In my tireless pursuit of the question raised this past summer by Keith Devlin in a series of columns in the MAA MONTHLY ("It Ain't No Repeated Addition" (June 2008); "It's Still Not Repeated Addition" (July-August 2008); and "Multiplication and Those Pesky English Spellings" (September 2008)) regarding the potential harm of leaving students with the belief that multiplication is nothing more than repeated addition, I am doing extensive reading about how children learn about, think about, and are taught to think about mathematics and the connection amongst and differences between the operations. One leg of this journey brings me to consider division. There are a number of ways to think about division, but the two "big ideas" looked at in school mathematics are the partitive or "fair sharing" model and the quotative or "measurement" model.
Partitive Division With Whole Numbers
As a quick overview or review, let's take some simple examples of both. Consider, for example, a simple sharing situation:
"Alexa has 30 cupcakes and wants to share them equally amongst six people. How many cupcakes does each person get?"
Note that this situation has several key elements, perhaps most important being the idea of "sharing equally" or "fair sharing." Without that stipulation, there's no clear-cut mathematical solution and instead any distribution one chooses that results in 30 cupcakes being shared in any manner whatsoever is "correct." But in this situation, there is only one right answer, which of course is that each person gets five cupcakes.
The other things to notice are that we have a fixed total (30), a known number of equal shares (six), and an unknown SIZE of each fair share which is to be determined.
How might a lower middle school student solve this problem? One way would be to model the problem with 30 counters of some sort. The student could keep placing one counter into each of six piles until all the counters are exhausted. She would then count how many there were in one pile and thus determine the size of the equal shares.
A similar approach would be for the student to count "one" during the placement of each counter on the first round, "two" during the second round, and so on through the fifth and final round, whereby the student would know that the last number that is repeated, "five," is the desired answer.
Students might all try subdividing strategies in which the original pile of 30 is subdivided into three equal groups of ten, and then each of those groups could be split into two equal groups of five, resulting in the final answer of five (in six groups). There are several variations on this sort of scheme depending on the factors of 30.
A more seat-of-the-pants approach would be to take a rough estimate of the share size, say "four," give that many to each of the six piles, and note that there are enough left to give one more to each resulting in a share of five. Or if one guessed too high, some chips could be redistributed until the piles were equal. Clearly, this sort of approach becomes less and less viable as the number of items to be shared and/or the number of shares goes up.
Quotative Division With Whole Numbers
Turning to the quotative or measurement model, let's take an example similar to the previous one, with appropriate differences:
"Zane has 30 cupcakes and wants to give six each to a number of his friends. How many friends can he do this for?"
Note that here we still have a known total (30) but now the size of a fair share is given (six). What is not known is the NUMBER of such shares possible of the given size with the original total. Though the solution here (five) is the same as that of the previous problem and the numbers given are the same, they do not represent the same things (in the case of the two smaller numbers). In the partitive case, we know the number of fair shares and seek to find the size of one fair share given the total; in the quotative case, we know the size of a desired share and need to find how many such shares can be made from the given total.
Possible solution strategies include taking a pile of 30 counters and subtracting six at a time until the pile is exhausted, then counting the number of piles to get the desired answer. Thus, it may be clear why the quotative or measurement model relates to the idea of division as repeated subtraction: one can repeatedly subtract a given number (the divisor) from the dividend until either zero or a number smaller than the divisor is reached. The number of such subtractions is the quotient, and any left over is the remainder. This relates the quotative model to the so-called division algorithm (more accurately a theorem):
The relationship and difference between partitive and quotative models of division is not always made clear in elementary mathematics education, and often their existence comes as news to elementary school teachers and education students. When asked to create division word problems, teachers and students often find it difficult to do so for the quotative model, even though the natural approach to solving such problems, repeated subtraction, would appear to be "obvious" and to link to the widely-held belief that multiplication is "nothing more than" repeated addition. The imbalance between ease of constructing partitive examples of real-world problems and difficulty of construction quotative examples becomes exacerbated for many teachers and students when dealing with rational numbers in fractional form. There is also an interesting conceptual break-down of the partitive and quotative models when dealing with division of integers, as we shall see shortly.
One notable point about division of rational numbers in fractional form must be made before turning to integer divisions. It has become increasingly well-known to those interested in elementary mathematics learning and mathematics teacher education that many Americans have difficulty with the concept of division of fractions. While fraction multiplication is a fairly straight-forward proposition (the product of a/b and c/d is equal to ac/bd, so that a standard error in fraction addition and subtraction, which is to add/subtract numerators and denominators from left to right, is correct when it comes to computing the product of two fractions: simply multiply numerators and denominators, respectively, from left to right.
When it comes to dividing fractions, however, there is often both procedural and conceptual difficulty (not that the simplicity of computing products with fractions is an indication that conceptual understanding is common): the old standby rhyme, "Ours is not to reason why, just invert and multiply," not only indicates that many teachers and learners either give up worrying about students' understanding of the mathematics or never cared about it to begin with, but also isn't even a guarantee that students will compute correctly: after all, it matters enormously WHICH fraction is inverted before doing the multiplication.
From a purely symbolic perspective, one approach is to write a/b divided by c/d as a complex fraction - a fraction that has one or more rational expressions in the numerator and/or denominator. One then shows how to simply this fraction by multiplying both numerator and denominator by the reciprocal of the latter. Thus, (a/b) / (c/d) = (a/b)(d/c) / (c/d)(d/c) =
(a/b)(d/c) / (cd/cd) = (a/b)(d/c) / 1 = (a/b)(d/c). Thus, we show a justification, though informally, for the "invert and multiply" procedure for fraction division.
A second commonplace is to ask students to construct real world situations that would be correctly addressed by solving something like 2/3 divided by 1/2, and/or to simply compute the correct quotient. A very significant percentage of both elementary students and elementary teacher education students will produce problems that represent 2/3 * 1/2 and give 1/3 as the desired answer. These failures have been cited as proof that the groups in question lack procedural knowledge of rational number division, conceptual knowledge of that process, or both.
I suggest that while it is true that many people have difficulty with rational number division, it is important to look at how the above sorts of questions are posed before drawing conclusions about any given individual or group. It is particularly risky to draw to strong of a conclusion if the question is posed orally with 1/2 as the divisor. It seems plausible and even likely that a significant number of people are so used to being asked to find 1/2 OF a number that they will mishear/misconstrue a question asking them to divide by 1/2. I think it would be important for formal research, classroom assessment, or instruction to be sure that students have the opportunity to respond to written as well as oral questions, and that unit fractions, particularly 1/2, are not the only divisors they are asked to work with. I would be more confident that there is a significant conceptual and/or procedural problem if someone were asked to construct a problem that would be solved by dividing 3/4 by 2/3 for which they gave something like, "Dominique has 3/4ths of a pizza and wants to give 2/3 of it to his brother. How much should he give him?" or gave the answer for the computation as 1/2, than if s/he gave similar responses to the previously mentioned problem, particularly if the first one were only given orally.
Take the simple example of dividing the four combinations of +/- 6 by +/- 2. First, consider dividing +6 by +2. It is to be hoped that individuals will conclude that both the partitive and quotative interpretations can be used sensibly. Six balloons can be shared equally by two friends, with each getting three balloons. Or six cups of flour can make three batches of cookies if it takes two cups of flour per batch.
Consider next -6 divided by +2. It does not make sense to use the quotative interpretation here. There are no positive 2's to be taken from -6. However, the partitive interpretation can be used sensibly, as there is no difficulty dividing -6 into two equal groups of -3. So, for example, if two friends wish to share equally a deficit of 6 dollars (-$6), each gets a share of -$3.
Next, consider dividing -6 by -2. Here, the partitive interpretation fails because it does not make sense to talk about -2 groups or sharing equally among -2 friends, etc. However, the quotative interpretation can be used sensibly: If you want to share a deficit of six dollars (-6) so that each person gets a share of -$2, then 3 people are needed.
Finally, we come to +6 divided by -2. If one attempts to use the partitive interpretation of division, the same problem mentioned in the previous example arises. If one tries the quotative model, we run into the problem raised in the second example: there are no -2's to remove from +6. So how does one interpret 6 / -2 intelligibly? One obvious solution is to talk about division as the inverse of multiplication. Thus, x = 6 / -2 asks the question, "What number times -2 equals 6?" and the answer is -3, which for the moment we will assume is something already established with students. There is no problem finding the correct answer.
Does this mean that partitive and quotative interpretations of division are wrong and should not be taught? Well, it's clear that they make perfect sense in whole number arithmetic. And they cover 3 of the 4 possibilities for integer division where the dividend is an integer multiple of the divisor. It seems reasonable to believe that by the time students reach integer arithmetic (typically 5th or 6th grade), most will be ready for some of the necessary algebraic concepts needed to make sense of all cases, with notions of inverses, identity elements, and so forth coming into play. It doesn't seem like too much of a stretch to think that many will be comfortable with the question given in the previous paragraph as a way to think about six divided by negative two.
No model or interpretation of a mathematical idea is likely to be perfect. But that should not be shocking. The model/interpretation is NOT the mathematics. It's a way of helping students to think about the mathematics. A model or interpretation may in fact be precisely what a given student will carry away with him from his education, though this is not necessarily "the" goal of mathematics teaching: certainly most educators hope that every student will have the opportunity to experience the beauty and power of mathematical abstraction for its own sake, as well as a way of thinking that can deepen as one encounters increasingly complex and challenging mathematical ideas. We should not leave students with the mistaken impression that the map is the territory, that the model or application or interpretation IS the mathematics. But absent evidence that exploring models and interpretations is harmful to student learning or that it inevitably limits their long-term mathematical growth, it seems reasonable for teachers to continue to use them thoughtfully.
The title of this piece suggests that there is something in all this talk about division, interpretations, models, and applications that can inform us about multiplication and whether it suffices to leave students thinking of it as "no more than repeated addition." In the next part of this piece, I hope to help shed some light on just that question.
"Zane has 30 cupcakes and wants to give six each to a number of his friends. How many friends can he do this for?"
Note that here we still have a known total (30) but now the size of a fair share is given (six). What is not known is the NUMBER of such shares possible of the given size with the original total. Though the solution here (five) is the same as that of the previous problem and the numbers given are the same, they do not represent the same things (in the case of the two smaller numbers). In the partitive case, we know the number of fair shares and seek to find the size of one fair share given the total; in the quotative case, we know the size of a desired share and need to find how many such shares can be made from the given total.
Possible solution strategies include taking a pile of 30 counters and subtracting six at a time until the pile is exhausted, then counting the number of piles to get the desired answer. Thus, it may be clear why the quotative or measurement model relates to the idea of division as repeated subtraction: one can repeatedly subtract a given number (the divisor) from the dividend until either zero or a number smaller than the divisor is reached. The number of such subtractions is the quotient, and any left over is the remainder. This relates the quotative model to the so-called division algorithm (more accurately a theorem):
given two integers a and d, with d ≠ 0
There exist unique integers q and r such that a = qd + r and 0 ≤ r < | d |, where | d | denotes the absolute value of d. The integer q is called the quotient, r is called the remainder, d is called the divisor and a is called the dividend.
The relationship and difference between partitive and quotative models of division is not always made clear in elementary mathematics education, and often their existence comes as news to elementary school teachers and education students. When asked to create division word problems, teachers and students often find it difficult to do so for the quotative model, even though the natural approach to solving such problems, repeated subtraction, would appear to be "obvious" and to link to the widely-held belief that multiplication is "nothing more than" repeated addition. The imbalance between ease of constructing partitive examples of real-world problems and difficulty of construction quotative examples becomes exacerbated for many teachers and students when dealing with rational numbers in fractional form. There is also an interesting conceptual break-down of the partitive and quotative models when dealing with division of integers, as we shall see shortly.
A Side Trip To Division of Fractions
One notable point about division of rational numbers in fractional form must be made before turning to integer divisions. It has become increasingly well-known to those interested in elementary mathematics learning and mathematics teacher education that many Americans have difficulty with the concept of division of fractions. While fraction multiplication is a fairly straight-forward proposition (the product of a/b and c/d is equal to ac/bd, so that a standard error in fraction addition and subtraction, which is to add/subtract numerators and denominators from left to right, is correct when it comes to computing the product of two fractions: simply multiply numerators and denominators, respectively, from left to right.
When it comes to dividing fractions, however, there is often both procedural and conceptual difficulty (not that the simplicity of computing products with fractions is an indication that conceptual understanding is common): the old standby rhyme, "Ours is not to reason why, just invert and multiply," not only indicates that many teachers and learners either give up worrying about students' understanding of the mathematics or never cared about it to begin with, but also isn't even a guarantee that students will compute correctly: after all, it matters enormously WHICH fraction is inverted before doing the multiplication.
From a purely symbolic perspective, one approach is to write a/b divided by c/d as a complex fraction - a fraction that has one or more rational expressions in the numerator and/or denominator. One then shows how to simply this fraction by multiplying both numerator and denominator by the reciprocal of the latter. Thus, (a/b) / (c/d) = (a/b)(d/c) / (c/d)(d/c) =
(a/b)(d/c) / (cd/cd) = (a/b)(d/c) / 1 = (a/b)(d/c). Thus, we show a justification, though informally, for the "invert and multiply" procedure for fraction division.
A second commonplace is to ask students to construct real world situations that would be correctly addressed by solving something like 2/3 divided by 1/2, and/or to simply compute the correct quotient. A very significant percentage of both elementary students and elementary teacher education students will produce problems that represent 2/3 * 1/2 and give 1/3 as the desired answer. These failures have been cited as proof that the groups in question lack procedural knowledge of rational number division, conceptual knowledge of that process, or both.
I suggest that while it is true that many people have difficulty with rational number division, it is important to look at how the above sorts of questions are posed before drawing conclusions about any given individual or group. It is particularly risky to draw to strong of a conclusion if the question is posed orally with 1/2 as the divisor. It seems plausible and even likely that a significant number of people are so used to being asked to find 1/2 OF a number that they will mishear/misconstrue a question asking them to divide by 1/2. I think it would be important for formal research, classroom assessment, or instruction to be sure that students have the opportunity to respond to written as well as oral questions, and that unit fractions, particularly 1/2, are not the only divisors they are asked to work with. I would be more confident that there is a significant conceptual and/or procedural problem if someone were asked to construct a problem that would be solved by dividing 3/4 by 2/3 for which they gave something like, "Dominique has 3/4ths of a pizza and wants to give 2/3 of it to his brother. How much should he give him?" or gave the answer for the computation as 1/2, than if s/he gave similar responses to the previously mentioned problem, particularly if the first one were only given orally.
Division of Integers
It is heuristic to have in-service and pre-service elementary teachers, as well as their students, to try to apply on their own the partitive and quotative interpretations of division that they employed successfully with whole numbers to division of integers.Take the simple example of dividing the four combinations of +/- 6 by +/- 2. First, consider dividing +6 by +2. It is to be hoped that individuals will conclude that both the partitive and quotative interpretations can be used sensibly. Six balloons can be shared equally by two friends, with each getting three balloons. Or six cups of flour can make three batches of cookies if it takes two cups of flour per batch.
Consider next -6 divided by +2. It does not make sense to use the quotative interpretation here. There are no positive 2's to be taken from -6. However, the partitive interpretation can be used sensibly, as there is no difficulty dividing -6 into two equal groups of -3. So, for example, if two friends wish to share equally a deficit of 6 dollars (-$6), each gets a share of -$3.
Next, consider dividing -6 by -2. Here, the partitive interpretation fails because it does not make sense to talk about -2 groups or sharing equally among -2 friends, etc. However, the quotative interpretation can be used sensibly: If you want to share a deficit of six dollars (-6) so that each person gets a share of -$2, then 3 people are needed.
Finally, we come to +6 divided by -2. If one attempts to use the partitive interpretation of division, the same problem mentioned in the previous example arises. If one tries the quotative model, we run into the problem raised in the second example: there are no -2's to remove from +6. So how does one interpret 6 / -2 intelligibly? One obvious solution is to talk about division as the inverse of multiplication. Thus, x = 6 / -2 asks the question, "What number times -2 equals 6?" and the answer is -3, which for the moment we will assume is something already established with students. There is no problem finding the correct answer.
Does this mean that partitive and quotative interpretations of division are wrong and should not be taught? Well, it's clear that they make perfect sense in whole number arithmetic. And they cover 3 of the 4 possibilities for integer division where the dividend is an integer multiple of the divisor. It seems reasonable to believe that by the time students reach integer arithmetic (typically 5th or 6th grade), most will be ready for some of the necessary algebraic concepts needed to make sense of all cases, with notions of inverses, identity elements, and so forth coming into play. It doesn't seem like too much of a stretch to think that many will be comfortable with the question given in the previous paragraph as a way to think about six divided by negative two.
No model or interpretation of a mathematical idea is likely to be perfect. But that should not be shocking. The model/interpretation is NOT the mathematics. It's a way of helping students to think about the mathematics. A model or interpretation may in fact be precisely what a given student will carry away with him from his education, though this is not necessarily "the" goal of mathematics teaching: certainly most educators hope that every student will have the opportunity to experience the beauty and power of mathematical abstraction for its own sake, as well as a way of thinking that can deepen as one encounters increasingly complex and challenging mathematical ideas. We should not leave students with the mistaken impression that the map is the territory, that the model or application or interpretation IS the mathematics. But absent evidence that exploring models and interpretations is harmful to student learning or that it inevitably limits their long-term mathematical growth, it seems reasonable for teachers to continue to use them thoughtfully.
What next?
The title of this piece suggests that there is something in all this talk about division, interpretations, models, and applications that can inform us about multiplication and whether it suffices to leave students thinking of it as "no more than repeated addition." In the next part of this piece, I hope to help shed some light on just that question.
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