The Quotient
Construct. We consider the quotient construct of rational numbers
from perspectives of both quotitive and partitive division. To get
a complete picture of a rational number as a quotient one needs
to consider several variables. Consideration should be given to
both continuous and discrete quantity and to different possible
unitizations of the numerator and denominator. In the rational number
x / y the numerator can be interpreted as x(1unit)s or as 1(xunit).
The denominator can be similarly interpreted as y(1unit)s or 1(yunit).
To date, our analysis for partitive division has considered the
two cases for the numerator but only the case for y(1unit)s for
the denominator.
PARTITIVE DIVISION.
Two interpretations of a rational number result from a partitive
interpretation of division. One interpretation leads to the concept
that threefourths is 3(1/4unit)s per [1unit]; the other, that
threefourths is 1/4(3unit) per [1unit]. We illustrate the first
interpretation using the bridging notation based on discrete (Figure
14.5) and continuous (Figure 14.6) quantity and follow this with
a corresponding mathematicsofquantity representation.
In the mathematicsofquantity
model that follows (Figure 14.7), we demonstrate the related nature
of the continuous, discrete, and mathematical models by noting in
parentheses, following each step in the mathematical derivation,
the corresponding steps in the discrete (Figure 14.5) and continuous
(Figure 14.6) models.
The notation
used in the continuous model (Figure 14.6) represents a true model
of how some children are known to solve such problems through sharing
(other partitions are given in the literature and can be modeled
by this notation equally well). They do this by first equipartitioning
each of the 3(1units) into four parts and then distributing these
parts equitably among the 4[1units] (Kieren, Nelson, & Smith,
1985). On the other hand, if 3 is interpreted as a composite 1(3unit)
instead of as 3(1units) as above, this gives 3/4 as 1/4(3unit)
per [1unit], or for the discrete case (Figure 14.8), as 1/4(3(4unit)sunit)
per [1unit].
A note to the
interested reader: The model for the continuous case can be constructed
by choosing for the numerator quantity one (3unit) in the form
of three circular regions and then carefully following the type
of partitioning that is done with the (3unit)s of four discrete
objects. In an investigation of partitioning behavior of children
in grades 6,7, and 8, Kieren and Nelson (1981) presented the task
of shading the amount one child gets if 3 candy bars are shared
equally by 4 children. The candy bars were presented as three rectangles
partitioned into 8 parts. Of 196 responses, 12.5% interpreted the
three candy bars as one composite unit — one (3unit) —
and (apparently) gave the following left to right partition and
solution (Figure 14.9). This solution suggests that threefourths
is 1/4(3unit). This partition given represents a lower level of
partitioning performance (Bigelow, et al., 1989) than one based
on successive halving as we illustrated for the discrete case above
(Figure 14.8). However, we suggest that children need the experience
of interacting with other children and teachers in making partitions.
Figure 14.10 presents another partition of a (3unit) that children
can accomplish with urging from a teacher.
QUOTITIVE DIVISION.
Each of the two representations that arise from the partitivedivision
interpretation involves the quotient of two extensive quantities,
and the result is a representation of the rational number 3/4 as
intensive quantity. Two additional representations for the quotient
subconstruct of rational numbers result from the quotitive (measurement)
meaning of division. We find four ways to look at this division
according to two different interpretations of the numerator, as
3(1units) or as a composite 1(3unit), and according to two similar
interpretations for the denominator. Because of space considerations,
we give demonstrations for the 3(1unit)s interpretation of the
numerator with the two interpretations of the denominator (Figures
14.11 and 14.13) respectively.
Figure 14.12
presents the mathematicsofquantity model that corresponds to the
demonstration in Figure 14.11. In this case, we are able to match
statements in the mathematicsof quantity model almost one for
one with steps in the diagram, so numbers from the diagram are indicated
in parentheses at the end of each algebraic statement to note the
correspondence.
The cognition
we hypothesize as necessary to go from step 7 to step 8 in Figure
14.11 and Figure 14.13 and the corresponding symbolic process deserves
comment. We hypothesize first (step 7, Figure 14.11) that in order
to assign a magnitude to the quantity to be measured  conceptualized
as 3(1unit)s  it needs to be reconceptualized as the composite
unit, or 1(3unit). To be sure, one can think of the measure of
a quantity as being the sum of the measures of its parts, as in
finding the total area of some irregular geometric regions. But
we hypothesize that even in this case the assignment of a single
number and unit as the measure of a quantity implies conceptualization
of the quantity as a single cognitive entity. This is consistent
with the interpretation of the difference in children's responses
to the directions of "count these objects" as compared
to "tell me how many things there are here." In the first
instance, most children count and say "1, 2, 3, 4, ..., n";
in the second case they will count in the same way, pause, and repeat
"n." Saying "n" a second time in the latter
case is interpreted to mean that the child distinguishes between
counting as a process of establishing a onetoone correspondence
and giving the cardinality (the measure) of the collection of things.
It is assumed (Markman, 1979) that cardinality is an attribute of
a collection, not of the individual elements in the collection.
Similarly, we assume measure of a quantity is an attribute of the
quantity conceptualized as an entity and not an attribute of its
constituent parts.
The assumption
that the quantity to be measured is cognized as a composite unit
introduces an additional requirement that the quantity to which
it is matched in the measurement process is also conceptualized
as a composite unit. Thus, in step 8 of Figure 14.11, it is suggested
that the three [1/4unit]s of step 7 are reunitized as 1[ 3/4unit]
composite unit.
These hypotheses
refer to the essential cognitive structures needed to complete the
measurement process at the level of manipulative materials. The
corresponding symbolic processing (steps 710, Figure 14.12) responds
to two sets of constraintsto model the manipulative processing
as closely as possible, and to obey the syntax rules of the symbol
system. The syntax rules for handling symbolic quantity units, in
particular the syntax operation of canceling units, requires that
the units be the same; this is represented in step 7 of Figure 14.12.
Here it is assumed that the unit represented by the ( )  that is,
the printover of ( on { and of ) on } is a common unit to the
units represented by ( ) and { }. This is analogous to the situation
in our motivational example in which units of "apple"
and "orange" were changed to the common unit of "fruit."
If our motive in Figure 14.12 had been to give the briefest symbolic
sequence, step 7 could have been transformed to the following alternate
step 8:
and then to
and then to
Note that the
reconceptualization of 3{1unit}s as 1{3unit} is not modeled in
this alternative symbolic sequence. It is in the interest of responding
to both sets of constraints that the more complicated steps 8 and
9 are given in Figure 14.12.
On the Operator
Construct. The operator concept of rational numbers suggests
that the rational number 3/4 is thought of as a function applied
to some number, object, or set. As such we can think of an application
of the numerator quantity to the object, followed by the denominator
quantity applied to this result, or vice versa. The basic notion
is that the naturalnumber numerator causes an extension of the
quantity, while the denominator causes a contraction. The question
of the nature of the extension and contraction leads to interesting
variations of this construct of rational number. Accordingly, we
give to the numerator and denominator the following paired interpretations:
1. Duplicator
and partition reducer,
2. Stretcher
and shrinker,
3. Multiplier
and divisor.
The analyses
of interpretations 2 and 3 that we have conducted suggest that certain
conditions lead to at least two hybrid interpretations:
4. Stretcher
and divisor,
5. Multiplier
and shrinker.
These hybrid
pairings arise because of different ways that a learner might cognize
units following the application of a stretcher in interpretation
2, or following a multiplier in interpretation 3.
THE DUPLICATOR/PARTITIONREDUCER
INTERPRETATION. The duplicator/partitionreducer interpretation
seems the most basic and closest to the application of the concept
in the domain of manipulative materials or real objects. Our ongoing
analysis has given attention to both discrete and continuous quantities
and to the question of the order in which the duplication and partitionreducer
operators are applied. Issues of units composition and recomposition
become very significant in these analyses.
THE STRETCHER/SHRINKER
CONSTRUCT. There is an important conceptual and mathematical difference
between a stretch of some continuous unit, or a set of discrete
objects conceptualized as a unit, and a repeatadd or duplicate
of either of these units. A repeatadd or duplication are iterative
actions on the entire conceptual unit. A stretch or shrink, on the
other hand, acts uniformly on any subset of discrete objects or
on a continuous subset of points of a continuous object to transform
it into one that measures n times the original subset. This raises
important considerations for providing experiences to help children
conceive the stretcher/shrinker operator as one interpretation of
rational numbers.
To interpret
rational numbers by way of the stretcher/shrinker operator, we consider
the numerator to be a stretcher and the denominator a shrinker.
With symbolic representation in terms of mathematics of quantity,
the outcome of applying a rational number to some unit does not
change regardless of the order of applying the stretcher and shrinker;
moreover, the process does not change substantially. In the domain
of physical representations, on the other hand, the process does
vary considerably.
THE MULTIPLIERDIVISOR
INTERPRETATION. For this interpretation we consider various meanings
of multiplierrepeatadder, times asmany (or greaterthan) factor,
first factor in a crossproduct  and also various meanings of divisorpartitive
divisor, quotitive divisor, timesasfew (smallerthan) factorand
finally two types of quantity  discrete and continuous. Our thinking
is that the timesasmany (greaterthan) factor and timesfewer
(less) combination is exactly the same as the stretcher/shrinker
interpretation, and that the first component of a crossproduct
is not a possible (at least not a reasonable) interpretation. Thus,
the analysis we have under way considers numerator as repeatadder,
denominator as partitive and quotitive divisor, numerator operator
applied first, and denominator operator applied first.
Section Summary
This section
of the chapter had several purposes. One was to call attention to
critical deficiencies in the mathematics curricula of elementary
and middle schools. We identified five areas where the teaching
of multiplicative structures is deficient:
 Composition,
decomposition, and conversion of units.
 Operations
on numbers from the perspective of mathematics of quantity.
 Constrained
models.
 Qualitative
reasoning.
 Variability
principles.
A second purpose
was to suggest that research and development leading to curricular
reform should be guided by an extensive content/semantic analysis
of the domain of multiplicative structures or, more specifically,
the portion of this domain concerned with rational number, ratio,
and proportion.
A goal of our
analysis is to provide a theoretical context that will guide research
into the cognition underlying students’ manipulations in transforming
physical representations. A second goal is to associate these manipulations
with the abstract representation of mathematics of quantity. We
claim that our bridging notation is essentially a onetoone map
between learners' cognitive structures and the mathematicsofquantity
representation.
The initial
results from this analysis suggest that the concepts in the multiplicative
structures domain are inextricably interrelated and exceedingly
complex. The purpose of a content/semantic analysis is to make use
of the knowledge gained from research into the knowledge structures
that children form in cognizing concepts in the content domain and
to make assumptions about the necessary cognitive structures where
research is lacking. These assumptions suggest further research
issues and questions. For example, numerous assumptions were made
in the demonstrations about unit formation and reformation. Can
these formations and reformations be conceptualized by children?
Research needs to address the issue of whether situations can be
developed that will help children construct implicit and intuitive
knowledge about unit conversion principles first, and then explicit
knowledge of them. Analyses of rational number constructs suggest
that their understanding depends on a grasp of these unit conversion
principles as well as rather deep knowledge about concepts of measurement.
We argue that
future research and curriculum development should be based on and
improve our analysis of multiplicative structures. A great deal
of effort will be necessary to develop situations from which children
can construct knowledge about these ideas. For example, research
has given us some information about children's ability to partition
both discrete and continuous quantity (Hunting, 1986; Kieren &
Nelson, 1981; Pothier & Sawada, 1983), but little is known about
instructional situations that might facilitate children's ability
to panition.
PRINCIPLES
FOR QUALITATIVE REASONING
IN FRACTION AND RATIO COMPARISON
Mathematical
VariabilityA Fundamental Issue
The flexibility
of thought that Resnick (1986) mentions as a characteristic of intuitive
knowledge is readily observable in children's protocols involving
intuitive reasoning. (Harel & Behr, 1988; Kieren, 1988; Pirie,
1988; Post, Wachsmuth, Lesh, & Behr, 1985; Resnick, 1982, 1986).
One can observe a flexible interaction among different levels of
representation or understanding (Kieren, 1988)  for example, between
thought directed at actual manipulative aids, or mental images of
the manipulation, and an oral or written symbolic representation
of the quantities involved. Moreover, while one does not often see
explicit reference to mathematical principles (for example, principles
of place value [Resnick, 1986]) in these protocols, consistent behavior
and uniform solution strategies suggest understandings that can
be abstracted from the protocols in the form of mathematical principles.
Germane to this section are the numerous observations that suggest
an implicit awareness of the invariance, or a compensation for variation,
of the value of a quantity under certain transformations. Awareness
of invariance (or variation) or the search for invariance under
certain transformations is a central concept, almost a defining
concept, of mathematical thinking. Invariance and compensation for
variation are basic to many areas of elementary mathematicsthe
development of basic fact strategies (Carpenter & Moser, 1982),
children's invention and use of alternative computation algorithms
(Harel & Behr, 1988; Pirie, 1988; Resnick, 1986), as well as
fraction equivalence and proportion problems (Lesh et al., 1988).
Research on
rationalnumber learning suggests the importance of fraction order
and equivalence to the understanding of a rational number as an
entity (that is, as a single number) and to the understanding of
the size of the number (Post et al., 1985; Smith, 1988). Fundamentally,
the question of whether two rational numbers are equivalent or which
is less is a question of invariance or variation of a multiplicative
relation (Lesh, et al., 1988). Two rational numbers a/b and c/d,
can be compared in terms of equivalence or nonequivalence by investigating
whether there is a transformation of a / b to c / d , defined as
changes from a to c and from b to d, under which the multiplicative
relationship between a and b is or is not invariant. In the current
mathematics curriculum, the issue of fraction order and equivalence
is treated as an isolated topic rather than as a special instance
of mathematical variability. We view the concepts of fraction order
and equivalence and proportionality as one component of this very
significant and global mathematical concept. Fraction equivalence
can be viewed in the context of the invariance of a multiplicative
relation between the numerator and denominator, or as the invariance
of a quotient.
Research and
development must address the need to provide children in early elementary
grades with situations that involve variability. There are two issues
here. First, children require adequate experience to understand
what is meant by the concept of change, or difference. For example,
the question of what change in 4 will result in 8 is one that very
young children can deal with. Moreover, as early as possible, children
should be brought to understand that the change in 4 to get 8 (or
the difference between 4 and 8) can be defined in two ways: additively
(with an addition or a subtraction rule) or multiplicatively (with
a multiplication or a division rule). The additive rule for changing
4 to 8 is to "add 4," while the multiplicative rule is
to "multiply by 2." The additivechange rule to get 8
from 4 is the same as to get 17 from 13; this is not true for multiplication.
The second issue is the investigation of change and the direction
of change in additive and multiplicative relationships and operations
under transformation on the components in the relation or operation.
While the additive relation between 4 and 8 is invariant under the
transformation of adding 9 to both 4 and 8, this is not true for
the multiplicative relation between the same two numbers. The ability
to represent change (or difference) in both additive and multiplicative
terms and to understand their behavior under transformation is fundamental
to understanding fraction and ratio equivalence.
The mathematics
curriculum needs to develop learning situations, problems, and computation
that will help children develop at least an implicit understanding
of the principles that underlie mathematical invariance. That is,
the curriculum should provide school experience to help children
construct intuitive knowledge about fraction and ratio equivalence.
Situations must be developed in which children systematically build
their understanding of principles that underlie the invariance and
the compensation for variation within additive, subtractive, multiplicative
and divisitive relations and operations. The goal is to help children
construct these principles as "theorems in action" (Vergnaud,
1988).
Qualitative
Reasoning. In our research on children's thinking strategies
applied to fraction and proportion tasks, we became interested in
children's ability to reason qualitatively about the order relation
between fractions and between ratios. Our interest was in children's
use of qualitative reasoning about situations modeled by a / b =
c, including reasoning such as, if a stays the same and b increases
then a / b decreases, or if both a and b increase, then qualitative
methods of reasoning are no longer adequate to determine whether
a / b increases, decreases, or stays the same in value, and quantitative
procedures are necessary.
The concern
in qualitative reasoning about a situation modeled by the equation
a / b = c is to determine the direction, as opposed to the amount,
of change (or no change) in c as a result of information about only
the direction of change (or no change) in a and b, or to establish
that the direction of change is indeterminate based only on qualitative
reasoning. Our thinking about the role of qualitative reasoning
was influenced by the work of Chi and Glaser (1982), who found that
expert problem solvers are known to reason qualitatively about problem
components and relationships among them before attempting to describe
these components and relationships in quantitative terms. Consensus
is that experts’ reasoning about a problem leads to a superior
problem representation that enables the expert to know when qualitative
reasoning is inadequate and quantitative reasoning is necessary.
It is not the
case that experts use qualitative reasoning and novices do not;
rather, experts’ qualitative reasoning is based on scientific
principles and involves formation of relationships among problem
components based on these principles. Novices, on the other hand,
reason about the surface structure of the problem. It appears that
the qualitative reasoning to which Chi and Glaser refer is not unlike
the schooled intuitive reasoning to which Kieren (1988) refers.
The issue in
this section is to present basic principles upon which to base intuitive
knowledge about fraction order and equivalence and proportional
reasoning. They are principles that need to become selfevident
to children through experience. These principles should then provide
the basis for flexible application of additive and multiplicative
notation and transformation rules to the solution of problems involving
fraction or ratio equivalence (Resnick, 1986).
There are two
aspects to our analysis of qualitative reasoning as applied to a
problem involving questions of fraction order or equivalence or
the proportionality of two ratios. One aspect of the problem is
determinability: Can the order relation requested in the problem
be determined through qualitative reasoning? The second aspect concerns
determination: What is the order relation requested in the problem,
if it can be determined? These questions are discussed in the next
two sections. The discussion is a recapitulation of an analysis
in Harel et al. (in press).
Principles
for Solving Multiplicative Tasks
Two Categories
of Multiplicative Tasks. In Harel et al., (in press) we showed
that, from the perspective of invariance of relations, proportion
tasks can be classified into two broad categories: invariance of
ratio and invariance of product. The orange juice task used by Noelting
(1980) and the balance beam task used by Siegler (1976) are representatives
of these two categories, respectively. In the orange juice task,
one of two pairs of ratios amount of water to amount of orange
concentrate, or amount of water to total amount of mixture  are
compared across two mixtures to determine which one tastes the more
orangy or if they taste the same. In the balance task, the expected
solution procedure is to compare the products of the values of the
distance and the weight of objects on each side of the fulcrum to
determine which side of the balance beam goes down. Strictly speaking,
a task would be classified according to the way a given subject
solved it. For the sake of discussion we classify these two tasks
according to expected solution procedures; most subjects do solve
the tasks as expected.
Quite different
mathematical principles, and thus different reasoning patterns,
are involved in the solution of the invarianceofratio and the
invarianceofproduct tasks. To describe principles forming the
basis for qualitative reasoning about these two types of tasks,
we introduce a refinement of each type. We identify two subcategories
of the invarianceof product category and illustrate the subcategories
with hypothetical balance beam or orange juice tasks.
FINDPRODUCTORDER
SUBCATEGORY. This subcategory consists of problems in which order
relations between values of corresponding task quantities are given,
and these order relations form the factors of the two products.
These problems ask about the relation between two values of the
quantity represented by the product. For example, let a1 and a2
denote the weights of objects placed on each side of the fulcrum
of a balance beam, and let b1 and b2 represent their respective
distances from the fulcrum. Further, suppose it is given that a1
< a2 and that b1 = b2. Which side of the fulcrum will go down?
That is, the task is to determine the relationship between the products
k1 = a1 x b1 and k2 = a2 x b2. We call attention to the fact that
the information given in this task is about the directionality of
the order relations, and it need not include specific numerical
values for the weights and distances. This is characteristic of
a qualitative proportional reasoning task. Different forms of the
task can be formulated by taking all possible combinations of the
three order relations between a1 and a2 and the three possible order
relations between b1 and b2. The order relation between k1 and k2
will in some cases be indeterminate through qualitative reasoning
alone.
FINDFACTORORDER
SUBCATEGORY. This category consists of problems in which an order
relation is given between values of quantities represented by two
products and an order relation is given between two corresponding
factors in the two products. The problem asks about an order relation
between the other two factors. If in the example above it were given
that a1 < a2 and k1 = k2, where k1 = a1 x b1 and k2 = a2 x b2
and the question was about the order relation between b1 and b2,
then this would be an exemplar of this subcategory.
Likewise, there
are two subcategories of the invarianceof ratio category
FINDRATEORDER
SUBCATEGORY. This subcategory consists of problems that give two
order relations between values of corresponding quantities in two
rate pairs and ask about the order relation between the values of
the quantities represented by the two rates.
An example can
be formulated from the orange juice context of Noelting (1980).
If two orange juice mixtures (1 and 2) are made from amounts of
water a1 and a2 with a1 < a2 and amounts of orange concentrate
b1 and b2 with b1 = b2, which of the two mixtures, 1 or 2, tastes
the more orangy, or do they taste the same? The decision of which
is more orangy would be based on the order relation between the
two ratios a1/b1 and a2/b2, or between a1/(a1 + b1) and a2/(a2 +
b2).
FINDRATEQUANTITY
SUBCATEGORY. This subcategory consists of problems that give an
order relationship between the value of quantities represented by
two rates and an order relation between the values of two corresponding
quantities in the two rate pairs, the problem asks about the other
order relation between the values of the two corresponding quantities
in the two rate pairs.
Again, an example
can be formulated from the orange juice context. If two orange juice
mixtures (1 and 2) were made so that mixture 1 tastes more orangy
than mixture 2 (that is, a1/ b2 > a2/ b2 and the amount of orange
concentrate in mixture 1 is greater than the amount of orange concentrate
in mixture 2 (that is, a1 > a2), then which of the mixtures has
more water, or do they both have the same amount?
Multiplicative
Determinability and Determination Principles. Each of the four subcategories
of problems involves two number pairs, a and b, c and d, and either
a pair of products a x b and c x d or a pair of ratios a/b and c/d.
For each of the three pairs in a given problem there are three possible
order relations; the structure of the problems in the subcategories
above is that the order relations between two of the three pairs
are given and the problem is to (a) decide if the third order relation
is determinable from the given information and if so (b) to ascertain
what that order relation is. The knowledge required to solve these
kinds of tasks relies on principles that we have placed into two
categories  multiplicative determinability principles and multiplicative
determination principles. The multiplicative determinability category
consists of principles that specify the conditions under which order
relations between factors in the product of the values of two quantities
can lead to declaring whether the order relation between the values
of these quantities is determinate or indeterminate (for example,
if a and b are equal but c and d are unequal, then the order relation
between the products a x c and b x d is determinate). The multiplicative
determination category consists of principles that specify the conditions
under which order relations between factors of two quantities can
lead to declaring that the relation between the two quantities is
less than, greater than, or equal to (for example, if a and b are
equal but c is greater than d, then the relation that holds between
a / c and b / d is that a/c < b/d).
The principles
in the first class of multiplicative determinability category give
information about the determinability of the order relation between
products. When order information is given about corresponding factors
within each product, we refer to these as product composition (PC)
principles.
PC1. 
The order
relation between the products a x c and b x d is determinate
if the order relation between a and b is the same as between
c and d or if one of them is the equalto relation. 
PC2. 
The order
relation between the products a x c and b x d is indeterminate
if the order relation between a and b conflicts with (is in
the opposite direction of) the order relation between c and
d. 
We offer some
examples, in this case from the balance beam context, to illustrate
problems in which these principles are the mathematical formulation
of the knowledge needed to solve the problem.
A classmate
knows that Billy is heavier than Jane, and he tells Billy to sit
farther from the center of a teetertotter than Jane. Before he
has Billy and Jane exert the force of their weight onto the teetertotter,
he tells the class about the relationship between their weights
and then asks the class to tell which end of the teetertotter 
Billy’s or Jane’s  will go down. In this case the direction
of movement of the teetertotter, off horizontal is indeterminate
from the information given. To be able to determine the order relation
of the two angular moments, numerical data on Billy and Jane's weight
and distance would be needed and the moments would need to be calculated.
Another example:
A classmate tells the class that Billy weighs 90 lbs and Jane 80
lbs. He has Billy sit 4 ft from the center of the teetertotter
and Jane 3 ft; he gives everyone in the class this information and
asks which side of the teetertotter will go down. In this case
(a) the direction of the order relation between moments (that is,
the order relation between the weight and distance products) is
determinate and (b) Billy's side will go down. Notice that the directionality
of the order relations of the respective weights and distances is
sufficient to determine the order relation between the moments;
there was not a need to compute the exact moments to determine this
order relation. This is what characterizes the solution as qualitative
or intuitive. If for some reason, say by teacher direction, calculation
of the moments was required, and if the solver actually compared
these computed values to determine the order relation, then the
solution would be characterized as quantitative. Nevertheless, one
can see how a prior qualitative analysis of the problem could guide
the quantitative calculations and serve as a check on them.
We also state
principles on which knowledge is based to decide the determinability
of the order of one pair of factors of a product when the order
relation between the other two factors and the product are given.
We refer to these principles as product decomposition (PD) principles.
PD1. 
The
order relation between the factors a and b in the products a
x c and b x d is determinate if the order relation between the
factors c and d is in the opposite direction from the order
relation between the products a x c and b x d or if one of the
order relations is the equalto relation. 
PD2. 
The
order relation between the factors a and b in the products a
x c and b x d is indeterminate if the order relations between
the other two factors c and d, and between the products a x
c and b x d are the same but neither is the equalto relation. 
An example where
PD1 can be applied, constructed from the balance beam context: A
classmate has Billy and Jane sit on a teetertotter so that Jane's
side would go down. Billy is heavier than Jane. Is Jane's distance
from the center equal to, less than, or greater than Billy's distance
from the center? In this case the requested order relationship is
(a) determinate and (b) Jane's distance from the center is greater
than Billy's. The structure of this problem in terms of the stated
principles is that the order relationship between the product of
weight and distance from the center is given for both Billy and
Jane and the order relation between their weights is given. The
question is about the relationship between the other corresponding
factorsdistances from the centerin the two products. Note that,
if the problem is changed so that Billy's weight is less than Jane's,
then the order relationship between the distances is indeterminate
based on the given qualitative data.
Two ratio composition
(RC) principles make up the third category of determinability principles.
RC1. 
The
order relation between the ratios a/c and b/d is determinate
if the order relations between a and b is in the opposite direction
from the order relation between c and d, or if one of them is
the equalto relation. 
RC2. 
The
order relation between the ratios a/c and b/d is indeterminate
if the order relation between a and b is the same as the order
relation between c and d, but neither of them is the equalto
relation. 
We offer an
example constructed from the orange juice context: If the amount
of water in mixture 1 is less than in mixture 2 and the amount of
orange concentrate in mixture 1 is less than in mixture 2, then
the relationship between the orangy tastes of mixtures 1 and 2 is
indeterminate (that is, the relationship between the two watertoconcentrate
ratios is indeterminate).
On the other
hand, if the amount of water in mixture 1 is less than in mixture
2 and the amount of orange concentrate in mixture 1 is greater than
in mixture 2, then (a) the order relation between the two watertoconcentrate
ratios is determinate and (b) the watertoconcentrate ratio for
mixture 1 is greater than for mixture 2, so mixture 1 tastes more
watery and less orangy.
Once a determinability
principle is applied and the requested order relation is found to
be determinable, then a determination principle can be applied to
ascertain whether that relation is the lessthan, equalto, or greaterthan
relation. There is a determination principle that corresponds to
the determinability principles; we denote them by [PC1], [PD1],
[RC1], and [RD1]. For example, [PD1] is as follows: if c ³ d and
a x c < b x d then the order relation between a and b is a <
b.
The entire set
of determinability and determination principles can be easily summarized
into a concise
as shown in
Table 14.1. Success on problems from the findrate and find product
subcategories can be achieved by reasoning as to how a qualitative
change in a1 to get a2 and b1 to get b2 affects the size and thus
the comparison of k1 and k2, where k1 = a1 x b1 (or a1/b1) and k2
= a2 x b2 (or a2/b2). The changes in a1 to get a2 and b1 to get
b2, and k1 to get k2 can be denoted by a,
b, and
k, respectively,
and the qualitative value (or directionality) of these changes can
be denoted by +, 0, or  according to whether the change is an increase,
no change, or decrease, respectively. In Table 14.1, selecting a
pair of values, one from the vertical and one from the horizontal
axis, and locating the corresponding value in the body of the table
gives information about how qualitative changes in rate or fractionquantities
(factors of a product, parentheses in text here correspond to parentheses
in Table 14.1) affect the qualitative value of the rate or fraction
(product). Note that the question marks in the body of the table
indicate that the value of k
is indeterminate, which means that instances can be found in which
k is
+, others for which it is , and still others for which it is 0.
Thus, each of the question marks could be replaced either mentallyor
physically in the tableby a disjunctive listing of the three possibilities
(+, , or 0). In this way, Table 14.1 describes the knowledge needed
to solve findrate and findproduct problems based on the qualitative
relationships among the problem quantities.
Selecting a
pair of values, one from an axis and another within the body of
the table along the row or column of the first, and then locating
the corresponding value on the other axis, Table 14.1 describes
the knowledge about the qualitative relationships among problem
quantities needed to solve problems from the findratequantity
and findproduct subcategories. We give two examples to illustrate
the information in the Table 14.1.
We have two
fractions and we know that the numerator of the first is greater
than the numerator of the second (that is, the change from the first
to the second is indicating a decrease) and that the two fractions
are the same size (the change in value of fraction 1 to get fraction
2 is 0, that is, no change). Is the denominator of the first fraction
less than, greater than, or equal to the denominator of the second
fraction? We can analyze the problem in this way: The change from
numerator 1 to numerator 2 is  (a decrease in the numerator), find
 on the horizontal axis of Table 14.1. The change from fraction
1 to fraction 2 is 0 (no change), so we look for 0 in the body of
the table in the column under . Since the only 0 in this column
appears implicitly, under the guise of ?, we move along that row
to the vertical axis and conclude that the direction of change from
denominator 1 to denominator 2 is a decrease. Again we note that
it is only the direction of change that we determine from the table,
not the amount of change.
Next consider
the situation of having two products, we know that one factor decreases
and that the product decreases. What happens to the other factor?
Using the table, we notice that under the column for a decrease
in the first factor, a minus sign (), indicating a decrease in
the product appears three times: twice explicitly, and once under
the guise of an indeterminate (?). Thus the change in the second
factor can be any one of +, , or 0 because anyone of three (horizontal
axis entry, table entry, vertical axis deduction) is possible(
, , + ), ( , , 0), and ( , , ).
Now suppose
we have two products, product 1 and product 2, and that the change
in the first factor of product 1 to the first factor of product
2 is an increase, and the same for factor 2. What is the order relation
between product 1 and product 2? We find + on the left of the table,
and + on the top; the qualitative change in the product, +, is given
in the body of the table at the intersection of the row and column
in parentheses, ( + ). If both factors of a product increase, then
the product increases as well.
Section Conclusion
Qualitative
reasoning is known to be a significant variable in problemsolving
performance. Expert problemsolvers are known to reason qualitatively
about problem components and relationships among them before attempting
to describe these components and relationships in quantitative terms
(Chi & Glaser, 1982). The consensus of research is that an expert's
reasoning about a problem leads to a superior problem representation
because it contains numerous qualitative considerations about problem
components and their interactions (Chi et al., 1981; Chi & Glaser,
1982; Chi, Glaser, & Rees, 1982). While some tasks used in traditional
studies of the proportion concept make it possible for subjects
to solve the task using qualitative reasoning (for example, Siegler
& Vago, 1978; Siegler, 1976; Karplus, Pulos, & Stage ,1983;
Noelting, 1980), it has only been very recently that qualitative
reasoning has become an object of study in this area of research.
Studies on
Qualitative RationalNumber and Ratio Reasoning. Harel, et al,
(in press) compared the tasks used by the researchers cited in the
conclusion above using several criteria, including whether the tasks
were solvable by qualitative reasoning alone. They found that some
variations of the balance scale task (Siegler, 1976) were solvable
by qualitative reasoning; for example when there is more weight
on one side of the fulcrum and the weight on the other side is further
from the fulcrum. Noelting (1980) gave 23 orange concentrate and
water mixture tasks, of which only two were solvable by qualitative
reasoning; for example, those where the differences in amounts of
orange concentrate and water between two mixtures could be defined
as ( + , ) and ( , + ) according to Table 14.1. All of the other
tasks were of the form (+, +) or (, ). The fullness tasks given
by Siegler and Vago (1978) require quantitative reasoning.
The question
of whether children can use qualitative reasoning in solving fractionequivalence
and proportion problems has been investigated by the Rational Number
Project. A study reported by Heller, Ahlgren, Post, Behr, and Lesh
(1989) investigated seventhgrade children's performance on numerica1
(quantitative) and qualitative problems, using both missingvalue
and comparisontype problems. Examples of qualitative missingvalue
and comparisontype problems, respectively, follow.
If Cathy ran
less laps in more time than she did yesterday, her running speed
would be: faster, exactly the same, slower, there is not enough
information to tell?
Bill ran the
same number of laps as Greg. Bill ran longer than Greg. Who was
the faster runner: Bill, Greg, they ran exactly the same speed,
there is not enough information to tell?
A conclusion
drawn from their work was that qualitative reasoning is helpful
(not completely necessary) but certainly not sufficient for successful
performance on quantitative proportionalreasoning problems. They
found that some subjects' performance on the qualitative problems
was low, while their performance on quantitative proportion problems
was high. They concluded that quantitative proportionalreasoning
problems can be solved without good qualitative proportional reasoning
by applying memorized procedures. They suggest that intuitive understanding
of the direction of change (qualitative understanding) in the value
of a ratio or fraction should precede quantitative exercises.
Another study
reported by Larson, Behr, Harel, Post, Lesh (1989) directly investigated
qualitativereasoning ability among seventhgrade children. The
tasks required the child to determine whether the value of a fraction
or ratio would increase, decrease, or stay the same under given
changes in the numerator or denominator as follows: Increase(I)/Decrease(D),
D/I, D/D, Same(S)/I, and I/S. Based on the protocol data collected,
numerous unsuccessful qualitativereasoning strategies were identified,
and several successful ones as well. We call attention to some of
the more interesting strategies. One category of strategies, observed
on tasks in which the changes in the numerator and denominator were
in the same direction, suggest that some children believe the value
of a fraction or ratio changes in the same direction as the changes
in its two components. Another category of responses suggests that
the child believes a greater change in one of the two components
will result in some change in the value of the fraction or ratio.
A refinement of this strategy by some children is that the direction
of change in the value of the fraction or ratio is in the same direction
as the directional effect of the component that has the greater
amount of change. For example, if it is given that both the numerator
and denominator increase, the child may reply that it depends on
which one changes most; if the denominator increases most, then
the value of the fraction or ratio will decrease. Some children
mulled over a change in the value of the fraction or ratio in a
sort of compositionofeffects strategy: For example, given that
the numerator increases and the denominator decreases, they would
reason that increasing the numerator causes an increase in the value
of the fraction, decreasing the denominator causes an increase in
the value of the fraction, the two increases together result in
an increase. These results were obtained at different times during
a teaching experiment, and they suggest that instructional situations
can lead children to reason qualitatively about the size of fractions
and ratios. It should be noted that these children, having attained
seventh grade, had already learned other quantitative strategies
for comparing fractions and ratios; a certain amount of unlearning
of existing incorrect strategies was necessary before the new strategies
could be successfully applied. Greater success might be possible
with a younger child whose thinking about fractions and ratios was
less affected by prior knowledge of quantitative strategies.
In a third study
(Harel et al., in press) a "blocks task" was developed
specifically to investigate children's qualitative reasoning in
a proportion context. The task involved two pairs of composite blocks
(A, B) and (C, D) with A and C constructed from the same kind of
smaller (unit) blocks, which were nevertheless larger than those
used to construct B and D. The number of unit blocks in A was less
than the number of unit blocks in C, and these numbers remained
constant across all variations of the blocks tasks used in the study.
Three instances of blocks B and D were used in which the number
of unit blocks was one less, the same, or one more, compared to
A and C, respectively. Given information about the weight relationship
between and A and B (<, = , >) in the context of a visually
observable number relationship among the four composite blocks,
seventhgrade children were asked to determine the weight relationship
between C and D. Of 27 possible task variations, 9 were used in
the study. Of particular interest in this study were the type of
problem representations that children formed in response to the
problem presentation, the solution strategies that were used, and
the relationship between the sophistication of the problem representation
and the solution strategy: Hierarchies of three problem representations
and three categories of solution strategies with two solution strategies
in the high category, three in the middle, and one in the low category
were identified. A high correlation was found between the level
of the problem representation and the level of solution strategy.
By matching a type of problem representation with its most highly
correlated solution strategy, a hierarchy of six solution processes
was identified. A high correlation was found between the level of
solution process and the level of the students’ mathematics
ability.
Van den Brink
and Streefland (1979) give evidence that children as young as 6
and 7 years old have intuitive knowledge about ratios and proportions
that suggests implicit knowledge of the principles given in this
section. They tell of a child who, during a discussion with his
father about how the propeller of a ship works, looked at a toy
boat, referred to a picture in his room of a large seagoing ship
with a man standing near the propeller, and inquired of his father
how big the propeller on such a ship really is. "It wouldn't
fit into your room," answered the father. After some moments
of reflection the child responded,
It is true.
In my book on energy is a propeller like this (shows a distance
of about 3 cm between thumb and forefinger) with a little man
like that (about 1 cm).
The authors
indicate that the child compared the relationship between the seagoing
ship's propeller (bigger than the boy's room) and his father to
the picture of the big ship with a man beside the propeller. By
qualitatively maintaining an invariant relationship between man
and propeller, the child confirms his father's statement that the
ship's propeller would be bigger than his room.
In her dissertation,
Larnon (1989) identified 16 proportion problem types by crossing
four problem dimensions with four semantic categories. The problem
dimensions were relative/absolute change, recognization of ratiopreserving
mappings, covariance/invariance, and construction of ratiopreserving
mappings; the semantic categories were wellchunked measures, partpartwhole,
unrelated sets, and stretchers/shrinkers. She notes that recognizing
relative change is likely an important prerequisite to moving beyond
additive relationships into multiplicative structures. Multiplicative
relationships arise from an evaluation of the size of a change by
considering its relationship to the starting values and not merely
in terms of its absolute amount. The problem dimensions of relative/absolute
change and covariance/invariance are particularly germane to the
issues raised in this section. She found that the performance of
sixthgrade students on relative/absolute change across the semantic
categories (in the order given above) was 6.8%, 57.4%, 83.3%, and
18.9%. Performance of sixthgrade students on the covariance/invariance
problem dimension across the four semantic types was 72.3%, 79.5%,
60.5%, and 50.0%. Tasks on this dimension involve ability to recognize
variability or invariance of the relationship between the two components
of a ratio (or fraction) under change in each component.
Teaching
for Qualitative Reasoning. While the evidence to date is sketchy,
there is a trend in the direction of supporting the notion that
qualitative knowledge can be constructed through school situations.
Moreover, though the evidence is again slight, a reasonable hypothesis
seems to be that the qualitative knowledge an individual has about
a situation is similar to what Resnick and Kieren have referred
to as intuitive knowledge. It is knowledge that "belongs"
to the individual, is constructed from real experience, and provides
for considerable flexibility in thought. The work of Chi and her
colleagues gives an important reason for stating principles on which
to base qualitative reasoning about proportional situations. A characteristic
of experts’ qualitative reasoning as compared to the qualitative
reasoning of novices is that it proceeds at a semantically deep
level and incorporates principles of the content domain. The qualitative
reasoning of novices, on the other hand, is at the surface level
and is directed at comparison of formulas and procedures for attempting
to isolate the problem unknown. The aim of initial instruction in
rational numbers and proportions should be to put children in situations
where they are able to construct principles to apply qualitatively
to questions of order, equivalence, and size of fractions and ratios.
The objective of helping children construct principles for qualitative
reasoning is based on the belief that this knowledge can guide their
quantitative thinking, as it does for experts, in a content domain.
Research needs
to determine how knowledge of principles for qualitative thinking
and the ability to think qualitatively will help children make connections
between this intuitive, informal knowledge and the symbol system
that is the basis for quantitative methods. While work in this area
has advanced in the domain of early numbers, very little has been
done in the area of rational numbers and proportions. Recent work
(Mack, 1990) represents a beginning. Mack points out that numerous
studies demonstrate that children possess a store of informal knowledge
about fractions. The issue she addresses is whether instruction
can build on this informal knowledge in a way that extends it to,
or connects with, the formal system of fraction notation. An instructional
move in her work that seems to help children make this connection
is to give a child a problem in symbolic form and ask the child
to reason about it in terms of a real situation. Mack (1990) reports
that while children were able to build on their informal knowledge,
the results also suggest that knowledge of rote procedures interferes
with their attempts to construct procedures that are meaningful
to them. Van den Brink and Streefland (1979) give suggestions for
the type of instruction that would help children develop intuitive
or qualitative knowledge about the principles stated in this section.
We give an example of an instructional situation that they suggest.
A story is told to the class about Liz Thumb, who once upon a time
became as small as a thumb. Liz Thumb is pictured on a worksheet,
and students are asked to draw common objects into the picture 
a flower, a stone, one of the child's own shoes, and the like. These
drawings will demonstrate children's conceptions of ratio, and discussion
among the students and the teacher can help the class to come to
agreement about how big the drawn objects should be and why.
The Semantic
Analysis of Rational Number: Some Implications for Curriculum
In section 2
of this chapter, "Rational Number Construct Theory: Toward
a Semantic Analysis," we showed that any rational number x/y
can be interpreted in anyone of four ways x/y(1unit), x(1/yunit),
1/y(xunit) and 1(x/yunit). Another analysis we have underway deals
with operations on rational numbers from the perspective of mathematics
of quantity. Each of the four different interpretations of rational
numbers can be shown to be embedded in real world, or at least in
conceivable textbook word problems. Our analysis of rationalnumber
operations from the perspective of mathematics of quantity has progressed
most with the operation of division. We will use problem examples
in this domain to illustrate the richness of problem situations
and alternative problem representations, as well as solution procedures
that arise from the analysis.
Division
Problem Examples. We will present one problem with some analysis
of its solution from the perspectives of the embedded rationalnumber
interpretations and the mathematics of quantity involved. In addition,
we will present other problems to represent their possible range,
but we give only the mathematical model that could be used to solve
them and the interpretations of rational number that are represented.
The numbers are the same in each of the example problems to make
comparison among models and rationalnumber interpretations easier.
We will give two solutions of a problem, based on different problem
representations. We call attention to the fact that solving problems
such as these depends on an understanding of the different interpretations
of rational numbers and on the different unitconversion principles;
these problems are not expected to facilitate initial learning of
these notions. They would provide situations in which they can be
applied in order to deepen the understanding, but knowing the rationalnumber
interpretations listed in the opening sentence of this section and
the unitconversion principles listed earlier are considered prerequisite
knowledge to problem interpretation and representation.
The following
problem is presented to illustrate.
Bob mixed
6 tubes of paint. He used 1/8 of the mixture to paint 2 pieces
of wood, each having an area of 1/8m2. How many tubes would he
need to paint 1m2?
We can interpret
this problem in more than one way. Traditionally, it would be interpreted
as a multistep problem involving multiplication and division. Using
this interpretation and notation, which is analogous to that which
we used in section 2 for the mathematics of quantity, the solution
of the problem could be as follows: First, identify the problem
quantities as 6(1tube)s of paint in 1(1mixture), 2(1piece)s of
wood, and 1/8(1m^{2}) measure of each piece of wood. The
question is to find the number of (1tube)s per each (1m^{2}),
then needed computations are carried out,
We offer some
observations about this problem representation and solution. Rather
than there being a holistic problem representation, the components
of the problem are represented separately and, later in the third
step of the solution, the solver must remember or determine (depending
on whether he has a problem plan) how these two components go together.
Moreover, the solver must remember or determine that the quantities
computed in steps 1 and 2 must be divided and must decide which
is the dividend and divisor, respectively. In the tree diagram that
we give later for this problem, the solution would be considered
bottomup rather than topdown. This solution represents all the
problem quantities in terms of singleton units. Does this give a
relatively strong cognitive representation of the problem, or would
other unitizations of the problem quantities lead to a more powerful
representation? The solution process illustrated might be characterized
as first interpreting units, then converting units to units of one,
followed by partitive division.
With an alternative
interpretation of the units, it is a partitive division followed
by units conversion. That is, if we identify the given quantities
as 1/8(6tube) and 2(1/8m^{2})s, it remains to find the
number of (1tube)s per each (1m^{2}). The steps in the
solution based on this problem representation would be as follows:
In contrast
to the first solution, this one proceeds from a holistic problem
representation and each subsequent step of the problem can be derived
from the one before it using previously learned principles of units
formation and conversion. This solution could be described as topdown
rather than bottomup. We see several interpretations of fractions
in this solution: 6/8 as 1/8(6unit), 2/8 as 2(1/8unit)s, and 3/8
as 3(1/8unit)s. Thus, while one might consider this a higherlevel
problem representation, it appears that the problem representation
and solution also requires a higher level of thinking. But the development
and use of higherorder thinking is something that we advocate for
mathematics curricula in middle school and earlier grades.
Another possible
advantage to our second interpretation is that it leads to uniformity
in problem representation across division problems. In terms of
a hierarchical treestructure for the problem (see Figure 14.14),
our interpretation represents the division (the central operation
in the problem) at the top level of the tree. The two multiplications
are at lower levels. The traditional approach (our first interpretation)
to solving the problem would perform the multiplications first and
then use these results as operands in the division, a bottomup
solution. A topdown solution would perform the division first and
then do a units conversion. The issue of problem representation
in this context is similar to the one Larkin (1989) discussed for
algebraic equations. Important research questions about problem
representation lurk here.
Our analysis
of partitive division problems has led us to identify three stages
in the solution of partitive division word or computational problems:
(a) units interpretation, (b) distributing and counting units (or
putting in and counting units), and (c) units conversion. It appears
that the units interpretation is the first step, while (b) and (c)
are interchangeable. In the first solution, (c) was done before
(b); in the second, the order was reversed.
Other Division
Problem Examples.
1. 
Oneeighth
of the heat consumed for the house provides 1/8 of what is
needed to heat the basement. If the house consumes 6 units
of heat and the basement has two equalsize rooms, how much
heat is needed to heat each of the basement rooms?


The topdown
representation using a mathematicsofquantity interpretation
for this is as follows: 1/8(6heatunit)s ÷ 1/8(2roomunit)s
= ? (1heatunit)/(1roomunit). In this representation 6/8
and 2/8, respectively, have the interpretation of 1/8(6unit)
and 1/8(2unit).

2. 
David
can save 1/8 of his monthly income. He found that 6 months
saving is enough for 2 payments, each of which is 1/8 of the
price of the car he wants to buy. In how many months can David
buy the car he wants?


The mathematics
of quantity interpretation: 6(1/8monthlyincome) ÷ 2(1/8car
payment)s = ? (1monthlyincome)/(1car payment). In this
case, 6/8 and 2/8 have the following interpretations, respectively6(1/8unit)s
and 2(1/8unit)s.

3. 
In 6 hours,
1/8 block of snow is melted into an amount of water that fills
1/8 of 2 cans of equal size. How many blocks are needed to
fill 1 can of water?


Interpretation:
6(1/8block)s ÷ 1/8(2can)s = ? (1block)/(1can); 6/8
and 2/8, respectively, have the interpretation 6(1/8 unit)s
and 1/8(2unit).

4. 
If the
value of a function at the point 3/4 is 2, what is the slope
of the function?


Interpretation:
1(2unit) ÷ 1[3/4unit] = ? (1unit)/[1unit]; the fraction
3/4 has the interpretation of 1[3/4unit].

A Wider Set
of Problem Situations. Restricted interpretations of arithmetic
operations and prescribed problem interpretations and representations
in the curriculum has led to a limited range of problem situations
and thus, in children, to constrained cognitive models for these
operations. Because to date our analysis has concentrated on division,
the thrust of our remarks on this issue will concern division, with
some references to addition.
CHILDREN'S MODELS
FOR DIVISION. Fischbein, Deri, Nello, and Marino (1985) indicate
that the dominant models children and (Graeber et al., 1989) teachers
use to solve multiplication and division problems have a very limited
range of applicability. Work by Kouba (1986), Fischbein et al. (1985),
and Greer (1987) clearly suggests that distribution is by far the
dominant model. The distribution model and the problem types to
which it is applicable leads to conceptions such as "the divisor
must be a whole number." Our analysis suggests the existence
of another model for partitive division, which we call the putin
model. This model is characterized by physically, or conceptually,
putting the objects represented by the dividend into the object(s)
represented by the divisor.
MORE APPROPRIATE
MODELS FOR DIVISION. Units interpretation interacts in an important
way with appropriate models or representations for division problems.
There are four question types for a division word problem with the
given quantities x(aunit)s and y[bunits]:
 How many
(aunit)s for each [bunit]?
 How many
(1unit)s for each [bunit]?
 How many
(aunit)s for each [1unit]?
 How many
(1unit)s for each [1unit]?
Mathematics
of quantity suggests at least two strategies to answer each of these
questions. One strategy is to compute the quotient of x divided
by y and then do appropriate conversion of units; that is, use the
problem representation x(aunit)s ÷ y[bunit]s and proceed
in a topdown order for the solution. The second is to convert units
as appropriate, so that the (given) units in the problem data statement
correspond to the (target) units in the problem question. The former
might be a more powerful problem representation; it provides for
a common problem representation for anyone of the four questions,
and it avoids the matter of classification of problems as multistep
(question 4 above) or as extraneous data (questions 1, 2, and 3).
Still a third method is to convert all the problem quantities to
units of one and then proceed. This representation seems to be the
least efficient and least accurate representation of some of the
problem forms. The necessary numerical computation resulting from
either strategy is essentially the same; the efficiency comes through
the understanding exemplified in the holistic problem representation
(Larkin, 1989). Important research issues about problem representation
are implicit in this discussion. Some examples of research issues
that can be investigated are:
 If a problem
solver first changes the units of the problem’s given quantities
to be the same as the units of the quantities in the problem question,
will problemsolving performance be improved?
 Do children
who are aware of the different interpretations of rational numbers
perform better on problem solving than those who don’t?
 Do children
who exhibit knowledge of the several unitsconversion principles
exhibit better problemsolving performance than those who don't?
Links Between
Additive and Multiplicative Structures. Considering the arithmetic
of whole and rational numbers from the joint perspectives of unitscomposition,
decomposition, and conversionand the mathematics of quantity provides
an essential link between wholenumber concepts (the additive conceptual
field [Vergnaud, 1983]) and multiplicative concepts, including rationalnumber
concepts, (the multiplicative conceptual field).
Firstgrade
children use a concretequantity representation for 2 + 5, such
as 2 apples plus 5 apples gives 7 apples. Abstractly, this has the
form 2(1unit)s + 5(1unit)s = 7(1unit)s; an analogous model holds
for 2/8 + 5/8: 2(1eighthunit)s + 5(1eighthunit)s equals 7(1eighthunit)s.
A situation of 2 stones plus 5 boys is cause for pause because stones
and boys do not add in the same way as apples and apples, unless
a common counting unit is found for stones and boys  objects, for
example. Similarly, for 2/4 + 5/6 there is the same need for a common
counting unit. A firm understanding of the need for a common counting
unit for addition situations, along with the recognition that 2/4
and 5/6 can be considered to be 2(1/4unit)s and 5(1/6unit)s, should
help to lay the conceptual base for avoiding the addition of 2/4
and 5/6 as 7/10.
Another type
of wholenumber problem that will help develop conceptual understanding
of the need for common counting units is the following (the problem
form is more important than its context):
Jane bas
2 bags with 4 candies in each, and 5 bags with 6 candies in each.
How many bags with 2 candies in each can she make?
This, again,
is traditionally a multistep problem; the first step, multiplication,
changes everything to units of one in spite of the fact that the
problem question asks about composite units of two. To our knowledge,
how children might solve this problem before school instruction
forces this solution model on them is not known; some recent but
limited pilot work suggests a likely solution to be to change the
bags of 4 candies and 6 candies to bags of 2 candies, that is, to
convert to the unit requested in the problem question and then count
or add to find the total number of bags of 2 candies. In terms of
our notation, this problem solution is as follows:
2(4unit)s
+ 5(6unit)s

= 4(2unit)s + 15 (2unit)s 

=
19(2unit)s. 
We call attention
to the conceptual similarity between this problem and 2/4 + 5/6:
(a) units interpretation indicates a need for a common unit, (b)
the magnitude of the common unit is a common divisor of the two
given units, (c) units conversion is needed before counting units
(that is, addition) can take place. After this, the strategy for
solving the wholenumber problem and the fraction addition problem
is conceptually and procedurally exactly the same.
We consider
the following questions to be of fundamental importance to research
and development in the area of multiplicative structures: (a) Do
firm cognitive links exist between the two systems? (b) If cognitive
links do exist, what are they? (c) If cognitive links exist between
the two structures, how can we help children develop these links?
(d) Does the acquisition of these links facilitate the transition
to the field of multiplicative structures so that learning of concepts,
operations, and relationships in this complex domain would be easier
than it currently is for children? (e) If such links are found to
exist, can multiplicative concepts be learned earlier in the curriculum
and be somewhat concurrent with learning about additive structures?
We believe that
some important links do exist, and part of our current analysis
seeks to identify them. Unfortunately, we have not progressed sufficiently
far to be able to elaborate at this point in time.
TEACHING
STUDIES AND RATIONAL NUMBER CONCEPTS
In the following
sections we examine current research on teaching and the implications
of these studies on the teaching of rational numbers in classroom
settings. We begin with programs that have not directly involved
rational numbers, and move progressively toward considering investigations
that give explicit attention to rational numbers.
Cognitively
Guided Instruction
The Cognitively
Guided Instruction (CGI) Research Paradigm acknowledges four fundamental
assumptions that appear to underlie much contemporary cognitive
research on childrens’ learning (Peterson, Fennema, & Carpenter,
in press; Cobb, Yackel, & Wood, 1988).
 Children
construct their own mathematical knowledge.
 Mathematics
instruction should be organized to facilitate children's construction
of knowledge.
 Children's
development of mathematical ideas should provide the basis for
sequencing topics of instruction.
 Mathematical
skills should be taught in relation to understanding and problemsolving.
The CGI model
basically assumes researchbased knowledge of children’s learning
within specific content domains. To date, the CGI model has been
used only with primary children’s addition and subtraction
concepts, although attempts are currently under way to provide implications
for other situations (Carpenter & Fennema, personal communication,
November 1989). In a recent study relating teachers’ knowledge
to student problemsolving behavior with onestep addition and subtraction
word problems, Peterson et al. (in press) suggested four CGI principles
for applying researchbased knowledge of children’s learning
from classroom instruction.
 Teachers
should assess not only whether a child can solve a particular
word problem, but also how the child solves the problem. Teachers
should analyze children’s thinking by asking appropriate
questions and listening to children’s responses.
 Teachers
should use the knowledge that they derive from assessment and
diagnosis of the children to plan appropriate instruction.
 Teachers
should organize instruction to involve children so that they actively
construct their own knowledge with understanding.
 Teachers
should ensure that elementary mathematics instruction stresses
relationships between mathematics concepts, skills, and problem
solving, with greater emphasis on problem solving than exists
in most instructional programs.
The primary
Wisconsin CGI study was conducted with 40 firstgrade teachers,
half of whom participated a fourweek summer CGI program. Participants
were urged to develop instructional sequences based on their interpretations
of the research literature on children’s learning of addition
and subtraction. Many of the successful CGI teachers adapted a loosely
structured discussion format where students were encouraged to solve
problems their own way and to look for alternative solution strategies.
Students of CGI teachers had encouraging achievement results, and
the CGI teachers appeared to have a better grasp of students’
capabilities and solution strategies.
It is important
to understand here that the CGI model attempted to capitalize on
children’s previous knowledge of addition and subtraction,
which they had essentially acquired prior to formal instruction.
In fact, didactic formal instruction in the traditional sense of
the word, was not an element in the Cognitively Guided Instructional
model. The basic research into children’s understanding of
addition and subtraction concepts involves the join, separate, combine,
and compare types of problems that are discussed frequently in the
literature (Moser, 1988; Carpenter, Heibert, & Moser, 1981;
Carpenter & Moser, 1982).
What implications
might the CGI model have for research into teaching and learning
rational number concepts? It is not at all clear that the basic
tenets of the CGI model are directly generalizable to the more complex
mathematical structures embedded in rationalnumber usage. The subtle
complexities within the domain are recurring themes in research
papers. With the introduction of the domain of rationalnumber concepts
comes a cognitively complex multivariate system requiring relativistic
thinking, a system where counting strategies and their variations
no longer form the basis of successful solution strategies.
The following
differences between additionandsubtraction studies and rational
number studies highlight difficulties involved in generalizing the
CGI model:
 The additionandsubtraction
studies depended on children’s informal concepts of those
operations and attempted to develop these informal notions through
informal discussion techniques. As yet, research has not established
that children have the same degree of informal rationalnumber
concepts. It is our position that these concepts must be developed
in classroom environments. Such environments do not, however,
preclude children’s construction of meaningful rationalnumber
concepts or contentorganization plans that build upon children’s
intuitions about mathematics.
 Primary teachers
basically understand the content of addition and subtraction and
their variations. The same cannot be said for teachers’ understanding
of rationalnumber concepts. In our recent survey of over 200
intermediatelevel teachers in Minnesota and Illinois, onequarter
to onethird did not appear to understand the mathematics they
were teaching (Post, Harel, Behr, & Lesh, 1988). Although
there is no indication that teachers cannot learn these concepts,
largescale inservice in these areas becomes a logical necessity.
It is our position that teachers must be generally wellinformed
about a content domain in order to provide appropriate instruction
for children.
 First and
secondgrade classes have traditionally spent a major part of
their time dealing with addition and subtraction concepts revolving
around basic facts, and their application in problemsolving settings.
This was precisely the context within which the CGI studies were
conducted. This will not be the case for rationalnumber instruction.
The content advocated for the intermediate grades will be very
different from what is currently in the mainstream curriculum,
with far less attention to symbolic operations and far more attention
to the underlying conceptual structureincluding order, equivalence,
concept of unit, and so on.
 The impact
of standardized tests on the curriculum are far more complex in
the rationalnumber domain. The issues transcend those concerned
with instructional paradigms, but they must be reconciled nonetheless
in any attempts to substantively change the nature of school curricula.
It appears,
then, that research on teaching rational numbers will be more complex
than research on teaching additive structures. We suspect that research
models will be firmly "situated" (Grenno, 1983) in specific
topics within the domains of rational numbers, proportional reasoning,
and other multiplicative structures. We envision a more direct approach
to instruction, one with less emphasis on traditional objectives
and more attention to complex conceptual underpinnings for the domain.
Attention to children’s knowledge construction will, of course,
be an essential element. Teachers must always be encouraged to learn
about student constructions and thinking strategies. It does not
follow, however, that students cannot or should not receive mathematical
knowledge from teachers, that mathematics instruction should not
be organized to facilitate the teachers' clear presentation of knowledge,
that the structure of mathematics should not provide a basis for
sequencing topics of instruction, or that mathematical skills cannot
be integrated and taught along with student understanding and problem
solving.
As we look to
the decade ahead, we are reminded of Gage's (1989) admonition to
once again make use of a variety of research paradigms in our attempts
to provide more viable and more informed research on teaching. The
Rational Number Project has employed one paradigm that appears to
hold promise for research on teaching rationalnumber concepts and
operation, proportions, geometry, and the like. Its theory base
is discussed and its implications for research on teaching is presented
in the next section.
Rational
Number Project Teaching Experiments
The Rational
Number Project (RNP) has been researching children's learning of
rationalnumber concepts (partwhole, ratio, decimal, operatorandquotient,
and proportional relationships) since 1979. The project has conducted
experimental studies (Cramer, Post, & Behr, 1989), surveys (Heller,
Post, Behr, & Lesh, 1990), and teaching experiments (Behr, Wachsmuth,
Post, and Lesh, 1984; Post et al., 1985).
The primary
source of RNP data has been four different teaching experiments
conducted with students in grades 4, 5, and 7. Our teaching experiments
focused on the process of mathematical concept development rather
than on achievement as measured by written tests. They were conducted
with 69 students and involved observation of the instructional
process by persons other than the instructor. Instruction was controlled
by detailed lesson plans (in some cases scripted), activities, written
tests, and student interviews. As in most teaching experiments,
our interest was to observe the learning process as it occurred
and to gauge the depth and direction of student understandings resulting
from interaction with carefully constructed, theorybased, instructional
materials.
The interview
was the primary source of data. The interview is ideally suited
to obtain detailed information about an individual’s acquisition
of new mathematical concepts. Our interviews began as structured
sets of questions but quickly became tailored to the specific responses
given by students. Consequently, we were able to probe student interest
and appeal, while at the same time assessing the depth of student
understandings and misunderstandings. Careful study of transcribed
interviews (protocols) resulted in insights about students’
evolving cognitions. The RNP conducted teaching experiments that
were 12, 18, 30, and 17 weeks in duration and that were conducted
simultaneously in the Twin Cities area and in DeKalb, Illinois.
Weekly interviews given to each student were transcribed and analyzed.
In this way, each individual’s progress was charted. Since
some of the questions were repeated from session to session, it
was possible to be quite precise in documenting individual development,
the stability of conceptual attainment, and areas of need. It was
also possible to contrast each student’s progress with the
others’, although that was not a primary concern. This contrast
was conducted not for grading purposes but to identify patterns
of growth and understanding across students.
Since statistical
assumptions of significantly large numbers of subjects were not
met, alternative analytic strategies were employed. These included
protocol and videotape analysis and the use of descriptive statistics.
Teaching experiments are not easy to generalize, but this shortcoming
is compensated for by the richness of the information provided.
In one case, the understanding unearthed in one of the teaching
experiments was tested in an experimental setting with welldefined
experimental and control conditions (Cramer et al., 1989).
Rational
Number Project Teaching Experiments:
Theoretical Model
The Rational
Number Project has relied on two basic theoretical models for the
development and execution of its four teaching experiments. We must
state initially that our position is squarely within the cognitive
psychological camp. We have been influenced by the work of Piaget
(1965), Dienes (Dienes & Golding, 1971), Bruner (1966) and a
host of more contemporary researchers. We, like Dienes (1960), believe
that learning mathematics can ultimately be integrated into one’s
personality and become a means of genuine personal fulfillment.
We have embraced the four basic components of his theory of mathematics
learning (the dynamic principle, perceptual variability, mathematical
variability, and the constructivity principle) and have tried to
embed their substance and spirit within our student materials. In
our materials we have utilized the play stage of student development
and have made provision for the transformation of play into morestructured
stages of fuller awareness (the dynamic principle). In addition,
we have actualized the notion that construction precedes analysis
(constructivity principle) by focusing heavily on individuals’
interaction with their environment. We have also provided opportunities
for students to talk about mathematics with their peers. In more
contemporary terms, we approached instruction from a constructivist
perspective. The two variability principles were used to guide the
construction of the teaching experiment lessons. The model employed
is essentially a twodimensional matrix with one of the variability
principles defining each dimension. The model as originally suggested
involved numeration systems, with various manipulative materials
contrasted to several number bases (Reys & Post, 1973; Post,
1974). It was quickly realized that the mathematical and perceptional
variability principles applied to a wide array of mathematical entities,
rational numbers included. In our initial model, the five rational
number subconstructs identified by Kieren (1976) constituted the
mathematical variability dimension, while a wide array of manipulated
materials made up the perceptual variability dimension. This original
model appears in Figure 14.15. At this point we had a "helicopter"
perspective of the teaching experiments. What remained was to develop
sequences of lessons within appropriate cells in the matrix. Dienes
contended that, psychologically, the perceptual variability provides
the opportunity for mathematical abstraction, while the mathematical
variability is concerned with the generalization of the concept(s)
under consideration. Certainly, both are important aspects of mathematical
conceptual development. Additionally, the variability principles
provide for some attention to individualized learning rates and
learning styles. The lessons would require very active physical
and mental involvement on the part of the learner. The scope and
sequence of the rationalnumber lessons appear in Table 14.2 (Behr
et al., 1984).
Having now the
basic orientation as to the broad parameters of our instructional
development, attention must be paid to the specific nature of the
ways in which individuals would interact with these mathematical
concepts. We found Bruner’s (1966) notion of modes of representation
useful in this regard. In his early work, Bruner suggested that
an idea might exist at three levels  or modes  of representation
(inactive, iconic, and symbolic). Although never specifically stated
by Bruner, these modes were interpreted to occur in a linear and
sequential order. Literally generations of curricula were developed
suggesting first inactive, then iconic, and then symbolic involvement
on the part of the student; a misinterpretation of Bruner’s
(1966) original intent (Lesh, personal communication, 1975). Realizing
the artificial nature of such linearity, Lesh (1979) extended the
model to two additional modes (spoken symbols and realworld situations),
eliminated the linearity, and stressed the interactive nature of
these modes of representation. Various analyses have shown that
manipulative aids are just one part of the development of mathematical
concepts. Other modes of representation  for example, pictorial,
verbal, symbolic, and realworld situations  also play a role (Lesh,
Landau, & Hamilton, 1983). The model suggests, and it has been
our contention, that the translations within and between modes of
representation make ideas meaningful for children. The Lesh translation
model appears in Figure 14.16.
The reader will
note the inclusion of Bruner's three modes of representation as
the central triangle in this model. Arrows denote translations between
modes and the concurrent ability to reconceptualize a given idea
in a different mode. For example, asking a student to draw a picture
of 1/2 plus 1/3 (first written on the blackboard) would be a translation
from written symbols to pictures, a translation between modes. Similarly,
asking a child to demonstrate 2/3 with Cuisenaire rods, given a
display of 2/3 with fraction circles, represents a translation within
modes  in this case, an instance of Dienes’ perceptualvariability
principle. Post (1988) elaborates on the cognitive function of these
translations.
Implications
for Teaching
Within the domain
of mathematics learning, perceptual variability is hypothesized
by Dienes (1960) to promote mathematical abstraction, while mathematical
variability provides for generalization and the opportunity for
expanded understanding of broader perspectives of the issues under
consideration. In a similar fashion, teachers need to be exposed
to various aspects of teaching in a wide variety of conditions or
contexts. For this reason, it is important to focus on a broad spectrum
of teacher roles (for example, as an instructor of large and small
groups, as a tutor, as a student, as an interviewer, as diagnostician,
as a confidant, and so forth.) and to relate these roles to specific
tasks teachers are expected to perform (Leinhardt & Greeno,
1986). Just as mathematical abstractions are not themselves contained
in the materials which children use, abstractions and generalizations
relating to the teaching profession are not necessarily embedded
in any single role which the teacher might assume. Such abstractions
and generalizations can only be extracted from consideration of
a variety of situational, contextual, and model activities, roles,
and tasks. In the same way that children are encouraged to discuss
similarities and differences between various isomorphs of mathematical
concepts, teachers should be encouraged to discuss similarities
and differences between pedagogically related actions in various
mathematical contexts. A wide variety of avenues should be exploited
to provide the foundation for these discussions. Clinically based
experiences, videotapes, demonstration lessons, and other types
of sharing experiences come immediately to mind. We hypothesize
that it is the opportunity to examine a variety of situations from
a number of perspectives and to simultaneously gain the perspectives
of other individuals that fosters the development of higherorder
understanding and processes of teaching. This is directly parallel
to our belief that it is the students’ ability to make translations
within and between modes of representation that makes ideas meaningful
for them (Post, Behr, & Lesh, 1986).
In the Rational
Number Project teaching experiments and the related Applied Mathematical
Problem Solving Project (AMPS) (Lesh, 1980), cooperative groups
of intermediatelevel children were asked to focus on a variety
of mathematical models, concepts, and problem situations and then
to discuss and come to agreement as to the intended meaning(s) (Figure
14.17, Diagram A). Individual students were also asked to focus
on several models or embodiments of a single mathematical idea and
to indicate similarities and differences in the different interpretations
(Figure 14.17, Diagram B). Later the group task was to reconcile
these interpretations in a way so as to arrive at the most probable
and widely agreed upon meaning(s). We believe that teachers can
also profit from discussing single pedagogical incidents and attempting
to reconcile the most probable meanings.
The models in
Figure 14.17 can be used directly with teachers and can also be
extended to include teachereducation instructional settings as
depicted in Figure 14.18. Notice that each of these situations is
in fact a variation of the perceptualvariability or multipleembodiment
principle as applied to various patterns of human interaction.
In our early
work with children, we continually attempted to stress higherorder
thinking and processing, defining higherorder thinking in part
as the ability to make these translations. It was important to us
to encourage children to go beyond a single incident and to reflect
about general meanings. This invariably involved intellectual processes
called metacognition: We were encouraging children to think about
their own thinking. In similar fashion, it seems reasonable to encourage
teachers to think seriously about their and others’ teaching
acts. The RNP and the AMPS projects determined that successful problem
solvers tend to think at more than one level. They think about the
problem at hand, but are also aware of their own thinking. The best
problem solvers also generalize problem approaches and problem and
solution types as described by Krutetskii (1976). The ability to
be simultaneously the "doer" and the "observer" is critical to the
solution of many multistage problems. Likewise, it is important
that teachers be able to identify and describe their own thoughts
about teaching at a number of levels.
Teachers not
only teach content; they also implicitly transmit attitudes, beliefs,
and understanding of mathematics. Whether desirable or not, students
think of teachers as models of correct problemsolving behavior.
As teachers act out or demonstrate solutions to problems, it is
especially important that they reflect on their own problemsolving
behavior so as to help students identify their own metacognitive
processes. The ability to accurately and insightfully observe one's
own problemsolving behavior is probably related to the ability
to accurately observe, describe, and critique the problemsolving
behavior of others (Post, et al., 1988).
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The development
of this paper was in part supported with funds from the National
Science Foundation under Grant No. OPE 8470077 (The Rational Number
Project). Any opinions, findings, and conclusions expressed are
those of the authors and do not necessarily reflect the views of
the National Science Foundation.
The authors
wish to thank Thomas E. Kieren, University of Alberta, and Leslie
Steffe, University of Georgia, for their helpful reviews of two
earlier drafts of this chapter.
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