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Harel, G., & Behr, M. (1995). Teachers' solutions for multiplicative problems. Hiroshima Journal of Mathematics Education, 3, 31-51.

 
TEACHERS' SOLUTIONS FOR
MULTIPLICATIVE PROBLEMS
 

Guershon HAREL
Purdue University

 

Merlyn BEHR
Louisiana State University

 
(Received July 19, 1994)
 
Abstract
 

Four solution strategies were used by inservice elementary school teachers to solve multiplicative problems that do not conform to Fischbein's intuitive models: (a) the Multiplicative strategy, involving the concept of proportionality, (b) the Pre-multiplicative strategy, reflecting an early stage toward proportionality, (c) the Operation-search strategy, based on a trial-and-error approach, and (d) the Keyword strategy. The main finding is that teachers who solved the problems correctly and relationally reasoned in terms of ratio and proportion concepts, whereas teachers who arrived at correct solutions without these concepts did so by a trial-and-error like method. This raises the question: Do multiplicative problems that do not conform to Fischbein's models require for their solution a scheme that includes the concepts of ratio and proportion? Another finding is that there is a striking similarity between the instrumental solutions offered by the teachers and those known to be used by children.

 
TEACHERS' SOLUTIONS FOR MULTIPLICATIVE PROBLEMS
 
During the first half of the last decade extensive research has been carried out to investigate children's conception of multiplicative problems (i.e., problems which traditionally were classified as multiplication, partitive division, and quotitive division problems; see, for example, Bell, Swan, & Taylor, 1981; Vergnaud, 1983; Fischbein, Deri, Nello, & Marino, 1985). The main finding of this research is that children's knowledge of multiplication and division includes beliefs which are incongruent to the multiplicative operations of rational numbers, and thus these beliefs block children's attempts to solve multiplicative problems correctly. Examples of such beliefs include "multiplication makes bigger" and "division makes smaller." Fischbein et al. (1985) put forth a theory to account for these limiting conceptions. They suggested three intuitive models which correspond to three basic multiplicative situations - multiplication, partitive division, and quotitive division-and argued that these models govern children's solutions. These models, which have been the research focus for many investigators in the last few years, are fully described and widely discussed in many research reports (e.g., Mangan, 1986; Nesher, 1988; Harel, Behr, Post, and Lesh, in press); they are briefly summarized in Table 1.
 
Table 1. Intuitive Rules Associated with Fischbein's Three Models
 
Multiplicative situation
Intuitive constraints
Multiplication

1 Multiplier must be a whole number

2 Multiplication makes bigger

Partitive division

1 Divisor must be a whole number

2 Divisor must be smaller than dividend

3 Division makes smaller

Quotitive division 1 Divisor must be smaller than dividend
 
Recent research has focused on adults' conceptions of multiplicative problems, mostly with college students, including preservice elementary school teachers (e.g., Mangan, 1986; Graeber, Tirosh, & Glover, 1989). It was found that many in this population possess the very same limiting conceptions identified in research with children. Further, similar results were found even with inservice elementary-school teachers. In a recent study with teachers of grades 4-6 we investigated the impact of the number type on the choice of operation to solve multiplicative problems (Harel et al., 1994). We found that inservice teachers, like children and college students, are also influenced by these intuitive models; this is despite the fact that the mathematics Curriculum they teach includes the concepts of multiplication and division with rational numbers. This disappointing and astonishing finding points to a possible source of children's limiting conceptions and misconceptions: the limited mathematical expertise of their teachers. Since this finding was obtained in a study which used a pencil-and-paper test, it does not tell much about teachers' reasoning while solving multiplicative problems. This paper reports on a clinical investigation which complements that study. Inservice teachers were interviewed to identify and classify their strategies in solving multiplicative problems. The study was designed to address the questions: What solution strategies do inservice teachers use in solving multiplicative problems? and, in particular, What solution strategies do teachers who are successful in solving multiplicative problems use? Little is known about these questions, especially the latter. Graeber et al., (1989) suggested the second question as one of the important, unanswered questions related to the influence of the primitive models.
 
To answer these questions, we interviewed teachers on their solutions to a set of multiplicative problems which according to Fischbein et al. (1985) violate the basic intuitive models associated with the concepts of multiplication and division. This set of problems is presented in Table 2 (see the Results and Discussion Section). Our analysis of these interviews calls for a reconsideration of the current instructional approach to multiplicative problems. Traditionally, problems of the kind that appears in Table 2 have been classified as multiplication and division, single-operation problems; that is, problems that are expected to be solved by one multiplication or division operation. Further, they have been distinguished from ratio and proportion problems and considered less complex than them; thus, in mathematics textbooks, problems of the kind appearing in Table 2 have always been introduced earlier than proportion problems. We found that only teachers who incorporated the concepts of ratio and proportion in their solution solved these problems relationally and correctly. As we will discuss later in this paper, this finding calls for a reconsideration of the traditional classification of, and the instructional treatment given to, these problems in the current mathematics curricula.
 
Another observation made in this study is that there is a striking similarity between the "immature" strategies (using Sowder's (1988) term) teachers use to solve multiplicative problems and those identified to be used by children.
 
METHOD
 
Thirty-two inservice teachers of grades 4-6 from several rural school districts in a Midwestern state in the US were interviewed about their solutions to multiplicative problems. These two-hour interviews consisted of two parts, one hour for each. The first part, reported in this paper, was devoted to teachers' solutions of multiplicative problems. The second part, reported in Post, Harel, Behr, & Lesh (1991), was on teachers' conception of rational number. All interviews were video-taped and transcribed.
 
The multiplicative problems which formed the basis for the interviews are presented k Table 2. They were selected from an instrument we developed and used to investigate different concepts of multiplication and division. This instrument (see Harel et al, 1994) controls for a wide range of confounding variables known to influence subjects' performance on multiplicative problems -the variables of number type, text, structure, context, syntax, and rule violation. The problems in Table 2, selected from a larger set of problems taken by the teachers prior to this interview, were found to be the most difficult; less than 50% of the teachers provided a correct mathematical expression for getting a solution to the problems (Harel et al., 1994).
 
Table 2. Problems Used in the Interview
 

M:

Each package of typing paper weighs 0.55 kg.

Adam used 0.35 of a package for his research paper.

How many kilograms of paper did he use?

P1:

11 blocks of cheese weighs 22.33 pounds.

How many blocks are there in each pound?

P2:

Marissa bought 0.46 of a pound of wheat flour for which she paid $0.83.

How many pounds of flour could she buy for $1?

Q:

Each bag of frozen peas weighs 154 grams.

You have prepared 65 grams of vegetables.

How many bags of vegetables have you prepared?

* The labels M, P, and Q denote the type of problem according to Fischbein's classification of multiplicative problems: Multiplication, Partitive division, and Quotitive division.
 
RESULTS AND DISCUSSION
 
Thompson (1994) offers a valuable distinction between two kinds of operation - quantitative operation and numeric operation.
 

A quantitative operation is a mental operation by which one conceives a new quantity in relation to one or more already-conceived quantities [e.g.J combine [or compare] two quantities additively, combine [or compare] two quantities multiplicatively ... It is important to distinguish between constituting a quantity by way of a quantitative operation and evaluating the constituted quantity [i.e., numeric operation]. One can conceive of the difference between your height and a friend's height without giving the slightest consideration to evaluating it . . . A quantitative operation is non-numerical; it has to do with the comprehension of a situation.... arithmetic notation has come to serve a double function. It serves as a formulaic notation for prescribing evaluation, and it reminds the person using it of the conceptual operations that led to his or her inferences of appropriate arithmetic.

 
We use this distinction in our discussion of the solutions given by the teachers to solve the problems in Table 2. Teachers' solutions were divided into four categories, or solution strategies:
 
1.
The Multiplicative strategy. In this strategy, teachers reason about the problem situation in terms of ratio and proportion, whereby they represent their quantitative operations to find the unknown quantity numerically by an algebraic equation.
2.
The Pre-multiplicative strategy. In this strategy, teachers derive an approximation of the problem unknown, but not the exact value thereof, either from a quantitative relationship they build up additively or from a unit-rate they construct multiplicatively.
3.
The Operation-search strategy. In this strategy, teachers arrive at a numeric representation of the problem unknown by mere process of elimination. That is, they choose one of the four formal arithmetic operations as a solution of the problem, and then they analyze the reasonableness of their choice. If it did not look to them reasonable they continued to try a different operation until they were satisfied.
4.
The Key-word strategy. In this strategy, teachers' solutions are based solely on keywords appearing in the problem statement.
 
A summary of the frequency distribution of these strategies is shown in Table 3.
 
Table 3. Distribution of the Strategies Used by Teachers to Solve Multiplicative Problems
 
Problem
Number of
Teachers Asked
Number of Strategies Used

Multiplicative
Pre-
multiplicative
Operation
search
Key-
word
Others**
MVP*
Unit-
rate
Pre-
MVP
Pre-Unit-
rate
M
21

3

4

9
5
P1
14
1



8

5
P2
15
8
2
3



2
Q
24
16

1

6

1
 
* Missing Value Proportion Problem
** This category consists of incorrect solutions without explanations. For example, some teachers offered the division expression 0.55+0.55 as their solution for Problem M without being able to explain their actions.
 
The Multiplicative Strategy
 
Here we discuss two instantiations of the Multiplicative strategy, a Missing Value-Proportion strategy and Unit-rate strategy. They are the only strategies that were not based on a trial-and-error approach (i.e., elimination process) and led to a correct solution of the problem.
 
MVP. When teachers used the MVP strategy they represented the problems as missing value proportion (MVP) problems by a quantitative operation of instantiating two ratios comparing them multiplicatively. They evaluated their quantitative operation by an algebraic expression to obtain the unknown quantity. The following response to Problem Q in Table 2 by one of the teachers exemplifies this strategy:
 

Okay ... If you have a bag of frozen vegetables that is 154 grams and you've only prepared 65 grams of the vegetables, you could look at this as a hundred percent of the bag being 154 gram and if 100 percent of the bag is 154, you could set this up as 154 over 100, 154 being 100 percent, and 65 over what percent.

 
This is MVP representation in which the quantities are expressed in terms of parts and wholes. Using Vergnaud's (1983) dimensional analysis, it can be described by the following scheme:
 
 
Percent
 
Grams
 
 
100
 
154
 
 
?
 
65
 
 
This teacher then went on to derive an expression that would give a solution to the problem by using the cross multiplying procedure, writing: "154n = 650".
 
Other teachers expressed their operation symbolically by writing down
 

1 bag = 154 grams

? bag = 65 grams,

 
as one teacher did, and then applying the cross multiplication procedure, "154 x ? = 65", from which she derived the expression, 65 - 154", as her solution of the problem.
 
A characteristic of the MVP strategy is the operation of coordinating the problem quantities into a proportional structure (Harel & Behr, 1989), such as equality of two ratios: a/b = c/d. In the Vergnaud's dimensional scheme the quantities were coordinated into a proportional structure where the numeric value "1" and the unknown value were identified as representing quantities in one measure space (bags) and the other two, 154 and 65, from another measure space (grams). In some cases, the coordination took place by pairing the "l bag" quantity with the "154 grams" quantity and the unknown quantity, "? bags," with the "65 grams" quantity, a between-measure - space strategy. This operation of coordination of quantities can be seen in the following response: After forming the proportional structure for Problem P2 in Table 2, saying,
 

I think this would be best to do it as a ratio 0.46 or 46 hundredths is to 0.83 as what the unknown is to $1 [then she wrote: 0.46/0.83 = n/1.00],

 
the teacher went on to check whether her coordination of the quantities was correct:
 

Let me just check to see if I did this right. Marissa bought 0.46 of a pound of wheat flour for which she paid $0.83. 0.46 over 0.83. How many pounds of flour could she buy for $1.00? I've got the money both on the denominator side. Okay, so what I need to do then is find out what the n would be (emphases added).

 
It must be indicated that from this protocol alone one cannot determine whether this coordination was done instrumentally, based on a memorized procedure, or relationally (Skemp, 1976), based on true understanding. The general impression from the overall interaction with this teacher on this and other problems was that he had acted relationally throughout.
 
MVP solutions included an additional act: estimation-a term which we use to refer to the judgment teachers made about the relative, size of the problem unknown with respect to one, or one-half, before or after they determined the algebraic sentence that represents their solution numerically. For example, as the following protocol of one of the teachers solving Problem Q (Table 2) indicates, some teachers began their solution by predicting that the value of the unknown is less than one, and then used this result to conclude that the problem solution must involve a ratio:
 

Well, you have prepared 65 grams, which is going to be less than a bag, so you' re going to have a fraction of a bag, 65 over 154 of a bag ... or about slightly less than 1/2 a bag. ... I would divide. I would set up a ratio. What you have done compared to the whole thing (emphasis added).

 
This teacher apparently abstracted a part-part-whole relationship that is advanced beyond consideration of only additive relationships among components of a part-part-whole situation to include multiplicative relationships among components in such a situation. She recognized that the value of the fraction of one part to the whole can be represented by the division of the measure of that part by the measure of the whole, and that this division involves the division of a number by a larger number.
 
Unit-rate. Teachers who used the Unit-rate strategy formed a unit-rate relationship from which they derived the value of the unknown quantity. Here are two examples of such an operation:
 
1.
After realizing that 22.33 ÷ 11 would not solve the given problem (Problem P1 in Table 2), one teacher said:
Uh, how many blocks in each pound. That one's really throwing me. I would probably start by finding out how many pounds in each block. I don't know why. ... (She divides 22.33 by 11) 2.03 pounds in each block.... Ratio [talks to herself] ... 2.03 pounds is one block, equals one pound to how many blocks? [wrote 2.03/1 = 1/x]. It's 1 over 2.03.
   
2.
After reading Problem P1 (Table 2), another teacher said:
Okay, wait a minute, something strange about that. So they weigh about 2 pounds apiece. Each block weighs about 2 pounds apiece. The question is just funny. Just looking at that, it is easy to see that each block weighs about 2 pounds. So you would only have half a block for one pound. One pound would only be half of a block.
 
When the interviewer asked her to write an expression that would solve the problem, she replied:
 

Okay, first of all, I would figure out how many pounds per block and that would be a division problem (writes 22.33 ÷ 11 and computes this division operation; she found its result 2.03). And then, of course, they would come up with a little bit over, 2.03 pounds. Then I would say, now if that's one block, how could you figure out ... one block would equal 2 and 3 hundredths to find out what percentage of one blocks one pound would be. You would have to divide by 2 and 3 hundredths to find out what percentage of blocks would be about half. ... You would have to divide 1 by 2.03 and you should come out with about half.

 
What was it about the problem question that seems to pull the teachers toward the division of 22.33 ÷ 11? Is it that a more often asked question would inquire about the weight of 1 block; that is, the number of pounds to a block? Or, is it that the intuitive model constraint of "divide big number by smaller number" is so strong that these teachers are unable to put it aside? Could it be that the division that the teachers wrote of 22.33 ÷ 11 was in the context of transforming the problem date statement "11 blocks of cheese weighs 22.33 pounds" to the equivalent "l block of cheese weighs 2.03 pounds"? In both protocols, we see the strength of the MVP representation to overcome an apparent predisposition to divide big number by smaller number. However, the application of a MVP representation is more apparent in the first protocol, where the teacher said: "2.03 pounds is one block, equals one pound to how many blocks?" and wrote 2.03/1=1/x. In the second protocol the application of a MVP representation can be derived from the teacher's statement, "one block would equal 2 and 3 hundredths [I block per 2.03 pounds] to find out what percentage [i.e., portion] of one block one pound would be [? blocks in 1 pound?] ."
 
The two strategies just discussed end up with a correct numeric operation formed by the teachers to represent their quantitative operations which, as was demonstrated earlier, we interpret to be multiplicative. All other solution strategies identified in this study are inferior to these multiplicative strategies. In the following sections, we discuss the rest of these strategies in order of this sophistication.
 
Pre-multiplicative Strategies
 
In this section we will discuss two instantiations of the Pre-multiplicative Strategy: the Pre-MVP strategy and the Pre-unit-rate strategy. In both, the teachers arrived at an approximation of the problem unknown, but not its exact value, by using lines of reasoning that can be identified with those in the MVP and Unit-rate strategies described in the previous section. As the labels Pre-MVP and Pre-unit-rate suggest, we conjecture that these strategies reflect a transition stage toward the more mature and sophisticated strategies in solving multiplicative problems: the MVP strategy and the Unit-rate strategy.
 

Pre-MVP. In the Pre-MVP strategy the teacher extracts a multiplicative relationship, which he or she uses to additively build up a new relationship for the purpose of determining the problem unknown. In all cases where this strategy was used, the building-up process did not lead to the exact value of the problem unknown. Here are 'two examples: The first is an attempt to solve Problem Q and the second, Problem P2.

 
1.
Dividing the bags into tenths. 10 pounds, that's not..(pause) ... okay, 15 pounds is about 10%, dividing 154 by 15 ... (pause)... 15 ... (pause)... 15 times 10 is 150, rounding off to the nearest tenth. 154 is closer to 150 than it is 160. So I rounded 154 to 150, so every 15 pounds represents 10%. So 15 would be 10%, 30 would be 20%, 45 would be 30%, 60 Would be 40%, 75 grams would equal 50%, 90 grams would equal 60% and so forth. And 65 would, 65 grams would represent between 40% and 50% and it would be closer to 40% than it would be to 50%. So I would estimate 40%.
   
2.
If she bought .46 of a pound of wheat, I think I'll round that off to ... I would round off 46 hundredths to 50 hundredths and make that into a fraction and make it into its lowest terms and getting one-half a pound of wheat flour for 83 cents, then rounding 83 to 80. How many pounds of flour could she buy for $1.00? If she bought a half a pound for 80 cents ... I'm rounding it off ... in order for her to buy another pound of flour ... if she bought a half pound for 80 cents, another half pound would cost her another 80 cents, so 1 pound would cost her approximately $1.60. So if she could not buy ... how many pounds could she buy ... she couldn't buy a pound, She would have to buy less than a pound. Then how much less than a pound ... a half a pound costs her 80 cents, so 1/4 pound would cost her 40 cents, want to know how many pounds could she buy for $1.00. You need to find something about half way between a half and three-fourths ... Because a dollar would be half way between 80 cents and $1.20 ... She could buy 3/8 of a pound of flour ... don't know how to solve it ....5/8, half way between 1/2 and 3/4 is 5/8. ... I'm having difficulty trying to come up with the exact mathematical calculation. I come up with 5/8, but I'm not sure whether it's division or multiplication. I know that for a dollar 3/4 of a pound is too much, 1/2 a pound is too little, and the difference is half way between that which would be 5/8.
 
These two protocols remind us of the Building-up strategy known to be used by children in ratio and rate problems (Hart, 1981). For example, a problem such as "8 pizzas per 6 kids, how many pizzas per 15 kids?" is approached by children in the following manner: They first extend the ratio "8 pizzas per 6 kids" to "16 pizzas per 12 kids." Then, realizing that "12" is 3 units short from "15," they reduce the ratio "8 pizzas per 6 kids" into "4 pizzas per 3 kids," and combine this ratio with the ratio "16 pizzas per 12 kids" to get "20 pizzas per 15 kids." In a similar manner, our two teachers created a ratio which they extended (additively) into a sequence of ratios to extrapolate for the unknown quantity. The teachers rounded off the problem quantities to whole numbers to make the building-up process easier. Neither teacher was able, however, to abstract their building-up approach into a MVP representation. In a similar manner, in the Pre-unit-rate strategy discussed below, teachers were able to from a Unit-rate representation, but stopped short of building from this representation a proportion structure to obtain the problem unknown.
 
Pre-unit-rate. In the Pre-unit-rate strategy, a unit rate is formed to estimate the value of the unknown quantity. For example, in solving Problem P1 (Table 2), some teachers estimated the result of 22.33 ÷ 11 to be "about 2," and interpreted the 2 correctly as indicating that one block weighs about 2 pounds; then they suggested that the problem unknown would be "about one-half" but they were still unable to give a problem representation that would lead to an exact numerical value. The following dialogue between the interviewer and one of the teachers is illustrative:
 

T:
22.33 divided by 11 (writes 22.33 ÷ 11). You have got the total weight.
I:
This (pointing to the expression 22.33 ÷ 11) is your expression to solve the problem. Explain why this expression would solve the problem.
T:
I know the total weight. I know that there are 11 blocks in that total weight. So if I know what the total is and I know there are 11 of them, how many 11' s in the total weight.
I:
Why is the answer to the question, how many 11's in the total weight, is the answer to the problem question, how many blocks are there in a pound?
T:
There's something fishy here. ... Here are the 11 blocks (draws 11 blocks). And together they all weigh this 22.33 pounds. And how much does this one weigh over here? How many blocks are there in each pound? There is no way you could find out how many blocks there are in each pound.
I:
Why not?
T:
Because I don't know the weight of one block. Yes, I do know the weight of one block ... Yes, I can find that out by doing this (writes 22.33 ÷ 11). ... then the number of blocks in a pound is going to be ... 11 blocks weighs 22.33 pounds, then one block weighs 2.33 pounds, so how many blocks are there in a pound? There isn't even going to be a whole block in a pound because a pound is 16 ounces and one block has 21/3 pounds in it.... It won't be a whole block in a pound.
I:
What part of the block will be in one pound?
T:
About 1/2 of a block.
   
From this point this teacher couldn't go further to get an expression which would give the exact value of the fractional part of a block in one pound.
 
In some respects the Pre-unit-rate strategy is a variation of the Building-up strategy, in that a ratio is formed and then reduced to a unit ratio which then is extended iteratively into a ratio to form a proportional structure. In the developmental process in the acquisition of proportionality, the Building-up strategy as well as the Pre-Unit-rate strategy may be intermediate stages.
 
Operation-search Strategy
 
The Operation-search category consists of solutions where the teacher would search for the appropriate operation to solve the problem by an elimination process. In most cases, the teacher first reads the problem and suggests an expression to solve it. Then he or she momentarily puts aside this expression, turns back to the problem, and estimates the expected value of the problem unknown. The teacher then compares this value to that expected from the operation initially suggested. When the teacher realizes that the estimates of the two values do not coincide, he or she reverses this operation (from x ÷ y to y ÷ x). The following are three examples to illustrate this strategy.
 
1. In solving Problem Q (Table 2), some teachers started out with the operation, 154 ÷ 65, but reversed this operation to 65 ÷ 154 when they realized that the unknown value must be less than one, as one teacher did:

Just to make sure of my thinking, I'd make it an easier problem ... So if I had 200 grams ... I used 50 grams ... How many bags? ... Four ... that would be four bags ... So it would be the same thing: 154 divided by 65 [writes: 65 ÷ 154]. ... Now, I'm thinking I did it wrong. What I'm seeing is that if this is 154 grams and I'm only going to need 65 grams ... there's something wrong. That's less than one bag and this (pointing to 65 ÷ 154) answer is more. It's going to have to be less than a bag. So then it means I'd have to reverse what I was doing. It would be 154 with 65 over it, because it's going to be less than one when you're done

 
2. In some cases the order of events was slightly different from the ones exemplified by the above protocol. Unlike those cases where the teachers first suggested their operation to solve the problem and then estimated the expected value for the problem unknown, this teacher started with the estimation first. This is worth mentioning because despite the fact that this teacher had a correct estimate for the unknown (in Problem Q, Table 2), she went on and suggested the operation 154 ÷ 65 (i. e., big ÷ small) as her solution for the problem. Only after she completed the computation for this operation, did she realize that her initial estimate conflicts with the result of her operation. Upon this observation she then reversed the operation as in the other cases:

We haven't even used a bag ... You've prepared less than 1/2 of a bag of vegetables. [She writes down 65 ÷ 154 = 224/651 . ... No, this indicates that you've used over 2 bags. What would I do? Somehow I'm going to have to make this decimal. I think I know to do it! (She writes down 65/154.) ... which would be less than 1/2. So that's the way you should do it. In looking at it, first of all, I knew I had to have a decimal somewhere (emphasis added).

Her statement, "Somehow I'm going to have to make this a decimal," demonstrates her instrumental approach in applying the Operation-search strategy to solve the problem.
 
3. In their search for a correct operation to solve the problem, some teachers went through four operations: subtraction, multiplication, division (the larger quantity by the smaller quantity), and inverse division (the smaller quantity by the larger quantity). Here is one protocol that illustrates such a solution process:

Okay. ... I would think that you would subtract 65 from 154 ... (long pause) doesn't make any sense. We haven't used more than a bag. So you'd have to say ... this is miserable ... 65 X 154 ... that gives too big a number. Okay, one bag is 154 grams, I used 65, a portion of a bag. Let's try 154 divided by 65. No ... How about 65 divided by 154? I've used 65, so 65/154.

 
In example 1, we notice that the teacher's first response was to give a division sentence which satisfies the "divide big number by smaller number" intuitive model constraint. Thus, the intuitive model definitely seems to be operative. However, this teacher, as contrasted to many children, is able to reason about this response and is able to overcome the apparent predisposition to satisfy this constraint. In example 2, the teacher had a correct estimate of the answer for the problem, but yet proceeded to write an incorrect division sentence consistent with the "big number divided by smaller number" constraint. What does this suggest about the teacher's understanding of division? It might be that the teacher's predisposition to satisfy this constraint was stronger than her realization that the answer was less than 1, or else possibly she didn't have a sense for the magnitude of the result of the division until it was actually carried out. In example 3, the teacher appears to demonstrate a random choice of operation on the two problem quantities (154 and 65). However, she does have a sense for the size of the answer from her comprehension of the problem. The teacher's difficulty appears to be with not having a number sense (Sowder, J., 1988) for operations, or using Thompson's (1994) terms, the teacher's numeric operation is not sustained by a quantitative operation.
 
A feature that is common to all solutions in the Operation-search category (and many solutions in the multiplicative strategies) is the act of estimation. Estimation was an important cue for the teachers in their decision of what operation to choose and in their reevaluation of the representation they formed. Sowder (1988; p.231) observed an analogous strategy with children which he labels, "Narrow the choices, Based on Expected Size of Answer."
 
[This strategy] involves an instrumental understanding of the operations, albeit one refined by classifying the operations according to their influences on numbers. It also involves an a priori consideration of the size of the expected answer, and thus requires some understanding of the problem.
 
  I: Why division [for a problem involving reduction in a photocopy machine] ?
  Fred:
(Grade 8)
Because it's reducing something. And I know that when you reduce something you're either taking it away or dividing it.
     
According to Bell, Greer, Gremision, & Mangan (1989) the act of estimation is necessary for problem comprehension and easier than the formation of a quantitative operation that can be represented numerically. In their investigation of adult (15year-olds and older) students' performance on multiplicative word problems, they asked half of the students to respond by choosing the operation that would solve the problem correctly, and the other half to make an estimate of the problem unknown. They found that the latter task was substantially easier than the former for division problems and for multiplication problems where the multiplier is smaller than one:
 

The making of a correct estimate depends on a correct perception of the operational structure of the problem. This does not necessarily require explicit identification of the numerical operation needed to calculate the exact result. We know from the numerical misconception MMBDS [Multiplication Makes Bigger, Division (makes) Smaller] that pupils must have an awareness of the size of the expected answer before making a choice of operation. We suggest that in the division problems and problems involving multiplication by numbers less than 1, the estimate is made directly by serniqualitative ratio comparison, without explicit identification of the division operation (van den Brink & Streefland, 1979) (P.447).

 

As it will be discussed in the next section, the more striking similarity between the solution strategies employed by teachers in this study and those used by children in other studies, is in the use of the Key-word strategy.

 
Key-word Strategy
 
Teachers who used the Key-word strategy basically did a direct translation of" the verbal problem statement to a symbolic statement using such words as "of", "gave away", "share", and "is" to suggest operations or relations of multiplication, subtraction, division, and equals, respectively. Here are four examples of responses by four different teachers:
 
1.
I can do this one (Problem M). ONe package weighs 0.55 kilograms. And Adam used 35 hundredths of that one package. How many did he use? In order to find out how many he used, I've got to subtract.
 

 

When the teacher was asked how she would explain this solution to a child, she, replied:

What I would say to them, I'd have them read the problem to me and my question would be, "Do you have to add, subtract, multiply, or divide?" All right. And from the last question, "How much did he use?" they know they have to take away. That's what they're looking at, is the last question. So then I'd say to them, "How much did he start out with?" 55 kilograms; "how much did he use up?" 35 kilograms. And then everything's set up all right so then we subtract.

   
2.
This [Problem Q] is a multiplication problem. 0.55kg times O.35. To explain this to a student I would tell them when they see the word 'of' they're taking a number of another number. It means multiplication. You're trying to find out how many kilograms of paper. The whole package weighed 0.55 kilograms and he only used part of the package. He only used part of that weight of paper, so ..
   
3.
One of the things that I always have my children do when we start multiplication is cue in for key words like 'of' and so on.
   
This teacher expected her students to be able to check their answers to such problems, so she teaches them that:

Multiplication by something less than one is going to give you something smaller, and multiplication by something greater is going to give me something larger.

 
4.
55 hundredths of a kilogram to begin with and he used. ... that would mean that we take this [pointing to 0.351 from this [pointing to 0.551 [She wrote 0.55kg - 0.35kg in columnar form].
   
When the interviewer asked, how would she explain to a student the thought process she used to solve this problem, she answered:

If we were having a hard time with this I would pretend that this stuff of decimal points wasn't here. I would ask them to pretend it wasn't here. This is just 55 and if some of it was taken away. So we're going to take away 35. So if you can do that [without decimal points] you can do that with decimal points and nothing changes. It wouldn't be 55 but 55 hundredths.

 
One wonders whether this teacher's interpretation of 0.35 package as 0.35 kg resulted from a misreading of the problem or whether the key word "used" (suggesting that a lesser amount would result) was such a strong indicator of subtraction that all other information in the problem became blurred.
 
As was noted by several researchers (e.g., Nesher & Teubal, 1975; Sherrill, 1983; Sowder, 1988), the Key-words strategy is known to be used by children and even older students in solving various types of word-problems. But, after seeing teachers' responses about their use of key-words both as their own solution strategy and as a method they teach their students, one cannot be surprised to encounter a response such as the following:
 
  Emmy:
[Grade 8]
…a lot of times you do word problems, and they're, usually word problems, you can find the words, certain words, and it tells you what to do.
  I: Like what?
  Emmy: Like, or certain words like "of" is equal [sic] , and "is" is to multiply, or something like that. And you learn to do that in certain word problems, such as percentage problems. That's how I learned last year. That "of" is equal. Certain words like that tell you what to do.
     
(Sowder, 1988; P.131).
 
SUMMARY AND IMPLICATIONS
 
The purpose of this paper was to identify and classify the kinds of strategies inservice teachers use in solving multiplicative problems. In particular, we wanted to know what strategies teachers who are successful in solving multiplicative problems use. In all, we found four strategies of problem solutions used by the teachers: the Multiplicative strategy, the Pre-multiplicative strategy, the Operation-search strategy, and the Key-word strategy.
 
The Multiplicative subcategory consists of ratio and proportion solutions. These were the only strategies that are not based on guess and check approach and which led to a correct solution of the problem. In all cases where the ratio and proportion solutions were used, the teachers were able to construct a quantitative operation and to represent it numerically to obtain the problem unknown.
 
The Pre-MVP and Unit-rate strategies reflect an early stage toward ratio and proportion conception in dealing with multiplicative problems. There is a strong similarity between the Pre-MVP used by the teachers and the Building-up strategy known to be used by children in ratio and proportional reasoning situations. Indeed, the Building-up strategy is viewed by many researchers as an indicator for pre-proportion stage (see Tourniaire and Polus, 1985).
 
The most important aspect of these findings is that teachers who solved the multiplicative problems correctly and, more importantly, relationally, reasoned about the problems in terms of ratio and proportion concepts. Further, teachers who arrived at correct solutions without the concepts of ratio and proportion did so by a trial-and-error like method. None of the teachers who solved the problems in Table 2 correctly did so with a one-step operation. In contrast, these teachers did give one-step operations as their solutions of other kinds of problems- problems which were syntactically isomorphic to the problems in Table 2 but differed from them in that their numerical data conformed to the Fischbein et al. intuitive models (see Harel et al., 1994). These findings raise an important question to mathematics education research: Does this mean that multiplicative problems that do not conform with the intuitive model for multiplication and division require for their solution such powerful schemes that include the concepts of ratio and proportion? If so, what does this imply for curriculum and instruction?
 
Greer (1987) raises the question of whether the intuitive models identified by Fischbein et al. (1985) are the result of teaching or of a natural cognitive development. Fischbein et al.'s view is that these models "correspond to features of human mental behavior that are primary, natural, and basic" (p.15). Greer (1987) pointed out that this is a pessimistic view which, as Fischbein et al., (1985) agree, restricts the role of instruction to merely providing "learners with efficient mental strategies to control the impact of these primitive models" (p.16). He suggested a more positive instructional approach, in which the aim is "to widen the range of models available to the pupils" (p.74). Neither Fischbein nor Greer, however, specified an instructional approach to teaching multiplication and division. Still the questions are, what mental strategies can help children control their misconceptions? and what other models should be taught to children that can "compete" with the primitive models? The findings of this research may suggest that the formation of ratio and proportion concepts can be powerful tools in dealing with multiplicative problems. The question whether these concepts are necessary tools for multiplicative problems of the kind presented in Table 2 is still open and needs further research. However, some evidence exists to support the view that instructional treatment of multiplicative problems, that involves the concepts of ratio and proportion can improve children's performance (see Selke, Behr, & Voelker, 1991).
 
We strongly believe that the "immature" strategies used by some of our teachers are inappropriate and must be discouraged and even condemned by mathematics educators. These are the Operation-search strategy and especially the Key-word strategy.
 
Indeed, the Operation- search solutions and the ratio and proportion solutions share the feature that they both result in a correct numeric representation of the problem unknown. Further, many of these two types of solutions involved the act of estimation. But, the two types of solutions differ in the most important feature: while the ratio and proportion solutions are based on relational, genuine understanding of the problem situation, the Operation-search strategy is based on a trial-and-error approach, reflecting an immature, low level problem solving skill.
 
Sowder (1988) indicated,

The use of key words is taught by some well-meaning teachers who are not aware of how students can abuse it.... The spirit of teaching key words -getting students to think about the situation -is all right, but students sometimes look only for the key words and ignore the whole context (pp.230-231).

 
The protocols presented in this paper indicate that (some) teachers also look for the key words and ignore the whole context; their teaching of key words to solve word-problems is the antithesis of getting students to think about the situation. The negative implications of this situation to students' education and mathematical ability is clear and needs no further elaboration. It is hoped that teacher-preparation and teacher-enhancement programs would take into consideration this situation seriously, and alert the teachers for the damaging consequences of teaching how to use keywords to solve word problems.
 
The use of "immature" solution strategies, such as the Operation-search strategy and the Key-word strategy, by inservice teachers is a direct consequence of deficiencies in the current teacher preparation programs. The deficiencies are in the three crucial accounts of teachers' ' knowledge: mathematics content, epistemology, and Pedagogy. (Harel, 1994). Teachers' mathematics knowledge is far from being satisfactory even in terms of the standards for elementary-school mathematics. The "Methods" courses for teachers on mathematics epistemology and pedagogy do not involve personal experiential basis of teaching, and thus they are in conflict with the well established principle that knowledge construction (and this includes mathematics knowledge as well as knowledge of mathematics epistemology and pedagogy) is a product of personal, experiential problem solving activity.
 
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