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وشكرا
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Introduction
A central issue in knowledge acquisition is how children and adults integrate their moment
to moment experiences into more general concepts. Children and adults are exposed
to a constant stream of regularities in the environment, some novel and some familiar. Over
time and experience, structural or relational information about the environment can be detected
in this stream of regularities. Eventually, much of this information becomes part
of children’s conceptual representation of the environment. For example, a child may be
repeatedly exposed to a correlation between a feature and category, such as facial hair and
gender. Eventually, facial hair may become part of the child’s concept of gender.
How regularities are detected and ultimately integrated into concepts is not well understood.
An important part of the issue is that people experience individual events (e.g., a
bearded man appears one moment, a non-bearded woman the next). They do not generally
experience summaries of those events. Rarely does any one point out that men tend
to have more facial hair than women. Yet, the relation eventually comes to be represented
semantically such that it can be used in reasoning, communication, and problem solving.
For example, a child may reason that a bearded monkey cannot be female because it has
too much facial hair.
There are two broad approaches to theorizing about this issue. These approaches differ
in their fundamental assumptions about the nature of children’s cognition when they
encounter a novel situation. One position proposes that children first use their associative
system to detect regularities in novel environments and only later consolidate that associative
representation into conceptual knowledge. The second position proposes that children
first generate an initial model for the new situation, a model that contains specific candidate
relations, and then evaluate the regularities in the environment relative to that model.
1.1. Associations lead, concepts follow
In the first account, associations “lead” and concepts “follow”. Children detect regularities
in the flowof experience and encode them as associations. Eventually, these associations
represent the relational structure of the situation. Through some process, the relational structure
represented in the associations crosses over into a child’s conceptual representation.
Consistent with the possibility that children detect relations in the environment, a large
body of work on category development in infants supports the idea that even very young
children can detect and represent relational structure. For example, 6-month-old infants
can quickly develop categories for the spatial relations “between” (Quinn, Norris, Pasko,
Schmader, & Mash, 1999) and “in” (Casasola, Cohen, & Chiarello, 2003). Cohen, Chaput,
and Cashon (2002) showed that even features that indicate physical causality in classic
launching events can be extracted through associative mechanisms. Cohen et al. suggested
that infants’ understanding of the world develops hierarchically with the extraction of regularities
among lower level features facilitating the extraction of regularities among higher
level properties.
Case and Okamoto (1996) suggested that regularities in the environment are detected and
encoded by associative mechanisms which then prepare the way for conceptual advances.
Conceptual advances, in turn, prepare the way for detecting further associative regularities.
J.A. Dixon, A.S. Bangert / Cognitive Development 20 (2005) 65–86 67
In this way, Case and Okamoto suggested that conceptual and associative processes are
involved in an iterative loop. The associative system initially detects new relations, but the
process through which these encoded relations become part of the conceptual system is not
explicitly addressed.
Recent work on the process of redescription (Dixon & Bangert, 2002; Dixon & Dohn,
2003; Kotovsky & Gentner, 1996) offers a potential explanation of how relations move
from an associative representation to a conceptual representation. This work draws from
Karmiloff-Smith’s theory of representational redescription (Karmiloff-Smith, 1992), but
emphasizes the process of redescription rather than intervening levels of representation.
Redescription of a relation occurs when the relation is repeatedly activated in close temporal
succession and when performance is highly accurate. For example, Dixon and Bangert
(2002) showed that children’s and adults’ discovery of a new relation in a gear-system
problem (i.e., that adjacent gears turn in opposite directions) was jointly predicted by (a)
having a history of accurate use with a strategy that contained embedded information about
the relation and (b) using that strategy repeatedly in tight temporal succession. They proposed
that having a history of accurate use created a strong associative link between the
problem and the relation in long-term memory. Subsequent repeated and successive use of
the strategy strongly activated the relation such that it became a new way of representing
the problem.
1.2. Concepts lead, associations follow
An alternative view is that on encountering a novel situation children propose an initial
conceptual model. This model provides an interpretation of the situation and strongly influences
attention and encoding. In this view, concepts “lead” and associations “follow”.
Regularities that are relevant to the current conceptualization are encoded into associations,
but other regularities are largely ignored. Rittle-Johnson and Alibali (1999) proposed that
conceptually driven differences in encoding may be an important source of developmental
change (see Logan, 2002 for a discussion of theories of the relationship between attention
and categorization). Given a new situation, children first propose models or hypotheses
about relational structure and then evaluate evidence relative to those hypotheses (Halford,
Brown, & Thompson, 1986; Klahr, Fay, & Dunbar, 1993; Kuhn, Garcia-Mila, Zohar, &
Andersen, 1995; Schauble, 1996). For example, Kuhn et al. (1995) showed that children
proposed reasonable and coherent models of multivariable situations. Although often incorrect,
these models tend to drive the types of evidence children seek and their interpretation
of that evidence (see, Dunbar, 1993; Klayman & Ha, 1987). Children’s ability to coordinate
evidence with their hypotheses is a central developmental dimension in this approach (see
Kuhn, 2002a for a recent discussion).
Similarly, Gopnik, Sobel, Schulz and Glymour (2001) showed that very young children
appear to propose plausible hypotheses about causal mechanisms. They also showed
that children have the ability to appropriately interpret the causal implications of patterns
of covariation relative to these hypothesized mechanisms. In these approaches previous
knowledge is used to construct a “best bet” model for a new relation or situation. Patterns of
covariation (i.e., regularities in the environment) serve as evidence about the adequacy of the
model; the model may be revised or replaced if the pattern of evidence conflicts with it. Note
68 J.A. Dixon, A.S. Bangert / Cognitive Development 20 (2005) 65–86
that the initial model may be proposed, and later evaluated, by either a heuristic or analytic
type of system (Klaczynski, 2001). Both types of processing can propose interpretations
of new situations and modify those interpretations in response to error (see Kuhn, 2002b
for a discussion of theory-revision under both types of systems and their developmental
relation).
Current evidence suggests that both theory-revision and redescription are important
in knowledge acquisition. The evidence supporting theory-revision is quite extensive in
both the developmental and adult literatures (e.g., Chen & Klahr, 1999; Dixon & Tuccillo,
2001; Halford, Brown, & Thompson, 1986; Klahr, Fay, & Dunbar, 1993; Kuhn, Garcia-
Mila, Zohar, & Andersen, 1995; Newstead, Thompson, & Handley, 2002; Schauble, 1996).
People generate candidate models and evaluate evidence relative to those models in many
situations. The evidence for redescription is more sparse. One goal of the current manuscript
was to provide additional data regarding the redescription process. A second goal was to test
whether redescription and theory-revision could help explain the acquisition of conceptual
understanding in arithmetic.
Both processes should be quite capable of extracting relational information in the domain
of arithmetic. The arithmetic operations have a highly regular relational structure. For
example, a child learning addition will experience a particular set of functional relations
between the addends and the sum as problems are correctly solved. Therefore, mathematical
structure might be easily detected by the associative system.
Similarly, initial hypotheses about the functional relations of arithmetic should be easy to
generate and evaluate. Because the arithmetic operations are used to model simple and well
understood physical events (i.e., transformations on sets of discrete objects), transferring
relations from the physical domain to the arithmetic domain (where they could serve as
initial hypotheses) should be quite straightforward (Dixon & Tuccillo, 2001).
1.3. Conceptual representation of arithmetic relations
The functional relations of the arithmetic operations can be seen by considering the signature
pattern of each operation. When the values of the operands vary within a particular
class of number (e.g., positive integers), each arithmetic operation produces a unique signature
pattern. For example, addition produces a set of parallel curves; multiplication produces
a diverging fan (see Fig. 1). Dixon, Deets, and Bangert (2001) proposed that with experience
children and adults come to represent the relationships or principles instantiated in
this signature pattern. For example, consider two principles, Relationship-to-Operands and
Direction-of-Effect as defined for positive integers. The Relationship-to-Operands principle
specifies whether the answer is greater or less than the operands. For addition and multiplication,
the answer is always greater than either operand. For subtraction and division, the
answer is less than the first operand, but unrelated to the second operand. The Relationshipto-
Operands principle, therefore, captures a fundamental aspect of the signature pattern,
whether it is shifted up or down relative to the operands.
The Direction-of-Effect (DE) principle specifies whether the value of the answer increases
or decreases, as the value of one operand changes (again, restricting ourselves to
positive integers). For addition and multiplication, as either operand increases, the value
of the answer increases. For subtraction and division, as the value of the first operand




increases, the value of the answer increases, but as the value of the second operand increases,
the value of the answer decreases. Therefore, the DE principle also captures an important
aspect of the signature pattern; whether it is increasing or decreasing across values of the
operands.
To test whether people represented these principles, Dixon et al. (2001) asked college
and 8th-grade participants to rate patterns of answers to completed problems. Each pattern
was comprised of nine completed problems presented in a 3×3 matrix. The value of
the first operand changed across the rows, the value of the second operand changed
across the columns. Each pattern was supposedly generated by a student who was just
learning the arithmetic operation. Participants were asked to rate how good or bad the
70 J.A. Dixon, A.S. Bangert / Cognitive Development 20 (2005) 65–86
fictitious student’s attempt at math had been. The distance of the presented pattern from
the correct answers and whether the presented pattern of answers violated a principle were
independently manipulated.
Participants were sensitive to violations of the principles, but this effect depended jointly
on the developmental history of the arithmetic operation and the principle. Specifically,
operations that had a longer developmental history [e.g., addition is introduced before
multiplication in standard curricula (National Council of Teachers of Mathematics, 2000)]
had more complete principle representations than those introduced later. Principles that
are in evidence more frequently were represented for more operations. For example, the
Relationship-to-Operands principle is in evidence each time an arithmetic problem is correctly
solved. The DE principle, however, requires comparing at least two problems; to
assess the direction of change one must compare the result of one problem to another.
College participants showed evidence of representing the Relationship-to-Operands and
DE principles for both addition and multiplication. Eighth-grade participants represented
the Relationship-to-Operands principle for addition and multiplication, but appeared to
represent the DE principle only for addition. They did not detect violations of the DE
principle for multiplication.
1.4. Development with experience: possible processes
The pattern of results across operations and principleswas consistent with the hypothesis
that the principles develop with experience, but very little is known about how this occurs.
The current manuscript attempts to provide initial evidence about the underlying processes.
Based on the literature reviewed above, we hypothesized that two types of processes could
explain the acquisition of the principles. First, participants may encode the relation through
associative mechanisms and then come to represent the relation conceptually through redescription.
Redescription should occur when performance is accurate and exposure to the
relation is concentrated, causing it to be highly activated (Dixon&Bangert, 2002;Kotovsky
& Gentner, 1996). Second, participants may propose an initial model or set of relations for
a new operation and evaluate evidence relative to that model. As errors accumulate, the
model would be modified until one that did not generate errors (or substantially reduced
them) was found. We refer to this second account as theory-revision.
We tested these two accounts for children’s acquisition of the DE principle for multiplication.
We focused on DE for multiplication, because previous work showed that 8th-grade
students did not represent this principle for multiplication, although they did for addition.
College students represented DE for both addition and multiplication. This suggests that
8th-graders, and perhaps even younger students, may be ready to acquire DE for multiplication.
Because theory-revision and redescription make predictions about how fairly complex
aspects of individual performance produce conceptual change (e.g., proposing an errorprone
model, correctly solving successive problems), we chose to measure, rather than try
to manipulate, these aspects of performance. The DE principle lends itself to a correlational
investigation, because it is likely to be quite difficult to acquire. Detecting the DE principle
requires coordinating the relationship between the direction of change in the operands and
the direction of change in the answers. This should be quite difficult for children. Therefore,
J.A. Dixon, A.S. Bangert / Cognitive Development 20 (2005) 65–86 71
we expected considerable variability in the effectiveness of the experience and, ultimately,
in children’s conceptual representation of DE. Variability in the outcome measure (i.e., the
measure of DE) is necessary for assessing the effects of predictor variables in correlational
designs.
In the current study, we first assessed 4th-, 6th-, and 8th-grade students’ representation
of the DE principle for multiplication. Participants were told that they were going to play a
game with a fictitious other player. The other player was thinking of one of the arithmetic
operations and would provide a hint about which one it was. Participants were asked to rate
how likely it was that the other player was thinking about each arithmetic operation. Each
hint described one value of one principle. For example, the hint, “When you increase the first
number, the answer always increases”, describes one value of the DE principle. Previous
work showed that these measures tap the conceptual representation of the principles (Dixon
et al., 2001) and predict important aspects of problem solving (Dixon & Bangert, 2004).
During a subsequent session, we presented students with 32 sets of experiential trials;
each set consisted of two multiplication problems. One problem had been correctly completed,
the other problem had a question mark in the space for the answer. Next to the
pair of problems was a number line divided into three regions. The regions were defined
numerically. Participants were asked to select the region that contained the answer to the
second problem. For all the multiplication problems, the correct region contained all numbers
greater than the presented answer (if the operands were increasing) or less than the
presented answer (if the operands were decreasing). We also presented 32 analogous pairs
of division problems.We did not expect substantial gains for division; even college students
do not appear to represent the DE principle for division. Finally, in a third session, we again
assessed participants’ representation of the DE principle for multiplication.1
1.4.1. Predictions
According to the theory-revision hypothesis, learning about the DE relation first requires
proposing an incorrect relation (participants might also propose the correct relation, but
this would simply give them a high score on the pre-test measure of DE and, therefore,
provide very little room for learning). As use of the incorrect relation generates errors on
the experiential task, participants should modify the relation or replace it with a new one.
This cycle of proposing a relation and testing it against the experiential material should
recur until a relation that produces little or no error is found. Therefore, conceptual gains
in understanding DE for multiplication (i.e., high scores on the post-experience measure of
DE) should be predicted by a pattern of early errors and later successes.
According to the redescription hypothesis, participants first encode the DE relation with
their associative system. After the relation has been encoded, each subsequent correct solution
activates it. When relevant trials occur successively (one immediately following
another) and those trials are solved accurately, the relation will become highly activated
1 The current design should not be confused with the obviously flawed, straight pretest–posttest design, in which
a unitary corpus of experience is evaluated without comparison. Rather, in the current study, individual differences
in performance and experience during experiential trials are used to predict posttest performance. The design
is the correlational analog of manipulating the types of experience in a pretest–posttest design; a valid type of
experimental design. See Achenbach (1978) for a discussion.
72 J.A. Dixon, A.S. Bangert / Cognitive Development 20 (2005) 65–86
and consequently redescribed into the conceptual representation. Therefore, higher scores
on the post-experience measure of DE should be predicted by runs of successive and accurate
solutions on the experiential task. Based on past work that showed that exposure to
multiple relevant trials facilitated redescription (Dixon & Bangert, 2002), we expected the
effect of correct runs to emerge after the completion of at least one block of experiential
trials.
In summary, the theory-revision and redescription hypotheses make very different predictions
about how performance during the experiential task will be related to improved
understanding of theDErelation for multiplication. Therefore, to test these two accounts, we
examined how these theoretically specified aspects of performance on the experiential trials
predicted later understanding of DE (we describe how these predictors were constructed in
detail in the results section).
2. Method
2.1. Participants
Children from grades 4 (N= 23), 6 (N= 27), and 8 (N= 19), were recruited from local
schools. Approximately half the sample was female. The mean ages were 9 years, 4 months;
11 years, 6 months; and 13 years, 7 months, for grades 4, 6, and 8, respectively.
2.2. Materials
2.2.1. Assessment of the DE principle
We assessed participants’ conceptual understanding of the DE principle for multiplication
with a rating task from Dixon et al. (2001). The rating task was presented in the
context of a communication game involving a fictitious other player. The other player was
supposedly thinking of one of the arithmetic operations. Participants were asked to rate
how likely it was that other player was thinking about each operation, given a statement or
hint about the operation. Two statements assessed the DE principle for the first operand and
two assessed the DE principle for the second operand. The statements that assessed the DE
principle for the first operand were: “When you increase the first number, the answer always
increases”, and “When you increase the first number, the answer always decreases”. The first
statement is consistent with multiplication, the second is inconsistent with it. The statements
for the second operand were: “When you increase the second number, the answer always
increases”, and “When you increase the second number, the answer always decreases”.
Again, the first statement is consistent with multiplication, the second is inconsistent with
it.
Each statement was presented at the top of a separate page. Participants were asked
to rate, “How likely is it that the other player is thinking about [addition, subtraction,
multiplication, division]?” for each operation separately. Participants responded by marking
a rating scale. The scales were 140mm horizontal lines. One end of each scale was marked
“Very Likely”, the other end was marked “Very Unlikely”. A separate scale was presented
for each operation.
J.A. Dixon, A.S. Bangert / Cognitive Development 20 (2005) 65–86 73
Note that children made their ratings for multiplication in the context of a game-like
setting in which the other arithmetic operations were also rated. However, the analyses in the
manuscript deal exclusively with ratings for multiplication. For each operand, we took the
difference between the multiplication ratings for the consistent and inconsistent statements
as our measure of the DE principle for multiplication; participants who understand DE
for multiplication should give high ratings to statements which are consistent with that
principle and low ratings to statements that are inconsistent with it. Participants completed
the assessment of the DE principle once pre-experience and once post-experience.
2.2.2. Experiential problems
Participants were presented with the experiential problems in the context of a game.
They were told that they had just found the remains of an alien spaceship buried below the
surface of the Earth. While exploring the ship, they discovered a message from the aliens.
The message explained that the ship contained 64 environmental tools, each capable of
reversing a negative effect of pollution. Each tool resided in a separate room inside one
of several sealed pods. To ensure that only highly intelligent life forms were able to use
these powerful tools, selecting an incorrect pod diminished the power of the tool. The aliens
had left a mathematical clue in each room to help identify the pod that contained the tool.
Therefore, correctly decoding the mathematical clue was very important.
The mathematical clues were pairs of multiplication problems. The problems were presented
one above the other (see Fig. 2). The top problem was correctly completed, but the
bottom problem had a question mark in the space for the answer. The set of experiential
problems comprised a 2 (Operand Position: First, Second)×2 (Direction of Change: Increasing,
Decreasing)×2 (Number of Digits in Changing Operand: 1, 2) factorial design.
The variables were defined as follows. Operand Position: For half the problems, the first
operand changed from the top to the bottom problem; the second operand did not change.
For the remaining problems, the second operand changed from top to bottom; the first
operand did not change. Direction of Change: For half the problems pairs, the changing
operand increased. For the remaining problems, the changing operand decreased. Number
of Digits in Changing Operand: One operand was always a single digit number, the other
operand was always a double digit number. Half the problems had the changing operand
as a single digit number and the other operand as a double digit number. The remaining
problems had the changing operand as a double digit number, the other operand was single
digit. Fig. 2 presents four examples. The left column shows two first-operand trials; in the
top panel the first operand increases, in the bottom panel the first operand decreases. The
right column shows analogous examples for the second operand (these examples all have
the two-digit operand changing).
Next to each pair of problems were three pods in a vertical stack. Each pod was labeled
with a portion of the number line. Participants were told that the tool was in the pod labeled
with the portion of the number line that contained the answer to the second problem. When
the operand increased, the correct pod was labeled with the portion of the number line that
contained all integers greater than the answer to the first problem (A1). One incorrect pod
contained A1 and the four integers belowthat answer: A1 to (A1 4). The other incorrect pod
contained all integers five or more below the correct answer: (A1 5) and lower. Consider
the upper left panel of Fig. 2 as an example. The answer to the first problem is 171.



upper (and correct) pod is labeled to include all integers 172 and greater. The middle pod
is labeled to include all integers 171–167. The lower pod is labeled to include all integers
166 and lower.
When the operand decreased, the correct pod was labeled with the portion of the number
line that contained all the integers less than A1. The incorrect pods were constructed
analogously to the description above. The lower panels of Fig. 2 show examples.
Each pair of problems and its accompanying pods were presented on a separate page.
A brief description of the tool was provided on the page. For example, the “Smelly
Smog Shield” was described as a tool that “cleans air, breaking down harmful pollutants”.
The “Landfill Lightening Laser” was able to increase “biodegradability of landfill
wastes . . .”. Each tool had a unique symbol (e.g., a shield or a lightening bolt, for
the tools described above, respectively). The symbol was drawn in commercially available
invisible ink in the correct pod. Participants were given marker pens that caused the
symbol to become visible when they colored in the pod. Marked incorrect pods indicated
errors.
Problem pairs were presented in blocks of eight trials. There were four multiplication
blocks and four division blocks. Multiplication and division blocks alternated. Whether multiplication
or division was presented first was randomly determined for each participant.
Within each block of eight trials, four were first-operand trials, four were second-operand
trials. The order of trials within blocks was randomized for each participant. Problem pairs
and accompanying pods for division were constructed analogously to those for multiplication.
Two sets of experiential problems were constructed. Participants were randomly assigned
to one set. We used two sets of experiential problems to allow us evaluate whether
experiential effects were dependent on a specific corpus of problems. No evidence of such
effects were found.
2.2.3. Procedure
Students participated in three separate sessions; consecutive sessions were separated
by no more than 2 days. Participants received the pre-experience assessment of the DE
principle in the first session, the experiential problems in the second session, and the postexperience
assessment in the third session. Participants worked individually; instructions
were presented in small groups of four to eight.
For the assessment of the DE principle, participants were asked to imagine that they
were playing a guessing game with another player. The other player was thinking of one
of the four arithmetic operations. The participant’s job was to rate how likely it was that
the other player was thinking about each operation. The other player would provide a hint
in the form of a statement about the operation. As an example, participants were given the
statement, “When you change the first number in the problem, the answer changes”. The
instructions showed two-operand arithmetic problems (e.g., X + Y = ) to provide a context
for referring to the first and second numbers.
For the experiential task, participants were read a cover story in which they were scientists
who had discovered an alien spaceship.Amessage left by aliens explained that the spaceship
contained advanced environmental tools in each of 64 rooms. The message also explained
that a mathematical clue would help them find the correct pod within each room. To ensure
76 J.A. Dixon, A.S. Bangert / Cognitive Development 20 (2005) 65–86
that only highly intelligent beings would have access to the full-strength tools, opening an
incorrect pod degraded the effectiveness of the environmental tool. Pods could be opened
using the special X-ray marker. The number line label was explained with an example.
Participants were told that if they made an error (i.e., opened an incorrect pod), they should
try again until they found the tool. An example problem was given to allow participants
practice at using the X-ray markers.
3. Results
3.1. Overview
First, we assess whether therewas sufficient variability in theDEscores for multiplication
to effectively address the predictions.2 Specifically, we examine the distribution of gains
in DE scores from pre-experience to post-experience. Next, we assess the pattern of errors
during the experiential task. Finally, we address predictions from the theory-revision and
redescription hypotheses. These hypotheses specify that different aspects of performance
on the experiential task should predict improvement in DE multiplication scores.
3.2. Pre-experience and post-experience representation of DE for multiplication
Recall that participants rated two DE statements for each operand; one statement was
consistent with multiplication, the other was inconsistent it. Those that were consistent with
multiplication stated that increasing the operand, increased the answer. Those that were
inconsistent with multiplication stated that increasing the operand, decreased the answer.
Ratings were measured on a 140mm scale; higher scores indicated that the statement was
more indicative of multiplication. The difference between the ratings of these statements
(consistent–inconsistent) indexes the degree to which participants represent theDEprinciple
for multiplication, henceforth the DE principle score.
Fig. 3 shows the meanDEprinciple scores for multiplication pre- and post-experience for
each age group separately. A 2 (Operand)×2 (Pre- versus Post-experience)×3 (Grade)
ANOVA showed that DE scores depended on grade, F(2, 62) = 4.84. None of the other
main effects or interactions was significant (an alpha level of 0.05 was used for all tests of
significance unless otherwise indicated). The means suggest that at the group level there
was no improvement in the representation of DE.
However, examining the individual patterns of scores shows considerable variability
from pre- to post-experience. Differences between pre and post scores ranged from 233
to 226.5 for the first operand and 179 to 147 for the second operand. It is perhaps worth
noting that approximately 25% of participants showed gains of 28 points or greater for the
first operand, and about 25% showed gains of 17 points or greater for the second operand.
These participants have substantial gains in their DE scores. Variability in the DE scores
2 Because we did not predict gains in the DE scores for division, we focus on the results for multiplication. At the
bivariate level, none of the predictors considered below predicted gains in the DE scores for division. Therefore,
we did not test the more complex models.


from pre-experience to post-experience does not in itself, of course, imply that learning
occurred. It is, however, an important prerequisite for assessing the effects of predictor
variables. Next we present descriptive information about errors during the experiential
trials. In the following section, we address whether predictor variables from the theoryrevision
and redescription hypotheses can explain differences in the acquisition of the DE
principle.
3.3. Errors on experiential trials
Recall that there were four blocks of multiplication experiential trials. Each block contained
eight trials, four that were relevant to the first operand and four that were relevant
to the second operand. The upper panel of Table 1 presents the mean number of errors by
block for each grade separately. The top three rows show errors for first-operand trials. The
bottom three rows show errors for second-operand trials. The number of errors was on average
quite low across all grades and blocks. There is little evidence for a reduction in errors
across blocks. A 2 (Operand)×4 (Block)×3 (Grade) ANOVA showed that first-operand
trials had significantly more errors than second-operand trials, F(1, 66) = 16.81. This effect
depended on which block was considered, F(3, 64) = 3.17. The main effect of grade
was not significant, F(2, 66) = 2.27, p > 0.11, nor was grade was involved in any significant
interactions.




3.4.1. Theory-revision
According to the theory-revision hypothesis, participants should learn about the DE
principle when they propose an incorrect initial hypothesis and modify that hypothesis in
response to error. Therefore, gains in the post-experience DE scores should be predicted by
a high error rate on the first block of experiential problems, indicative of having proposed an
incorrect hypothesis, and a low error rate on the last block, indicative of having successfully
modified the hypothesis.
As a first test of this hypothesis, we examined whether the number of errors in each
block was associated with post-experience DE scores. Because neither error rates nor gains
in DE scores varied across grades, we collapsed across grades initially, but included grade
in subsequent models. The top row of Table 2 shows the correlations between number of
first-operand errors for each block and the post-experience DE score for the first operand.
The number of errors was negatively associated for all blocks, the correlation is significant
for blocks 2 and 4. The second row of Table 2 shows the analogous results for the second
operand. Again, the correlations are all negative. None of the second-operand correlations is
significant. Because participants should propose an initial model when they first encounter
the task, the lack of a positive correlation between the block 1 errors and post-experience
DE scores is contrary to the theory-revision hypothesis.
To further test the theory-revision hypothesis, we conducted two multiple regression
analyses, one for first-operand variables, one for second-operand variables. The logic of
these analyses, alluded to above, is as follows. According to theory-revision, when participants
first encounter a new problem, they should generate an initial hypothesis about
the relevant relations. If this hypothesis is incorrect, it will result in errors, leading to its
subsequent modification. If the correct relation is eventually generated, errors will radically
decline and remain low. Therefore, a pattern of early (block 1) high errors and later (block
4) low errors should predict gains in DE. This prediction is captured by the interaction
between block 1 errors and block 4 errors; the initial incorrect hypothesis should lead to
J


block 1 errors, revising that hypothesis should drastically lower the error rate (from the
time it is proposed through the last block).We entered grade, pre-experience DE scores, the
number of block 1 errors, block 4 errors, and the interaction between blocks 1 and 4 errors,
as predictors of the post-experience DE scores. The model for the first-operand DE scores
was significant, R = 0.59, F(5, 59) = 6.42. Pre-experience DE scores for the first operand
were positively associated with post-experience DE scores for the first operand, B = 0.391,
t(59) = 3.84. The number of block 4 errors (for first operand trials) was negatively associated
with DE scores, B =14.03, t(59) =2.41. Neither the number of block 1 errors for
first operand trials, nor the interaction between block 1 and block 4 errors were significant
predictors, t’s(59) = 0.40, 0.10, respectively.3 Grade did not contribute significantly to the
model, t(59) = 1.52, nor was the three-way interaction involving grade, block 1 errors, and
block 4 errors significant, t(56) = 0.17.
The model for the second-operand DE scores was also significant, R = 0.47, F(5,
59) = 3.34. However, only pre-experience DE scores for the second operand significantly
predicted post-experience DE scores for the second operand, B = 0.403, t(59) = 3.52. None
of the other coefficients was significantly different from zero, largest absolute value of
t(59) = 0.72. The three-way interaction among grade, block 1 errors, and block 4 errors was
not significant, t(56) =1.15.
The predictors from the theory-revision model did not explain post-experienceDEscores.
There was no evidence that errors were positively associated with the post-experience DE
3 Variables indexing interactions were computed with centered variables and with uncentered variables. The
substantive conclusions were the same in both analyses. Results using centered variables are reported.
80 J.A. Dixon, A.S. Bangert / Cognitive Development 20 (2005) 65–86
scores. Further, the interaction between blocks 1 and 4 errors, which should capture the
pattern of initial errors leading to later success, did not explain the post-experience scores.
The inability of this pattern to predict post-experience DE scores did not vary across grades.
3.4.2. Redescription
Recall that according to the redescription hypothesis, concentrated runs of correctly
solved experiential problems should lead to higher DE scores. Therefore, for each block
we calculated the number of successive, correct solutions for experiential problems of each
type of trial (i.e., first or second operand). For example, suppose a participant received
first-operand problems on trials 1, 2, 3, and 6. He or she would receive one point if trials 1
and 2 were correctly completed; trials 1 and 2 would constitute one correct “run”. He or she
would receive another point if trial 3 were correctly solved; trials 2 and 3 would constitute
a second correct run. Because trial 6 is not adjacent to another first-operand trial, solving it
correctly would not increase the number of correct runs for first-operand trials. Considering
both first- and second-operand trials, the mean number of correct runs per block ranged
from 0.96 to 1.39 for fourth grade, 1.07 to 1.52 for sixth grade, and 1.09 to 1.58 for eighth
grade.4 The number of correct runs was negatively related to error, but far from perfectly
collinear. Correlations per block ranged from 0.50 to 0.66.
Paralleling the analysis above, we first examined the bivariate correlations between the
number of correct runs for each block and post-experience DE scores. The third row of
Table 2 shows the results for the first-operand variables. The number of correct runs for
first-operand trials for blocks 2 and 4 were significantly related to post-experienceDEscores
for the first operand. The fourth row of Table 2 shows that for second-operand variables, the
number of correct runs for block 2was significantly related to post-experience DE scores for
the second operand (a Bonferroni correction for multiple comparisons was used to hold the
family-wise error rate below 0.05. The four correlations for each operand were considered
a family of comparisons).
To test the redescription hypothesis, we conducted two multiple regression analyses,
one for the first-operand variables and one for the second-operand variables. Of particular
interest, was whether the number of correct runs would predict post-experience DE scores
when pre-experience DE scores were in the model. We also tested whether the number
of correct runs was a significant predictor when error was included in the model. This
allows us to statistically separate the effects of correct runs from the more general effects of
accurate performance. For example, one might expect children who are more skilled with
mathematics to both improve their understanding of the DE principle more easily and to
make fewer errors during the experiential trials. As a result of making fewer errors overall,
they would also, on average, have a greater number of correct runs—each correct response
can potentially contribute to a correct run, each error potentially interrupts a correct run.
Therefore, we were concerned that the number of correct runs might simply be related to
4 Note that two factors contribute to the number of correct runs: the random ordering of the trials within the block
and whether performance was accurate. We already showed that error, the simple inverse of correct performance,
was not related to grade. Randomization of the trials is, of course, not related to grade either. Therefore, it follows
that the number of correct runs will not be related to grade. The results of an ANOVA confirmed this logically
implied prediction.
J.A. Dixon, A.S. Bangert / Cognitive Development 20 (2005) 65–86 81
the post-experience DE scores through the differences in error rates predicted by individual
variation in mathematical ability.
Because correct runs of first-operand trials in blocks 2 and 4 predictedDEpost-experience
scores for the first operand, we included both these variables in the model. We also included
grade and pre-experience first-operandDEscores. The overall modelwas significant,
R = 0.66, F(4, 60) = 11.30. Pre-experienceDEscores significantly predicted post-experience
scores, B = 0.36, t(60) = 3.59, as did grade, B = 10.22, t(60) = 2.07. Correct runs in blocks
2 and 4 positively predicted post-experience DE scores, B’s = 18.15, 23.95, t’s(60) = 2.24,
3.15, respectively. When number of errors in blocks 2 and 4 were added to the model, their
coefficients were not significantly different from zero, t’s(58) = 0.48, 0.64, respectively.
The estimated effects of correct runs in blocks 2 and 4 were largely unchanged, B’s = 19.76,
20.72, t’s(58) = 1.97, 2.09, p’s < 0.054, 0.041, respectively. The effect of the number of correct
runs did not depend on grade, B’s =1.78, 9.05, t’s(56) =0.35, 1.65, for blocks
2 and 4, respectively.
For the second operand, we included the pre-experience DE scores, grade, and number
of correct runs (on second operand trials) in block 2 as predictors. The overall model was
significant, R = 0.52, F(3, 61) = 7.37. Pre-experience DE scores and the number of correct
runs significantly predicted post-experience scores, B’s = 0.39, 16.92, t’s(61) = 3.60, 2.29,
respectively. The coefficient for gradewas not significantly different from zero, t(61) = 0.33.
When the number of errors on block 2 was added to the model, its coefficient was not
significantly different from zero, t(60) = 0.69; the estimated effect of number of correct
runs remained significant, B = 20.41, t(60) = 2.27. The effect of the number of correct runs
did not depend on grade, B = 7.93, t(59) = 1.63.
4. Discussion
The results suggest that theory-revision was not responsible for changes in children’s
understanding of the DE principle. The evidence against the theory-revision account is quite
strong. First, errors during the early experiential blocks (or later blocks) were not positively
associated with understanding of the DE principle after the experience. If the acquisition
of the DE principle depended on initially proposing an incorrect model, then the errors
generated by that model should be positively associated with change. Further, these early
errors should lead to later successes. That is, as errors accumulate the model should be
revised such that later blocks have few errors. This high to low pattern of errors, captured
by the interaction term, should be indicative of having successfully revised one’s initial
inappropriate model. Put another way, the interaction term should allow us to differentiate
individuals who showed the predicted high-to-low error pattern from those who had high
errors early on, but did not modify their hypothesis and, therefore, did not decrease their
error rates. However, the interaction between errors on the first block and errors on the last
block, which should index the high-to-low pattern, did not predict understanding of DE
after the experience.
The redescription hypothesis appears to fit the results quite well. According to the
redescription position, repeated and successive activation of the relation should result in
its becoming part of the conceptual representation. In the experiential task, successfully
82 J.A. Dixon, A.S. Bangert / Cognitive Development 20 (2005) 65–86
completing an experiential problem should expose the relation (e.g., the answer to the second
problem is greater than the answer to the first problem), activating it in memory. Runs
of correct solutions should result in concentrated activation and, therefore, redescription.
The multiple regression results provide fairly strong evidence for redescription. First,
consider the results for the first-operand variables. Children’s post-experience understanding
of the DE principle for the first operand was predicted by the number of correct runs of firstoperand
trials in blocks 2 and 4. In addition to grade, the model also contained children’s
pre-experience scores for DE. Therefore, the number of correct runs explained variance in
post-experience DE scores above and beyond that explained by pre-experience scores. The
ability of correct runs from both blocks 2 and 4 to contribute to the same model suggests some
participants redescribed the DE relation based on concentrated experience relatively early
in the task, but others redescribed later. This may be caused by differences in the degree to
which the associative regularity of theDErelationwas represented prior to experience. Some
participants may have already had reasonably strong associations between the direction of
change in the first operand and the answer and, therefore, could redescribe the relation early
in the experiential task. Others may have needed to first construct those associations during
the early experiential blocks and, therefore, redescribed the relation later.
The results for the second-operand variables were similar. The number of correct runs
of second-operand trials in block 2 predicted children’s post-experience understanding of
the DE principle for the second operand. This model also contained grade and the relevant
pre-experience DE scores. The degree to which participants correctly solved experiential
problems in succession predicted understanding of the DE principle, above and beyond
their initial understanding of the principle. Only the number of correct runs on block 2
were predictive, suggesting that participants may have come to the task with fairly strong
associative links between changes in the second operand and the answer. This is consistent
with previous work which showed that, when faced with a variable that changes value,
children often prefer placing it in the second-operand position (Bell, Greer, Grimison, &
Mangan, 1989). Children may, therefore, be more likely to encode the effects of changing
the second operand or may have more experience with the second operand as the changing
variable.
For both the first- and second-operand models, the predictive effects of correct runs were
largely unchanged when the numbers of errors committed in the relevant blocks were added
to the model. This argues quite strongly against two more trivial interpretations of the results.
One interpretation is that the positive relationship between the number of correct runs and
post-experience DE scores might simply be driven by general mathematics ability. High
ability students would solve more experiential trials correctly and, therefore, be more likely
to solve runs of experiential problems correctly. Their better mathematical ability would
also make learning the DE principle easier. This explanation does not fit the current results
because the number of correct runs explains variance in DE scores beyond that explained
by the number of errors.
A second somewhat less trivial interpretation of the results is that some participants
learned about the DE principle early during experiential trials, but not through redescription.
This learningwould improve later experiential task performance such that these participants
would have more correct runs on subsequent experiential blocks. This learning would also
improve post-experience DE scores. In this explanation, as in the previous one, a third factor
J.A. Dixon, A.S. Bangert / Cognitive Development 20 (2005) 65–86 83
(i.e., early learning and general math ability, respectively) produces higher post-experience
DE scores and low error rates. Low error rates, in turn, increase the number of correct runs.
Neither of these hypotheses can explain the current data, because correct runs explains
variance in DE scores beyond that explained by errors.
For both operands, the model estimates suggest that each correct run (i.e., correctly solving
a pair of temporally adjacent problems) increases post-experience DE scores by about
20 points. This is a fairly substantial gain in conceptual understanding of multiplication.
Our measure of the DE principle taps conceptual understanding of how multiplication functions.
It relies on a communication-based rating task that requires children to evaluate the
semantic information contained in the “hint” relative to the semantic information they have
in long-term memory about the operation. In recent work, we showed that when children
try to generate a mathematical solution in a physical problem domain, they align the DE
principle with structurally analogous principles of the physical domain (Dixon & Bangert,
2004). Therefore, the DE principle plays an important conceptual role in mathematical
problem solving.
We note that the DE principle, as described, holds for multiplication with positive numbers,
but not for multiplication with negative numbers. For example, if one operand is
positive and the other is negative, increasing the positive operand decreases the value of
the answer. Children, of course, first learn multiplication with positive integers and only
employ negative numbers later (National Council of Teachers of Mathematics, 2000). That
is, they first have considerable exposure to instances that contain the dominant form of the
regularity (i.e., the DE principle, as we have presented it), but later encounter instances that
do not share that regularity. This is analogous to many other conceptual domains in which
exposure to a dominant regularity occurs first, but is later followed by exposure to exceptional
cases. For example, as children learn about mammals, they are first exposed to a large
number of mammals that are land-dwelling. Later, they are exposed to marine mammals, a
class of mammals that obviously does not share the “land-dwelling” regularity. Therefore,
we suggest that conceptual development of multiplication (and other arithmetic operations)
parallels that of other conceptual domains; early regularities come to be represented as part
of the concept, later exposure to instances that do not share that regularity may cause a
new subcategory to form or a modification of the concept for the entire category. These
rich issues might be very fruitfully explored within mathematics, because exposure to the
instances that share or violate a regularity is tightly controlled by the curriculum.
The number of correct runs for the four blocks were not equally predictive of gains in
understanding DE. Similarly, the number of errors for the blocks were not equally predictive
of DE. This is consistent with both the redescription and theory-revision processes because
both are thought to be temporally specific; they occur directly after the requisite conditions
have been met. For example, the redescription hypothesis first requires the formation of
associations between the relation and the problem, and then concentrated experience with
the relation. It may well be the case that some participants redescribed earlier or later than
others in the sample. The current design does not allow us to evaluate this possibility and,
therefore, the performance of these participants contributes to the error term. That said, the
current models explain reasonable proportions of the variance, suggesting that there was
similarity at least among a substantial number of participants in the strength of their initial
associative representations and the timing of their concentrated experience.
84 J.A. Dixon, A.S. Bangert / Cognitive Development 20 (2005) 65–86
Although we did not find evidence for theory-revision in the current study, previous
research has demonstrated that theory-revision is an important process in knowledge acquisition
(e.g., Chen & Klahr, 1999; Dixon & Bangert, 2002; Kuhn et al., 1995; Schauble,
1996). We believe that, given the evidence for both processes, the question is not which
process constitutes the sole or central mechanism in knowledge acquisition, but rather what
conditions recruit one process rather than the other. In the current context, either process
should have been available. Children should be able to generate reasonable initial hypotheses
about the DE relation for multiplication and evaluate those hypotheses based on the results
from the experiential problems. Similarly, they should be able to detect the regularities that
arise from the DE principle and redescribe those into a more abstract form. Indeed, there
was no restriction in our analytic approach that would have prevented observing evidence
for both models.
We propose that redescription, rather than theory-revision, was recruited in the current
context because participants had a means of solving the experiential problems without
generating a newhypothesis about the relation. By estimating or computing the answer to the
second problem, participants could locate the tool. In thisway, itwas possible for participants
to complete the experiential problems without proposing any specific hypotheses about
how changes in the operands were related to changes in the answers. Based on the current
results and previous work, we suggest that theory-revision will predominate when one’s
current representation of the task is very weakly invoked by the problem context. When
the problem context does not strongly cue a unique representation, multiple representations
may be available and one’s subjective feeling of confidence in the representation may be
low. Both these factors, the availability of alternative hypotheses and limited confidence in
the current hypothesis, may make one more receptive to error as an impetus for modifying
the representation.
In the current context, participants had a compelling representation of the experiential
problems, as examples of the multiplication operation. This representation does not contain
the DE relation explicitly, but it does allow participants to complete the experiential
problems. Each time they did so successfully, the relation was available. We suggest that
early presentations of the relation were encoded by the associative system; later presentations
activated the associative representations. If this activation occurred in a tight temporal
sequence, the relation became redescribed into their conceptual representation.
We found that the process of redescription did not appear to show developmental differences
across grades. Grade did not significantly interact with the number of correct runs for
either operand. Dixon and Bangert (2002) reported a similar finding. Given this developmental
invariance, we suggest that the redescription process might be available for knowledge
acquisition at very young age. However, the associative relations on which redescription
operates must be constructed over time, sometimes, as in the case of mathematical relations,
over considerable periods of time. Therefore, the redescription process may tend to operate
on different types of relational information at different points in development. As different
types of relations are encoded, they can become grist for the redescription process.
Rittle-Johnson, Siegler, and Alibali (2001) proposed that conceptual knowledge and procedural
skill in mathematics developed in an iterative fashion. That is, gains in conceptual
knowledge led to gains in procedural skill and vice versa. The current results provide a
potential explanation for how individual differences in the ability to accurately execute a
J.A. Dixon, A.S. Bangert / Cognitive Development 20 (2005) 65–86 85
procedure may impact conceptual knowledge. Redescription works when relational information
is activated by correctly executing procedures. Rittle-Johnson et al. also suggested
that the problem representation, resident in short-term memory, was the crucial link that
connected procedural and conceptual knowledge. Consistent with this idea, we suggest that
the concentrated activation of the relation changes the representation in short-term memory,
such that it includes relational information. In this way, regularities become represented in
concepts.
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