OBSTACLES IN APPLYING A MATHEMATICAL MODEL: THE
CASE OF THE MULTIPLICATIVE STRUCTURE
Irit Peled - University of Haifa
Leonid Levenberg, Ibby Mekhmandarov, Ruth Meron Center for Educational Technology, Tel-Aviv
Alex Ulitzin - Ministry of Education, Jerusalem
Third and forth grade children, who started learning about
multiplication in second grade, do not tend to use multiplication
in multiplicative situations. This study looks at children’s
choice of mathematical models in different non contextual
displays: equal groups, rectangular arrays, and a rod model
(which they use in class). The findings show that many children
do not perceive the displays as representing multiplicative
situations.
Even children who exhibit knowledge of
multiplication facts do not apply their knowledge in these tasks.
Instead, they use addition and counting strategies.
An important goal in teaching mathematical operations is the
development of schemes, such as the additive scheme or the
multiplicative scheme. These schemes, or structures can act as
mathematical models of given situations. Usually the situation does not
uniquely determine the structure, nor does it easily hint that a certain
mathematical model can be used (although conventional school word
problems might imply that it does). We teach children basic operations
with the intention that they apply them in different situations. We also
expect to see progress over time in the choice of a model, i.e. in the
ability to apply more efficient models in the relevant cases.
Given a rectangular array of objects with the task of finding their total
number, it is expected that a young child will count the objects one by
one. An older child will count the elements in each row and then add up
the rows. An even older child is expected to realize the size equivalence
of the rows, i.e. the repeated addition structure and then add or multiply,
or recognize the rectangular array structure, and multiply the number of
rows by the number of columns.
The above description talks about different structures, which have
been identified by several researchers. A lot of the research on
multiplicative structures deals with the categorization of multiplicative
situations (Vergnaud, 1983; Schwartz, 1988; Nesher, 1988) and with
children’s word problem solving (Fischbein et al., 1985; Kouba, 1989;
and many other significant works reviewed by Greer, 1992).
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Some of theses researchers have focused on younger children’s work.
Carpenter et al. (1993) show that even kindergarten children can solve
some types of multiplicative word problems. Kouba (1989) looks at
solution strategies of children in grades 1-3, and is interested more in the
nature (and quality) of their solutions than in the question whether they
can answer a given problem correctly. Kouba uses equivalent set
problems (later termed repeated addition problems or mapping rule
problems) and finds that children use a variety of counting strategies.
She also observes that the intuitive model that children seem to have for
equivalent set problems is linked to the intuitive model for addition as
both involve building sets and then putting them together.
Similarly, Mitchelmore and Mulligan (1996) show that during their
second and third grades children use many different strategies in solving
multiplicative problems. These strategies include quite a lot of addition
and counting calculations. However, they also find that over time the
strategies are chosen more efficiently.
These different research findings suggest that children do not
necessarily use multiplication in solving multiplicative word problems.
Many children use addition and some choose to use a long counting
process.
It is often claimed that children are efficient and use a more effective
tool or a shorter route once they possess it. Such a behavior is described
by Woods, Resnick, and Groen (1975) in the case of choosing between
solution strategies in subtraction (e.g. going two steps backwards in 9-2
while counting up from 7 to 9 in 9-7). This behavior is also evident in
Siegler and Shrager strategy choice model (1984), where children choose
to retrieve an addition fact rather than count, when they reach a
reasonable degree of confidence.
If children tend to be efficient, why do they not use multiplication but
instead do quite a lot of counting or adding? Several explanations can be
suggested: They are not able to identify the structure of a multiplicative
word problem as a multiplicative structure, or they do recognize a
multiplicative structure but do not know the relevant multiplication facts.
In this work we differentiate between these two obstacles by looking not
only at children’s solution strategies but also at the way they perceive
different situations.
Most of the existing research, including the works described above,
involves word problems. In solving word problems children are engaged
in text interpretation, a stage which might contribute to the difficulty in
identifying the efficient structure (although in some word problem types
the verbal description contains clues for identifying a multiplicative
structure). In this work we avoid this stage by looking at children’s
behavior in different non-contextual displays of objects. As we show
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below, the use of a multiplicative structure is scarce even in these noncontextual tasks.
PROCEDURE
Fifty four third and forth grade Israeli children were individually
interviewed. The children were chosen from regular classes that use the
same curriculum, called “One-two-and three”. They were identified by
their teacher as having some minor difficulties in mathematics. This
curriculum usually introduced beginning ideas about multiplication at the
end of first grade, and further develops the concept in second grade. A
sequence of concrete models is used in order to represent multiplication:
a “train” of equal rods, the same rods in a rectangular configuration, and
eventually an array of dots. Using the array model the children discuss
the number of rows and the number of dots in each row, and apply the
model in different problem solving situations.
The purposes of the interview were: To find what mathematical model
children use spontaneously in perceiving a given display, which has a
multiplicative structure, to observe which strategy they apply
spontaneously in calculating the amount represented in the display, and
to find how they calculate multiplication facts in multiplication number
problems. Eventually these findings were used to investigate the
relation between knowing multiplication facts and recognizing and using
multiplication in different situations.
In the course of the interview each child was presented with different
displays, asked to describe what she sees, and then requested to find
“How many there are”. The order was: Equivalent sets, a “train” of
equal rods, a rectangular array, and some contextual situations (involving
eyes and fingers). Each display was presented several times with
different numbers (4x5, 10x5, 4x2, 10x2). Here we present the results
for the case of 4x5, (complete results to be presented in an extended
article).
The request to describe what they see was intended for investigating
how children perceive the display. The child was given a card with a dot
configuration and asked to tell the interviewer, who supposedly could not
see the card, what to draw.
Several additional tasks included: representing a given expression,
such as 4x5, using rods, inventing a story problem to a given expression,
and finding some multiplication facts, e.g. 4x5. The questions that
mentioned multiplication came only at the end of the interview in order
to avoid any hints about the choice of an operation in the different
displays.
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RESULTS
Children’s conceptions were deduced from their display descriptions
and explanations during the interview. In several cases a child was
considered to be perceiving the display as a multiplicative structure even
when an additive or a counting strategy was used in the calculation of the
total amount of objects. In the following excerpt a third grader, Lorry,
tries to figure out the total amount represented by a “train” consisting of
four ‘5-rod’-s, as shown below:
yellow
yellow
yellow
yellow
I: What do you see?
L: Yellow rods of 5.
I: And what else? (no answer) How many rods?
L: 4.
I: How much is it?
L: Should I do 4 multiplied by 5 [note: In Hebrew 4X5 can be read as ’4
multiplied by 5’, or as ‘4 times 5’. Here she used the word ‘multiplied’.
Further on when she calculates the amount by addition, she uses
‘times’].
I: Yes.
L: (Thinks for a while) 25.
I: How did you do it?
L: I did four times five, 5 plus 5 that’s 10, and another 5 that’s 20, and
another 5 that’s 25.
Table 1 presents the percentages of students who perceived the
different displays of 4X5 as multiplicative situations. It also shows these
percentages separately for students who could do a mental calculation of
4X5 (using either fact retrieval or repeated addition), and for those who
could not do a mental calculation (e.g. had to use objects). It should be
noted that the columns are not disjoint. A child who perceived a
multiplicative structure in one display, could also see it in another
display.
Table 1: Children exhibiting multiplicative display conceptions.
display
calculation
mental
n=32
non-mental
n=22
all students
n=54
equal sets
yes
no
7
25
(22%)
(78%)
0
22
(0%)
(100%)
7
47
(13%)
(87%)
rods
yes
5
(16%)
2
(9%)
7
(13%)
no
27
(84%)
20
(91%)
47
(87%)
array
yes
no
10
22
(31%)
(69%)
2
20
(9%)
(91%)
12
42
(22%)
(78%)
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The data in Table 2 shows the different calculation strategies in each
of the displays for children who could figure out 4x5 mentally. This
subgroup of children includes those who could potentially utilize their
knowledge in the different displays.
Table 2: Strategy choice in different situations for children who could
mentally multiply 4x5.
Strategy
Situation
equal sets
rods
rectangular
array
contextual
cases
multiplication
6
(19%)
4
(13%)
8
(25%)
7
(22%)
addition
counting
18
(56%)
19
(59%)
9
(28%)
20
(63%)
6
(19%)
-
other
9
(28%)
3
(9%)
2
(6%)
9
(28%)
6
(19%)
2
(6%)
total
32
(100%)
32
(100%)
32
(100%)
32
(100%)
As can be seen in Table 2, less than a quarter of the children who
calculated 4x5 mentally, used multiplication in each of the different
display calculations. Our data (not presented here) details this
distribution separately for children who did the calculation of 4x5 by
retrieval, and those who used addition. Most of the children who used
multiplication in the display calculation were those who used it in the
calculation of 4x5.
Children’s answers and explanations contribute some interesting
information on the way they perceive the given representations. The
following are two of these examples:
1. Post hoc identification of a multiplicative structure:
Given a rectangular array of 4x5 X-s , Ron (a forth grader) draws it
correctly from memory. The task is followed by this dialog:
How many rows are there? 4
How many X-s are there in each row? 5
How many X-s altogether? (Ron thinks for quite a while) 20
How could you tell? I did 4x5.
Did you do it in your head? No. I counted the X-s.
So why did you say 4x5? Because it’s 20.
But 2x10 is also 20. (Ron hesitates a moment) Ah! But here
(in the array) we have 4 and also 5.
So why did you count earlier rather than do 4x5? Because
I was in a hurry...
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This dialog might initially suggest that Ron identified the
multiplicative structure, or at least the repeated addition structure of the
array. However, the time it took him to figure out the total amount, his
own account on his counting, and his surprised discovery of the
connection to the 4 and 5 lead to a different interpretation. Ron
suggested the expression 4x5 after counting 20 X-s in the display. He
might have chosen 4x5 because it is an expression that yields 20. It is
probably during the discussion that he suddenly saw how the array
dimensions related to the expression.
2. A selective application of multiplication:
Some children used addition or even counting for small amounts, while
applying multiplication for larger amounts. Other children had different
reasons for the choice of a strategy, such as: I counted [even though I
used multiplication in another situation] because I was in a hurry....
[and would have wasted time if I stopped to analyze the situation].
DISCUSSION
Third and forth grade children participating in this study showed very
little use of multiplication strategies in non-contextual displays, while
multiplication was the more efficient strategy. These findings could only
partially be attributed to the fact that most of these children did not know
the relevant multiplication facts. This was revealed by investigating the
way children perceive the displays. Only a small proportion of children
perceived the displays as multiplicative structures. Even among those
who could figure out a multiplication fact mentally, only about a third
identified the multiplicative structures. The conclusion that the blame
does not lie in absent knowledge of facts is also manifested by the choice
of strategies in figuring the amount in different displays. Less than a
quarter of the children who could figure out the facts mentally used
multiplication, a large proportion of them used addition, and some even
counted.
The children in this study were identified by their teachers as having
some difficulties. Yet the displays presented to them were familiar
representations, the same ones through which multiplication was defined
to them in first and second grades. If fact retrieval is not the main
obstacle in applying multiplication, perhaps the difficulties involve the
nature of the displays or the nature of instruction.
The identification of a multiplicative structure is quite complex. In
equivalent set situations, for example, one has to be able to perceive all
the given sets at the same time and recognize their equivalence. In
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deriving the number expression one of the factors is an intensive
quantity, appearing not just in one set but in each of the sets. The second
factor is not even represented as a simple set. Rather, it is the number of
the sets in the display. This complexity makes it difficult for children to
identify and apply a multiplicative structure in a given situation.
The interviews disclose some of the display difficulties. In additional
tasks, where children were asked to use manipulatives and represent an
expression, such as 3x4, some of them tended to represent both numbers.
Thus, for example, one of the children used rods and built a train
consisting of a single 4-rod and three 3-rod -s, as follows:
(4)
(3)
(3)
(3)
When realizing that something was wrong because it does not measure
12, as expected, he changed it to one 3-rod and three 4-rod -s. He was
frustrated upon realizing that it still does not measure 12. Another child
represented this expression by building an array consisting of four rows,
with three 3-rod -s in each row.
In the course of class instruction children are directed to those
elements in the display which represent the multiplication factors. If we
want them to develop the ability to look at a given display and choose an
efficient mathematical model, we need to teach them to analyze situation
structures and detect relevant features of these situations. Our instruction
should include tasks that give them the opportunity to develop the ability
to analyze and apply the mathematical models which are available to
them.
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