Effect of Strategy on Problem Solving

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The Journal o Problem Solving volume 3, no. 1 (Fall 2010)

1

The Effect of Strategy on Problem Solving: An fMRI Study

Sharlene D. Newman1, Benjamin Pruce 1, Akash Rusia1,and Thomas Burns Jr.1

Abstract:

MRI was used to examine the dierential eect o two problem-solving strategies. Par-

ticipants were trained to use both a pictorial/spatial and a symbolic/algebraic strategy

to solve word problems. While these two strategies activated similar cortical regions, anumber o dierences were noted in the level o activation. These dierences indicate that

the algebraic strategy is more demanding than the spatial strategy, which was particularly

true or the anterior insula and the parietal cortices. In addition, an exploratory analysis

was perormed that examined eects o strategy preerence. These results revealed that

participants who preerred the algebraic strategy, while having a similar mathematics

background, elicited less activation and had higher working memory capacity (as mea-

sured by the reading span task) than those participants who preerred the spatial strat-

egy. These data have implications or MRI, as well as behavioral studies o higher-order

cognitionthe use o dierent strategies by participants within one study could alter

the nal results and, thereore, the conclusions drawn.

Keywords:

problem solving, MRI, strategy, algebra

Acknowledgments: We would like to thank John Greco, Kristen Ratli, and Tara Muratore or all o their help

with data collection. This research was supported in part by the Indiana METACyt Initiative o Indiana Uni-

versity, unded in part through a major grant rom the Lilly Endowment, Inc.

1Indiana University, Bloomington


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The Journal o Problem Solving

2 SharleneD.Newman,BenjaminPruce,AkashRusia,andThomasBurnsJr.

Introduction

When conronted with a problem we develop an action plan or strategy to solve it. But

how do we develop that strategy and how do we choose which one o multiple strategies

to select or the current problem? There appears to be a number o actors that infuencestrategy ormation and selection. For example, the ability to think about a problem in

multiple ways (cognitive fexibility) seems to enhance the ability to learn more about the

problem (Graham & Perry, 1993; Siegler, 1995), which in turn allows or the development

o more ecient problem-solving strategies. Also, the more inormation (relevant rules,

strategies, conceptualizations) an individual has about a given problem the more likely she

or he is to t a strategy to it, or to select the most appropriate strategy (Siegler et al., 1996).

All o this goes to show: (1) that there is typically more than one way to solve a problem,

and (2) that ability and experience both impact the strategy chosen.

The use o unctional neuroimaging techniques to study more complex cognitive

unctions has grown signicantly. Although it has been demonstrated that there is vari-

ability in the strategies used in complex tasks (Kwong & Varnhagen, 2005; LeFevre et al.,

1996; Rogers, Hertzog, & Fisk, 2000), this variability is seldom accounted or. However, an

understanding o how strategy dierences may impact the underlying neural network is

extremely important. It can, or example, help to explain the increase in variance observed

when examining higher-order cognitive tasks compared to perceptual tasks. Studying

the eect o strategy dierences can also be expected to provide a clearer picture o the

unctional properties o neural networks.

In order to begin to address this issue we have examined problem solving. The prob-

lem used here is a verbal reasoning problem that we have used previously (Newman etal., 2002). The problems were patterned ater the ollowing prototype, which has proven

to be enigmatically dicult (Casey, 1993): Imagine that a man is looking at a photograph

while saying,

Brothers and sisters have I none. That mans ather is my athers son.

Who is in the photograph?

In a study with 101 adult participants, 77% o the respondents chose the same incor-

rect answer (the man himsel ), while only 13% chose the correct answer (it is a photograph

o the mans son). Casey suggested that the various processing demands o this riddle

may exceed verbal working memory capacity. One such demand arises rom the ordero the two phrases That mans atherand my athers son.These demands are related to

those observed in sentence comprehension. Earlier comprehension studies indicated that

processing is slower when the given inormation ollows rather than precedes the new

inormationwhich is the case or the hard problems where the avorite (the new inor-

mation) precedes the given inormation (Haviland & Clark, 1974; Clark 1977). Furthermore,

it has been suggested that the reason or the slower processing is due to a reordering,


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The Effect of Strategy on Problem Solving: An fMRI Study 3

volume 3, no. 1 (Fall 2010)

or manipulation, o the sentence constituents (Carpenter & Just, 1977). Taking Caseys

original problem you would have My athers son is that mans ather and it was shown

that signicantly more people correctly answered this version o the problem (accuracy

increased to as high as 90%).

In the current event-related MRI study we developed two versions o the brothers

riddle that varied the order o the two noun phrases that demonstrate dierences in the

timing o cortical responses that subserve part o the working memory network. In the rst

problem type (easy: The month ater April is the month beore my avorite month), compre-

hending the rst phrase (the given inormation) o the sentence requires a computation

(computing the reerent o the rst month ater April), whereas no such corresponding

computation is required or the second phrase (the newinormation). By contrast, in the

second problem type (hard: The month beore my avorite month is the month ater April),

the reverse is true. Based on the new/given ordering it is expected that the hard problems

will place a larger load on working memory processes. Also, as suggested by Carpenterand Just (1977), it is to be expected that to solve the hard problems the problem will be

reordered in a ormat similar to that o the easy problems.

In a small pilot study using these problems in which the two phrases were presented

separately it was revealed that participants generated two strategies rather consistently

(some participants were unable to articulate their strategy)a symbolic strategy and a

spatial strategy. The symbolic strategy reported by participants involved the conversion o

the word problem into an equation. Participants described converting avorite to a variable,

most oten x, so using the easy example above, they reported adding one to April and

holding that inormation until the second phrase was presented and then i the second

phrase contained beore they would add the number o months beore to what was be-

ing held in memory. The interesting point is that those who reported using this strategy

used math terms like adding and subtraction and x(as a variable) when describing how

they perormed the task. All o these terms suggest the use o a symbolic mathematical

algorithm that we will reer to here as an algebraic strategy.

The spatial strategy was qualitatively very dierent. Individuals using the spatial

strategy reported imagining a number line o sorts with the months, in the case o the

example, as bins. They reported beginning by starting at the April bin and moving let

(beore) or right (ater); they would then hold the number line in memory with the new

position being the ocus. Ater reading the second phrase they would then, in the case othe easy example, look to see which month the current location is beore. The description

o the spatial strategy, unlike the algebraic strategy, contained no mathematical terms and

used the spatial relations provided in the problem (e.g., beore and ater).

In the current study, we have used these two, participant-developed strategies to

explore how they dierentially aect problem-solving processes. These two strategies are

quite interesting in that they represent two types o strategies to solve word problems that


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have been investigated or many yearspictorial/model and symbolic/equation methods.

The pictorial representations make explicit the important relationships and may, thereore,

aid in problem comprehension; this is not necessarily true in the symbolic representation.

In act, it has been ound that participants perorm better when provided with a picto-

rial representation (Koedinger & Terao, 2002; Lewis, 1989). However, here, participants are

not provided with a pictorial representation. Either they are to use a strategy in which

they generate one mentally to aid in problem solving or they are told to generate a more

symbolic representation mentally.

In order to help ground the study, we used the ACT-R model o problem solving/

cognition developed by Anderson and colleagues (described most recently in Anderson

et al., 2008) as a basis o comparison. The latest version o the ACT-R model described by

Anderson and colleagues (2008) involves our major modules: an imaginal module respon-

sible or constructing internal representations that is linked to parietal cortex; a declarative

memory retrieval module linked to the lateral prerontal cortex; a module responsible orthe setting o controlling goals linked to the anterior cingulate; and a procedural execu-

tion module linked to the head o the caudate nucleus, a part o the basal ganglia. These

modules are thought to be central to most o cognition and according to the ACT-R theory

these basic operations are cycled through in order to complete a cognitive task. Here we

ocus on three o these modules, the imaginal, the declarative memory retrieval, and the

setting o controlling goals modules.

The imaginal module is thought to be involved in the maintenance and transorma-

tion o internal representations. There is extensive neuroimaging and neuropsychological

data to support the posterior parietal cortices involvement in these processes (Carpenter

et al., 1999; Newman et al., 2007, 2003; Zacks et al., 2008). For example, in studies examin-

ing the Tower o London task both the let and right posterior parietal cortex have been

ound to be involved in the spatial processing necessary to solve the task, including spatial

working memory and spatial attention (Newman et al., 2003). In addition, Danker and

Anderson (2007) examined the neural bases o transormation and retrieval processes

during algebra problem solving and ound the posterior parietal cortex was closely linked

to these transormation processes. Based on the previous studies, the imaginal module

may be predicted to be involved in both the algebraic and spatial strategies. One question

that is addressed here is whether one strategy relies more heavily on these transormation

processes than the other.The declarative memory retrieval module, located in the lateral prerontal cortex, is

involved in both the retrieval and the selection o inormation rom memory stores. It has

been ound that how long inormation must be held as well as the diculty in selecting the

appropriate inormation rom memory drives the processing o this module. For example,

Gold and Buckner (2002) ound that the lateral inerior prerontal cortex was involved in

controlled retrieval rom memory or both semantic and non-semantic inormation. In that


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study a semantic task (deciding whether a visually presented word is abstract or concrete)

and a non-semantic task (deciding whether visually presented words and pseudowords

contain a long or short vowel) requiring controlled retrieval o inormation rom memory

were ound to elicit the involvement o the lateral prerontal cortex. In the current study

this region is thought to be involved in the retrieval o inormation such as arithmetic acts

or months o the year rom long-term memory stores.

The setting o controlling goals module, associated with the anterior cingulate cortex

(ACC), is thought to be responsible or indicating the state transitions during problem

solving. These processes are analogous to the cognitive control and error likelihood pro-

cesses that have been previously linked to the ACC (Botvinick et al., 2001; Brown & Braver,

2005; MacDonald et al., 2000). For example, Brown and Braver (2005) showed that the ACC

learned to predict the likelihood o committing an error or a given condition during a

stop-signal task using both MRI and a computational model o the region. In addition, the

ACC has been ound to respond to task diculty, particularly when diculty is dened bythe increase in the number o mental steps (Newman et al., 2009; Anderson et al., 2008).

However, it can be argued that when task diculty increases the likelihood o commit-

ting an error also increases. In the current study the ACC is expected to be more involved

when solving the hard problems, particularly i we assume, as is suggested above, that

during the hard problems participants reorder the phrases beore solving. This reordering

is an additional step that could increase the likelihood o error and, thereore, increase

the activation within the region. It is not clear at this point i dierences as a unction o

strategy are expected in this region. There is no reason to assume that the error likelihood

is dierent as a unction o strategy.

In sum, the primary aim o this study is to better characterize how strategy dier-

ences impact the underlying neural network that supports problem solving by examining

two qualitatively dierent strategies. As suggested above, while the algebraic and spatial

strategies appear to be quite dierent, they are expected to rely on very similar cognitive

processes and, as a consequence, may be expected to rely on similar brain structures. For

example, or both strategies there is the conversion o the word problem into a dierent

internal representation (one a number line and one an equation), the maintenance o that

representation in working memory, and the transormation o that representation (moving

the ball in one and isolating thexin the other). While both strategies are expected to rely

on overlapping brain regions, subtle dierences in how much they rely on these regionsare expected (i.e., the amount o activation in these regions are expected to vary with

strategy). For example, even though both strategies rely on working memory it may be that

the algebraic strategy relies heavily on verbal working memory while the spatial strategy

does not; thereore, greater involvement o regions linked to verbal working memory

such as the anterior insula or the inerior parietal cortex (Awh et al. 1996; Carpenter, Just,

& Reichle, 2000; DEsposito et al. 1999; Postle, Berger, & DEsposito, 1999; Smith and Jonides


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1999) may be expected when using the algebraic strategy. Another dierence may be in

the amount o declarative memory retrieval necessary. Long-term memory retrieval is re-

quired or both strategies; however, the algebraic strategy may be predicted to rely more

on retrieval processes i we assume that the strategy requires the association between the

terms beore and aterwith the arithmetic symbols o addition and subtraction.

Methods

Participants. A total o 23 individuals rom the Indiana University community participated

in the study. Data rom six participants were not used in the data analysis due to exces-

sive errors (> 40%), or an inability to consistently use the instructed strategy. The data

rom the remaining participants, 17 young adults (mean age = 24.1, 18-44; 7 emales, 10

males), are reported here. All participants were right-handed, native English speakers and

all participants gave written inormed consent approved by the IRB committee o IndianaUniversity prior to their participation.

Materials.The stimuli consisted o two levels o diculty o verbal problems (easy

versus hard) as shown in the examples above. Three additional variables were manipulated

to introduce supercial variation among the problems: (1) distance rom the reerence

point (e.g., the rst, second, or third month ater); (2) direction rom reerence point (i.e.,

beore or ater), and (3) problem domain (e.g., days, months, or letters).

Two problem-solving strategies were examined. The rst was a spatial strategy. This

strategy required participants to solve the problems by imagining moving backward and

orward along a number line. The second strategy examined was an algebraic strategy in

which participants were required to convert the verbal problem into an equation, substitut-

ing avorite orxand then solve orxas described in the introduction (see Figure 1).

A mixed event-related design was employed in which blocks o our trials were pre-

sented with the strategy to be used being displayed at the beginning o the block or one

second. Each block contained two easy and two hard problems randomly ordered. The

sentences were projected onto a transparent screen and viewed by the participant via

a mirror attached to the head coil. The rst hal o the sentence, phrase 1, was presented

alone on the screen or 4 sec (with a 500 msec delay ater). Aterward the second hal o

the sentence, phrase 2, was presented alone or 4 sec (with a 500 msec delay ater). The

probe, which consisted o two possible targets and Other, was then presented alone on

the screen or 3 sec (see Figure 2). The timing o presentation was obtained by taking the

time that corresponded to 75% o the trials in pilot data. Each trial was ollowed by a 12

sec rest period to allow or the hemodynamic response to approach baseline. Four 6.7

minute runs were presented with a total o 64 problems. Each run contained two 24 sec

xation periods, one at the beginning and one at the end, in order to obtain a common

baseline or comparison.


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Figure

1.Strategydescription.


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Procedure. Participants took part in a training session prior to scanning.During this

session participants completed a paper-and-pencil training task that prepared them or

the actual computer-based experimental task. During this training the experimenter

presented the participants example problems and explained the two strategies. Ater

explaining the strategies, the experimenter then demonstrated, on paper, how to use both

the spatial and algebraic strategies with both the easy and hard problems (see Figure 1).

Ater the demonstration, participants then completed 16 problems using paper and pencil

and were required to write out each step or draw the number line. The rst eight were

solved using the spatial strategy while the second eight were solved using the algebraic

strategy. There were an equal number o easy and hard problems in this practice. Their

perormance was checked to determine that they could solve the problems using the

appropriate strategy. I they ailed to understand either o the strategies ater the paper-

and-pencil problem-solving portion o the training they did not go on to the next part

o the training session but instead participated in another study being conducted in the

lab. Participants then were administered the experimental computer training that had

the same timing restrictions as the actual imaging version o the experiment in order toexpose the participants to the task and to ensure that they were able to perorm the task

under the time constraints required and without paper and pencil. Only participants who

could perorm the task, who understood both strategies, and who could perorm the task

under the computer-based conditions were scanned. Other participants were directed

to other ongoing studies. These steps were taken to ensure that only participants who

could use both strategies and who could adequately perorm the task were scanned. The

Figure 2. Timing o each trial.


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ancillary behavioral study described below, however, did not have a perormance criterion

and includes a wider range o participants.

A debrieng was perormed ater the training session and ater the scanning ses-

sion. The debrieng questions were designed to determine how well the participants

understood and were able to use both strategies when instructed. In addition, inormation

regarding their math background (the highest-level math class taken) was obtained as

well as measures o spatial processing ability, via the Vandenberg mental rotations task

(Vandenberg, 1971), and working memory capacity, via the reading span task (Daneman

& Carpenter, 1980). During the debrieng participants were also asked which strategy,

either the algebraic or spatial strategy, they preerred to use or that they considered

easier to use.

MRI Acquisition and Analysis. The images were acquired on a 3T Siemens TRIO scanner

with an 8-channel radio requency coil located in the Imaging Research Facility at Indiana

University. The unctional images were acquired in 18 5 mm thick oblique axial slices us-ing the ollowing parameters: TR = 1000 msec, TE = 25 msec, fip angle = 60, voxel size =

3.125 mm x 3.125 mm x 5 mm with a 1 mm gap.

The data were analyzed using statistical parametric mapping (SPM5 rom the Well-

come Department o Cognitive Neurology, London). Images were corrected or slice acqui-

sition timing, and resampled to 2 x 2 x 2 mm voxels. Images were subsequently smoothed

in the spatial domain with a Gaussian lter o 8 mm at ull-width at hal maximum. The

data were also high-pass ltered with 1/128 Hz cuto requency to remove low-requency

signals (e.g., linear drits). The images were motion-corrected and the motion parameters

were incorporated in the design estimation. The unctional, EPI images were registered

and normalized to the Montreal Neurological Institute (MNI) EPI template. At the individual

level, statistical analysis was perormed on each participants data by using the general

linear model and Gaussian random eld theory as implemented in SPM5. Each event

(trial) was convolved with a canonical hemodynamic response unction and entered as

regressors in the model (Friston et al., 1995). Although there were three phases or each

trial (phrase 1, phrase 2, and response) only one regressor that encompassed all phases

was used in this analysis.

Analysis was perormed based on a set o predened regions o interest (ROIs). The

ROIs were based on a series o studies by Anderson and colleagues (Anderson et al., 2004;

Qin et al., 2004; Rosenberg-Lee, Lovett, & Anderson, 2009; Stocco & Anderson, 2008) andincluded the let prerontal cortex (the declarative module), the let and right parietal

cortices (the imaginal module), the bilateral anterior cingulate cortex (the setting o con-

trolling goals module), and the head o the caudate (the procedural execution module).

These ROIs were dened by using the coordinates reported in Anderson et al. (2008). In

addition to these regions the let anterior insula was also examined. The center coordi-

nates or the anterior insula ROI were dened using the conjunction map, which revealed


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common activation across all conditions with a threshold op < 0.05, corrected or multiple

comparisons using amilywise error correction. All ROIs were dened as a sphere with a

radius o 5 mm and center dened by the coordinates listed in Table 1. Using the Mars-

bar toolbox (Brett et al. 2002), averaged timecourse data rom voxels within an ROI were

computed or each individuals imaging dataset and sorted by experimental condition.

The averaged timecourses across all trials were converted into percentage signal change

(PSC) using the ormula (signal - baseline/ baseline) 100 or each time point, where the

baseline constant was the mean signal o the xation periods. Then, the PSC timecourses

were baseline corrected to 0.

In addition to the ROI analysis, the random eects analysis on group data was per-

ormed using a one-sample t-test. Activated brain areas were dened using an uncorrected

threshold op < 0.001 and a cluster extent threshold o greater than 20 voxels and were

rendered on a template brain in SPM5. An exploratory analysis was also perormed to

examine strategy preerence eects.

Results

Behavioral Results. Even though much o the actual problem solving takes place when

processing the rst and second phrases, the response time to the probe was examined.

The response time to the probe, as well as the error rate, was subjected to a 2 x 2 within-

participant ANOVA. For response time, main eects o diculty [F(1,16) = 24.12,p < 0.001]

and strategy [F(1,16) = 8.7,p < 0.01] were observed, with the hard problems and problems

solved using the spatial strategy taking longer to respond to. There was no interaction

[F(1,16) = 1.92,p > 0.1]. The error rate analysis revealed main eects o diculty [F(1,16) =

30.06,p < 0.0001] and strategy [F(1,16) = 6.03,p < 0.05], with the spatial strategy having

ewer errors. There was no signicant interaction [F(1,16) = 2.75,p > 0.1] (see Figure 3).

A between-subjects ANOVA was perormed to examine the eect o strategy preer-

ence. Reaction time was ound to show a signicant eect o preerence [F(1,6) = 11.29,

p < 0.005], with the algebra preerence group showing a slower reaction time. Error rate

ailed to show an eect o strategy preerence [F(1,6) = 3.53,p = 0.067], but did reveal an

interaction between diculty and preerence [F(1,6) = 4.2,p


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Figure 3. Behavioral data.

Table 1. Predened ROIs

ROI Eects: F(1,16)

MNI coordinates

x,y,z

Let Superior Parietal Lobe 7 Strategy: 6.1*Diculty: 1.36

Interaction: < 1

-24, -62, 40

Right Superior Parietal Lobe 7 Strategy: 4.5*

Diculty: 5.94*

Interaction: < 1

26, -54, 41

Let Prerontal Gyrus 46 Strategy: 12.61*

Diculty: 5.51*

Interaction: < 1

-42, 30, 24

Let Anterior Insula 13 Strategy: 12.59*

Diculty: < 1

Interaction: < 1

-33, 21, 6

Let Anterior Cingulate 32 Strategy: 4.37*

Diculty: < 1

Interaction: 1.84

-5,8,47

Right Anterior Cingulate 32 Strategy: < 4.89*

Diculty: < 1

Interaction: 2.69

5,8,47

Note: *p < 0.05.


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1999; Postle, Berger, & DEsposito, 1999; Smith & Jonides, 1999), the results also suggest

that the algebraic strategy has greater working memory demands.

None o the predefnedregions revealed greater activation or the spatial strategy

compared to the algebraic strategy. Within the predened regions, diculty eects were

observed in the let prerontal cortex and right parietal cortex, with the hard problems

eliciting greater activation. None o the predened regions revealed a signicant interac-

tion.

Eect o Strategy Preerence. In addition to the above analysis, dierences as a unc-

tion o strategy preerence were also analyzed (see Table 2 and Figure 4). Participants

were asked during a debrieng ollowing scanning which strategy, i any, they preerred.

Seven participants reported preerring the algebraic strategy, seven the spatial strategy,

and three reported having no preerence. As a result, the 14 participants with a preerence

were examined. Within the predened ROIs, the let prerontal, let anterior insula, and

right parietal regions revealed signicant eects o preerence, with the spatial preerencegroup eliciting greater activation while the let ACC ROI revealed greater involvement or

the algebra preerence group. Interestingly, these two groups were very similar in terms

o their math background (they were asked the highest-level math course they took) and

their spatial ability as measured by the Vandenberg mental rotation task (p > 0.6). However,

there was a dierence in working memory capacity as measured by the reading span

task (p < 0.001), with the algebra preerence group having a higher span (3.5 and 2.7 or

algebra and spatial preerence, respectively).

Whole Brain Analysis. In addition to the ROI analysis, whole brain analysis was per-

ormed. Using an uncorrected threshold op < 0.001 and an extent threshold o greater

than 20 voxels, the main eect o strategy revealed three clusters o activation, one in

the rontal cortex that extended rom the anterior insula anteriorly and superiorly to the

middle rontal gyrus, as well as clusters in bilateral superior parietal cortex (see Table 3

and Figure 5). The main eect o diculty revealed that let prerontal and let temporal

cortices showed greater activation or the hard problems, while regions including the right

rostral ACC (which lies more anterior and inerior to the predened ACC ROI) and bilateral

middle/posterior insula revealed greater activation or easy problems. The interaction

between the two actors revealed that a number o regions, including bilateral rostral

ACC, temporal, and cerebellar regions, showed greater activation or the spatial/hard and

algebra/easy problems compared to the spatial/easy and algebra/hard problems.Ancillary Behavioral Study. A total o 44 individuals participated in the study (24 emales

and 20 males; mean age = 20.8 4.3). All participants gave inormed, written consent ap-

proved by the Indiana University IRB. The experimental session consisted o three phases:

psychometric testing (the Daneman and Carpenter [1980] reading span [working memory

capacity] and the Vandenberg [1971] mental rotations tests); a paper-and-pencil training

and debrieng; and the experiment.


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Table 2. Predened ROIs: between-subjects preerence analysis

ROI Eects: F(1,6)

MNI coordinates

x,y,z

Let Superior Parietal Lobe 7 Preerence: 1.73Strategy: 3.94*

Diculty: 1

Preerence*Strategy: < 1

Preerence*Diculty: 1.53

3-way: < 1

-24, -62, 40

Right Superior Parietal Lobe 7 Preerence: 10.46*

Strategy: 3.05

Diculty: 5.33*

Preerence*Strategy: 1.23

Preerence*Diculty: 4.78*

3-way: < 1

26, -54, 41

Let Prerontal Gyrus 46 Preerence: 60.91**Strategy: 18.38**

Diculty: 5.32*


Preerence*Diculty: < 1

3-way: < 1

-42, 30, 24

Let Anterior Insula 13 Preerence: 17.24**

Strategy: 15.58**

Diculty: 1.51



3-way: < 1

-33, 21, 6

Let Anterior Cingulate 32 Preerence: 19.85**

Strategy: 3.88*

Diculty: < 1


Preerence*Diculty: 1.05

3-way: < 1

-5,8,47

Right Anterior Cingulate 32

Preerence: < 1

Strategy: 4.43*

Diculty: < 1



3-way: < 1

5,8,47

Note: *p < 0.05, **p < 0.0001.


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Figure

4.M

aineectsostrategyanddifculty.


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Table 3. Exploratory Analysis

ROI (Brodmanns Area) k z MNIx,y,z

Strategy eect: Algebra > Spatial

Let Inerior Frontal/Insula 13 315 4.18 -34, 22, 6

Let Middle Frontal Gyrus 46 4.18 -44,32,22

Let Superior Parietal Cortex 7 549 3.95 -20, -68, 42

Right Superior Parietal 7 70 3.92 28, -54, 40

Let Cerebellum 27 3.66 -10, -76, -12

Let Precentral Cortex 6 45 3.46 -40, -4, 34

Right Cerebellum 41 3.41 30, -64, -22

Diculty: Hard > Easy

Let Prerontal Cortex 46 47 3.56 -46, 24, 26

Let Temporal Cortex 37 37 3.58 -52, -46, -6

Diculty: Easy > Hard

Let Hippocampus 208 4.79 -32, -44, 2

Right Insula 13 326 4.44 42, 10, -6

Let Insula 13 650 4.43 -44, 0, 2

Let Cerebellum 56 4.30 0, -38, -28

Let Caudate Nucleus (Tail) 171 4.25 16, -32, 20

Right Caudate Nucleus (Tail) 75 3.89 -12, -30, 22

Right Superior Frontal/ACC 9/32 68 3.65 20, 40, 28

Interaction between Strategy and Diculty

Cerebellum 140 4.3 -6,-48,-16Let Superior Temporal 39 769 4.27 -38,-54,30

Let Middle Temporal Gyrus 37 199 4.23 -46,-66,8

Let Hippocampus 286 4.05 -30,-32,-8

Let Middle Frontal Gyrus 6 135 3.98 -38,14,44

Let Medial Frontal/ACC 9/32 101 3.97 -14,42,22

Let Anterior Insula 13 70 3.87 -42,0,-2

Right Anterior Insula 13 68 3.61 32,-8,16

Let Precentral Gyrus 4 92 3.55 -32,-20,52

Medial Frontal 6 109 3.51 0,-28,54

Let Fusiorm Gyrus 37 54 3.98 -48, -56, -16Right Cerebellum 36 3.73 8, -64, -30

Inerior Frontal Gyrus 47 26 3.71 -48, 42, -8

Right Medial Frontal/ACC 10/32 53 3.7 14, 46, 16

Let Caudate Nucleus (Body) 33 3.55 -10, 8, 18

Let Parietal 2 31 3.49 -44, -28, 54

Let Insula 13 41 3.46 -34, -6, 18

Right Middle Temporal 37 23 3.42 50, -60, 6


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Figure

5.EectostrategypreerenceinpredefnedROIs


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The experiment was similar to the imaging study in terms o stimuli used, the order

o presentation, and the training procedures, with three exceptions. First, the behavioral

study was a sel-paced design in which each individual trial was presented in the ollowing

manner: phrase 1 (button press), phrase 2 (button press), probe (respond)in order to

obtain processing times or each portion o the problem. Because problem solving is not

thought to occur only during the presentation o the probe but actually when reading each

phrase and because the timing needs to be controlled during imaging, it is important to

obtain these data. The second dierence here is that there were no perormance cut-os.

Ater the paper-and-pencil training i participants perormed poorly, they were given an

opportunity to redo that portion instead o being directed to a dierent study. Finally, there

was no timed computer training session prior to the perormance o the experiment.

The reaction time to the probe and the error rate both revealed signicant diculty

eects [RT: F(1,43) = 14.11,p < 0.001; error: F(1,43) = 20.85,p < 0.0001], both ailed to show

a main eect o strategy [Fs < 1], the error data revealed a signicant interaction [F(1,43)= 8.55,p < 0.01]. This appears to contradict the results rom the scanner; however, when

the phrase reading times are examined it is clear that problem solving is taking place

during problem presentation (see Figure 6). In act, phrase 2 revealed a signicant eect

o strategy [F(1,43) = 13.58,p < 0.001] and phrase 1 revealed a similar trend such that the

algebraic strategy took longer to read. This suggests that the solution may have been

generated during phase 2 instead o during the probe phase in this sel-paced task due

to the lack o time constraints.

Strategy preerence was also examined. Here 27 participants reported preerring the

algebraic strategy and the remaining 17 reported preerring the spatial strategy; no one

reported having no preerence. The between-subjects analysis ailed to show signicant

eects o preerence; however, each measure revealed an eect o diculty (p < 0.01).

This lack o an eect o preerence may be due to the unequal group size and the greater

variance observed in each group possibly due to the lack o restrictions placed on peror-

mance or to participants randomly choosing a preerence when they did not have one.

In an exploratory analysis, within-participant ANOVAs were perormed on each group

separately. This analysis revealed that the algebra preerence group showed an interac-

tion between strategy and diculty [F(1,26) = 8.82,p < 0.01] primarily due to no eect o

diculty when using their preerred strategy. Additionally, the spatial preerence group

revealed an overadditive interaction or the reading time o phrase 2 [F(1,16) = 6.84,p

Documents

Effect of Strategy on Problem Solving