Non-adaptive strategy selection in adults with high mathematical anxiety

Abstract

Participants with mathematical anxiety (MA) tend to show particular difficulty in mathematical operations with high working memory (WM) demands compared to operations with lower WM demands. Accordingly, we examined strategy selection to test the cognitive mechanism underlying the observed weakness of high MA participants in mathematical operations with high WM demands. We compared two groups of college students with high or low MA, in the solution of simple non-carry addition problems (e.g., 54 + 63) and complex carryover addition problems (e.g., 59 + 63). The results indicated that high MA participants showed particular difficulty in the harder carry condition. Testing the strategy selection mechanism among high MA participants, we found in the carry condition 1) they used the common strategy less often compared to low MA participants and 2) employed unusual strategies more often compared to low MA participants. Therefore, high MA participants were less efficient in their strategy selection, which may be due to weaker spatial representations, numerical difficulties, or less experience solving complex problems. These primitive representations are not adaptive, and can negatively impact performance in math tasks with high WM demands.

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Non-adaptive strategy selection
in adults with high mathematical
anxiety
Sarit Ashkenazi & Deema Najjar
Participants with mathematical anxiety (MA) tend to show particular diculty in mathematical
operations with high working memory (WM) demands compared to operations with lower WM
demands. Accordingly, we examined strategy selection to test the cognitive mechanism underlying
the observed weakness of high MA participants in mathematical operations with high WM demands.
We compared two groups of college students with high or low MA, in the solution of simple
non-carry addition problems (e.g., 54 + 63) and complex carryover addition problems (e.g., 59 + 63).
The results indicated that high MA participants showed particular diculty in the harder carry
condition. Testing the strategy selection mechanism among high MA participants, we found in
the carry condition 1) they used the common strategy less often compared to low MA participants
and 2) employed unusual strategies more often compared to low MA participants. Therefore, high
MA participants were less ecient in their strategy selection, which may be due to weaker spatial
representations, numerical diculties, or less experience solving complex problems. These primitive
representations are not adaptive, and can negatively impact performance in math tasks with high
WM demands.
Mathematical anxiety (MA) is a feeling of tension and anxiety that interferes with the manipulation of numbers
and the solution of mathematical problems in a wide variety of everyday life and academic situations1. ere is
an ongoing debate in relation to the origin of MA, and the situational inuences of MA in the case of high anx-
iety2–4. Some studies found that MA originates from a weakness in basic numerical processing that is mediated
by impaired spatial abilities5,6. While others have found that MA inuences working memory (WM), and math
performance2–4. According to this line of studies, poor math performance among people with MA is a result
of limited WM resources, which are required for the solution of math problems, due to anxiety induced verbal
ruminations2–4. e aective drop in performance is observed when math is performed under timed, high-stakes
conditions2,7. Hence, when people with MA perform arithmetic tasks that place high demands on WM their
performance is worse compared to tasks with lower WM demands. Similar to the aective drop in performance
theory, the attentional control theory proposes that MA, similar to anxiety disorders, aects central executive
WM, thus reducing inhibition and shiing abilities8.
Contrary to the limited WM resources theory regarding MA2,7,8, studies have found that participants with
high MA (HMA) had lower performance compared to participants with low MA (LMA) in basic numerical
tasks (such as symbolic and non-symbolic comparison tasks) related to number sense, which have minimal WM
demands9–11. Number sense is an understanding of approximate quantities and the relation between quantities,
grounded in a spatial representation12–14. Some studies found that a weakness in spatial processing was the origin
of the number sense decit in HMA participants. Ferguson et al.15 found a strong negative correlation between
spatial abilities and MA. e same study also found that participants with HMA reported a worse sense of direc-
tion and more spatial anxiety than their LMA peers15. Moreover, participants with HMA had lower mental rota-
tion abilities than LMA participants15. Last, participants with HMA had a stronger mental representation that
binds numbers and space (abnormal numerical distance eect and a spatial-numerical association of response
codes eect)5. Weakness in spatial abilities can result in abnormal basic numerical representation or weakness in
number sense, and thus may contribute to MA.
Learning Disabilities, the Seymour Fox School of Education, the Hebrew University of Jerusalem, Mount Scopus,
Jerusalem, 91905, Israel. Correspondence and requests for materials should be addressed to S.A. (email: sarit.
ashkenazi@mail.huji.ac.il)
Received: 1 September 2017
Accepted: 29 May 2018
Published: xx xx xxxx
OPEN
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One way to understand the origin of MA, and the situational inuences of MA in the case of high anxiety, is to
test strategy selection (i.e., choice of appropriate strategies in relation to task characteristics) and strategy execu-
tion (i.e., performance of the cognitive processes involved in each strategy) during the solution of math problems
(for an explanation of the common strategies employed during complex arithmetic problem see our explanation
of the task below). From early childhood, children constantly learn and acquire new strategies. e new strategies
result from conceptual understanding of the requisites of appropriate strategies for a specic problem16. Flexibility
in problem solving will result from knowledge of (a) multiple strategies and (b) the relative eciency of these
strategies17,18. Research on mathematical strategy selection and strategy execution has dierentiated between
groups based on arithmetical capabilities, WM capacity, age and counting knowledge19–21. For instance, children
with mathematical learning disabilities used strategies that were developmentally below age level during the solu-
tion of simple arithmetic problems (e.g., 7 + 8), such as counting all (count both addends) and using ngers as
counting aids. Moreover, their strategy selection was poor and they continued to use the same strategy regardless
of problem type22.
Among adults, mathematical experts tend to use a larger variety of strategies and more advanced strategies
compared to non-experts23. Hodzik and Lemaire24 found that young and older adults diered in how many strat-
egies they used, as well as in inhibition and shiing capabilities during complex arithmetical problem solution,
which mediated age-related dierences in strategy repertoire and strategy selection (for a similar nding see25).
Using a similar methodology to Hodzik and Lemaire24, with the addition of WM manipulation, we found that
strategy selection was modulated by WM demand, number sense ability and central executive abilities26.
Very few studies directly examined strategy selection among HMA participants27,28. One study found that
children with HMA used retrieval less oen and less frequently than their low MA peers27. Hence, similar to
participants with learning disabilities, individuals with HMA tended to employ backup procedural strategies
rather than direct retrieval19, which is less ecient. In a study with children with MA, Ramirez et al.29 grouped the
HMA participants based on WM abilities. ey found that participants with high WM abilities tended to rely on
advanced strategies (such as direct retrieval or decomposition), while participants with low WM abilities relied
on strategies with minimal WM demand (such as counting all). e use of advanced memory based strategies
partially mediated the relationship between MA and math achievement: advanced strategies are prone to suer
from the fact that WM capacity is co-opted by MA (for similar results in adults see30–32). Hence, children with MA
and high WM capabilities have poorer performance in mathematics tasks with high WM demands because they
cannot access strategies they would normally rely on.
Si, et al.28 tested the eect of MA on strategy choice in computational estimation and mental arithmetic tasks,
and examined age-related dierences. e results indicated that MA had a greater eect on computational esti-
mation than on mental arithmetic. Moreover, MA had a greater aect on 6th grade students compared to 4th
graders and adults.
e goal of the present study was to explore strategy selection and repertoire in participants with HMA,
in order to understand the origin of MA and the influence of MA on the solution of math operations. MA
was expected to interfere with the solution of operations that involved high WM load but not low WM load.
Accordingly, we manipulated WM resources by comparing operations with no carryover (low WM load) to oper-
ations with carryover (high WM load). Ashcra and Kirk3 found that participants with HMA had particular
diculty with carryover operations. Hence, we expected to nd larger group dierences in solution eciency
and strategy selection for carryover problems compared to non-carryover problems. It was reported that both
numerical and executive function abilities aect strategy selection and solution eciency26. Hence, we explored
the respective roles of non-symbolic comparison and central executive abilities as possible explanatory factors for
individual dierences in eciency of solution and strategy selection. We believe that the eects of non-symbolic
comparison and central executive abilities will be mediated by the MA group (we found in a previous study that
these abilities aect strategy selection in the normative population26).
ere is an ongoing debate in relation to the origin of MA, and the situational inuences of MA in the case
of high anxiety. One approach suggests that MA is based on an impairment in quantity representation that is
mediated by spatial weakness, which resulted in poorer math performance among individuals with HMA5,10.
While others posit that anxiety itself is the origin of reduced math performance among individuals with HMA;
anxiety reduces executive functions abilities (attentional control capacity)2,3,8. Hence, seemingly contradictory
predictions can be hypothesized based on these theories. If the decit in numerical processing is the heart of MA
then we should expect to nd that the HMA participants will select inappropriate, unusual strategies5. However,
if MA reduces cognitive control2,3,8, then participants with HMA should use a smaller number of strategies and
use them less eectively compared to participants with LMA24 (but see Dowker23 for contradictory predictions
related to larger strategic variability with high proportions of non-sophisticated strategies among math novices
compared to math middle range of expertise).
Results
e data analysis included several stages. First, we compared the HMA and LMA participants performance in
the non-symbolic comparison task and an executive function task. en, we analyzed strategy selection in each
group. Aerwards, we examined solution eciency for complex addition problems by group and problem type,
and tested proportion of use of the ten common strategies by problem type and group. Finally, to test strategic
variation we analyzed the number of strategies used by problem type and group.
• Group comparison of general abilities
e two groups had comparable performance in non-symbolic comparison and the Tower of Hanoi (see
Table1).
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• Strategy selection
Global analysis of the favorite strategy (the strategy used with the greatest frequency) (see Table2 for all of
the strategies) showed that columnar retrieval was used in more than half (56%) of all trials. Analysis of the
favorite strategy in each group showed that columnar retrieval was used the most oen in HMA (51%) and
LMA (62%), followed by vertical imagery in HMA (16%) and rounding the two operands down (10%) in
LMA. Finally, the HMA group used rounding the second operand up and rounding the two operands down
(8% for both) while the LMA group used vertical imagery and rounding the second operand (6% for each
strategy).
Accuracy rate. A two-way analysis of covariance (ANCOVA) was performed on the accuracy rates with
problem type (carry or no-carry) as the within subject factor and group (HMA or LMA) as the between subject
factor. Tower of Hanoi and accuracy in non-symbolic comparison were the covariates. e only signicant eect
was Tower of Hanoi, F(1, 44) = 18.09, partial η2 = 0.29, p < 0.01. e interaction between Tower of Hanoi and
problem type also reached signicance, F(1, 44) = 4.36, partial η2 = 0.09, p < 0.05. Better performance in the
Tower of Hanoi was associated with higher accuracy both in carry and no carry problems r(48) = −0.56, p < 0.001
and r(48) = −0.37, p < 0.01 (for carry and no carry respectively). However, the correlation was larger in the carry
condition (see Fig.1).
Reaction time (RT). A two-way analysis of covariance (ANCOVA) was performed on the RTs with problem
type (carry or no carry) as the within subject factor and group (HMA or LMA) as the between subject factor. Tower
of Hanoi and accuracy in non-symbolic comparison were the covariates. e eect of group reached signicance,
F(1, 44) = 10.26, partial η2 = 0.19, p < 0.01, the HMA participants had longer RT’s compared to the LMA group
HMA LMA
pMean S.D.Mean S.D.
Tower of Hanoi 2.90 1.98 2.30 2.01 0.24
Weber fraction 0.36 0.23 0.37 0.47 0.86
Non-symbolic comparison accuracy 74.87 7.67 76.86 10.33 0.46
Non-symbolic comparison RT 914.59 327.54 863.08 318.56 0.58
Accuracy addition non-carry 0.81 0.11 0.84 0.12 0.35
Accuracy addition carry 0.73 0.14 0.78 0.16 0.19
RTs addition non-carry 6,982.25 2,751.23 5,050.12 2,126.05 0.001
RTs addition carry 10,850.56 4,402.35 6,878.74 3,076.36 0.001
Table 1. Executive functions, quantity discrimination and complex addition performance by group.
Strategy 1 2 3 4 5 6 7 8 9 10 11 12 13
HMA 0.5 1.0 7.8 50.6 5.0 7.9 0.1 0.3 0.2 4.1 16.0 2.8 3.8
LMA 0.5 1.3 9.5 61.9 4.8 6.1 0.4 3.0 0.0 0.1 6.1 0.5 5.9
Table 2. Average percentage use of each strategy in each group. 1 = rounding the rst operand down
2 = rounding the second operand down 3 = rounding both operand down 4 = columnar retrieval 5 = rounding
the rst operand up 6 = rounding the second operand up 7 = rounding the two operands up 8 = unit addition
9 = retrieval 10 = others 11 = vertical imagery 12 = decomposition of the units 13 = decomposition of the
decade.
Figure 1. High executive function ability associated with high accuracy in math problem solution. e gure
presents the correlation between tower of Hanoi scores and accuracy rates in math operations. Better executive
function ability was associated with high accuracy. e correlation was stronger in the carry condition (A)
compared to the no carry condition (B).
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(M = 8,916 ms SD = 3,576 and M = 5,964 ms SD = 2,601 for the HMA and LMA groups respectively). Importantly,
there was a signicant group and problem type interaction, F(1, 44) = 7.08, partial η2 = 0.15, p < 0.01. e dierences
between the time to respond to carry and no-carry problems was much larger in the HMA group than the LMA
group (M = 3,868 SD = 3,211 and M = 1,828 SD = 1,527, for the HMA and LMA groups respectively, t(46) = 2.85,
p < 0.01. ere were no other signicant main eects or interactions (See Fig.2).
Proportion of use of the ten common strategies. A three-way analysis of covariance (ANCOVA) was
performed on the proportion of use of the ten common strategies with problem type (no carry vs. carry) and
strategy (rounding the rst operand down, rounding both operands down, columnar retrieval, rounding the rst
operand up, rounding the second operand up, unit addition, others, vertical imagery, decomposition of the units,
decomposition of the decade) as within participant factors and group (HMA or LMA) as between participant
factors. Tower of Hanoi and accuracy in non-symbolic comparison served as the covariates.
e only signicant main eect was strategy use, F(9, 396) = 2.53, partial η2 = 0.054, p < 0.01. Participants,
across groups, used the columnar retrieval strategy signicantly more frequently than the other strategies. e
interaction between problem type and strategy was signicant, F(9, 396) = 1.98, partial η2 = 0.043, p < 0.05.
Columnar retrieval and rounding the rst operand down were less frequent in the carry condition (M = 50%,
SD = 36% and M = 0.23%, SD = 1%) compared to the no carry condition (M = 74%, SD = 30% and M = 0.8%
SD = 2%), t(47) = 9.02, p < 0.01 and t(47) = 2.19, p < 0.05. Yet, four other strategies, rounding the two operands
down, rounding the rst operand up, rounding the second operand up and decomposition of the units, were used
more frequently in the carry condition (M = 13%, SD = 25%, M = 9% SD = 9%, M = 12% SD = 12% and M = 3%
SD = 8% respectively) compared to the non-carry condition (M = 4% SD = 15%, M = 1% SD = 3%, M = 1 %
SD = 3%, and M = 0.2% SD = 1% respectively) t(47) = −3, p < 0.01, t(47) = −5.73, p < 0.01, t(47) = −7.4, p < 0.01
and t(47) = −2.3, p < 0.05.
Importantly, the interaction between problem type and strategy was modulated by group, F(9, 396) = 5.41,
partial η2 = 0.11, p < 0.01. In the non-carry condition participants with high and low MA used the columnar
retrieval strategy very frequently (M = 74%, SD = 27% and M = 74%, SD = 33%, for HMA and LMA respectively).
While both groups employed this strategy signicantly less in the carry condition (M = 27%, SD = 28% and
M = 50%, SD = 37% for HMA and LMA respectively), this dierence was signicantly larger in the HMA group
compared to the LMA group, t(46) = −2.28, p < 0.05. Moreover, in the non-carry condition participants with
high and low MA used the other strategies in similar frequencies (but the other strategies were more frequent in
the HMA group compared to the LMA group t(46) = 1.85, p = 0.07). However, in the carry condition, the HMA
group used decomposition of units more frequently, t(46) = 2.1, p < 0.05, and tended to use vertical imagery
more frequently, t(46) = 1.92, p = 0.06, and other strategies, t(46) = 1.83, p = 0.07, than the LMA group (vertical
imagery: M = 20% SD = 7% compared to M = 5% SD = 4% for the other frequencies see Fig.3) (see Fig.3).
Interestingly, accuracy in the non-symbolic comparison task (accuracy rates of the ANS task) and strat-
egy interacted F(9, 396) = 2.7, η2 = 0.39, p < 0.01. Hence, we tested the correlation between accuracy of the
non-symbolic comparison and strategy use for each of the strategies. Non-symbolic comparison accuracy neg-
atively associated to vertical imagery r(48) = −0.40, p < 0.01 (see Fig.4). ere were no additional signicant
correlations with any of the other strategies. ere was no main eect or interactions with Tower of Hanoi score.
Number of strategies used. A two-way analysis of covariance (ANCOVA) was performed on the number of
used strategies with problem type (carry or no carry) as the within subject factor and group (HMA or LMA) as the
between subject factor. Tower of Hanoi and accuracy in non-symbolic comparison served as the covariates. None
of the eects or interactions reached signicance. e only marginally signicant eect was group, F(1, 44) = 2.86,
partial η2 = 0.06, p = 0.098. e HMA group tended to use more strategies than LMA group (M = 3.26, SD = 1.54,
M = 2.68, SD = 1.44 for HMA and LMA respectively).
Discussion
e current study explored arithmetical strategy selection among young adults with HMA. e complex addition
problems included two levels of diculty: 1) problems with carryover, which had higher WM demands, and, 2)
non-carry problems, which had lower WM demands. ere is an ongoing debate on the origin of MA, and the
situational inuences of MA in the case of high anxiety. While some studies found that MA originates from a
Figure 2. HMA participants showed longer (reaction times) RT in math problem solution; this tendency was
stronger in complex carry operations than simple non-carry operations. RTs to solve mathematical operations
by problem type (carry or no carry) and group (HMA or LMA).
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weakness in basic numerical processing mediated by impaired spatial abilities5,6, others have suggested that MA
results in situated impairment in executive functions such as inhibition and shiing33,34. Examination of strategy
selection allowed us to compare these theories among HMA participants. If impaired numerical abilities and
spatial weakness characterizes MA, then participants with HMA would have diculty selecting ecient strate-
gies and would employ atypical strategies instead. While, executive function impairment would result in smaller
variability in the number of strategies selected among HMA participants.
We found that HMA participants took longer to solve the arithmetic problems across problem type, but the
dierence between the groups response times was greater for carryover problems. In addition, HMA participants
employed unusual strategies more oen than LMA participants. Some of these strategies involved rigid spatial
representation, such as vertical imagery. Similar to the RT results, we found larger group dierences in strategy
selection for carryover problems compared to non-carry problems. Specically, both groups used the common
strategy less oen for carryover problems compared to non-carry problems; however, this dierence was much
larger among the HMA participants compared to the LMA participants. Instead of using the common strategy,
the HMA participants used atypical strategies such as vertical imagery. In addition, vertical imagery negatively
associated with innate quantity discrimination ability.
e present study found that HMA individuals were much slower in solving carry operations compared to
LMA individuals, group dierences were reduced in the non-carry condition. It has been widely documented
that the eect of MA on mathematical task performance is modulated by task demand2–4. For example, a pre-
vious study found that performance on simple addition and multiplication operations (e.g., 2 + 4 or 6 × 4) was
minimally impacted by MA2–4, while it had a greater impact on complex two-column addition problems (e.g.,
21 + 18), and more so for carryover problems. e main dierence between these kinds of arithmetic problems
is that the solution for smaller, one-digit operations are typically retrieved from long term memory and therefore
require minimal WM resources; while two-column addition problems, and more specically carry problems, are
calculated and require greater WM resources2–4.
Figure 3. Proportion of use by strategy, problem type and group; HMA- A and LMA- B. In the easy no carry
condition, most of the participants, regardless of group, used the common columnar retrieval strategy. However,
in the harder carry condition, both of the groups presented a reduction in the proportion of use of the common
strategy. e reduction was larger in the HMA participants compared to the LMA participants. Moreover, the
strategies that were chosen instead of the common strategy in the carry condition were dierent between the
groups: HMA tended to use decomposition of the units, vertical imagery and other strategies. VI = vertical
imagery. Du- decomposition of the units, RSU- rounding second operand up. RFU- rounding rst operand up.
RTD- rounding two operands down. RFD- rounding rst operands down. CR- columnar retrieval.
Figure 4. Proportion of use of the strategy vertical imagery by non-symbolic comparison abilities. Individual
quantity discrimination abilities were negatively associated to the percent of usage of the strategy vertical
imagery.
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One explanation for diculty in mathematics tasks with high WM demands among HMA participants is that
their WM resources are tapped out due to verbal ruminations, and therefore, have reduced WM resources avail-
able for the task at hand2–4. e present study suggests an alternative explanation that HMA participants select
unusual, non-adaptive strategies for mathematical tasks with higher WM demands; as was seen in the dierence
between strategies selected for carryover versus non-carry problems.
e current study expands upon previous studies that have found that HMA participants have a particular
weakness in carryover problems2–4, because it is the rst to explore the strategy selection and cognitive processes
underlying this weakness. Hodzik and Lemaire24 compared younger and older adults’ strategy selection for arith-
metic problems, and found that high central executive abilities among the younger adults resulted in larger strate-
gic variations and selection of the best available strategy. is nding sheds light on the relationship between MA
and strategy selection due to reduced central executive abilities during situations of high anxiety8. Accordingly,
HMA participants should act as older adults, and display lower strategic variations, and impaired selection of the
best strategy. In line with this view, a study that examined strategy selection among children found that HMA
children used direct retrieval (the best strategy) less oen than LMA children during the solution of simple arith-
metical operations27. However, in the current study, HMA adults displayed a tendency towards higher strategic
variability compared to LMA. Please note that for adults direct retrieval is not the most sophisticated strategy
for solving complex problems, and the best strategy relates to specic characteristics of each problem16,18,24,26,28.
Dowker23 compared college students based on their major and found that English majors were less accurate in
a computational estimation task compared to psychology students. In addition, English majors showed larger
strategic variations and reduced ability to select the best strategy compared to psychology students.
e results of the present study were in line with Dowker’s23 ndings. HMA participants in the current study
showed a tendency towards larger strategic variations compared to LMA participants, especially in the harder
carry condition. In addition, in line with the previous results, HMA participants tended to use less sophisticated
strategies compared to LMA participants. Specically, HMA participants tended to use “other” strategies and ver-
tical imagery more oen. e HMA group used vertical imagery in 20% of the carry trials compared to only 5%
in the LMA group. In the vertical imagery strategy, participants spatially imagine writing a vertical representation
of the operation. is strategy is based on spatial representation in a very primitive form.
Mature representation of number originates from preverbal spatial representation of quantity in the form
of the mental number line12,13. e connection between quantities and space emerges during infancy35. Hence,
reduced spatial abilities could result in a weak mental number line and later weakness in the mature representa-
tion of numbers, which can contribute to MA. Accordingly, a new line of studies found a connection between MA
and reduced spatial abilities5,6,10,15. For example, we found that spatial WM span negatively associated to MA6.
Moreover, Maloney et al.10 found that weakness in spatial abilities among women was the origin of higher rates of
MA in women compared to men.
Interestingly, Georges et al.5 examined numerical distance eect and SNARC eects as indicators of the spatial
representation of the number line in HMA participants compared to LMA participants. A steeper distance eect
and a stronger SNARC eect were observed in HMA participants. e authors concluded that the stronger asso-
ciations between number and space in HMA participants resulted in a reliance on concrete spatial representation
preventing an understanding of abstract and complex mathematical concepts. In line with this view Cipora et al.36
found that professional mathematicians, unlike matched controls, did not reveal a SNARC eect. ey suggested
that professional mathematicians have more abstract or exible spatial representations of numerical information
than matched controls. In line with Georges et al.5 conclusions and supported by Cipora et al.’s36 nding, here we
found a reliance on concrete spatial representation in some of the strategies that were reported in HMA partici-
pants, such as vertical imagery. is reliance may prevent HMA participants from selecting the best strategy and
inuence how eciently they solve mathematical operations.
In line with the view that selection of unusual strategies characterizes MA, we found an association between
basic weakness in quantity representation abilities and the tendency of using unusual strategies. For complex
problem solving among adults, the best, most sophisticated strategy relates to the problem characteristics and
retrieval is uncommon16,18,24,26,28. In order to select the best strategy, one approximates the quantity of each oper-
and and then selects an appropriate strategy for those approximate quantities. For example, in order to solve
69 + 71, the best strategy would be rounding up the units of 69 and rounding down the units of 71. is is the best
strategy for the problem due to the operands proximity to decade numbers. However, high MA individual tended
to select unusual strategies regardless of problem characteristics.
In line with this view, we found in a previous study, a relationship between strategy selection and quantity
discrimination26: better quantity discrimination negatively associated to the percent use of the common strategy
(columnar retrieval) in the carry condition. at previous study found for the non-carry condition, regardless of
quantity discrimination ability, most of the participants used columnar retrieval. However, in the harder carry
condition, a reduction was found in the percent use of the common strategy, the reduction was larger in partic-
ipants with high quantity discrimination ability than in participants with low quantity discrimination ability. A
similar reduction (i.e., in the percent use of the common strategy) was found in the present study; this tendency
was larger in the HMA participants compared to the LMA participants. However, the main dierence between
the previous study and the tendency of the HMA participants in the current study is the alternative strategy that
was chosen instead of the common strategy. In the present study, a large percent of the HMA participants used
non-adaptive strategies, such as vertical imagery and unusual strategies, in the carry condition. However, in
Ashkenazi et al.26 a large percentage of the participants in the carry condition chose more adaptive strategies such
as rounding the second operand up. In line with this view, in the present study, better quantity discrimination
ability negatively associated to the percent use of the unusual strategy (vertical imagery). ese results empha-
size that strategic variations alone are not indicative of superior math abilities23. Future studies should look at
the sophistication and nature of strategy selection, as well as the strategic variations in order to examine math
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capabilities. Spatial weakness is one possible explanation for the unusual strategies that selected in the HMA
group. Other explanations can relate more directly to numerical diculties or less practice solving complex addi-
tion problems.
In addition, low math performance can be the origin of MA, which can then result in a further drop in math
performance9,37. Hence, one cannot fully rule out the explanation that HMA participants in the current study
selected unusual non-adaptive strategies due to low math abilities. Moreover, future studies should further
explore the direct relationships between the strategies frequently selected by high MA individuals (such as vertical
imagery) and weakness in numerical- space associations in MA. e ndings of the current study suggests weak
numerical space associations among MA participants based on the unusual strategies selected for the carryover
problems, but did not directly relate that tendency to spatial weaknesses.
Conclusion
Over the last two decades it has been reported that MA impairs performance specically in mathematical oper-
ations that involve high WM demands, such as carry problems2–4. Here, for the rst time we uncovered the cog-
nitive mechanism underlying the decits in operations that require carryover by examining strategy selection.
We found that the HMA participants used primitive, non-adaptive strategies grounded in space. e use of these
unusual strategies was mostly observed in the harder operations that required carryover.
e innate representation of quantity found in infancy is related to space35; however, over the course of devel-
opment and formal education, more mature symbolic numerical representation is built upon this primitive
representation and creates more abstract numerical representations. Spatial weaknesses among HMA partici-
pants5,6,10,15 can prevent the mental construction of abstract numerical representation. us, they continue to use
primitive numerical representations, based on spatial representations, that are not adaptive and can inuence
performance in operations with high WM demands.
Material and Methods
Participants. Forty-eight students from the Hebrew University of Jerusalem, 42 women and 6 men, par-
ticipated in the experiment. ey were between the ages of 18 and 30-years-old (M = 23.45 S.D = 2.38). Four of
the participants were le-handed and the rest were right-handed. None of the participants reported diagnosis of
ADHD or learning disabilities and all had normal or corrected-to-normal visual acuity. ey were compensated
with course credits or 30 NIS (approximately $8.60).
All the methods of the study were performed in accordance with the relevant guidelines and regulations.
e study was approved by the local Ethics Committee of the Seymour Fox School of Education at the Hebrew
University of Jerusalem. Written consent was obtained according to the Declaration of Helsinki. e participants
provided written informed consent.
e participants were divided to two groups according to their scores in e Mathematics Anxiety Rating
Scale - Shortened (MARS-S). Participants with scores lower than 75 were included in the LMA group. Participants
with scores higher or equal to 81 were included in the HMA group. Participants with a score between 76 to 80
were not included in the present study.
Two of the participants were excluded due to long RT’s (more than 3 SD from the average), another participant
was eliminated due to accuracy rates less than 50% in each of the conditions.
Apparatus. e experimental task (complex calculation), was controlled by a Genuine-Intel compatible PC
1.73 GHz using E-prime experimental soware, 2.1 version (Schneider, Eschman, & Zuccolotto, 2002). e Tower
of Hanoi was controlled by PEBL (Mueller, 2012). e ANS task was controlled by the Panamath soware38.
Instructions and stimuli were presented on a 14″ monitor. e computer monitor was located approximately
50 cm in front of the participant.
MARS-S. e Mathematics Anxiety Rating Scale (MARS) has been widely used since 1972, the MARS-S is
a 30 item self-rating scale to assess MA39, a short version of the MARS. Participants are asked to indicate on a
5-point Likert scale from 1 (low anxiety) to 5 (high anxiety) how anxious they feel in various math-related situa-
tions. Adequate internal consistency (Cronbach’s α = 0.96), test-retest reliability (r = 0.9) and construct validity
have been reported for this instrument. It was translated into Hebrew by our lab.
Complex calculations: main task. Each participant solved 48 complex addition problems. A typical trial
began with a xation mark presented in the center of the computer screen for 300 ms, which was followed by a
blank screen for 500 ms. en the problem appeared in the form of a + b = , the participant had to orally report
the answer. e problem was presented until the participant responded. Aerwards, the experimenter recorded
the participants’ response. en a screen with the question “How did you solve the problem?” appeared. e
question appeared until the participant answered the question. e experimenter recorded participants’ response.
e trial was nished with a blank screen for 1500 ms.
e problems in the task were composed of two two-digit numbers with a mean sum of 135.63 (S.D. = 17.89,
range = 104–169). To test problem diculty eects, problems were categorized as either “carry problems” or
“non-carry problems” on the basis of the presence/absence of a carryover in the unit’s digit; all the problems
required carryover in the hundreds digit (e.g. 64 + 66 = 130). Following previous ndings in arithmetic (see26),
problems were selected with several constraints: (a) half the problems had their larger operand on the le position
(e.g., 68 + 37); (b) none of the operands had the tens or units digits equal to 0; (c) none of the operands had unit
digits equal to 5; (d) none of the pairs of operands had the same ten and unit digits (e.g., 64 + 68, 64 + 54), the
same ten and unit digits (e.g., 33 + 88), or the same operands (e.g., 71 + 71); (f) none of the problems were the
reverse of another problem (i.e., if 72 + 64 was used, 64 + 72 was not). Participants were not allowed to use paper
and pencil, and no feedback was provided. e experimenter could classify the reported strategy to one of the
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following (using the example (72 + 46)): (1) rounding the rst operand down (e.g., (70 + 46) + 2), (2) rounding
the second operand down (e.g., (72 + 40) + 6), (3) rounding both operands down (e.g., (70 + 40) + (2 + 6)), (4)
columnar retrieval (e.g., (2 + 6) + (70 + 40)), (5) rounding the rst operand up (e.g., (80 + 46) − 8), (6) rounding
the second operand up (e.g., (72 + 50) − 4), (7) rounding the two operands up (e.g., (80 + 50) − 8 − 4), (8) unit
addition (e.g., (78 + 40)), (9) retrieval (e.g., 118), and (10) others (none of the other strategies) (11) vertical
imagery (e.g.,
+12
46
), (12) decomposition of the units (e.g., (2 + 6) + 2 + (70 + 40) − 2), (13) decomposition of the
decade (similar to the previous one but in the decade unit).
Tower of Hanoi. In a computerized version of this task, three rods and a number of disks of dierent sizes,
which can slide onto any rod, were displayed on the screen. e participant was instructed to move all the discs
from a start position to the end position (task from the PEBL40). e dependent variable was the average dier-
ence between the shortest number of steps available in a specic trial and the actual number performed by the
participant.
ANS task. Two sets of overlapping dots, one in yellow and one in blue, appeared on the screen briey (for
300 ms.). e participants were requested to indicate if the blue cloud or yellow cloud was larger in quantity.
Ratios between sets were manipulated between 1:2, 3:4, 5:6 or 7:8. For every set, the number range was between
5 and 16 dots. To control for possible intervening variables such as total area and dot size, the trials were split in
half: half of the trials were controlled for area, the total area of the two sets were identical and the remaining trials
were controlled for size, such that the size of the dots in both sets were equal (n = 120) for the full details see38.
Materials and procedure. First, participants completed the MARS-S followed by 1) the main arithmetical
task, 2) Tower of Hanoi task and 3) ANS task. Participants with a score of less or equal to 75 (minimum = 32,
maximum = 75, average = 60.28, S.D., 11.42) were dened as LMA (n = 25, mean age = 23.67 S.D = 1.77), partic-
ipants with a score greater or equal to 81 (minimum = 81, maximum = 118, average = 104.14, S.D. = 7.78) were
dened as HMA (n = 23, mean age = 23.42 S.D = 2.9). Out of the LMA participants, 19 were females and 22 of the
high MA were females. Two participants in each group were le handed.
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Author Contributions
D.N. contributed to the study design, and collected and analyzed the data. A.S. took part in designing the study,
analyzed the data and wrote the manuscript. is work was funded by the Marie Curie CIG grant. Grant number
631731.
Additional Information
Competing Interests: e authors declare no competing interests.
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