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Implicit Understanding of Arithmetic with Rational Numbers: The Impact of Expertise

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Abstract

Recent work has shown that undergraduates at a major public university demonstrate implicit understanding of inverse relations between multiplication problems with fractions, as evidenced by the fact that solving one problem facilitates solving its inverse. The present study investigated whether such implicit understanding of mathematical relations is related to overall math ability. We found that low performers showed relational facilitation only when it was supported by perceptual similarity, whereas high performers showed relational facilitation on both perceptually similar and dissimilar problems. These findings are interpreted in terms of novice-expert differences in the representation of mathematical relations.
Implicit Understanding of Arithmetic with Rational Numbers:
The Impact of Expertise
Melissa DeWolf (mdewolf@ucla.edu)
Department of Psychology, University of California, Los Angeles
Los Angeles, CA, USA
Ji Y. Son (json2@calstatela.edu)
Department of Psychology, California State University, Los Angeles
Los Angeles, CA, USA
Miriam Bassok (mbassok@u.washington.edu)
Department of Psychology, University of Washington
Seattle, WA, USA
Keith J. Holyoak (holyoak@lifesci.ucla.edu)
Department of Psychology, University of California, Los Angeles
Los Angeles, CA, USA
Abstract
Recent work has shown that undergraduates at a major public
university demonstrate implicit understanding of inverse
relations between multiplication problems with fractions, as
evidenced by the fact that solving one problem facilitates
solving its inverse. The present study investigated whether
such implicit understanding of mathematical relations is
related to overall math ability. We found that low performers
showed relational facilitation only when it was supported by
perceptual similarity, whereas high performers showed
relational facilitation on both perceptually similar and
dissimilar problems. These findings are interpreted in terms
of novice-expert differences in the representation of
mathematical relations.
Keywords: mathematical reasoning, rational numbers,
relational reasoning, expertise
Introduction
Research on expertise has highlighted differences between
the mental representations of experts and novices. Experts
and novices not only approach problems differently, but
also differ in how they allocate attention and relate problems
to one another (e.g., Chi, Feltovich & Glaser, 1981; Novick,
1988; Chase & Simon, 1973). These differences have been
observed in a variety of problem-solving contexts, including
chess, physics, and mathematics. One important
consequence of differences between expert and novice
processing involves transfer between problems or situations.
For example, Chi et al. (1981) found that expert physicists
tended to group certain problems together based on the
physical laws involved in solving each problem. In contrast,
novices grouped problems based on perceptual similarity,
rather than on underlying principles. By attending to the
relational structure governing the problems, expert
physicists were able to transfer problem-solving strategies
effectively between problems involving the same abstract
principles. The perceptual similarities on which novices
focus are much less effective in supporting transfer. It is
often difficult for beginning students to ignore surface
features and encode relational structure, a fact that likely
contributes to the difficulty in obtaining transfer between
problems that are analogically similar but perceptually
dissimilar (e.g., Gick & Holyoak, 1980, 1983, Hayes &
Simon, 1977; Holyoak & Koh, 1987; Ross, 1987). In
general, expertise or deep understanding is characterized by
mental representations that go beyond perceptual similarity.
In the current study we investigated the extent to which
perceptual similarity between two relationally-similar math
problems facilitates the performance of solvers who differ in
their level of math expertise.
Expertise in Mathematics: The Case of Rational
Numbers
The general pattern of differences between expert and
novice understanding is found within the realm of
mathematics (Novick, 1988; Schoenfeld & Hermann, 1982).
Expert mathematicians (e.g., math professors, or those who
achieve high scores on a math proficiency test) are more
likely to judge problems embodying the same mathematical
structure to be similar, are more likely to apply the same
problem-solving strategies to relationally-related problems,
and demonstrate greater transfer after a delay (Novick,
1988; Novick & Holyoak, 1991). Expertise in mathematics
is often associated with more rapid solution times (e.g.,
Kellman, Massey & Son, 2010; Stevenson et al., 1990).
Understanding of rational numbers (fractions and
decimals) provides a particularly interesting context for
investigating differences in expert and novice
understanding. Novice students often inappropriately
transfer characteristics of whole numbers to fractions, and
therefore expect fractions to be countable and discrete (Ni &
Zhou, 2005; Stafylidou & Vosniadou, 2004). These
In R. Dale, C. Jennings, P. Maglio, T. Matlock, D. Noelle, A. Warfaumont & J. Yoshimi (Eds.),
Proceedings of the 37th Annual Conference of the Cognitive Science Society (2015). Austin, TX:
Cognitive Science Society.
misconceptions persist in adults with lower overall
competence in mathematics. For example, based on oral
explanations of fraction concepts, Stigler and colleagues
(Givvin, Stigler & Thompson, 2011; Stigler, Givvin &
Thompson, 2010) found that community-college students
exhibit many of the same misconceptions about fraction
magnitudes and fraction arithmetic as do middle-school
students. In contrast, other studies conducted with students
at highly competitive universities (e.g., Schneider & Siegler,
2010; DeWolf, Grounds, Bassok & Holyoak, 2014) have
found that these students, who tend to have greater
mathematical expertise, are able to represent fraction
magnitudes with little difficulty.
Expert-novice differences have been observed in fraction
arithmetic. In the realm of (positive) whole numbers,
multiplication yields a larger result and division a smaller
result, but this is not generally true of rational numbers,
such as fractions or decimals less than 1. Siegler and Lortie-
Forgues (in press) asked participants from a wide range of
math backgrounds to perform a simple task involving
fraction multiplication and division. Participants simply had
to decide whether problems like 31/56 * 17/42 > 31/56 are
true. The investigators found that while math and science
students from a highly competitive university were
consistently correct, pre-service teachers and middle-school
students performed at below-chance levels.
A central issue is that beginning students (and many
elementary-school teachers) have difficulty understanding
mathematical operations and how they relate to one another
(Ma, 1999; Siegler et al., 2011, 2013). For example, fraction
addition and subtraction require finding common
denominators, whereas fraction multiplication and division
do not. Students are often unclear as to when and why it is
necessary to find common denominators. This lack of
understanding of procedures may reflect a lack of deep
conceptual understanding of the relations between operators,
and of how the division operation within a fraction relates to
other operations in the problem.
Understanding Fraction Multiplication and Inverse
Relations
This lack of understanding is especially evident in the case
of the “invert and multiply” strategy in fraction division.
Early on, students are taught that to complete a fraction
division problem, all that is required is to invert the second
fraction in the problem and then proceed with the fraction
multiplication procedure. But understanding why this
strategy works is not simple. Tirosh (2000) found that even
pre-service teachers have little understanding of this
strategy.
The reason why the invert-and-multiply strategy works
involves the reciprocal relationships between the two factors
in a multiplication problem and their relationship to a
product. For example, 5 ÷ 10 is the same as 5/10, which is
the same as 5 X 1/10 since 10 and 1/10 are reciprocals. In
addition, students have to understand that the “bar” in the
fraction expression denotes a division operation; and as
such, it can be used to represent a relation within the
multiplication operation itself. Thus, the same strategy
applies when multiplying either whole numbers or fractions.
Performing the operation 2 ÷ 3is equivalent to 2/3and
also 2 X 1/3”. Furthermore, 2 ÷ 3 represents the same
proportional relation as 4 ÷ 6. A deep understanding of
this type of relational structure would allow students to
move flexibly between any of these equivalent notations.
Overview of Current Study
In the current study we sought to better understand the
differences in understanding of multiplication with rational
numbers and inverse relations across high- and low-
performing math students. We tested whether adults from
varying math backgrounds are sensitive to different types of
similarities between multiplication problems. DeWolf and
Holyoak (2014) found that participants from a highly
competitive university showed facilitation in solving a
multiplication problem when it was preceded by its inverse.
For example, participants were faster to solve 3 X 4/3 = 4 if
it was preceded by 4 X 3/4 = 3. College students were
sensitive to the inverse relation between problems when the
second multiplier was expressed as a fraction, but not as a
decimal (e.g., the pair 3 X 1.33 = 4 and 4 X .75 = 3 yielded
no facilitation of the second problem). Importantly,
facilitation was found for fraction problems even when the
inverse relation was less perceptually apparent (e.g., 4 X 6/8
= 3 preceded by 3 X 4/3 = 4). These college students thus
showed implicit understanding of the inverse relation
between fraction multiplication problems.
This apparent relational transfer might seem surprising in
light of the evidence discussed above indicating that many
students have difficulty in understanding fraction
multiplication, let alone transferring relational knowledge
between inverse problems. One hypothesis is that this type
of implicit understanding of inverse relations only emerges
for relatively expert (or in our study, relatively high-
performing) students.. In order to assess transfer
performance across a wide range of math ability that would
span relatively novice and expert levels of performance, we
recruited participants from two universities in one American
city. We administered a general math ability test to obtain a
measure of participants’ overall math ability, and used this
measure to separate students into high- and low-performing
groups. Our sample included students who ranged widely in
overall mathematical ability.
We varied the degree of perceptual similarity between
inverse fraction problems. In the high-similarity condition,
the relationship between the fraction problems was
perceptually salient (e.g., 3 X 4/3 = 4; 4 X 3/4 = 3). In the
low-similarity condition, the two fraction multipliers were
perceptually different but still maintained their relational
similarity (e.g., 3 X 4/3 = 4; 4 X 6/8 = 3, where 4/3 and 6/8
are reciprocals of one another). We hypothesized that low-
performing participants may show facilitation in the high-
similarity case, for which a perceptual strategy supports
relational similarity, but not in the low-similarity case where
reliance on perceptual similarity is not possible.
Method
Participants
A total of 89 undergraduates participated in the study for
course credit. Thirty-four participants were undergraduates
from California State University, Los Angeles (CSULA) (19
females) and 55 were undergraduates from University of
California, Los Angeles (UCLA) (44 females).
Design, Materials, and Procedure
Speeded Multiplication Task. This task was an adaptation
of the multiplication-priming paradigm used by DeWolf and
Holyoak (2014), which demonstrated implicit relational
transfer. Participants were shown a series of multiplication
problems that were either true (correct) or false (incorrect).
They were simply asked to verify whether the problems
were true or false. A quarter of the problems were true
“primed pairs” in which a prime problem was inversely
related to the successive target problem (e.g., 3 X 4/3 = 4
primes 4 X 3/4 = 3). Participants were randomly assigned to
one of three between-subjects conditions in this task: high-
similarity fraction pairs (N = 30), low-similarity fraction
pairs (N = 29), and decimal pairs (N = 30).
The high-similarity fractions were identical to the
“matching fractions” used by DeWolf and Holyoak (2014),
which afford a variety of perceptually-driven strategies. The
low-similarity fractions were constructed by mixing the
prime and target problems from the “matching fractions”
and “non-matching fractions” conditions in Experiment 2 of
DeWolf and Holyoak (2014). That is, these primed pairs
included an expression in which the fraction and whole
number components matched (e.g., 3 X 4/3 = 4), and an
expression in which the fraction components did not match
the whole numbers (e.g., 4 X 6/8 = 3). Priming in this
condition thus depends on appreciating the inverse relation
between the successive problems despite the low perceptual
similarity between them. The prime and target assignments
within the primed pairs were counterbalanced across
participants. Importantly, the two fraction conditions were
identical except for this difference in primed pairs.
As in similar previous studies of rational numbers (e.g.,
DeWolf, Bassok & Holyoak, 2015a), a decimal condition
was included for comparison with fractions. The decimal
condition used the same values as the fraction conditions, in
that the second term in the multiplication problem was
simply the fraction converted to its equivalent decimal
rounded to the nearest hundredth (e.g., 3 X 1.33 = 4).
A total of 240 multiplication problems were used, half
true and half false. Sixty of the 120 true problems were true
primed problems (30 true primed pairs). Sixty of the 120
false problems were false primed problems (30 false primed
pairs); these shared the inverse relation between successive
problems, but were false. The remaining 120 problems were
foil problems that were not related to each other in any way.
These problems were designed to obscure the similarity
between the primed problems. Besides the pairing of the
problems within the primed pairs, the overall order of the
problems was random for every participant.
The multiplication task was administered using Superlab
4.5 (Cedrus Corp., 2004), which was used to collect
accuracy and response time data. Participants were told that
they would see multiplication problems. They were told to
press the “a” key if the problem was true or the “l” key if
the problem was false. Participants were told that the
answers were shown rounded to the nearest whole number.
As we were interested in potentially subtle response time
differences, participants were instructed to respond as
quickly as possible while maintaining high accuracy. They
were first given four practice trials that used only whole
numbers. After the practice trials, participants were given a
chance to ask questions before starting the test trials.
Explicit General Math Test A second task that participants
completed was an explicit measure of general math
knowledge, which was used to split the participants into
relatively low- and high-performing groups. This task
involved a total of 25 multiple-choice problems, and eight
problems requiring a solution (either equations or word
problems). The test comprised three subsections, each
designed to assess a domain of mathematical understanding:
algebra, fractions, and multiplicative understanding. The
algebra questions (adapted from Booth et al., 2014) included
basic equation solving questions, word problems, and
evaluations of algebraic expressions. Fraction problems
queried participants about equivalent fractions, ratio
relationships, and the relation between the size of the
numerator and denominator. Multiplicative questions asked
about the greatest common factor of two numbers, the
reciprocals of certain numbers, and lowest common
multiples of two numbers. These problems were adapted
from released questions from the 2008 California State
Standards exam for Algebra I. Successful performance on
this test would only require a level of understanding
corresponding to basic high school math.
This test was administered with paper and pencil.
Participants were randomly assigned to one of three
different random orders. They were encouraged to use space
on the page to write out their work, and were told not to use
a calculator.
Results
Explicit General Math Test
Because the explicit test consisted of questions that were
multiple choice, or questions for which there was only one
correct answer, questions were scored on a 0, 1 basis. Final
scores for subtests were averaged across questions in a
relevant subset.
Overall accuracy on the test across all participants was
79% (SD = 15; minimum score = 36%, maximum score =
100%). For the subset of algebra problems, overall accuracy
was 77% (SD = 17, minimum score = 41%, maximum score
= 100%). For the subset of fraction questions, overall
accuracy was 77% (SD = 23, minimum score = 13%,
maximum score = 100%). Finally, for the subset of
multiplicative questions, overall accuracy was 77% (SD =
17, minimum score = 50%, maximum score = 100%). The
participants thus ranged considerably in overall math ability,
providing a high-variance sample to examine differences in
performance between low- and high-performing
participants.
Relation Between Speeded Multiplication and Math
Expertise
Accuracy on the speeded multiplication task was computed
for true1 prime and target trials after dividing participants
into low- and high-performance groups, based on a median
split performed separately for each of the three conditions.
For the high-similarity fractions condition, the high-
performance group achieved significantly higher accuracy
than the low-performance group (.97 vs. .87; t(24) = 2.53, p
= .02). For the low-similarity fraction condition, the high-
performance group showed a non-significant trend for
higher accuracy than the low-performance group (.91 vs.
.84; t(27) = 1.33, p = .19). For the decimals condition, the
high-performance group also showed a non-significant trend
for higher accuracy relative to the low-performance group
(.67 vs .54, t(27) = 1.36, p = .18). There was no evidence of
a priming effect for any of the conditions on the accuracy
measure, and no evidence of a difference in priming across
low- and high-performing groups.
Figure 1 shows response times for true prime and target
trials, separated into low- and high-performance groups
based on the same median split of the explicit task used for
the accuracy analysis. Response times for error trials and
those more than three standard deviations from the mean of
accurate trials were excluded from analyses. The change in
RT from the average prime RT to the average target RT
(prime target) was calculated to assess the speed-up
attributable to priming for each participant. For the high-
similarity fractions, the average speed-up across participants
in the low-performing group was significantly larger than
that for the high-performing group (.73 s vs. .23 s, t(24) =
3.18, p = .004). Thus when pairs of mathematical problems
were perceptually similar, as in the high-similarity
condition, the priming effect held for both high- and low-
performing students, and was actually largest for low-
performing students.
The priming difference observed in the high-similarity
fraction condition may in part be related to general
differences in RT between the low- and high-performing
groups. Average prime RT for the low-performing group
was considerably slower than for the high-performing
group. Also, average target RT for the low-performing
group was slower than RT for either the prime or target in
1 As in DeWolf and Holyoak (2014), we find that the priming
effect only held for the true primed trials (high-similarity false
primes: 4.5 s vs. 4.3 s, t(25) = .91, p = .37; low-similarity false
primes: 4.9 s vs. 4.5 s, t(28) = 1.93, p = .07). The same pattern of
results for false primed trials was found after conducting the
median-split analysis.
Figure 1. Average response times for true prime and target
trials for each condition, separated by low- versus high-
performing math students.
the high-performing group. Thus, although the speed-up for
the low-performing group was much larger than that for the
high-performing group, these differential gains may be
related to the longer overall response times of the former
group.
In stark contrast, for the low-similarity fraction condition,
the high-performing group had a significantly greater
average speed-up than did the low-performing group (.27 s
vs. -.06 s, t(27) = 2.33, p = .03). In fact, the latter group
showed no priming effect, and, their RTs were slower
overall compared to those of the high-performing group.
For the decimals condition, neither high- nor low-
performing students showed any evidence of priming. The
average speed-up for the low-performing group did not
differ reliably from that of the high-performing group (.03 s
vs. .09 s, t(27) = .13, p = .89). For both groups, RTs on
decimal trials were slower than on either type of fraction
trials.
Correlations between Multiplication Task and Math
Expertise In order to better understand how the subparts of
the explicit math test related to performance on the speeded
multiplication task, we correlated performance on each of
the subparts of the explicit test (algebra, fraction,
multiplicative) with overall accuracy and response times for
each of the three conditions on the multiplication task, for
all participants (i.e., combining high- and low-performing
participants). Table 1 shows the correlations between each
of the multiplication-task conditions and accuracy on the
explicit math test (overall and for each subtest).
For the high-similarity fraction condition, both overall
accuracy and RT on the multiplication task were
significantly correlated with overall accuracy on the explicit
test and with each of the subtests. A similar pattern of
correlations was observed for the low-similarity fraction
condition, except that correlations with the multiplicative
subtest were not reliable. For the decimals condition,
accuracy was significantly correlated with all but the
multiplicative test, whereas RT was correlated only with the
algebra subtest (and overall score).
0.0#
0.5#
1.0#
1.5#
2.0#
2.5#
3.0#
3.5#
4.0#
4.5#
5.0#
5.5#
6.0#
6.5#
Low#
Performing#
High#
Performing#
Low#
Performing#
High#
Performing#
Low#
Performing#
High#
Performing#
High7Similarity#Frac?ons# Low7Similarity#Frac?ons# Decimals#
Response'Time'(s)'
Prime'Condi1on'
Prime#
Target#
*
*
*
Table 1: Correlations between performance on the speeded
multiplication task (accuracy and RT) and score on explicit
math test (overall and each subtest).
Multiplication
Condition
Overall
Test
Fractions
Subtest
Multipli-
cative
Subtest
High-similarity Fraction
Accuracy
RT
.70***
-.60***
.54**
-.51**
.75***
-.40*
Low-similarity Fraction
Accuracy
RT
.53**
-.54**
.50**
-.57**
.25
-.36
Decimals
Accuracy
RT
.55**
-.37*
.46*
-.15
.25
-.19
* p<.05; ** p<.01; *** p<.001
The observed pattern of correlations between speeded
multiplication and explicit math performance suggests that
across all participants, multiplicative knowledge is not a
reliable predictor when perceptual cues to relational
structure are lacking (i.e., in the low-similarity fraction and
decimal conditions). Especially in a speeded task,
multiplicative strategies may not be employed without
obvious perceptual supports (at least for students with lower
math ability).
Discussion
The present study demonstrates clear differences in
understanding of inverse relations and fraction
multiplication between low- and high-performing math
students. In the high-similarity fraction condition,
perceptual cues guide attention to relational structure, as the
numbers in the first problem are simply rearranged in the
inverse manner in the second problem (e.g., 3 X 4/3 = 4; 4
X 3/4 = 3). For this pair type, the low-performing group
showed greater facilitation on response time for target
problems than did high-performing participants. Low-
performing students were thus able to capitalize on
perceptual similarity between problems to facilitate transfer.
Despite their large decrease in response time on target
trials, low-performing participants were still slower in
responding than high-performing participants. Moreover,
the two groups performed very differently in the low-
similarity fraction condition. In such problems the numbers
in the successive fractions were different, even though they
were reciprocals (e.g., 3 X 4/3 = 4; 4 X 6/8 = 3). For these
low-similarity problems, for which perceptual cues did not
strongly guide attention to relation structure, only high-
performing students demonstrated facilitation on target
problems.
The present findings are consistent with previous work on
novice versus expert transfer in mathematics and other
domains, but go beyond previous studies by using the
speeded multiplication task to provide an implicit measure
of relational transfer. This implicit measure of transfer
indicates that individuals with greater math expertise
process relationally-similar problems more effectively than
do novices.
In general, low math performers were slower and less
accurate on the speeded multiplication task. In addition, we
found that while the magnitude of priming for the RT
measure correlated with all of the subparts of the explicit
math test, RT in the decimal condition only correlated with
the algebra subset of questions. Because performance on
decimal problems largely depends on correctly estimating
the magnitude of the decimal and the resulting product (i.e.,
there is no possible simplification of the problem as there is
for fraction problems), the decimal condition is largely an
estimation task. Thus, the decimal version of the task is
likely measuring something akin to decimal magnitude
understanding. Previous work has shown that, at least for
middle-school Algebra-I students, decimal magnitude
understanding is a strong predictor of algebra performance
(DeWolf, Bassok & Holyoak, 2015b).
Several potential mechanisms may contribute to the
priming effect we observed in the speeded multiplication
task. Perceptual similarity between problems accounts for
part of the effect, as evidenced by the robust priming effect
observed in the high-similarity fraction condition for both
low- and high-performing participants. As observed in other
relational tasks, reasoning is facilitated by salient semantic
or perceptual cues that are correlated with more abstract
relations (Bassok, 1996).
More interesting, perhaps, are the possible mechanisms
by which relatively expert adults are able to exploit shared
inverse relations between fraction problems that lack simple
perceptual cues to the relational correspondences. Our
findings suggest that low-performing participants
understand the inverse relation at some level, but lack the
abilities in pattern recognition or in simplifying fractions
that are required to obtain priming in the low-similarity
condition. It appears that experts have an advantage in
recognizing equivalent fractions based on different
constituent numbers, likely due to the greater fluency with
which experts are able to reduce or simplify common
fractions. These high-performing participants may in some
sense “see” fractions differently. High-performing adults
may have greater perceptual expertise with fractions, and in
connecting alternative simplified or reduced forms of
fractions (cf. Kellman et al, 2008). Thus, an expert (or high-
performing student) may have a deeper appreciation for
relevant relations and operations even at an implicit level.
Low- and high-performing participants showed little
difference in accuracy on the simple multiplication
problems we tested; however, the large differences we
observed in response times suggest that more expert adults
benefit from more direct access to inverse relations,
allowing them to make mappings between problems with
greater ease.
Acknowledgments
Preparation of this paper was supported by NSF Fellowship
DGE-1144087 to Melissa DeWolf. We thank Michael
Ambrosi, Austin Chau, Queenie Cui, Eugene Goh, Kaitlin
Hunter, Andrew Molica, and Jennifer Talton for help
collecting and analyzing data at UCLA. We also thank
Louis Lopez, Abigail Solis, and Magnolia Cedeno for help
collecting data at CSULA.
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... Because previous work has found relational understanding of fractions to be related to algebra understanding ( DeWolf et al., 2015b), a measure of algebra understanding was added in Experiment 2. A 27-question paper-and-pencil assessment provided a baseline measure of participants' algebra understanding ( DeWolf, Son, Bassok, & Holyoak, 2015;adapted from DeWolf et al., 2015b). This assessment included algebra problems that were either taken from the California State Standards for Grade 8 or adapted from Booth, Newton, and Twiss-Garrity (2014). ...
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