Behavioral equivalence, but not neural equivalenceâ•flneural evidence of alternative strategies in mathematical thinking

representation of problem information regardless of representation formats8,9. If so, we would not observe differences in brain activations. In contrast, if they use different representations, we would observe activations in different brain regions depending on representation format. We present evidence here that story and equation problems are solved in similar ways but that they do differ in terms of the brain regions involved. Of the 24 participants, 12 memorized eight prototypes of algebra problems in a verbal, story format (for example, “Brian earns $7.00 an hour and gets $9.00 tips for his earnings.”) and another 12 memorized the same prototypes in a symbolic, equation format (for example,“7H + 9 = E”). Each prototype was given an identifier. The prototypes required two arithmetic operations, one of which was either division or multiplication and the other either addition or subtraction. To assure that any differences were due to the mental representation of the problems and not physical differences, we tested participants with identical stimuli. Participants were asked either to figure out the result given a start value (for example, “hours = 3, earnings =?”) or to figure out the start value given a result (for example, “earnings = 29, hours =?”). Thus, the two groups of students solved exactly the same problems presented in the same format, while accessing the prototypes memorized in different formats. Half of the algebra problems involved single-digit answers (simple problems) and the other half involved double-digit answers (complex problems). Participants first practiced the problem-solving phase out1Department of Psychology and 2Human Computer Interaction Institute, Carnegie-Mellon University, Pittsburgh, Pennsylvania, USA. 3Department of Radiology, University of Pittsburgh Medical Center, Pittsburgh, Pennsylvania, USA. 4Department of Psychology, University of Pittsburgh, Pittsburgh, Pennsylvania, USA. 5Departments of Psychiatry and Psychology, University of California, Davis, California, USA. Correspondence should be addressed to M.-H.S. ([email protected]). Published online 10 October 2004; doi:10.1038/nn1337 NATURE NEUROSCIENCE VOLUME 7 | NUMBER 11 | NOVEMBER 2004 1193 Figure 1 Activation changes in each representation format condition. (a) Changes in left prefrontal cortex. (b) Changes in bilateral parietal cortices. © 20 04 N at u re P u b lis h in g G ro u p h tt p :/ /w w w .n at u re .c o m /n at u re n eu ro sc ie n ce B R I E F COM M U N I C AT I O N S side a scanner and then solved problems in a scanner. In both practice and scanning sessions, participants solved 128 problems presented in 8 blocks of 16 algebra problems. In the scanning session, event-related fMRI data were collected by using a single-shot spiral acquisition on a GE 3T scanner, 1,500 ms TR, 18 ms TE, 70° flip angle, 20 cm field of view, 27 axial slices/scan with 3.2-mm-thick, 64 × 64 matrix, and with anterior commisure–posterior commisure (AN–PC) on the seventh slice from the bottom. Scanning was synchronized with each trial, with 14 scans of 1.5 s each per trial. Images were motion corrected and crossregistered to a Montreal Neurological Institute brain using the 12parameter rigid-body model of image registration nonlinear warping procedure. Functional images were set to a standard mean intensity and smoothed (8 mm full-width half-maximum 3D Gaussian kernel). We expected that students in the story condition would access the problem information using a verbal representation, whereas students in the equation condition would do so using a nonverbal, symbolic representation. Behaviorally, there was a problem difficulty effect both on accuracy (0.68 for complex and 0.85 for simple problems, F1,22 = 59.99, P < 0.0001) and on latency (8.2 s for complex and 6.5 s for simple problems, F1,22 = 418.56, P < 0.0001). However, there was not a main effect of representation format nor of its interaction with problem complexity, Ps > 0.30. Despite this behavioral equivalence between the two formats, it is possible that participants were solving problems differently depending on how they were framed. The behavioral equivalence may have resulted because participants could bypass the initial representation differences8,9. Alternatively, the equivalence could be achieved by equally efficient use of different representations. To examine these possibilities, we conducted voxelwise (voxel size 3.125 × 3.125 × 3.2 mm) ANOVAs with representation format and scan as variables. A region of interest was defined as a cluster of >7 contiguous voxels10 showing representation-by-scan interaction at the significance level of F13,288 = 4.35, P = 0.00001, Bonferroni corrected for multiple comparisons. The left anterior dorsolateral prefrontal cortex (Brodmann area (BA) 9, –49, 25, 28) showed greater activation increase in the story condition than in the equation condition (Fig. 1). In contrast, bilateral posterior parietal regions (BA 7, –11, 73, 46; 18, –73, 43) showed the reversed pattern. If the prefrontal region mediates accessing problem representation in the verbal format, its activation should affect the latency from the story condition but not the latency from the equation condition. Similarly, if the parietal regions truly involve the equation format, their activation should affect the latency only from the equation condition. Trials were median-split in terms of summed activation change over a trial. A three-way ANOVA was conducted with the representation format, region (prefrontal and parietal) and activation change (high and low) as variables. Figure 2 shows the threeway interaction (F1,22 = 8.09, P < 0.01). The latency in the story condition was faster on the trials with higher rather than lower prefrontal activation change (7.5 s and 7.1 s, t11 = 2.86, P < 0.05). The equation condition latency did not differ depending on the prefrontal activation (P > 0.90). The latency in the equation condition was faster on the trials with higher rather than lower parietal activation change (7.6 s and 7.2 s, t11 = 3.99, P < 0.01). The story condition latency did not differ depending on the parietal activation (P > 0.80). In another analysis, we identified regions showing significant problem difficulty effect in both representation conditions. The left anterior cingulate (BA 24, –3, 6, 48), left precentral gyrus (BA 4, –33, –18, 48), left midfrontal gyrus (BA 6, 29, –5, 49), left prefrontal cortex (BA 9, 49, 5, 32) and left precuneus (BA 7, –8, –60, 51) showed greater activation change for the complex problems. The problem difficulty effect was not modulated by the representation format in any of these regions. Moreover, no regions were identified to show a significant three-way interaction involving representation format, problem complexity and scan. The lack of a significant three-way interaction suggests that the representation format effect and the problem difficulty effect are associated with different brain regions. Our results show that alternative neural pathways can underlie equivalent behavioral performance. The observed prefrontal-parietal dissociation is consistent with previous studies showing verbalsymbolic distinction in these regions1,2. The dissociation we showed is particularly notable because it demonstrates the capacity to choose between alternative neural routes to represent and manipulate the same information with the same extent of efficiency. This notion resonates with a recent demonstration that different brain activities during encoding could result in similar behavioral performance in a subsequent memory test11. Behavioral equivalence, therefore, does not necessarily imply neural equivalence. ACKNOWLEDGMENTS This research is supported by US National Science Foundation grant BCS-9975220 (to J.R.A., K.R.K. and C.S.C.) and National Institute of Mental Health Independent Scientist Award MH64190 (to C.S.C.). We thank Y. Qin for his technical help. COMPETING INTERESTS STATEMENT The authors declare that they have no competing financial interests. Received 19 July; accepted 9 September 2004 Published online at http://www.nature.com/natureneuroscience/ 1. Dehaene, S. et al. Science 284, 970–974 (1999). 2. Dehaene, S., Piazza, M., Pinel, P. & Cohen, L. Cogn. Neuropsychol. 20, 487–506 (2003). 3. Anderson, J.R. et al. Psychon. Bull. Rev. 10, 241–261 (2003) 4. Qin, Y. et al. Proc. Natl. Acad. Sci. USA 100, 4951–4956 (2003). 5. Reichle, E.D., Carpenter, P.A. & Just, M.A. Cognit. Psychol. 40, 261–295 (2000). 6. Nathan, M.J., Kintsch, W. & Young, E. Cogn. Instr. 9, 329–389 (1992). 7. Koedinger, K.R. & Nathan, M.J. J. Learn. Sci. 12, 129–164 (2004). 8. Dehaene, S., Bossini, S. & Gairaux, P. J. Exp. Psychol. Gen. 3, 371–396 (1993). 9. McCloskey, M., Harley, W. & Sokol, S.M. J. Exp. Psychol. Learn. Mem. Cogn. 17, 377–397(1991). 10. Forman, S.D. et al. Magn. Reson. Med. 33, 636–647 (1995). 11. Morcom, A.M. et al. Brain 126, 213–229 (2003). 1194 VOLUME 7 | NUMBER 11 | NOVEMBER 2004 NATURE NEUROSCIENCE Figure 2 Effects of activation change on behavioral latency. © 20 04 N at u re P u b lis h in g G ro u p h tt p :/ /w w w .n at u re .c o m /n at u re n eu ro sc ie n ce