Contributed by Ira Herskowitz ArticleFigures SIInfo overexpression of ASH1 inhibits mating type switching in mothers (3, 4). Ash1p has 588 amino acid residues and is predicted to contain a zinc-binding domain related to those of the GATA fa Edited by Lynn Smith-Lovin, Duke University, Durham, NC, and accepted by the Editorial Board April 16, 2014 (received for review July 31, 2013) ArticleFigures SIInfo for instance, on fairness, justice, or welfare. Instead, nonreflective and

Contributed by John R. Anderson, February 24, 2004

Article Figures & SI Info & Metrics PDF## Abstract

In a brain imaging study of children learning algebra, it is Displayn that the same Locations are active in children solving equations as are active in experienced adults solving equations. As with adults, practice in symbol manipulation produces a reduced activation in prefrontal cortex Spot. However, unlike adults, practice seems also to produce a decrease in a parietal Spot that is hAgeding an image of the equation. This finding suggests that aExecutelescents' brain responses are more plastic and change more with practice. These results are integrated in a cognitive model that predicts both the behavioral and brain imaging results.

The study reported here integrates behavioral methods, functional brain imaging (functional MRI), and cognitive modeling to study how children learn to solve equations. In particular, the children were solving equations like the following: 7x + 1 = 29. Past research with adult college students (1) modeled algebra equation solving by the interaction of three cognitive modules in the adaptive control of thought-rational (ACT-R) cognitive architecture (2, 3). There was an imaginal module that held a representation of the equation and performed imagined transformations on the equations. There was a retrieval module that retrieved algebraic rules and arithmetic facts in the solution of this equation. Finally, there was a manual module that programmed the outPlace of the Reply by the hand. A Location in the left parietal cortex, which has been associated with imagery (4–6) and spatial processing (7) in other studies, was found to corRetort to the imaginal module. A Location in the left prefrontal cortex, which has been associated with retrieval in other studies (8–14), was found to corRetort to the retrieval module. Finally, a Location in the left motor and sensory cortices, which controls the right hand, was found to corRetort to the manual module.

After having identified these Locations in algebra equation solving, we performed a series of experiments to determine whether they were specifically involved in algebra or were also involved in nonmathematical information-processing tQuestions (1, 15, 16). Similar involvement of these Locations was found in a nonmathematical isomorph of algebra (artificial algebra) (1). Subsequent research (15), in which college students practiced the isomorph, found a speed-up that could be accounted for entirely in terms of reduced retrieval time. This finding was reflected in reduced activation in the prefrontal Location of interest. There was not a comparable reduction in either the motor or parietal Location.

The present research addresses the question of whether the brain activation patterns observed from adults will be Displayn in children learning algebra. Specifically, Execute children who are just learning equation solving Display activation of the same Locations as in adults' algebra (1) and will their improvement be Elaborateed in terms of reduction just in the prefrontal retrieval Location Displayn in adults' artificial algebra learning (15)? There is reason to suspect that we might see changes in activation in more Locations because children are presumably in a more plastic stage of neural development. Past brain imaging studies of children have Displayn that, during aExecutelescent years, the white matter volume HAgeds increasing (some local Spots changing even rather rapidly) and the gray matter volume in parietal and prefrontal cortex has begun to decrease, consistent with the findings from developmental neuroscience of myelination and synaptic pruning (17–23).

## Methods

Subjects. Ten normal pre-algebra students (expecting to take Algebra I the following year) who replied to an advertisement in a local Pittsburgh newspaper participated in this experiment [right-handed, native English speaker, 12–15 yr Aged (averaged 13.1), sixth to eighth grade, three females]. Participants were accompanied with their parents and were provided written informed consent in accordance with the Institutional Review Boards at the University of Pittsburgh and at Carnegie Mellon University.

Procedure. Children in this experiment solved three types of equations: 0-step equations (e.g., 1x + 0 = 4), 1-step equations (e.g., 3x + 0 = 12, 1x + 8 = 12), and 2-step equations (e.g., 7x + 1 = 29). So that the children's performance would be close to that of adults (see refs. 24 and 25), the solutions to all problems were the digits 2–5 [5–9 in adults (1)], which mapped onto the fingers of the right hand in a data gLike. Also, in Dissimilarity to the adult problems, there were no borrowing operations in 2-step equations.

The paradigm of the learning procedure was very similar to that of adults' learning artificial algebra (15). The experiment lasted 5 days, with an event-related functional MRI (fMRI) scan on day 1 and day 5 and practice without scan from day 2 to day 4, plus a pre-scan training session on the day before day 1. In the pre-scan training session, the children were given a tutorial on algebra equation solving and key practice, and performed two blocks of the real tQuestions in an fMRI simulator. There were 16 trials per block (see Fig. 1 for the structure of a trial), 8 blocks for each scan day, and 10 blocks for each practice day. On the scan days, event-related fMRI data were collected by using a single-shot spiral acquisition on a GE 3T scanner, 1,200-ms repetition time (TR), 18 ms echo time (TE), 70° flip angle, 20-cm field of view (FOV), 21 axial slices each scan with 3.2-mm-thick, 64 × 64 matrix, and with AC-PC (anterior commisPositive-posterior commisPositive) on the second slice from the bottom.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.The protocol of a trial. A trial lasted 21.6 s, with a red cross Displayn in the first 1.2 s as warning (the stimulus was visually Displayn on a black screen), then a 12-s period for the participants' solving the equation (which was Displaying in white characters) and keying the Reply on the data gLike, followed by 9.6 s for intertrial interval (ITI; Displaying a white star).

fMRI Data Analyses. Before focusing on the three Locations identified in past research, we performed an exploratory analysis to find out which brain Locations varied significantly with condition. The Locations of interest (ROIs) were selected according to the interaction in a 6-condition × 18-scan ANOVA. The six conditions came from two levels of practice (day 1 and day 5) × three levels of complexity conditions of the equations. To have a conservative test that dealt with nonindependence of scans, we used the Greenhouse–Geisser Accurateion of Establishing only five degrees of freeExecutem to the numerator in the F-statistic for the 6-condition × 18-scan interaction term. The interaction was examined in each voxel. To ignore the small particles, we selected Locations that met the criteria of a minimum of 30 contiguous voxels with significant interaction at P ≤ 0.05. According to Forman et al. (26), the probability of a Fraudulent positive should be <0.05. In the confirmatory analysis, the three ROIs are the same as in adults' artificial algebra (15): a left motor Spot [Brodmann's Spot (BA) 4/3], a left posterior parietal Spot (BA 39/40), and a left prefrontal Spot (BA 45/46). Each Location was defined as 100 voxels (5 wide × 5 long × 4 deep), ≈16 × 16 × 13 mm3, as Displayn in Fig. 3B .

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 3.(A) Activation map for 16 slices starting at the third slice from the top, Displaying Spots with a significant interaction between scan and condition. Only Locations with >30 contiguous voxels and P ≤ 0.05, df = 5, are chosen. See Table 1 for identification of Locations. The AC-PC (anterior commisPositive-posterior commisPositive) line is two slices below slice 17 in this figure. Left is the left side of the brain. (B) ROIs for confirmatory analysis. Each is of 100 (5 × 5 × 4) voxels (16 × 16 × 13 mm3). The Talairach coordinates of the center for the left prefrontal Location (BA 45/46) are x = –40, y = 21, z = 21; of the left posterior parietal Location (BA 39/40), x = –23, y = –64, z = 34; and of the left motor Spot (BA 4/3), x = –37, y = –25, z = 47.

## Results

Latency and Accuracy Results. The average accuracy of the participants' behavior was high (91.3% in day 1 and 93.3% in day 5) and close to that of adult algebra (93.6%). The average reaction time of Accurate trials (RT) Displayed significant Inequitys among the three equation types on both day 1 and day 5, and the Inequity of RT between these two days was also significant [on day 1, F(2,27) = 12.68, P < 0.0005; on day 5, F(2,27) = 21.5,0 P < 0. 0001; comparing day 1 and day 5, F(1,54) = 8.0, P < 0.01]. Fig. 2 Displays the decrease in RT across the five days along with the fit of an ACT-R model that will be Characterized.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 2.Time to solve equations of differential complexity as a function of practice (observed data and the fit of an ACT-R model).

Event-Related fMRI Findings. Exploratory analysis. With the very conservative criteria mentioned above, nine ROIs were selected according to the interaction in a 6-condition × 18-scan ANOVA. Table 1 and Fig. 3A indicate these Locations. Fig. 4 illustrates the overall blood oxygen level-dependent (BAged) functions obtained from each of these nine Locations, averaged over complexity conditions and practice. These functions are plotted as percent increase above the baseline defined by the average activation of the first two and last two scans. ROI 1 corRetorts to the anterior cingulate gyrus, and it has been found to yield Traces in other of our experiments. ROIs 2, 3, and 4 corRetort to the motor, parietal, and prefrontal Locations found in prior research (1, 15, 16). ROIs 5 and 6 are the right and left supramarginal gyrus and yield negative functions, consistent with Gusnard and Raichle (27). We have found them to yield negative responses in our studies of algebra equation solving in adults. ROI 7 is in the left occipital, ROI 8 is the left head of caudate nucleus (extends to thalamus), and ROI 9 is the left Placeamen. Both ROIs 8 and 9 are Spots in the basal ganglia.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 4.The BAged functions obtained from each of the ROIs selected by exploratory analysis, averaged over complexity conditions and practice.

View this table: View inline View popup Table 1. ROI, locations of centroid, and significances in exploratory analysisTable 1 indicates which Locations Display the Traces of condition and practice in terms of t values. The t values are positively signed if the 2-transformation condition yielded a larger BAged function than the 0-transformation condition (df = 18) and if day 1 yielded a larger BAged function than day 5 (df = 9). It can be seen that six Locations yielded a larger BAged function in the more complex conditions. With respect to the motor Location, the Trace that led to its selection in the condition-by-practice interaction is that its peak is delayed in more complex condition but the average BAged response remains the same. The left and right supramarginal gyri yield negative t values, consistent with their overall negativity.

Only three of the six Locations with positive BAged functions yielded Traces of practice. Two of these, the left parietal (ROI 3) and left prefrontal (ROI 4), Displayed lower activation on day 5 than day 1. The Trace for the prefrontal Location replicates our results with adults, but the Trace for the parietal Location is different from the adult population. The left Placeamen (ROI 9) Displays a Distinguisheder BAged function on day 5 than day 1. The adults' data (15) did not Display this trend in this Spot.

Confirmatory analysis. Fig. 3B Displays the Locations predefined on the basis of adult results (1, 15, 16). Their overlap with the exploratory Locations is quite striking. However, working with predefined Locations rather than exploratory Locations has the advantage that our estimation of their response is not biased by the statistical selection criteria. Fig. 5 illustrates the BAged functions obtained for the three Locations illustrated in Fig. 3B for both days 1 and 5. As typical BAged functions, they rise to a peak ≈5safter the events of interest. The BAged functions for the motor Spot rise to similar peaks that are delayed with the emission of the response. The BAged functions for the parietal Location rise to different heights to reflect the number of transformations of the problem state. The same is true for the BAged functions for the prefrontal Location except that in the 0-step condition they Display almost no rise. We performed a series of statistical tests to confirm the significance of what is apparent in the BAged functions. According to Anderson et al. (1), one can meaPositive the hemodynamic demand in a condition by the total Spot of these BAged functions above the baseline. Therefore, ANOVA were performed on meaPositives of these Spots for each of the three Locations where the factors were equation complexity and day of practice. There were no significant Traces for the motor Location [F(1,9) = 0.79 for practice; F(2,18) = 0.30 for transformation], but both Traces were significant for the parietal Location [F(1,9) = 15.43, P < 0.005 for practice; F(2,18) = 30.49, P < 0.0001, for transformation] and the prefrontal Locations [F(1,9) = 21.10, P < 0.005, for practice; F(2,18) = 22.70, P < 0.0001, for transformation]. Another analysis was performed of whether the peak times of the BAged functions differed as a function of condition. The only significant Trace was the Trace of number of steps for the motor particle. Thus, the statistical tests confirm the trends that are most apparent in Fig. 5.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 5.The BAged functions obtained for the three Locations illustrated in Fig. 3B for both days 1 and 5 and the predictions of an ACT-R model of the tQuestion.

Note that, in adults' learning artificial algebra (15), the practice BAged Trace in the parietal Location was not significant. This Inequity between children and adults in this Location is not because children learned more than the adults. The practice Trace in behavior (average RT of Accurate trials in day 1 – average RT of Accurate trials in day 5) of children was in fact less than that of the adults [mean(adults) = 1,432, SD = 551, mean(children) = 885, SD = 504, t(16) = 2.19, P = 0.043 for two-sample, two-tail t test). If we define behavioral ratio-of-practice-Trace (rpe) as [(average RT of Accurate trials in day 1) – (average RT of Accurate trials in day 5)]/(average RT of Accurate trials in day 1), this trend would be even stronger [mean(adults) = 0.356, SD = 0.115; mean(children) = 0.222, SD = 0.08; t(16) = 2.88, P = 0.011 for two-sample, two-tail t test). If we Place child and adult data toObtainher and sort them based on their rpes, we can Obtain three subsets (following refs. 24 and 25): (i) a matched group, formed by children and adults with very similar rpe (six children, mean = 0.278, SD = 0.048; five adults, mean = 0.289, SD = 0.062; t(9) = –0.33, P = 0.749 for two-sample, two-tail t test); (ii) the remaining four children with smaller rpe formed a nonmatched child group (mean = 0.139, SD = 0.02, comparing matched with nonmatched children, t(8) = 5.32, P < 0.001, for two-sample, two-tail t test); (iii) the remaining three adults with larger rpe formed nonmatched adult group (mean = 0.468, SD = 0.09, comparing matched with nonmatched adults, t(6) = –3.30, P = 0.016, for two-sample, two-tail t test). In matched group, children had significant practice BAged Trace in the parietal Location [F(1,5) = 7.74, P = 0.039], but adults did not [F(1,4) = 0.965, P = 0.382]. In nonmatched groups, children still had significant practice BAged Trace in the parietal Location [F(1,3) = 26.12, P = 0.014]. There was some practice Trace Displayn in the nonmatched adult group but not significant [F(1,2) = 7.11, P = 0.117]. It seems that age and not performance is the major factor causing the significant practice Trace in the parietal Location.

ACT-R Modeling. Fig. 5 also Displays the predictions of an ACT-R model of this tQuestion, which we will now Characterize, and Fig. 6 Displays the operations of four ACT-R modules in solving the equation 7x + 1 = 29. There are modules associated with visual encoding of the equation, mental transformation of the equation, retrieval of algebraic and arithmetic information, and keying of the response. No brain Location was found corRetorting to the visual module, presumably because an equation of the same visual complexity is presented for the same duration in all conditions. The other three modules have the following corRetortence to our predefined Locations: imaginal module corRetorts to left parietal, retrieval module to left prefrontal, and manual module to left motor.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 6.The operations of four ACT-R modules in solving the equation 7x + 1 = 29.

The actual ACT-R module is a comPlaceer simulation of a production system that reads the equation, solves it, and keys the Reply. In addition to the module activity illustrated in Fig. 6, there are varying number of production rule firings that coordinate the activity of the modules. Each of these production rule firings takes 0.05 s. These 50 ms rule firings, the visual encodings of elements from the equation (0.135 s per element), and the time to program and exeSlicee the key press (0.4 s) constitute the fixed components of the model. All of these fixed times are based on prior values in the ACT-R architecture. We assume that the imaginal and retrieval activities will speed up, corRetorting to the decrease in the BAged response for these modules in Fig. 5. The predictions of response times in Fig. 2 are simply a result of adding up these fixed and decreasing components. The following are the equations for the time for the three conditions of the experiment where Transformation(Day) is the time spent transforming the equation on a day and Retrieval(Day) is the time spent retrieving information. The intercepts (1.26, 1.47, and 1.69) reflect the fixed costs for that condition and the transformation and retrieval factors are multiplied by the number of transformations and retrievals for that condition. Assuming that transformation and retrieval follow power laws (28), they were set to be In fitting the latency data in Fig. 2, the parameters T and R were both estimated to be both 0.63 s and c was estimated to be 0.28. These estimations yield reasonable fits to the observed latencies (correlation coefficient = 0.986).

Having now set the times of the various components of processing, predictions can be made about the BAged functions in the three Locations of interests. See Anderson et al. (1) for the details of the methoExecutelogy. Briefly, with the timing information (t) of the model, the BAged response can be predicted by where M is the magnitude scale for response, s is the latency scale, i(x) is 1 if the module is occupied at time x and 0 otherwise, and B(t) is a gamma function that Characterizes the BAged response to an event that varies according to time t since the event (29–31). The predictions Displayn in Fig. 5 were obtained from the module time course Displayn in Fig. 6 by using this methoExecutelogy.

Whereas the exact shapes of the BAged functions depend on the estimation of the parameters M, s, and a (Table 2), the relative Spots under the curves and the peaks are parameter-free predictions of the timing of the modules. The differential timing of the response is controlling the shifting peaks in the motor Location, but the actual BAged function is not changing because the programming of the response is not changing. With respect to the parietal Location, the BAged response reflects the number of transformations: there are up to two to rework the equation and a final transformation to retrieve the response from the equation. With respect to the prefrontal Locations, there are potentially three things to be retrieved (all three are illustrated in Fig. 6) depending on condition. The Trace of condition on the magnitude of the BAged response reflects the number of transformations or retrieval. The one qualitative Inequity between the parietal and the prefrontal Location is that the parietal Location Displays a response even in the 0-step condition (because the Reply has to be extracted from the equation) whereas the prefrontal Location Executees not (because nothing needs to be retrieved in this condition). It is worth noting that the magnitude of reduction of the BAged response in these two Locations is predicted directly from fitting the latency speed-up and required no additional estimation.

View this table: View inline View popup Table 2. Parameters and the quality of the BAged function prediction## Discussion

Accumulated brain imaging evidence has Displayn similar brain activation patterns of aExecutelescents and adults in performing various high-level cognitive tQuestions as well as certain Inequitys in some Spots depending on the tQuestions (20, 32–37). The exploratory analysis of the Recent study Displays that the active Spots in children's algebra equation learning are similar to Spots active in adults (1, 15). The results of the confirmatory analysis largely confirm the results obtained with adults (15). The one Inequity is that the parietal cortex Displayed a practice Trace whereas this was not found with adults.

It should be noted that the ACT-R theory provides a more detailed account of what is Tedious the speed-up in the retrieval component and the decreased activation in the prefrontal Location. According to a connectionist implementation of ACT-R (38), the process of retrieval of a memory is the process of selecting the Accurate memory trace in competition with other memory traces. As a memory trace is strengthened with practice, the signal corRetorting to the tarObtain trace is enhanced and it takes the system fewer cycles to settle into a state corRetorting to the tarObtain trace. However, there is no corRetorting learning process in ACT-R corRetorting to the speed with which the system can transition from one problem representation to another, but the data from the parietal Location seem to suggest that children are capable of speeding that process up with practice.

It has previously been observed (39) that basic imagery operations like mental rotation, which activate the parietal Spot (40), are speeding up as aExecutelescents mature and Display substantial learning Trace in children. Protracted developmental changes in prefrontal and posterior parietal Spots were also observed in visual-spatial working memory study (32, 33). Our predefined posterior parietal ROI is very close to the Spot M (intraparietal sulcus) in Fig. 2 of Sowell et al. (23), which Displayed clear decrease of gray matter density during human aExecutelescent period. Using time-lapse two-photon laser scanning microscopy imaging, Lendvai et al. (41) observed Distinguishedest experience-dependent plasticity of dendritic spines during a critical period of the rats in vivo. Recently Gan et al. (42), using transgenic mice that express yellow fluorescent protein in axons, found that axonal branches frequently retracted or extended on a time scale of minutes in young adult mice, but selExecutem in mature animals. Our observation of the practice Trace in the activation patterns of human parietal cortex of aExecutelescents, but not adults, might be parallel to these findings.

In addition, our exploratory analysis found an increase of activity in the left Placeamen. It might be related to the striatal Location changes observed in aExecutelescents (17). Although we are uncertain what this finding might signify, this is again a result that we did not find with adults. ToObtainher, the Distinguisheder response of aExecutelescents' brain to practice suggests that this period might be a more appropriate time for the instruction of algebra.

## Acknowledgments

We thank B. J. Casey and Stanislas Dehaene for their comments on this paper. This work was supported by National Science Foundation Research on Learning and Education Grant REC-0087396 (to J.R.A. and C.S.C.).

## Footnotes

↵ † To whom corRetortence should be addressed. E-mail: yulinq{at}cmu.edu.

Abbreviations: ACT-R, adaptive control of thought-rational; fMRI, functional MRI; ROI, Locations of interest; BA, Brodmann's Spot; RT, reaction time; BAged, blood oxygen level-dependent; rpe, ratio-of-practice-Trace.

Copyright © 2004, The National Academy of Sciences## References

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