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Children Learn Spurious Associations in Their Math Textbooks: Examples From Fraction Arithmetic

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Abstract

Fraction arithmetic is among the most important and difficult topics children encounter in elementary and middle school mathematics. Braithwaite, Pyke, and Siegler (2017) hypothesized that difficulties learning fraction arithmetic often reflect reliance on associative knowledge-rather than understanding of mathematical concepts and procedures-to guide choices of solution strategies. They further proposed that this associative knowledge reflects distributional characteristics of the fraction arithmetic problems children encounter. To test these hypotheses, we examined textbooks and middle school children in the US (Experiments 1 and 2) and China (Experiment 3). We asked the children to predict which arithmetic operation would accompany a specified pair of operands, to generate operands to accompany a specified arithmetic operation, and to match operands and operations. In both countries, children's responses indicated that they associated operand pairs having equal denominators with addition and subtraction, and operand pairs having a whole number and a fraction with multiplication and division. The children's associations paralleled the textbook input in both countries, which was consistent with the hypothesis that children learned the associations from the practice problems. Differences in the effects of such associative knowledge on US and Chinese children's fraction arithmetic performance are discussed, as are implications of these differences for educational practice.
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Children Learn Spurious Associations in Their Math Textbooks:
Examples from Fraction Arithmetic
David W. Braithwaite
Carnegie Mellon University
Robert S. Siegler
Carnegie Mellon University and Beijing Normal University
Fraction arithmetic is among the most important and difficult topics children
encounter in elementary and middle school mathematics. Braithwaite, Pyke, and
Siegler (2017) hypothesized that difficulties learning fraction arithmetic often
reflect reliance on associative knowledgerather than understanding of
mathematical concepts and proceduresto guide choices of solution strategies.
They further proposed that this associative knowledge reflects distributional
characteristics of the fraction arithmetic problems children encounter. To test these
hypotheses, we examined textbooks and middle school children in the US
(Experiments 1 and 2) and China (Experiment 3). We asked the children to predict
which arithmetic operation would accompany a specified pair of operands, to
generate operands to accompany a specified arithmetic operation, and to match
operands and operations. In both countries, children’s responses indicated that they
associated operand pairs having equal denominators with addition and subtraction,
and operand pairs having a whole number and a fraction with multiplication and
division. The children’s associations paralleled the textbook input in both countries,
which was consistent with the hypothesis that children learned the associations
from the practice problems. Differences in the effects of such associative
knowledge on US and Chinese children’s fraction arithmetic performance are
discussed, as are implications of these differences for educational practice.
Fractions are crucial to numerical development. Individual differences in fractions
knowledge in earlier grades predict not only later success in algebra (Booth & Newton, 2012;
Booth, Newton, & Twiss-Garrity, 2014) but also overall math achievement in high school, even
after controlling for IQ, reading achievement, whole number arithmetic skill, and familial SES
(Siegler et al., 2012). Fractions are also important for occupational success: 68% of American
white-collar and blue-collar employees report using fractions, decimals, or percentages in their
work (Handel, 2016). Reflecting their importance, fractions are a major focus of US mathematics
instruction in third through sixth grade (CCSSI, 2010).
Despite this prolonged instruction, many children in the US fail to master fractions,
especially fraction arithmetic (Byrnes & Wasik, 1991; Fuchs et al., 2014; Hecht & Vagi, 2010;
Jordan et al., 2013; Lortie-Forgues, Tian, & Siegler, 2015; Newton, Willard, & Teufel, 2014;
Siegler, Thompson, & Schneider, 2011). For example, in one recent study that presented all four
fraction arithmetic operations with both equal denominator operands (e.g., 3/5+1/5, 3/5×1/5) and
unequal denominator operands (e.g., 3/5×1/4, 3/5÷1/4), percent correct was only 46% for sixth
graders and 57% for eighth graders (Siegler & Pyke, 2013).
ASSOCIATIONS IN FRACTION ARITHMETIC 2
Certain types of problem are especially challenging. On addition and subtraction problems in
Siegler and Pyke (2013), unequal denominator problems elicited many more errors than equal
denominator ones (45% vs. 20%), but on multiplication problems, unequal denominator
problems elicited far fewer errors than equal denominator ones (42% vs. 63%). Errors on
unequal denominator addition and subtraction problems often involved using a strategy that
would be appropriate for multiplication, that is, performing the arithmetic operation separately
on the numerators and denominators, as in 3/5+1/4 = (3+1)/(5+4) = 4/9. Conversely, errors on
equal denominator multiplication problems often involved using a strategy that would be
appropriate for addition or subtraction, that is, performing the operation on the numerators and
maintaining the common denominator, as in 4/5×3/5 = (4×3)/5 = 12/5. Children in other studies
of fraction arithmetic have displayed similar patterns of accuracy and specific errors (e.g.,
Siegler et al., 2011).
To explain the poor overall fraction arithmetic performance and the specific patterns of
accuracy and errors, Braithwaite, Pyke, and Siegler (2017) hypothesized that children’s strategy
choices rely on associative knowledge, rather than correct mathematical rules or conceptual
understanding. They further proposed that children acquire this associative knowledge from the
statistical distribution of practice problems they encounter. To test these hypotheses, they
analyzed all fraction arithmetic problems stated in numerical form (i.e., not as word problems)
from the fourth, fifth, and sixth grade volumes of three major US mathematics textbook series
Pearson Education’s enVisionMATH (Charles et al., 2012), Houghton Mifflin Harcourt’s GO
MATH! (Dixon, Adams, Larson, & Leiva, 2012a, 2012b), and McGraw Hill Education’s
Everyday Mathematics (University of Chicago School Mathematics Project, 2015a, 2015b,
2015c). Results of those analyses are presented in Table 1, which includes similar data to those
described in Braithwaite, Pyke, and Siegler (2017), except that the new Table includes problems
from third grade textbooks as well as fourth to sixth grade ones, averages the input over the three
textbook series, and only includes problems with two fraction operands (problems with a whole
number and a fraction were included in the earlier analysis).
Table 1. Percent problems classified by arithmetic operation and denominator equality. Data
averaged over 3rd-6th grade volumes in three US textbook series (only problems with no whole
number operands are included, N = 1359 problems).
Arithmetic Operation
Operand Denominators
Addition
Subtraction
Multiplication
Division
Equal denominators
18
20
1
2
Unequal denominators
18
14
17
9
The analyses revealed strikingly non-random relations between arithmetic operations and
features of operands in the textbooks. As shown in Table 1, 93% of problems with equal
denominators involved addition or subtraction (e.g., 3/5+1/5), whereas unequal denominator
problems involved multiplication or division almost as often as addition or subtraction. Viewed
from another perspective, addition and subtraction problems involved equal and unequal
ASSOCIATIONS IN FRACTION ARITHMETIC 3
denominator operands with about the same frequency, whereas 90% of multiplication and
division problems had unequal denominator operands.
To understand the effects that this unbalanced distribution might have on learning,
Braithwaite, Pyke, and Siegler (2017) constructed FARRA, a computational model of fraction
arithmetic learning. This model formalized the assumptions that a) children associate operand
features with arithmetic operations, b) these associations derive in large part from problems
encountered in textbooks, and c) the associations guide children’s choices of solution strategies.
In particular, distributions of problems in textbooks are hypothesized to lead children (and
FARRA) to associate equal denominators with addition and subtraction and to associate unequal
denominators at least as much with multiplication as with addition or subtraction.
FARRA displayed all eight phenomena identified in Braithwaite et al.’s review of the
literature on children’s fraction arithmetic, including the patterns of accuracy and specific errors
described above. The similarity of children’s performance to that of FARRA suggested that they,
like FARRA, might learn associations between operand features and arithmetic operations from
textbook input, and rely on these associations to select solution strategies. The textbook
problems might well not be the only source of the children’s associations; another source might
be the requirement of the standard addition and subtraction algorithms that equal denominators
always be present at some point in the solution process. However, the distribution of textbook
problems does seem likely to be one important source of children’s associations between
arithmetic operations and operands.
Although consistent with the hypothesis that children learn associations between problem
features and operations that are present in textbooks, the evidence is relatively indirect. The
present study is an attempt to provide more direct evidence regarding this hypothesis by
assessing the associations directly, rather than inferring them from children’s arithmetic
performance.
To assess whether children learn associations between operand features and operations in
textbook problems, we presented three tasks: one that specified the arithmetic operation and
required children to generate pairs of operands to accompany it; one that specified the operands,
and required children to predict the arithmetic operation that would accompany them; and one
that specified two pairs of operands and two operations and required children to match an
operation to a pair of operands.
From a mathematical perspective, arithmetic operations are independent of operands: any
arithmetic operation can be performed on any pair of operands (with the exception that if the
second operand is zero, the operation cannot be division). However, if children learn associations
between operations and operands, they should follow the textbook patterns in predicting which
arithmetic operation will accompany particular pairs of operands and which types of operands
will accompany each arithmetic operation. For example, if presented equal denominator
operands (e.g., 4/5 and 3/5), children would be expected to more often predict that the operation
is addition or subtraction than that it is multiplication or division.
This theoretical position also implies that children’s associations should parallel other
patterns that appear in the distribution of textbook problems. Another pattern that Braithwaite et
al. found in the textbooks was an association between the arithmetic operation and whether the
operands were two fractions or a whole number and a fraction. In the three textbook series that
they analyzed, 94% of problems that included both a whole number operand and a fraction or
mixed number operand involved multiplication or division (Table 2). In contrast, in the same
textbooks, 71% of problems in which both operands were fractions or mixed numbers involved
ASSOCIATIONS IN FRACTION ARITHMETIC 4
addition or subtraction (Table 2). Thus, we predicted that children would associate operand pairs
that included a whole number and a fraction more strongly with multiplication and division than
with addition and subtraction.
Table 2. Percent problems classified by arithmetic operation and whether operands were both
fractions or had one whole and one fraction (mixed numbers classified as fractions.) Data
averaged over 3rd-6th grade volumes in the three US textbook series cited above (N = 1972
problems).
Arithmetic Operation
Operand Number Type
Addition
Subtraction
Multiplication
Division
Fraction-fraction
25
24
13
7
Whole-fraction
0
2
17
12
It was far from a foregone conclusion that children would learn such spurious associations
between problem features and operations simply because the associations exist in textbooks.
Mathematics is a formal system in which frequency of various types of problems and problem
features is irrelevant to the rules that should be used. Consistent with the content being learned,
mathematics education emphasizes learning the explicit rules that specify the conditions under
which each solution strategy should be used and ignoring irrelevant features of problems. To the
extent that students learn what their teachers and textbooks are trying to teach them, there is no
reason for children to learn relations that are irrelevant to that mathematical content.
On the other hand, implicit learning of statistical patterns in the environment, sometimes
termed “statistical learning” (Saffran, Aslin, & Newport, 1996), is a fundamental learning
mechanism present throughout the lifespan (Perruchet & Pacton, 2006). For example, it plays an
important role in infants’ language development (Pelluchi, Hay, & Saffran, 2009) and in school-
age children’s learning of orthographic regularities (Pacton, Perruchet, Fayol, & Cleeremans,
2001; Treiman & Kessler, 2006). In the context of mathematics learning, rather than explicit
learning of rules entirely replacing statistical learning, both types of learning mechanism may
operate simultaneously.
If this proposal is correct, children may also detect statistical patterns other than operand-
operation associations. To test this possibility, we examined another type of statistical
informationthe frequencies with which specific fractions appear as operands. Some fractions
appear often, others rarely, in textbook problems. For example, 9 of the 10 fractions appearing as
operands most often in the textbooks analyzed in Tables 1 and 2 were the same in all three series:
1/2, 1/3, 2/3, 1/4, 3/4, 2/5, 1/6, 5/6, and 3/8. Other fractions with single digit numerators and
denominators, including 3/6, 1/7, 2/7, 3/7, 6/7, 1/9, 2/9, and 8/9 were not among the 20 most
common fractions in any of the series. The most commonly presented fractions tended to have
small numerators and denominators, but that is not the only consideration. Unsimplified fractions
(e.g., 3/6) and fractions with prime denominators (e.g., 3/7) appeared less often than their
ASSOCIATIONS IN FRACTION ARITHMETIC 5
numerator and denominator sizes would suggest (e.g., both were rarer than 3/8)
1
. To test whether
children learn such frequency information, we calculated correlations between the frequencies
with which fractions appeared as operands in textbooks and the frequencies of children
generating those fractions as operands.
Experiment 1
Participants in Experiment 1 performed three tasks: the “choose-operation,” “generate-
operands,” and match-operands-with-operationstasks (Table 3).
Table 3. Tasks used in Experiments 1 and 3. (A modified version of the choose-operation task
was used in Experiment 2.)
Instructions
Example Trial
“Look at the arithmetic operation, and
try to write numbers that you think
probably would appear in problems
with that operation.”
“For each problem, please guess what
the arithmetic operation probably was.
It could be ×, -, ÷, or +. Draw a circle
around the one you guess.”
“Try to guess which problem was the
[e.g., addition] problem. In other
words, in which problem would you
guess the missing sign is a [+] sign?”
In the generate-operands task, children were shown eight arithmetic problems in which only
the operation and two empty boxes were visible. They were asked to write numbers in the boxes
that would be likely to appear with that operation: two fractions on half of trials, and a whole
number and a fraction on the other half. Based on the textbook input, we expected that children
would write two fractions with equal denominators more often when the operation was addition
or subtraction than when it was multiplication or division and that children would write a whole
number and a fraction more often when the operation was multiplication or division than when it
was addition or subtraction.
1
These patterns may reflect influences of the Common Core State Standards in mathematics. In the Standards,
expectations for fractions in grades 3 and 4 are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100
(CCSSI, 2010). All of these denominators except for 100 appeared more often than any other denominator in all
three textbook series. The patterns may also reflect textbook designers favoring fractions that are easy to use in
calculations. Arithmetic problems involving fractions with large denominators may be difficult because they are
likely to require relatively difficult whole number arithmetic calculations (e.g., calculating 1/18+1/15 requires
identifying 90 as the least common denominator of the operands and retrieving the facts 18×5 = 90 and 15×6 = 90,
or alternatively, calculating 18×15 = 270). Problems with unsimplified fractions (e.g., 3/6) as operands may be more
difficult to solve because children may try to simplify the fractions (e.g., 3/6 = 1/2) before calculating the answer.
Problems with prime numbers as denominators (e.g., 3/7) may be difficult to calculate with because adding or
subtracting such fractions to other fractions with unequal denominators requires conversion to a large common
denominator (e.g., compare 3/7+1/4 = 19/28 to 3/8+1/4 = 5/8).
(a)
(b)
ASSOCIATIONS IN FRACTION ARITHMETIC 6
The generate-operands task also afforded a test of our hypothesis that children are sensitive
to the frequencies with which fractions appear as operands. We predicted a positive correlation
between the frequency with which children generated specific fractions and the frequency with
which fractions appeared as operands in the three US mathematics textbooks.
In the choose-operation task, children were shown 12 pairs of operands with an empty box
between them where an arithmetic operation would be. Children were asked to guess which
arithmetic operation would most likely appear in the boxes and to choose each operation equally
often over the 12 problems. Based on the textbook input, we predicted that when the operands
were fractions with equal denominators, children would guess addition or subtraction more often
than multiplication or division, and when the operands were a whole number and a fraction,
children would guess multiplication or division more often than addition or subtraction. We did
not predict any preference among arithmetic operations for trials in which the operands were
fractions with unequal denominators, because textbook problems with such operands involved
addition and subtraction about as often as multiplication and division (Table 1).
In the match-operands-with-operations task, eight trials were presented, each with two pairs
of operands and a box between them. Children were told that one problem involved a certain
operation (e.g., addition) and the other a different operation (e.g., multiplication), and were asked
to connect the first operation to one pair of operands. We predicted that when choosing between
an equal denominator and an unequal denominator operand pair, children would connect the
equal denominator pair more often to addition or subtraction than to multiplication or division.
We also predicted that when choosing between two fraction operands and a whole number and a
fraction, children would connect the pair with the whole number operand more often to
multiplication or division than to addition or subtraction.
These predictions are summarized in the leftmost two columns of Table 4.
Method
Participants. Participants were 137 children, 66 sixth graders (mean age = 11.4 years) and
71 eighth graders (mean age = 13.4 years), attending a middle school in Pittsburgh, PA. The
percent of children eligible for free or reduced price lunch at this school was 34% (the state
median was 54%; Pennsylvania Department of Education, 2016). The experimental sessions
were administered by the first author and two female research assistants. The Carnegie Mellon
University Institutional Review Board approved this experiment and Experiment 2.
Materials. Two sets of operand pairs were created for the choose-operation and match-
operands-with-operations tasks. Each child was randomly assigned to receive one of the two sets.
(No operand pairs were created for the generate-operands task, because only arithmetic
operations were presented in that task.)
Each set of problems consisted of 12 pairs of operands, four groups of three pairs each (Table
S1 in the online supplemental materials). The three pairs of operands in each group had the
second operand in common; the first operand was varied to produce one equal denominator
fraction-fraction pair, one unequal denominator fraction-fraction pair, and one whole-fraction
pair. The first operand was always larger than the second, so that combining the two operands
with any of the four arithmetic operations would yield a positive answer. In the whole-fraction
pairs, the whole number was always the first operand. All fractions were between 0 and 1.
ASSOCIATIONS IN FRACTION ARITHMETIC 7
Table 4. Summary of predictions and outcomes in Experiments 1, 2, and 3: “” indicates that a prediction was consistent with the
results (p < .05), “(6)” or “(8)” that it was consistent with the results for sixth or eighth graders only (p < .05), “†”that a marginal
effect in the predicted direction was found (p < .1), and “×that a prediction was not consistent with the results.
Task
Prediction
US Students
(Experiment 1)
US Students
(Experiment 2)
Chinese Students
(Experiment 3)
Generate-
Operands
1A. Children will generate two fractions with equal denominators more
often for addition and subtraction than for multiplication and division.
1B. Children will generate one whole number and one fraction more often
for multiplication and division than for addition and subtraction.
×
1C. The frequency with which children generate a fraction will be
positively correlated with the fraction’s frequency in textbook problems.


Choose-
Operation
2A. Children will choose addition or subtraction more than multiplication
or division on equal denominator fraction-fraction trials.

2B. Children will choose multiplication or division more than addition or
subtraction on whole-fraction trials.

Match-
operands-
with-
operations
3A. Children will choose equal denominator fraction-fraction operand pairs
more often for addition and subtraction than for multiplication and division.
3B. Children will choose whole-fraction operand pairs more often for
multiplication and division than for addition and subtraction.
(6)
×
ASSOCIATIONS IN FRACTION ARITHMETIC 8
Procedure. The tasks were presented in a fixed order: generate-operands, choose-operation,
match-operands-with-operations.
Generate-operands. On each trial, children were shown an arithmetic operation with an
empty box on either side and asked to generate operands with numbers that would be likely to
appear with that operation. The first page displayed four problems in which the order of
operations was +, -, ×, ÷, and the second page displayed four problems in which the order of
operations was ÷, ×, -, +. Children were instructed to insert a pair of fractions for two of the four
problems on each page, and a whole number and a fraction for the other two problems on each
page. Mixed numbers (e.g., 3 1/4), which children generated on 4.9% of trials, were counted as
fractions; classifying these numbers as whole numbers did not change the results of the analyses.
Choose-operation. On each of 12 trials, children were shown an arithmetic problem with the
two operands visible and asked to predict which arithmetic operation would appear with each
operand pair.
Compared to multiplication and division, addition and subtraction are conceptually more
basic and are introduced earlier in the mathematics curriculum. Thus, it seemed possible that
without any constraint on their choices, some children might choose addition or subtraction on
most or all of the trials, without regard to the features of the operands. To prevent this outcome,
children were asked to choose each operation equally often, that is, on 3 of the 12 trials. To make
it easier for children to check how often they had chosen each operation, all 12 trials were shown
on the same page. Children were randomly assigned to receive the trials in either a fixed random
order or the reverse of that order.
Match-operands-to-operations. On each of eight trials, children were presented two
operations and two pairs of operands; the task was to choose which pair of operands would be
more likely to involve the first operation. On the first two trials, children were told: “In each row
below, there are two problems with the operation missing. One of them was an addition problem,
and the other was a multiplication problem. For each row, try to guess which problem was the
addition problem. In other words, in which problem would you guess the missing sign was a +
sign? Please circle (a) or (b) in each row.” The word “addition” and the “+” sign were replaced
by “subtraction” and “-” on the third and fourth trials, by “multiplication” and “×” on the fifth
and sixth trials, and by “division” and “÷ on the seventh and eighth trials. In the instructions for
the multiplication and division trials, the instruction “the other was a multiplication problem”
was replaced with “the other was an addition problem.”
On one trial for each operation, the two response options were the unequal denominator
operand pair and the equal denominator operand pair from one of the four groups of number
pairs in the stimulus set (e.g., 3/4 1/6 and 4/6 1/6). On the other trial, the response options
were the whole-fraction operand pair and the unequal denominator operand pair from the same
stimulus group (e.g., 3 1/6 and 3/4 1/6). Each group of operand pairs was used for one
arithmetic operation. The order in which these two types of trials were presented was determined
randomly for each child; for a given child, the order was the same for each arithmetic operation.
Administration of tasks. The experiment was administered in a paper-and-pencil, whole
class format. Instructions for each task were read aloud to the class to avoid the possibility of
reading difficulties interfering with performance. To allow this procedure, children waited until
everyone finished a given task before starting the next one. Children were told to complete each
task within 2.5 minutes, but were permitted to finish even if they exceeded this time limit.
ASSOCIATIONS IN FRACTION ARITHMETIC 9
Results
We present results from the three tasks in the order in which the tasks were presented. In all
cases, we examined effects involving grade (sixth or eighth) on the dependent variable in all t-
tests and ANOVAs. Most main effects and interactions involving grade were not significant;
exceptions are reported below.
Children produced invalid responses on a small percentage of trials (1-2%) on two of the
three tasks in Experiment 1 and on fewer than 1% of trials on each of the tasks in Experiments 2
and 3. Data from these invalid trials were excluded from analysis; details regarding the
exclusions are provided in the online supplemental materials. On the generate-operands task in
Experiment 1, invalid responses were produced on a greater percentage of trials (10%). Reasons
for these more frequent invalid responses are described in the next section along with other data
from the task.
Generate-operands Task. Children completed an average of 7.16 valid trials (of a possible
8): 3.54 addition/subtraction trials and 3.62 multiplication/division trials. The other 10% of trials
were excluded from analysis; the most common reason for exclusion was that children generated
whole numbers for both operands (5% of trials). Eight children (five sixth graders and three
eighth graders) were excluded from the analyses for predictions 1A and 1B because they did not
generate a valid response for any addition/subtraction trial, for any multiplication/division trial,
or both.
As predicted (Table 4, prediction 1A), percent fraction-fraction operand pairs in which the
fractions generated by children had equal denominators was higher on addition/subtraction trials
(46.1%) than on multiplication/division trials (29.0%), F(1, 97) = 17.75, p < .001,
 .
(Here and in Experiment 3, this percentage could only be calculated for trials on which children
generated fraction-fraction pairs; 16 sixth graders and 14 eighth graders were excluded because
they did not generate a fraction-fraction pair on at least one addition/subtraction and one
multiplication/division trial. The effect remained when the analysis was performed without
excluding these children, using percentage of all responses instead of percentage of fraction-
fraction responses as the dependent variable, F(1, 127) = 22.17, p < .001,
 ) Percent
fraction-fraction responses with equal denominators was higher among eighth graders (46.1%)
than sixth graders (25.5%), F(1, 97) = 10.26, p = .002,
 .
Consistent with another prediction (Table 4, prediction 1B), children tended to generate
whole-fraction operand pairs on more than half of multiplication/division trials (54.5%, SE =
2.5%), one-sample t(128) = 1.82, p = .070, d = 0.16, and therefore tended to generate whole-
fraction operand pairs on fewer than half of addition/subtraction trials (45.1%, SE = 2.5%). (The
first percentage was compared to chance, rather than to the second percentage, because the two
percentages were not independent of each other.)
Finally, as predicted (Table 4, prediction 1C), the frequency with which children generated a
fraction was positively correlated with that fraction’s frequency as an operand in the textbook
problems analyzed in Tables 1 and 2, r(796) = .778, p < .001. The correlation did not merely
reflect fractions with smaller numerators and denominators being more common in both
children’s responses and textbooks: The partial correlation remained significant, r(796) = .773, p
< .001 after controlling for the inverse of the sum of numerator and denominator. (The inverse of
the sum, rather than the sum, was used as a control because it explained a larger portion of the
variance in fractions’ frequencies in textbooks than the sum did (37.6% vs. 0.2%) and thus
constituted a more stringent control for problem size.)
ASSOCIATIONS IN FRACTION ARITHMETIC 10
Choose-operation Task. Children completed an average of 11.85 valid trials (of a possible
12): 3.94 equal denominator trials, 3.96 unequal denominator trials, and 3.96 whole-fraction
trials. As predicted (Table 4, prediction 2A), when presented with equal denominator operands,
children chose addition or subtraction on more than half (68.2%, SE = 2.2%) of trials, one-
sample t(136) = 8.24, p < .001, d = 0.70. Also as predicted (Table 4, prediction 2B), when the
operands were a whole number and a fraction, children chose multiplication or division on more
than half of trials (60.6%, SE = 2.3%, one-sample t(135) = 4.65, p < .001, d = 0.40. On unequal
denominator trials, children also chose multiplication or division on more than half (58.8%, SE =
2.1%) of trials, one-sample t(136) = 4.18, p < .001, d = 0.36.
Match-operands-with-operations Task. On this task, children generated an average of 7.82
valid trials (of a possible 8), including 3.94 addition/subtraction trials and 3.88
multiplication/division trials. As predicted (Table 4, prediction 3A), children chose equal
denominator operand pairs in preference to unequal denominator operand pairs more often on
addition/subtraction trials (73.3%, SE = 2.7%) than on multiplication/division trials (43.7%, SE =
3.3%), F(1, 133) = 48.15, p < .001,
 . Also as predicted (Table 4, prediction 3B),
children showed a marginal tendency to choose whole-fraction operand pairs in preference to
fraction-fraction pairs more often on multiplication/division trials (58.2%, SE = 3.2%) than on
addition/subtraction trials (50.0%, SE = 3.0%), F(1, 132) = 3.05, p = .083,
 .
Although no main effect of grade was present in either analysis, grade interacted with
arithmetic operation in the latter analysis, F(1, 132) = 7.10, p = .009,
 . Sixth graders
chose whole-fraction operand pairs over fraction-fraction operand pairs more often on
multiplication/division trials (62.1%, SE = 4.6%) than on addition/subtraction trials (40.9%, SE =
4.3%), F(1, 65) = 8.76, p = .004,
 . In contrast, eighth graders showed no effect of
arithmetic operation (multiplication/division: 54.4%, SE = 4.5%; addition/subtraction: 58.8%, SE
= 3.9%), F(1, 67) = 0.47, ns. Thus, prediction 3B was consistent with the results for sixth but not
eighth graders.
Discussion
The findings of Experiment 1 indicated that US children form spurious associations between
arithmetic operations and operand features and that these associations parallel relations in
textbook problems. As predicted, children associated equal denominator operands more strongly
with addition and subtraction than with multiplication and division. Also as predicted, children
associated whole-fraction operand pairs more strongly with multiplication and division than with
addition and subtraction, though evidence for this latter association was less consistent. Further
as predicted, the frequency with which children generated particular operands was closely related
to the frequency with which the operands appeared in textbook problems. These findings indicate
that children detect a variety of spurious associations and other distributional features of
textbook input, as hypothesized by Braithwaite et al. (2017).
Experiment 2
The choose-operation task is particularly relevant to the assumptions of Braithwaite, Pyke,
and Siegler’s (2017) FARRA model of children’s fraction arithmetic. FARRA uses numeric
features of operands as cues for selecting solution strategies. Similarly, in the version of the
ASSOCIATIONS IN FRACTION ARITHMETIC 11
choose-operation task in Experiment 1, children appear to have used numeric features of
operands as cues to predict the most likely arithmetic operation.
However, results of that choose-operation task may have partially reflected the task
constraint that children choose each arithmetic operation equally often. For example, children
who chose addition or subtraction on more than half of equal denominator trials would
necessarily have chosen multiplication or division on more than half of the remaining trials. This
fact could explain why children chose multiplication or division more often than chance for
whole-fraction operand pairs and unequal-denominator operand pairs.
To test this alternative explanation, we administered a modified version of the choose-
operation task in Experiment 2. On this version, children were not required to choose each
arithmetic operation equally often, or at all. We predicted that children still would choose
addition or subtraction more often than chance for equal denominator operand pairs, and would
choose multiplication or division more often than chance for whole-fraction operand pairs (Table
4, predictions 2A and 2B).
Method
Participants. Participants were 168 sixth graders (mean age = 11.2 years), attending the
same middle school in Pittsburgh, PA where Experiment 1 was conducted. None of the children
had participated in Experiment 1, and the teachers from whose classes the children were drawn
were different from the teachers from whose classes the children in Experiment 1 were drawn.
Half of the experimental sessions were administered by each of the two female research
assistants who administered Experiment 1.
Materials. The same two sets of operand pairs used in the choose-operation task in
Experiment 1 were used as stimuli (Table S1 in the online supplemental materials). Each child
was randomly assigned to receive one of the two sets.
Procedure. As in Experiment 1, on each of 12 trials, children were shown an operand pair
with an empty box between the operands and were asked to choose which arithmetic operation
they thought would appear with the operands. Children were randomly chosen to receive the
trials in either a fixed random order or the reverse of that order. The only differences between
this version of the task and that used in Experiment 1 were that in this version, children were
given no instruction regarding whether, or how often, to choose each operation, and rather than
circling their chosen arithmetic operation, children wrote the operation directly in the empty box
between the two operands.
Results
As predicted (Table 4, prediction 2A), on problems with equal denominator operands,
children chose addition or subtraction more often than the chance level of 50% (84.3%, SE =
1.9% of choices), one-sample t(167) = 18.55, p < .001, d = 1.43. Also as predicted (Table 4,
prediction 2B), when the operands were a whole number and a fraction, children chose
multiplication or division more often than chance (83.5%, SE = 2.2% of choices), one-sample
t(167) = 15.57, p < .001, d = 1.20. On unequal denominator trials, children chose multiplication
or division more often than chance (58.6%, SE = 2.3% of choices), one sample t(167) = 3.72, p
< .001, d = 0.29.
ASSOCIATIONS IN FRACTION ARITHMETIC 12
Discussion
Findings from the choose-operation task in Experiment 1 replicated even when children were
not required to choose operations equally often. In fact, when presented equal denominator
operand pairs, children predicted addition or subtraction more often in Experiment 2 than in
Experiment 1 (84.3% vs. 68.2%), and when presented whole-fraction operand pairs, children
predicted multiplication or division more often in Experiment 2 than in Experiment 1 (83.5% vs.
60.6%). In Experiment 1, the need for children to attend to the number of times they had chosen
each operation may have distracted them from attending to numeric features of the operands, and
thereby weakened the effects of those features on children’s predictions.
Experiment 3
The results of Experiments 1 and 2 raised the question of how anyone learns fraction
arithmetic. Chinese middle school students display much higher accuracy on fraction arithmetic
problems than US middle school students (92% vs. 46% correct in Bailey, Zhou, Zhang, et al.,
2015; see also Torbeyns, Schneider, Xin, & Siegler, 2015). Does this high accuracy reflect
Chinese children not forming spurious associations, either because the associations are not
present in Chinese textbooks or because they ignore spurious associations in practice problems?
Or do Chinese children form spurious associations but override them in their arithmetic strategy
choices? This could occur through strong conceptual knowledge allowing them to understand
and choose appropriate procedures for each arithmetic operation, through substantial practice
allowing them to recall correct procedures for each operation, or through both processes.
One reason to suspect that strong conceptual knowledge of mathematics prevents learning of
spurious associations is that children who understand the conceptual bases of correct solution
procedures do not need to rely on mathematically irrelevant associations to select solution
procedures. Consistent with this possibility, when presented a mathematics word problem after
studying a structurally similar analogue and a superficially similar distractor, mathematical
experts were less likely than novices to show negative transfer from the superficial distractor to
the test problem (Novick, 1988). Similarly, people with greater understanding of non-
mathematical domains, ranging from radiology to baseball, often do not recall irrelevant
information that novices and people with moderate amounts of knowledge do recall (e.g., Arkes
& Freedman, 1984; Hagen, 1972; Myles-Worsley, Johnston, & Simons, 1988; V. Patel & Groen,
1991; Voss, Vesonder, & Spilich, 1980).
On the other hand, children with stronger mathematical knowledge are often more, not less,
likely to encode numerical features of problems. When comparing fractions, mathematically
proficient students often solve difficult problems by relying on subtle relations among the
numbers involved to choose strategies; less proficient students rarely rely on these subtle but
useful features (Fazio, DeWolf, & Siegler, 2016). Similar findings have been obtained with
whole number mental arithmetic (Braithwaite, Goldstone, van der Maas, & Landy, 2016). These
findings suggest that children with strong mathematical knowledge could learn spurious
associations involving numerical features of problems, but override the associations to choose
correct fraction arithmetic procedures.
To distinguish among these several possibilities, we examined in Experiment 3 whether
spurious associations between fraction arithmetic operations and operand features are present in
Chinese textbooks, and if so, whether Chinese children learn the spurious associations. If
ASSOCIATIONS IN FRACTION ARITHMETIC 13
Chinese textbooks include spurious associations, and children display knowledge of them, those
findings, together with previous findings of Chinese children’s superior knowledge of fraction
arithmetic, would suggest that forming spurious associations between operations and operand
features does not necessarily prevent mastery of fraction arithmetic. Such a finding would also
suggest that high mathematical proficiency does not prevent acquisition of irrelevant associative
knowledge. On the other hand, finding that Chinese textbooks do not include the spurious
operation-operand feature associations would suggest that absence of such associations could be
one factor contributing to Chinese students’ success in learning fraction arithmetic. A third
possibilitythat Chinese textbooks do include the spurious associations but children do not learn
themwould suggest that superior conceptual understanding or more extensive practice prevents
them from learning the associations.
To determine whether Chinese textbooks, like US textbooks, exhibit spurious associations
between fraction arithmetic operations and operand features, we analyzed the 3rd-6th grade
volumes of one textbook series from each of the three major Chinese publishers of primary
school mathematics textbooks: Beijing Normal University Press (Beijing Normal University
Press, 2014; New Century Primary Mathematics Curriculum Writing Group, 2015), People’s
Education Press (“Helping You Learn Mathematics Classroom Workbook” Writing Group,
2014; People’s Education Press Curriculum Research Center Primary Mathematics Curriculum
Research and Development Center, 2014), and Phoenix Education Publishing (Nanjing Oriental
Mathematics Education Scientific Research Center & Jiangsu District Primary and Middle
School Education Research Center, 2014; Suzhou Education Press Primary School Mathematics
Curriculum Writing Group, 2014).
The criteria for inclusion of problems were identical to those employed with the US
textbooks. Problems had to have one arithmetic operation and two operands, one of which was a
fraction or mixed number. The other operand could be a fraction, mixed number, or whole
number. We excluded problems not in numerical form, such as story problems, as well as
problems that did not require calculation of an exact answer. Problems were selected according
to these criteria by the first author, who is fluent in Chinese. The number of problems meeting all
of these criteria for the textbook and workbook was 442 in the Beijing Normal University series,
507 in the People’s Education Press series, and 391 in the Phoenix Education Publishing series.
It should be noted that many Chinese children solve the problems in more than one workbook, so
that the smaller number of problems obtained from the Chinese textbooks does not mean that
Chinese children encounter fewer fraction arithmetic problems than US children do.
In all three Chinese textbooks, more than 85% of equal denominator problems involved
addition or subtraction (Table 5), and more than 85% of problems with one whole number and
one fraction operand involved multiplication or division (Table 6). These patterns were similar to
those observed in US textbooks (Tables 1 and 2).
ASSOCIATIONS IN FRACTION ARITHMETIC 14
Table 5. Percent problems classified by arithmetic operation and denominator equality in 3rd-6th
grade volumes of three Chinese textbook series (only problems with no whole number operands
are included, N = 835 problems).
Arithmetic Operation
Operand Denominators
Addition
Subtraction
Multiplication
Division
Beijing Normal University (N = 249)
Equal denominators
14
7
0
3
Unequal denominators
17
17
28
14
People’s Education Press (N = 333)
Equal denominators
17
13
2
3
Unequal denominators
16
16
21
12
Phoenix Education Publishing (N = 253)
Equal denominators
12
11
1
2
Unequal denominators
16
15
21
21
Table 6. Percent problems classified by arithmetic operation and whether operands were both
fractions or had one whole and one fraction (mixed numbers classified as fractions) in 3rd-6th
grade volumes of three Chinese textbook series (N = 1340 problems).
Arithmetic Operation
Operand Number Type
Addition
Subtraction
Multiplication
Division
Beijing Normal University (N = 442)
Fraction-fraction
17
14
16
9
Whole-fraction
0
4
24
16
People’s Education Press (N = 507)
Fraction-fraction
22
19
15
9
Whole-fraction
0
4
16
14
Phoenix Education Publishing (N = 391)
Fraction-fraction
18
17
15
15
Whole-fraction
0
3
14
18
Chinese children performed the same three tasks as US children did in Experiment 1.
Because the Chinese textbooks displayed associations very similar to those in US textbooks, we
tested the same predictions for the Chinese students as for the US students in Experiment 1.
ASSOCIATIONS IN FRACTION ARITHMETIC 15
Method
Participants. Participants were 126 children, 65 sixth graders (mean age = 11.2 years) and
61 eighth graders (mean age = 13.4 years), attending a middle school in Beijing. The first author
and two research assistants, one male and one female, administered the experiment. The Beijing
Normal University Institutional Review Board approved the experiment.
Materials. The materials used in Experiment 3 were a Chinese language version of the
English language materials used in Experiment 1. The materials were translated into Chinese by
the first author; all translations were checked by a native Chinese speaker.
Procedure. The procedure in Experiment 3 was the same as in Experiment 1, with the
exception that children read the instructions on their own and were allowed to work at their own
pace. This was done because the children’s teachers indicated that all of the children had
sufficiently high reading levels to understand the instructions; the teachers’ impression appeared
accurate, based on children’s low percentage of missing or invalid responses. In the generate-
operands task, mixed numbers were again counted as fractions; children generated such numbers
on 2.5% of trials, and classifying them as whole numbers did not change the results of the
analyses.
Results
As in Experiment 1, we tested for effects and interactions involving grade (sixth or eighth) in
all t-tests and ANOVAs. None of the main effects or interactions involving grade were
significant; they therefore are not described further.
Generate-operands Task. Children completed an average of 7.82 valid trials (of a possible
8): 3.90 addition/subtraction trials and 3.93 multiplication/division trials. As predicted (Table 4,
prediction 1A), percent fraction-fraction operand pairs with equal denominators was higher on
addition/subtraction trials (42.1%) than on multiplication/division trials (20.6%), F(1, 99) =
20.92, p < .001,
 . The effect remained when the analysis was performed using
percentage of all responses instead of percentage of fraction-fraction responses as the dependent
variable, F(1, 123) = 17.00, p < .001,
 
Children generated whole-fraction operand pairs on 46.7% (SE = 2.2%) of
addition/subtraction trials and on 52.9% (SE = 2.3%) of multiplication/division trials. Contrary to
our prediction (Table 4, prediction 1B), the percentage of multiplication/division trials on which
children generated whole-fraction operand pairs did not differ from chance (i.e., 50%), one
sample t(124) = 1.27, p = .205.
Finally, as predicted (Table 4, prediction 1C), the frequency with which children generated a
fraction as an operand was positively correlated with that fraction’s frequency in problems drawn
from the three Chinese textbooks, Pearson’s r(432) = .927, p < .001. As in Experiment 1, the
correlation remained significant after controlling for the inverse of the sum of the numerator and
denominator of each fraction, r(432) = .915, p < .001.
Choose-operation Task. Children completed an average of 11.97 valid trials (out of a
possible 12): 3.99 equal denominator trials, 4.00 unequal denominator trials, and 3.98 whole-
fraction trials. As predicted (Table 4, prediction 2A), the Chinese children chose addition or
subtraction as the operation on more than half of trials with equal denominator operands (72.9%
of trials, SE = 2.2), one-sample t(125) = 10.36, p < .001, d = 0.92. Also as predicted (Table 4,
prediction 2B), the Chinese children chose multiplication or division on more than half (57.4%,
ASSOCIATIONS IN FRACTION ARITHMETIC 16
SE = 2.1%) of trials in which one operand was a whole number and the other was a fraction, one-
sample t(125) = 3.58, p < .001, d = 0.32. As in Experiments 1 and 2, these children also chose
multiplication or division on more than half (65.5%, SE =2.1%) of unequal denominator trials,
one sample t(125) = 7.22, p < .001, d = 0.64.
Match-operands-with-operations Task. Children completed an average of 7.94 valid trials
per child (of a possible 8): 3.97 addition/subtraction trials and 3.97 multiplication/division trials.
As predicted (Table 4, prediction 3A), children chose equal denominator operand pairs in
preference to unequal denominator operand pairs more often on addition/subtraction trials
(79.8%, SE = 3.0%) than on multiplication/division trials (46.0%, SE = 3.4%), F(1, 124) = 52.26,
p < .001,
 .
Contrary to our prediction (Table 4, prediction 3B), but consistent with the performance of
US eighth graders in Experiment 1, Chinese children tended to choose whole-fraction rather than
fraction-fraction operand pairs more often on addition/subtraction trials (55.2%, SE = 3.6%) than
on multiplication/division trials (48.0%, SE = 3.9%), F(1, 122) = 3.07, p = .082,
 .
Discussion
Both Chinese textbooks and Chinese children showed spurious associations between fraction
arithmetic operations and operand features like those observed in their US counterparts in
Experiments 1 and 2. Thus, children in both countries extract associations between arithmetic
operations and operand features, even when these associations do not reflect a mathematical rule
or principle. Given that Chinese middle school students are highly accurate on fraction arithmetic
problems (Bailey et al., 2015; Torbeyns, et al., 2015), the findings indicate that learning spurious
associations does not preclude mastery of fraction arithmetic procedures, and that mathematical
expertise does not preclude learning spurious associations.
General Discussion
Below, we discuss implications of the present findings for understanding whether children
learn spurious associations between operations and operands in the fraction arithmetic problems
they encounter, the effects of learning such associations on their fraction arithmetic performance,
and how these findings might be used to improve mathematics instruction.
Learning of Spurious Associations in Fraction Arithmetic Input
The present study tested a central hypothesis of Braithwaite, Pyke, and Siegler’s (2017)
theory of fraction arithmetic learningthat children learn associations between arithmetic
operations and features of operands in the practice problems they receive, even when those
associations have no mathematical basis. The present findings were largely consistent with that
hypothesis and therefore with the theoretical assumption on which it was based.
Braithwaite, Pyke, and Siegler (2017) documented associations between arithmetic
operations and operand features in US textbooks and provided indirect evidence that US children
detect these associations. The present study yielded much more direct evidence that US children
form associations paralleling those in the textbook problems, and extended the analysis to
Chinese textbooks and children, demonstrating that learning of mathematically irrelevant
associations between fraction arithmetic operations and operand features is not idiosyncratic to
ASSOCIATIONS IN FRACTION ARITHMETIC 17
US children. Of particular interest, the greater mathematical expertise of Chinese children did
not prevent them from learning such relations.
The choose-operation task was most directly relevant to our model of fraction arithmetic
learning, because predicting the arithmetic operation based on numeric features of the operands
strongly resembles the model’s use of numeric features of operands as cues to which solution
strategy to use. Findings from this task were invariably consistent with our predictions, for both
US and Chinese children. When presented with two equal denominator fractions, children
expected the arithmetic operation to be addition or subtraction; when presented with a whole
number and a fraction, children expected the operation to be multiplication or division.
In principle, findings in Experiments 1 and 3 that children expected that whole-fraction
problems would involve multiplication/division might have reflected task constraints. That is, if
children disproportionately chose addition or subtraction when presented with equal denominator
operands, the requirement to choose each operation equally often could have led them to choose
multiplication and division more often than chance when presented other types of operands.
However, children displayed an even stronger preference for choosing multiplication/division on
whole-fraction pairs in Experiment 2, where there was no requirement or even encouragement to
choose operations equally often. Thus, the preference for choosing multiplication/division on
whole-fraction problems did not depend on requirements to choose operations equally often.
In all three experiments, children chose multiplication or division more often than chance
when presented with two unequal denominator fractions. This result was not predicted, because
problems with unequal denominator operands involved multiplication or division about half the
time in both US textbooks (44.4%) and Chinese textbooks (53.7%). However, among problems
with two fraction operands, multiplication and division problems involved unequal denominators
far more often than did addition and subtraction problems, in both the US textbooks (89.9% vs.
46.2%, Table 1) and the Chinese textbooks (90.6% vs. 56.6%, Table 5). This large difference
may have led children not only to associate multiplication and division with unequal
denominators, but also to associate unequal denominators with multiplication and division.
An alternate explanation of our results is that, rather than children’s responses reflecting
associations learned from textbook problem distributions, both children’s responses and the
textbook distributions might reflect differences among fraction arithmetic procedures. Adding
and subtracting fractions with unequal denominators require converting the fractions to a
common denominator and then following the procedure that would be used if the original
problem had equal denominators. The fact that equal-denominator addition and subtraction is a
component of the procedure for unequal-denominator addition and subtraction could explain
why textbooks present large numbers of equal-denominator problems for addition and
subtraction; the fact that equal-denominator addition and subtraction is easier could explain why
children often paired equal denominators with addition and subtraction in the present
experiments.
However, at least three considerations argue against this alternate explanation being the sole
source of the present findings (though it could have been one source). First, given that children
would not be performing the arithmetic problem regardless of the operation or operands they
chose, there was no reason for them to choose the easiest operand-operation pairing. Second, if
ease of executing procedures were critical, children should have associated whole-fraction
operand pairs with addition, since adding such operands is trivial (e.g., 3+1/6 = 3 1/6). The actual
pattern was the opposite; addition was rarely predicted on such problems. Third, the difference in
difficulty between equal and unequal denominator addition and subtraction problems is much
ASSOCIATIONS IN FRACTION ARITHMETIC 18
smaller among Chinese children (e.g., 93% vs. 88% correct in Bailey et al., 2015) than among
US children (e.g., 80% vs. 55% correct in Siegler & Pyke, 2013), but Chinese children appeared
to associate equal denominators with addition and subtraction at least as strongly as US children.
In the generate-operands task, the operands children generated paralleled textbook problems
not only with respect to the association of equal denominators with addition and subtraction, but
also with respect to the frequencies of specific fractions. The correlation between frequency of
particular fractions in textbooks and children was very strong in both the US (r = .778) and
China (r = .927). Thus, besides learning spurious associations between operations and operand
features, children also learn the frequency with which specific fractions appear as operands. This
finding supports the general perspective that children learn statistical patterns in math practice
problems, even when these patterns do not reflect any mathematical principle.
Effects of Learning Spurious Associations on Fraction Arithmetic Performance
Braithwaite et al. (2017) argued that reliance on irrelevant associative knowledge partially
explains US children’s poor mastery of fraction arithmetic procedures. US children display
especially low accuracy on types of problems that are rarely presented in textbooks. Moreover,
they often err on such problems by using strategies that would be appropriate for more frequently
encountered types of problems (Siegler & Pyke, 2013; Siegler et al., 2011). For example, they
often err on equal denominator multiplication problems by using a strategy that would be
appropriate for the much more common equal denominator addition or subtraction problems,
leading to errors such as 3/5×4/5 = 12/5. Such errors appear to reflect children associating
operand features with arithmetic operations (in this case, associating equal denominator operands
with addition and subtraction).
This perspective on potential negative effects of associative knowledge dovetails with
previous research on spurious correlation effects in mathematics learning (Ben-Zeev & Star,
2001; Chang, Koedinger, & Lovett, 2003). For example, Ben-Zeev and Star (2001) trained
university students to use two algorithms for comparing algebraic fractions. On a subsequent
test, students used each algorithm more often for problems similar to the example problems
shown for that algorithm during training, although the two algorithms were equally valid for all
test problems. Ben-Zeev and Star (2001) dubbed such influences of formally irrelevant problem
features on students’ strategy choices a “spurious correlation effect.”
The present findings suggest that spurious correlation effects also occur in the context of
fraction arithmetic. Together with prior data on the high frequency of fraction arithmetic errors
that seem to reflect the effects of spurious correlations, the new findings suggest that reliance on
such spurious correlations can hinder mastery of fraction arithmetic.
However, the findings of Experiment 3 in the present study also suggest that learning
spurious associations between operations and operand features does not inevitably lead to high
frequency of incorrect arithmetic strategies. The fraction arithmetic problems in the Chinese
textbooks were distributed roughly like those in US textbooks, and children in the two countries
formed similar spurious associations, yet Chinese students typically solve fraction arithmetic
problems very accurately. For example, in Torbeyns, Schneider, Xin, and Siegler (2015),
Chinese sixth and eighth graders scored above 90% correct on the same set of fraction arithmetic
problems on which US peers scored below 50% correct. Thus, although Chinese children appear
to learn the same spurious associations as US children, Chinese children less often choose
incorrect fraction arithmetic strategies based on those associations.
ASSOCIATIONS IN FRACTION ARITHMETIC 19
One possible explanation is that Chinese children have better conceptual understanding of
fraction arithmetic operations than US children, and this superior understanding allows them to
override the influence of the spurious associations on arithmetic strategy choices. Consistent
with this hypothesis, although most US sixth and eighth graders incorrectly judge that the
product of two positive fractions smaller than one is larger than either operand alone (Siegler &
Lortie-Forgues, 2015), most Chinese sixth and eighth graders do not commit this error (Tian &
Siegler, in preparation). Further, a far higher percentage of primary school mathematics teachers
in China than in the US can explain the rationales for fraction arithmetic procedures (Ma, 1999),
making it possible for them to teach the rationales to their students. Chinese children’s superior
understanding of fraction arithmetic could override their spurious associations between
operations and operand features, thereby avoiding detrimental effects of the spurious
associations.
Another possibility, not exclusive from the first, is that Chinese students receive more
practice with fraction arithmetic, resulting in stronger learning of correct procedures. Primary
and high school students spend much far time per week doing mathematics homework in China
than in the US (Fuligni & Stevenson, 1995). This finding suggests that the quantity of students’
fraction arithmetic practice, like other types of math practice, is greater in China. This greater
practice may enable children to overcome effects of spurious associations, due to it creating
stronger associations between operations and correct problem-solving procedures.
Instructional Implications
Either or both of these possibilities may account for why Chinese children accurately solve
fraction arithmetic problems despite their spurious associations between operations and operand
features on such problems. Further, both interpretations suggest possible directions for
improving US children’s mastery of fraction arithmetic procedures: improve children’s
conceptual understanding, provide substantially more practice using the procedures, or both.
Less clear, however, is how to improve children’s conceptual understanding of fraction
arithmetic. Conceptual difficulties in this area have proven remarkably resistant to instruction.
For example, children’s ability to estimate a sum of two fractions without calculating an exact
answer has shown almost no improvement despite decades of effort, most recently codified in
the Common Core (CCSSI, 2010). When US eighth graders were asked in 1978 to choose the
best estimate of 12/13+7/8 from among the options 1, 2, 19, 21, and “I don’t know,” the correct
answer, 2, was chosen by only 24% of children (Carpenter, Corbitt, Kepner, Lindquist, & Reys,
1980). By 2014, percent correct on this problem had only risen to 27% (Lortie-Forgues et al.,
2015). Similarly, in 2016, when sixth and eighth graders were asked to estimate the magnitudes
of each operand and the sum of the operands in fraction addition problems, half of the estimated
sums were smaller than the estimate of one or both operands (Braithwaite, Tian, & Siegler,
2017). This unimpressive performance in no way implies that we should abandon the goal of
improving conceptual understanding of fraction arithmetic, but it does illustrate the challenge of
doing so.
Giving children substantially more practice with fraction arithmetic would likely be
beneficial, but it also is easier said than done. Increasing fraction arithmetic practice without
increasing total time spent on mathematics would require decreasing time spent on other areas of
mathematics. Increasing total time spent on mathematics would require decreasing time spent on
ASSOCIATIONS IN FRACTION ARITHMETIC 20
other subjects or increasing total time spent in school. None of these changes would be easy to
implement.
An alternative approach to improving US fraction arithmetic instruction would be to present
children with more problems of the types to which they currently are rarely exposed, such as
equal denominator multiplication problems and addition and subtraction problems involving a
whole number and a fraction. This approach could prevent children from forming spurious
associations and therefore from relying on such associations when solving problems. In contrast
to the approaches described above, this approach would not require far-reaching changes to
methods or amount of instruction. Instead, it could be implemented by instructional designers
quite easily. The approach would likely be most effective if implemented not only by traditional
textbook publishers, but also by designers of alternative resources on which many mathematics
teachers rely for practice problems, such as online resources (e.g., Khan Academy, Illustrative
Mathematics, education.com).
It remains to be tested whether this approach is effective, and if so, how many additional
“underrepresented” practice problems would be needed to achieve the desired effect. A small
number of examples can be sufficient to correct misconceptions in mathematics. For example,
when placing whole numbers on a number line, second graders’ responses are usually distributed
logarithmically, but often shift to a linear response pattern after children are shown the correct
response on a single trial (Opfer & Siegler, 2007). Similarly, adding a small number of currently
underrepresented fraction arithmetic problems to existing curricular materials could prevent
children from forming, and relying on, spurious associations. If so, there would be no obvious
downside to this approach, as it would not require significant reduction in the number of other
problems or increased time in school. On the other hand, if many additional problems are
required, the benefits of adding them, relative to the costs of requiring children to solve a greater
total number of problems, would need to be assessed. We hope to investigate these issues in
future research and hope that others will as well.
Summary
In both the US and China, children learn mathematically irrelevant associations between
fraction arithmetic operations and operand features that parallel associations between operations
and operand features in mathematics textbooks. The findings were consistent with Braithwaite,
Pyke, and Siegler’s (2017) hypothesis that children form such associations and thus can use them
to choose solution strategies. However, the data from Chinese children demonstrate that forming
spurious associations does not predestine children to use them to choose incorrect strategies.
How best to avoid the potential drawbacks of forming such spurious associations is an important
issue for improving both theory and practice in this crucial area of mathematics learning.
Author Note
This work was supported in part by the Institute of Education Sciences, U.S. Department of
Education, through Grant R305A150262 to Carnegie Mellon University, and by the Advanced
Technology Center and Siegler Center for Innovative Learning, Beijing Normal University. The
authors would like to thank Xinlin Zhou for his assistance with data collection, and Tian Jing and
Jing Shao for checking the Chinese translations of the materials used in Experiment 3.
ASSOCIATIONS IN FRACTION ARITHMETIC 21
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