USING A MULTIPLICATIVE APPROACH TO CONSTRUCT
DECIMAL STRUCTURE
Irit Peled*, Ruth Meron** and Shelly Rota**
*University of Haifa / **Center for Educational Technology, Israel
This study suggests an alternative instructional sequence intended to promote
children's construction and understanding of decimal structure through a
multiplicative perspective. Using a constructivist approach 3rd grade children are
engaged in tasks that call for challenging investigations to determine what orders can
be met in a Cookie Factory where cookies come in a limited number of box types. In the
article we demonstrate the power of this didactical model in eliciting rich strategies
and in facilitating the emergence of decimal structure understanding through
reasoning with number multiples.
INTRODUCTION
In this study we present an alternative instructional trajectory to introducing decimal
structure that takes a constructivist view of learning. This alternative approach is based
on children's earlier knowledge of multiplication, introducing the new structure as a
special case of multiplicative structures. This study describes the teaching trajectory,
demonstrating how the planned sequence enables conceptual change while still leaving
room for children's own knowledge construction.
THEORETICAL BACKGROUND
Place value and decimal system instruction
Children have a lot of trouble in constructing their decimal system and place value
number concepts, carrying these difficulties further on into their learning of multidigit
operations. Kamii (1986) analyses the complexity of place value knowledge and Ross
(1989) shows that even fourth and fifth graders lack good understanding of place
value.
Hiebert and Wearne (1992) and Wearne and Hiebert (1994) demonstrate the
importance of learning mathematics with understanding, claiming that learning
numbers with an emphasis on place value meaning rather than on symbol
manipulations, has a positive effect and proves to be beneficiary in the long run. While
their suggestion emphasizes the importance of teaching multi-digit operations with
understanding right from the start, Segalis and Peled (2000) show that it is not too late
to develop conceptual understanding of multidigit procedures at a later point by
making the right connections.
The extent of the problem and topic importance are self evident in an international
study detailed by Fuson et al. (1997), comparing four different projects on teaching and
learning multidigit number concepts and multidigit number operations. All four
2007. In Woo, J. H., Lew, H. C., Park, K. S. & Seo, D. Y. (Eds.). Proceedings of the 31st Conference of
the International Group for the Psychology of Mathematics Education, Vol. 4, pp. 65-72. Seoul: PME.
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projects support learning number concept and operations with understanding. In some
of the projects children learn place value concepts using base-ten blocks, in others they
use different kinds of base-ten materials such as Montessori Cards, number charts, or
frames with many rows of moveable beads (ten in a row). In one of the projects (CBI),
children are given word problem situations involving packaging in tens with the
intention to assist them in constructing meaning for the written symbol. In this project,
and in the other projects as well, children are given word problems and are encouraged
to invent procedures for multidigit addition and subtraction. One of the projects
(STST) involves urban Latino children. These children learn to represent tens and
hundreds as new units with special symbols and apply their symbolic representations
in real-world problems involving grouping activities.
The acts of grouping units into groups of tens and tens into groups of hundreds have
often been supported and constructed by various packaging activities and regrouping
of physical objects. The STST project uses contexts such as a doughnut store or money
expenditures. The store context has single doughnuts, boxes of ten doughnuts, and
baking trays of 100 doughnuts (or 10 boxes of ten).
In a computer based project Champagne and Rogalska-Saz (1984) let children pack
and unpack bundles of sticks or use a special version of “messy” Dienes Blocks. In this
messy version a long box holds ten unit cubes and a square box hold ten long boxes or
100 unit cubes. This modification replaces the act of trading (using "the bank") with
acts of grouping and regrouping with no need of “external” help. The computer
environment enables children to use these acts in mapping between number operations
and physical representations.
A constructivist perspective
While the computer assisted instruction was structured, aiming towards a specific
traditional algorithm, other projects (including the abovementioned four projects) give
children more room for invention. With a constructivist view on learning, Cobb et al.
(Cobb, Yackel, & Wood, 1992; McClain, Cobb, & Bowers, 1998) conducted a nine
week teaching experiment with third graders, during which the researchers designed a
sequence of instructional activities in collaboration with Gravemeijer in the spirit of
Realistic Mathematics Education (RME) as described by Gravemeijer (1997).
The sequence is built around yet another packaging situation called “The Candy
Factory” and was designed “to support third graders’ construction of increasingly
sophisticated conceptions of place value numeration and increasingly efficient
algorithms for adding and subtracting three-digit numbers”. The researchers
emphasize that “the goal was not to ensure that all the students would eventually use
the traditional algorithm.” According to McClain (who was also the teacher) et al.
(1998), initial whole-class discussions started with the students and teacher negotiating
“the convention that single pieces of candy were packed into rolls of ten and ten rolls
were packed into boxes of one hundred.” Following this initial agreement, children
were engaged in estimations involving looking at drawings of rolls and pieces of candy.
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Further activities involved packing and unpacking activities using Unifix blocks and
developing a coding system to record the actions. An important part of the activity
involved the symbolic description with pictures or tally marks or numerals. The final
phase consisted of using an “inventory form” to record addition and subtraction
operations corresponding to acts of filling orders or increasing inventory.
The construction of new units and its connection to multiplication
Understanding decimal structure is a process that involves the construction of new
units. Fuson (1990) details difficulties involved with this process and investigates
conditions that affect it. She demonstrates the positive effect of instructional models
that use a representation of tens and hundreds as units (e.g. Dienes blocks) and,
similarly, the effect of having a language that uses tens and hundreds as units in
number names, on developing new decimal unit conceptions.
The conception of a three digit number as consisting of three different types of units
(e.g. view a number such as 432 as 4 of a new unit called hundred, 3 of a new unit
called ten and 2 of the unit one) involves a combination of multiplicative
understanding, with place value knowledge. While the amount taken of each unit is
shown, the unit itself is hidden and coded by place value. Obviously this is not a simple
extension of multiplicative structure. However, as we will show, through this partial
similarity, multiplicative structure can offer a bridging trajectory to the further
construction of decimal structure.
In developing their multiplicative conceptions children have to undergo some
transitions from counting by ones to counting by an emerging new counting unit, a
complex process which is thoroughly investigated and described by Steffe (1988). The
operations of multiplication and division involve coordination between creating
groups or measuring with the new unit, while at the same time keeping a count of the
number of groups using a different counting unit (the original ones). By the time
children start third grade, which is when our curriculum extends decimal structure
knowledge beyond 2-digit numbers, they have been introduced to multiplication.
Thus, the instructional unit that we have designed has, in fact, a double purpose. It is
aimed at strengthening the understanding of multiplicative structure while at the same
time using these structures to create new insights of decimal structure.
THE INSTRUCTIONAL SEQUENCE
In the present study we use a context and a constructivist view similar to those used in
the Candy Factory. However, our approach to introducing the decimal structure is very
different. We start with a long process of investigations, focusing on multiplicative
structures. We attribute more importance to the process of re-inventing the base ten
grouping, and to perceiving the base-ten grouping as a special case of other possible
multiplicative groupings. The purpose of this study is to investigate whether children
manage to make relevant and meaningful discoveries in this designed instructional
trajectory. In our broader study we have conducted teacher workshops to explore
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whether teachers are able to comprehend this didactical model and appreciate its
potential effect on children. This part will not be reported here.
The context: The Cookie Factory. The factory has cookie boxes that can hold a certain
fixed amount of cookies. At some point the factory only has 2 types of boxes, at a later
point it might have 3 types of boxes. People come to the factory to buy cookies. The
constraints: They can only buy a quantity that can be supplied using the current factory
boxes. For example, if the factory uses only boxes containing 15 or 6 cookies, the
sellers would be able to give 36 cookies (they might deliberate on whether to use 2
boxes of 15 and 1 box of 6 or 6 boxes of 6, an efficiency criterion of using a smaller
number of ready-made boxes can be discussed). They would have to investigate if and
how they can give 33 cookies, and would find out that they are unable to supply an
order for 25 cookies.
Figure 1: An example of available box types (cardboard cut-outs).
Children are given the current factory constraints and told that the workers are
interested in investigating which quantities can be supplied. That is, what quantities
can be generated by current box types. Figure 1 shows the boxes available in one such
case, where the box types are 25 and 10.
Following several class sessions with investigations of this kind, children are told that
the factory engineers need help in deciding which 3 box types to use. The children's
task is to come up with suggestions that would have the following features: Cover as
many orders as possible, supply the order using as little boxes as conditions allow, and
decide quickly how the order should be supplied.
It is expected that class discussion will lead to the idea that the choice of boxes with
100, 10, 1, has many advantages, although with some numbers it is not ideal. Even if
some children will not agree on making it their own choice, they will be able to get the
feel of the nature of using this choice.
The discovery of the power and meaning of the 100,10,1 option is expected to come as
a surprise involving an “aha” reaction. In the following section we describe some
episodes from our grade 3 implementation of the instructional trajectory, paying
special attention to identifying the moment of discovery.
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IMPLEMENTATION OF THE INSTRUCTIONAL TRAJECTORY
The initial activities were expected to engage children in finding linear combinations
of multiples of given quantities. They were meant to promote reasoning about efficient
investigations, about data recording and about data representation. As expected,
children found several different ways to generate what orders can be supplied, and
record how the orders can be supplied.
Two main ways to generate possible orders involved the use of a hundreds table and
the use of a linear combinations table. The first table depicted the number sequence,
where children circled amounts that could be supplied, the second table had multiples
of one box type on one dimension, and another box type on the other dimension, as
depicted in Figure 2 for the case of boxes with 6 or 10 cookies.
Figure 2: Ido's generation of some of the possible orders for 6 and 10.
In a reflection on his work, Ido wrote in his notebook: When I got to task number 14, I
wanted to know what quantities can be supplied not just between 60 and 75 but beyond
that. So I decided to make a table like a multiplication table only one side has 6 and the
other 10 according to the quantities. So if I will take the number 38, for example, it is 3
boxes of 3 (he meant to say 6) and 2 boxes of 10 according to the table. This way I
could tell what amounts can be generated and supplied. (Ido added a comment
underneath the table depicted in Fig. 2 saying that: I did more. Meaning he actually
generated a bigger table and only gave a partial example in his notebook.)
The investigations had benefits beyond their intended aim in the decimal instruction
sequence to promote decimal system understanding. The use of number multiples in a
rich variety of tasks resulted in emergent fluency of number multiples. We even started
hearing comments from parents who expressed their surprise in children’s fluency with
multiples such as 25 and 20. The time “investment” in these activities also paid off
later when children got to another topic studied in third grade dealing with divisibility
criteria. Once they realized that the question “Is this number divisible by 6?” is
equivalent to the more familiar question “Can this number order of cookies be supplied
using only boxes of 6 cookies?” the task became clear.
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The investigation on the choice of 3 new box types was a crucial point, where children
were expected to discover the power of the decimal expression. As it turned out, this
activity achieved its goal. The following section details the events leading to this
discovery.
Children were working in groups. Each group had a list of orders as examples and
could choose to consider any additional orders. When they were done, each group
presented and explained its choice. It is interesting to note that one of the groups had
chosen 1,10,100, but at the point of presenting the different suggestions no one saw
anything special with their offer. In the following excerpt two of the groups present
their choices and then the teacher asks the whole class to select the best choice.
Orit (presents the choice of her group): We decided on 1, 2, 100.
Teacher: Why 100?
Orit: Because if they order 200, you can take 2 boxes of a 100.
Teacher: Why 2?
Orit: With 2 we can do many numbers 2+2=4, 2+2+2=6, we can supply orders of even
numbers.
Teacher: Why 1?
Orit: With 1 we can build all the numbers. If we need an odd number, we can add 1 to the
even number.
Benny (presents the choice of his group): We suggest 1, 25, 10.
Teacher: Why 10?
Benny: With 10 we can supply tens and hundreds.
Teacher: Why 25?
Benny: With 25 we can supply hundreds using less boxes than we need with 10.
Teacher: Why 1?
Benny: To be able to supply the units.
Teacher: Is there an order you could not supply?
Benny: No.
Teacher (turning to the whole class and starting a discussion): What, in your opinion, is
the best suggestion?
Matan (voting for a choice made by another group): 1, 10, 25, because it has 1 with which
you can supply both even and odd numbers.
Teacher: What would happen if I will have to supply an order for 124?
Matan: I will take 4 boxes of 25 for the 100, 2 boxes of 10 and 4 boxes of 1.
Shahar (at this point only realizing that instead of taking 4 boxes of 25, they might take 1
box of 100): So then the suggestion of 1, 10, 100 is better because I can take
1 box of 100, 2 of 10 and 4 of 1, and it's less boxes.
Aviv (suddenly noting the power of Shahar's suggestion, while Shahar herself was not
aware of it): With 1, 10, 100 you can make anything! Any number you give,
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it (the number itself) immediately tells you how many boxes of each type
you need! For example – give me a number.
Oren: 973.
Aviv: So it's 9 boxes of 100, 7 boxes of 10, and 3 boxes of 1.
Shahar: There is another advantage to 1, 10 and 100 - the boxes can be [well] organized.
As can be seen in Orit's and Benny's presentations, each of these groups had very good
argumentations to support their choice of box sizes. Similarly good arguments were
presented by the other groups as well.
The discussion that followed these presentations demonstrated how the realization of
the power of the decimal expression can emerge in the course of a whole class
discussion following investigations that make such a discovery possible. It is also clear
from this excerpt that the discovery is not a product of one mind but a result of the
accumulation of many ideas. In this specific example it started with the fact that
Benny's group suggested a certain choice, which was appreciated by Mathan. Then
came Shahar with her idea of exchanging 4 of 25 with 1 box of hundred, triggering
Aviv's final realization of the more complete and powerful picture.
DISCUSSION
This study investigates a new approach to learning the meaning of the decimal
structure. Specifically, it uses a multiplicative approach that is based on viewing the
decimal expression of a number as a special combination of multiples.
This approach was realized through the design of a Cookie Factory story context. The
sequence of activities was constructed in the spirit of the Realistic Mathematics
Education approach with a constructivist view on learning and was tried with 3rd
graders.
The data we presented from this implementation showed that the instructional
sequence was successful from several perspectives. As seen in the class excerpt,
children were able to discover a meaningful connection between their investigations
with boxes of cookies and the decimal representation of a number.
In addition to that, the tasks elicited investigations that were characterized by deep
mathematical thinking, good argumentation, development of strategies for recording
data, and development of search strategies.
The focus on multiplicative structures created a connection to previous knowledge of
multiplication and further expanded this knowledge, creating computational fluency. It
also enabled connections and transfer to subsequent topics in a way that made them
more meaningful.
We started this article by mentioning another teaching approach that was rejected by
teachers. We can end it by saying that our experience with teachers has shown that they
can appreciate the benefits of this instruction. Teachers who participate in a workshop
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and perform the Cookie Factory tasks undergo a similar discovery experience as that
encountered by children and thus "feel" what children go through.
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