in most everyday situations we only need to be confident of approximately what the correct answer is. People develop personal approximate techniques for dealing with these situations. But these techniques are almost never a part of formal curriculum in elementary classrooms, as mental calculation is not recognized as a valued activity. Wrong answers are treated equally, whether almost correct or wildly inaccurate. In the adult world the consequences of a gross error are far more serious than those of minor deviations.
For more accurate estimation, people often use a "rounding to whole numbers" approach, sometimes with an "adjustment" afterwards. So, if they are to multiply 589 by 49.3, they will instead multiply 600 by 50, and then adjust the answer of 30,000 downward a bit, say to 29,000.
Seldom are these techniques formally taught in schools, and almost never at the elementary or secondary levels. Yet a common complaint among college teachers is that students don't seem to have a "feel" for numbers and their computation. So one approach to the challenge raised by calculators and computers would be to explicitly teach children estimation techniques.
But don't estimation techniques require a previously acquired skill with the current right-to-left computational techniques? How could elementary school students be taught estimation techniques before they learn to master the conventional computation techniques? Before answering this, let us briefly explore the basic concepts underlying the conventional techniques and then consider a different basis for mental estimation.
It is a bit hard to imagine doing addition, subtraction, multiplication, or division without using the standard techniques and the tables on which they depend. Let us consider a case in which people spontaneously used a mental estimation technique that doesn't require the addition or multiplication tables.
These subjects were shown two poker hands. They were told that their hand had a 70% chance of beating one hand, and a 50% chance of beating the other. What was the chance of beating both hands? This is a situation requiring the subjects to multiply 0.70 by 0.50 to find the joint probability. Some of the people in an experiment to examine how people combined this kind of information reported using the following mental technique: they imagined a unit quantity. Then they imagined taking 70% of this quantity. Next they imagined taking 50% of this smaller quantity. Then they determined approximately how much this remaining quantity was of the original unit quantity. This is then the product of multiplying the two probabilities, which is the probability of beating both hands.
Let us use a line as our unit quantity, to make this technique clearer. Following is a line, with 0 at one end and 1.0 at the other:
0 1.0 .------------------------------------------------->
Now the first step is to take 70% of the unit quantity, which in this case is a smaller line, 70% as long as the original:
.---------------------------------->
The next step is to take 50% of this smaller line:
.----------------->
We can now see that this is about 30% to 35% of the original line. This is the product of the two fractions (0.7 x 0.5), giving us the joint probability of beating both hands.
Notice that this technique does not require the use of the traditional algorithms for multiplication, nor the use of the multiplication table facts. Instead it draws on several other kinds of number facts and skills. To use this approach for multiplication, you need to know how to go from a number to a quantity, such as a position on a number line. To determine what 70% of a unit line is, you need to be able to specify where 0.7 is on a number line going from 0.0 to 1.0. You need to be able to apply this skill to lines of different sizes. And then you need to be able to say what number is represented by a particular position on a number line.
The two basic skills are required: (1) translating from a number to a position on a number line, and (2) translating from a position to a number. Since these are basic components of knowledge about number, there are exercises for teaching these two skills. We have developed two computer programs specifically designed to teach this knowledge, one called Harpoon and one called Sonar.
Harpoon: A program for teaching position-to-number skills. Harpoon is a computer game (written in Pascal for an Apple II Computer) that presents the players with a drawing of a shark's fin on the computer screen, with two perpendicular lines intersecting over the shark. Each line has its endpoints labeled with numbers. The program asks the players to specify the position of the shark left and right and then its position up and down. After they enter the two numbers, a "harpoon" flies across the screen to the position they have specified. If that spot is close enough to the shark, then the harpoon hits the shark, and the shark sinks out of view. If the harpoon misses, than a "splash" occurs on the screen to mark the spot, and the players can try again, using the splash mark as feedback.
The Harpoon game was inspired by another computer game called Darts (Dugdale & Kibbey, 1975), created explicitly to teach children the number line. Harpoon has been extensively tested with ten year old children, who find it challenging and motivating (Levin & Kareev, 1980). Initially they have to work together to hit the shark, dividing up the responsibility for parts of the task, but after some practice they can hit the shark with a high degree of accuracy. They acquire the skills of translating from position on number lines to numbers. When working with only one dimension (the version most analogous to the original Darts game) children reach proficiency within ten "games", moving from random performance to high accuracy (deviation of less than +/- 3% on their first guess).
Sonar: A program for teaching Number-to-Position skills. Sonar is another game program (also written in Pascal for the Apple II Computer) that teaches math skills within the framework of a game. This game is similar to Harpoon, as the players have the goal of hitting a shark with a harpoon. But in Sonar, the shark doesn't initially appear on the screen, but is hidden underwater. The player's "sonar" readout tells where the shark is hiding, giving the X and Y coordinate numbers. The players try to move the "crosshairs" to that spot on the screen. Then the harpoon flies to that spot and if it is close enough to the shark's position, the shark surfaces and then is harpooned. Otherwise, the harpoon splashes into the water, and the coordinate numbers of their guess are displayed as feedback.
This game has been tested with ten year old children, who have found it also challenging and motivating. It teaches the mathematical skill of specifying a position on a number line, given a number. With this Sonar game and the Harpoon game, children can learn to have an "intuitive feel" for numbers, freely converting between numbers and number line lengths. Given these lower level skills, people can then apply the "serial fractionation" techniques for multiplication described above to compute the products of fractional numbers. Note that this technique doesn't require the use of the multicolumn "long" multiplication technique, nor the use of the elementary multiplication facts ("the times tables").
Questions of precision. Current paper and pencil computational techniques are designed to provide absolute accuracy. Children are evaluated on the basis of producing exactly correct answers in their computations. However, in most non-school uses of addition, subtraction, multiplication and division, there is no need for absolute accuracy. In fact, for scientific uses of computation, students are taught explicitly about the limits of accuracy, and are trained in techniques for keeping track of the degree of precision in their computations. People checking the results of a calculator or computer often only need to know that the answers are within a reasonable range of accuracy.
An important characteristic for mental estimation techniques is that they be able to provide a whole spectrum of precision, from very rough approximation to high precision. The rough approximation techniques should be easy to apply, while higher precision uses can require more effort. The techniques should provide partial results that are more and more accurate approximations of the precise answer.
Note that these are not characteristics of the current paper and pencil techniques. The current multicolumn multiplication techniques, for example, produce partial results that are far from approximations to the accurate answer. Some advanced university students are taught approximation techniques, but these are separate processes from the normal computational techniques. There is a need for an integrated process, that has the following characteristics:
Next, the two fractional numbers are multiplied, using the "serial fractionation" technique described above. The precision of representing the numbers and performing the fractionation is a function of the mental effort put into the process by the person. Rough estimates can be performed with less mental effort than more precise estimates.
In the second step, the two numbers are represented as relative line lengths, and the proportion that the numerator is of the denominator is determined. For example, suppose we wanted to estimate the price per unit quantity of peanut butter in a supermarket. The price is $1.57 for 28 ozs. We first represent these as 0.157 x 10 to the 1 and 0.28 x 10 to the 2. So the answer will have an exponent of -1. Next we represent mentally 0.157 and 0.28 as relative line lengths:
0 1.0 .------------------------------------------------->
.-------> 0.157
.--------------> 0.28
We can now estimate that 0.157 is approximately 0.5 of 0.28. So the result is 0.5 x 10 to the -1 or approximately $0.05/oz. If we were comparing a set of peanut butter jars of approximately the same price range and sizes, then we could skip the first part, and just do the division of the fractional portion.
This process may seem long, complex, and effortful, but that is, to some extent, due to its novelty. Imagine the length and complexity of description you would have to give to someone unfamiliar with the paper and pencil division algorithm so that s/he could divide $1.57 by 28. Also note that neither of the mental estimation techniques discussed so far requires knowledge of the "times" tables (the memorization of one hundred single digit multiplication results).
0 500 .------------------------------------------------->
.-------------------------> 256
.----> 43
Then we mentally translate the shorter line to the end of the longer line. The resulting line represents the sum of 256 and 43, which we can estimate to be approximately
.------------------------->----> 300
Note that using this technique requires only the skill of converting numbers into relative line lengths (trained by the Sonar game described previously), the skill of converting relative line lengths to numbers (trained by the Harpoon game), and the mental skill of mental translation. There is strong psychological evidence for the abilities of people to perform mental translation in a linear way (Kosslyn, 1975).
-100 0 100 <------------------------.------------------------>
-45 <-----------.
Then, subtraction is performed in the same way as Estimation Addition: the relative line length for the second number is mentally translated to the end of the line length for the first, and the resulting line length is the result. So, 53 - 45 is represented by
-100 0 100 <------------------------.------------------------>
.--------------> 53
-45 <-----------. (translated)
.--> Result: 10 (approximately)
The negation operation, within this framework, is expressed very concretely as reflection across the zero point. The negation of a number represented as a relative line length can be accomplished by mentally rotating the line length 180 degrees. There is solid psychological evidence for the ability of people to perform mental rotation in a reliable and linear way (Cooper, 1975; Shepard & Metzler, 1971).
At the other end of the spectrum, Brown, Burton, and Van Lehn have been mapping out in detail the arithmetic cognitive processes (particularly place value subtraction) (Brown & Burton, 1978; Brown & Van Lehn, in press; Burton, 1980). They have taxonomized a large number of subtraction errors, and developed a theory of computational "bugs" and their generation. One of the implications of their descriptive studies of the ways that people perform arithmetic is to highlight individual variation between the ways that people have for carrying out the arithmetic operations.
Both for generating mathematics curricula and for constructing cognitive theories of arithmetic, individual variation has been largely ignored in previous work. But some of the earliest work focused in an interesting way on the astoundingly large differences between the ways that different people think about and operate on numbers. Galton (1907) collected "number forms" from people, drawings that people made to express where numbers are spatially located mentally for themselves. Most people have a strong notion of spatial location of numbers ("ONE is up here; TWO is next to it in this direction; THREE is below and to the right; . . . "). The variety of number forms he collected is striking, especially given the implicit assumption underlying most mathematics curricula that all people think about numbers and operate on them in the same way (the "right" way). This study of "number forms" has been carried forward recently by Petitto (1975), with much the same results: a considerable individual variation in the ways that people think about numbers.
More importantly, we have examined in this paper a series of estimation techniques as alternates to the standard algorithms for computation. Each of these techniques has advantages over the current techniques, especially in light of current technology for computation. The particular set of techniques for performing mental calculation were presented in enough detail to demonstrate that there are viable alternate techniques that we can consider, evaluating them in light of the multiple functions served by elementary mathematics instruction.
In light of the evidence for individual variation, perhaps the most fruitful course will be to provide students with a wide variety of approaches for computation, rather than any one canonical technique. One advantage of using computers in mathematics instruction is that they can support a wide variety of approaches to manipulating numbers. Rather than reacting to the new technology for calculation as a threat, we should consider it a valuable opportunity to reconsider the assumptions underlying the mathematics curriculum, a chance to discard the mechanical, mind-stultifying elements that are better turned over to dumb machines, a chance to focus on the creative qualitative aspects that make mathematics exciting as well as useful.
University of California, San Diego
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