Sources of mathematical thinking: Behavioral and brain-imaging evidence

Science

Volumn 284, Issue 5416, 1999, Pages 970-974

  Dehaene, S ^a   Spelke, E ^b   Pinel, P ^a   Stanescu, R ^a   Tsivkin, S ^b  

^a DIF   (France)

^b MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ARTICLE; BRAIN MAPPING; COMPETENCE; HUMAN; LANGUAGE; LEARNING; LINGUISTICS; MATHEMATICAL COMPUTING; NERVE CELL NETWORK; PRIORITY JOURNAL; SPATIAL DISCRIMINATION; THINKING;

ADULT; BRAIN MAPPING; EVOKED POTENTIALS; FEMALE; FRONTAL LOBE; HUMANS; INTUITION; LANGUAGE; MAGNETIC RESONANCE IMAGING; MALE; MATHEMATICS; PARIETAL LOBE; THINKING;

EID: 0033532353     PISSN: 00368075     EISSN: None     Source Type: Journal    
DOI: 10.1126/science.284.5416.970     Document Type: Article

References (71)

1. J. Hadamard, An Essay on the Psychology of Invention in the Mathematical Field (Princeton Univ. Press, Princeton, NJ, 1945).
2. A. Einstein, cited in (1), p. 142.
3. L. E. J. Brouwer, Cambridge Lectures on Intuitionism (Cambridge Univ. Press, Cambridge, 1981).
4. N. Bourbaki, J. Symb. Logic 14, 1 (1948); D. Hilbert and W. Ackermann, Principles of Mathematical Logic (Chelsea, New York, 1950); A. N. Whitehead and B. Russell, Principia Mathematica (Cambridge Univ. Press, Cambridge, 1910).
5. N. Bourbaki, J. Symb. Logic 14, 1 (1948); D. Hilbert and W. Ackermann, Principles of Mathematical Logic (Chelsea, New York, 1950); A. N. Whitehead and B. Russell, Principia Mathematica (Cambridge Univ. Press, Cambridge, 1910).
6. N. Bourbaki, J. Symb. Logic 14, 1 (1948); D. Hilbert and W. Ackermann, Principles of Mathematical Logic (Chelsea, New York, 1950); A. N. Whitehead and B. Russell, Principia Mathematica (Cambridge Univ. Press, Cambridge, 1910).
7. H. Poincaré, Science and Hypothesis (Walter Scott Publishing Co., London, 1907).
8. J. I. D. Campbell and J. M. Clark, J. Exp. Psychol. Gen. 117, 204 (1988); J. I. D. Campbell, Cognition 53, 1 (1994); S. Dehaene, ibid. 44, 1 (1992).
9. J. I. D. Campbell and J. M. Clark, J. Exp. Psychol. Gen. 117, 204 (1988); J. I. D. Campbell, Cognition 53, 1 (1994); S. Dehaene, ibid. 44, 1 (1992).
10. J. I. D. Campbell and J. M. Clark, J. Exp. Psychol. Gen. 117, 204 (1988); J. I. D. Campbell, Cognition 53, 1 (1994); S. Dehaene, ibid. 44, 1 (1992).
11. S. Dehaene and L. Cohen, Math Cogn. 1, 83 (1995).
12. Participants were three female and five male bilingual Russian-English speakers, aged 18 to 32 years (mean age, 22.5 years), who began to learn English at a mean age of 15.3 years, had been in the United States for an average of 4.9 years and demonstrated fluent comprehension of both Russian and English on formal testing. All were undergraduate or graduate students, and all performed accurately on a variety of elementary arithmetic problems administered in Russian and in English during a pretest. Participants were trained on 12 sums of two two-digit numbers, totaling between 47 and 153. On each trial, an addition problem and two candidate answers were presented on a computer screen in word form, either in English or in Russian. Subjects selected one of the two answers, which appeared left and right of center, by pressing a corresponding key with the left or right hand. For exact addition, the candidate answers were the exact answer and a distractor in which the tens digit was off by one unit. For approximate addition, they were the multiple of ten closest to the correct sum, and another multiple off by 30 units. Each subject participated in 2 days of training with a fixed language and task (for example, exact addition in Russian) which was randomized across subjects (six repetitions of the 12 problems per day). Subjects also were trained for 2 days in a multiplication task (E. Spelke and S. Tsiukin, data not shown) with the same range of numbers in their other language, thus equalizing exposure to the two languages. On the fifth day, subjects were tested twice on each trained problem and twice on 12 similar untrained problems. Testing was done both in the original language of training and in the other language in different blocks.
13. The fact that subjects can estimate the solution of simple addition problems does not necessarily entail that the underlying mental representations are approximate. The present experiments only allow us to conclude that these representations are language-independent and encode numerical proximity.
14. Participants were four male and four female native Russian speakers, aged 18 to 24 years (mean age, 19.8 years), who were introduced to English at a mean age of 15.4 years, had lived in the United States for an average of 3.8 years, and fluently comprehended both Russian and English. All were undergraduate and graduate students who performed accurately in Russian and English pretests of elementary arithmetic skills. Each subject was trained in one language on 12 two-digit base-10 addition problems with addend of 54, 12 base-6 addition problems with two-to three-digit answers, and 12 cube-root estimation problems for numbers under 5000. The same subject was trained in the other language on 12 two-digit base-10 addition problems with addend of 63, 12 base-8 addition problems with two-to three-digit answers, and 12 base-2 logarithm estimation problems for numbers under 8500. Subjects received 2 days of training in each language, with six training trials per problem per training day. Order of training problems and pairings of problems and training languages were counterbalanced across subjects. Subjects received 1 day of testing in each of their languages on all the trained problems, plus an equal number of novel problems within the same six categories (two test trials per problem per testing day).
15. Anecdotal reports have suggested that when bilinguals calculate, they often revert to the original language in which they acquired arithmetic facts [P. A. Kolers, Sci. Am. 218, 78 (March 1968); B. Shanon, New Ideas Psychol. 2, 75 (1984)]. Furthermore, bilinguals solve arithmetic problems with greater speed and accuracy when the problems are presented in their first language [C. Frenck-Mestre and J. Vaid. Mem. Cogn. 21, 809 (1993); L G. Marsh and R. H. Maki, ibid. 4, 216 (1976); L McClain and J. Y. Shih Huang, ibid. 10, 591 (1982)]. These observations, however, might simply reflect easier word comprehension and production processes in the first language, rather than a language-dependent encoding of arithmetic knowledge itself [M. McCloskey, P. Macaruso, T. Whetstone, in The Nature and Origins of Mathematical Skills, J. I. D. Campbell, Ed. (Elsevier, Amsterdam, 1992), pp. 493-537]. Our results, by contrast, showed a language-switching cost for exact calculation regardless of whether training was in the first or second language. Indeed, the cost of switching from the subjects' first language (Russian) to their second language was no greater than the cost of switching in the reverse direction (519 and 810 ms, respectively, for all exact tasks combined).
16. Anecdotal reports have suggested that when bilinguals calculate, they often revert to the original language in which they acquired arithmetic facts [P. A. Kolers, Sci. Am. 218, 78 (March 1968); B. Shanon, New Ideas Psychol. 2, 75 (1984)]. Furthermore, bilinguals solve arithmetic problems with greater speed and accuracy when the problems are presented in their first language [C. Frenck-Mestre and J. Vaid. Mem. Cogn. 21, 809 (1993); L G. Marsh and R. H. Maki, ibid. 4, 216 (1976); L McClain and J. Y. Shih Huang, ibid. 10, 591 (1982)]. These observations, however, might simply reflect easier word comprehension and production processes in the first language, rather than a language-dependent encoding of arithmetic knowledge itself [M. McCloskey, P. Macaruso, T. Whetstone, in The Nature and Origins of Mathematical Skills, J. I. D. Campbell, Ed. (Elsevier, Amsterdam, 1992), pp. 493-537]. Our results, by contrast, showed a language-switching cost for exact calculation regardless of whether training was in the first or second language. Indeed, the cost of switching from the subjects' first language (Russian) to their second language was no greater than the cost of switching in the reverse direction (519 and 810 ms, respectively, for all exact tasks combined).
17. Anecdotal reports have suggested that when bilinguals calculate, they often revert to the original language in which they acquired arithmetic facts [P. A. Kolers, Sci. Am. 218, 78 (March 1968); B. Shanon, New Ideas Psychol. 2, 75 (1984)]. Furthermore, bilinguals solve arithmetic problems with greater speed and accuracy when the problems are presented in their first language [C. Frenck-Mestre and J. Vaid. Mem. Cogn. 21, 809 (1993); L G. Marsh and R. H. Maki, ibid. 4, 216 (1976); L McClain and J. Y. Shih Huang, ibid. 10, 591 (1982)]. These observations, however, might simply reflect easier word comprehension and production processes in the first language, rather than a language-dependent encoding of arithmetic knowledge itself [M. McCloskey, P. Macaruso, T. Whetstone, in The Nature and Origins of Mathematical Skills, J. I. D. Campbell, Ed. (Elsevier, Amsterdam, 1992), pp. 493-537]. Our results, by contrast, showed a language-switching cost for exact calculation regardless of whether training was in the first or second language. Indeed, the cost of switching from the subjects' first language (Russian) to their second language was no greater than the cost of switching in the reverse direction (519 and 810 ms, respectively, for all exact tasks combined).
18. Anecdotal reports have suggested that when bilinguals calculate, they often revert to the original language in which they acquired arithmetic facts [P. A. Kolers, Sci. Am. 218, 78 (March 1968); B. Shanon, New Ideas Psychol. 2, 75 (1984)]. Furthermore, bilinguals solve arithmetic problems with greater speed and accuracy when the problems are presented in their first language [C. Frenck-Mestre and J. Vaid. Mem. Cogn. 21, 809 (1993); L G. Marsh and R. H. Maki, ibid. 4, 216 (1976); L McClain and J. Y. Shih Huang, ibid. 10, 591 (1982)]. These observations, however, might simply reflect easier word comprehension and production processes in the first language, rather than a language-dependent encoding of arithmetic knowledge itself [M. McCloskey, P. Macaruso, T. Whetstone, in The Nature and Origins of Mathematical Skills, J. I. D. Campbell, Ed. (Elsevier, Amsterdam, 1992), pp. 493-537]. Our results, by contrast, showed a language-switching cost for exact calculation regardless of whether training was in the first or second language. Indeed, the cost of switching from the subjects' first language (Russian) to their second language was no greater than the cost of switching in the reverse direction (519 and 810 ms, respectively, for all exact tasks combined).
19. Anecdotal reports have suggested that when bilinguals calculate, they often revert to the original language in which they acquired arithmetic facts [P. A. Kolers, Sci. Am. 218, 78 (March 1968); B. Shanon, New Ideas Psychol. 2, 75 (1984)]. Furthermore, bilinguals solve arithmetic problems with greater speed and accuracy when the problems are presented in their first language [C. Frenck-Mestre and J. Vaid. Mem. Cogn. 21, 809 (1993); L G. Marsh and R. H. Maki, ibid. 4, 216 (1976); L McClain and J. Y. Shih Huang, ibid. 10, 591 (1982)]. These observations, however, might simply reflect easier word comprehension and production processes in the first language, rather than a language-dependent encoding of arithmetic knowledge itself [M. McCloskey, P. Macaruso, T. Whetstone, in The Nature and Origins of Mathematical Skills, J. I. D. Campbell, Ed. (Elsevier, Amsterdam, 1992), pp. 493-537]. Our results, by contrast, showed a language-switching cost for exact calculation regardless of whether training was in the first or second language. Indeed, the cost of switching from the subjects' first language (Russian) to their second language was no greater than the cost of switching in the reverse direction (519 and 810 ms, respectively, for all exact tasks combined).
20. Anecdotal reports have suggested that when bilinguals calculate, they often revert to the original language in which they acquired arithmetic facts [P. A. Kolers, Sci. Am. 218, 78 (March 1968); B. Shanon, New Ideas Psychol. 2, 75 (1984)]. Furthermore, bilinguals solve arithmetic problems with greater speed and accuracy when the problems are presented in their first language [C. Frenck-Mestre and J. Vaid. Mem. Cogn. 21, 809 (1993); L G. Marsh and R. H. Maki, ibid. 4, 216 (1976); L McClain and J. Y. Shih Huang, ibid. 10, 591 (1982)]. These observations, however, might simply reflect easier word comprehension and production processes in the first language, rather than a language-dependent encoding of arithmetic knowledge itself [M. McCloskey, P. Macaruso, T. Whetstone, in The Nature and Origins of Mathematical Skills, J. I. D. Campbell, Ed. (Elsevier, Amsterdam, 1992), pp. 493-537]. Our results, by contrast, showed a language-switching cost for exact calculation regardless of whether training was in the first or second language. Indeed, the cost of switching from the subjects' first language (Russian) to their second language was no greater than the cost of switching in the reverse direction (519 and 810 ms, respectively, for all exact tasks combined).
21. Participants were right-handed French students aged between 22 and 28 years (three men and four women in the fMRI version, five men and seven women in the ERP version). The project was approved by the regional ethical committee, and all subjects gave written informed consent. Stimuli were addition problems with addends ranging from 1 to 9 and sums ranging from 3 to 17. Ties such as 2 + 2 were excluded. For the exact task, the two candidate answers were the correct result and a result that was off by, at most, two units. In 90% of exact problems, the wrong result was of the same parity as the correct result, thus preventing the use of a short-cut based on parity checking [L. E. Krueger and E. W. Hallford, Mem. Cogn. 12, 17 (1984)]. For the approximation task, the two alternatives were a number off by one unit, and a number off by at least four units. A third control task of letter matching was also introduced, in which digits were replaced by the corresponding uppercase letter in the alphabet, and subjects depressed the button on the side on which a letter was repeated from the initial pair. Tasks were presented in runs of alternating blocks of trials with a 4-s intertrial interval, separated by resting periods of 24 s. Four such runs were presented in semi-random order, two runs alternating exact calculation (three blocks of 18 trials each) with letter matching (three blocks of nine trials each), and two similar runs alternating approximation with letter matching. For fMRI, we used a gradient-echo echo-planar imaging sequence sensitive to brain oxygen-level dependent contrast (30 contiguous axial slices, 5 mm thickness, TR = 4 s, TE = 40 ms, angle = 90°, field of view 192 mm by 256 mm, matrix = 64 by 64) on a 3-T whole-body system (Bruker, Germany). High-resolution anatomical images (three-dimensional gradient-echo inversion-recovery sequence, TI = 700 ms, TR = 1600 ms, FOV = 192 mm by 256 mm, matrix = 256 × 128 × 256, slice thickness = 1 mm) were also acquired. Analysis was performed with SPM96 software (www.fil.ion.ucl.ac.uk/spm). Images were corrected for subject motion, normalized to Talairach coordinates using a linear transform calculated on the anatomical images, smoothed (full width at half maximum = 15 mm), and averaged across subjects to yield an "average run" in each condition (the results were replicated when the same analysis was applied to individual data with 5-mm smoothing). The generalized linear mode was used to fit each voxel with a linear combination of two functions modeling early and late hemodynamic responses within each type of experimental block. Additional variables of noninterest modeled long-term signal variations with a high-pass filter set at 320 s. Because approximate and exact calculation blocks were acquired in different runs, the statistics we report used the interaction term (exact calculation - its letter control) - (approximate calculation - its letter control), with a voxelwise significance level of 0.001 corrected to P < 0.05 for multiple comparisons. In a separate session, ERPs were sampled at 125 Hz with a 128-electrode geodesic sensor net reference to the vertex [D. Tucker, Electroencephalogr. Clin. Neurophysiol. 87, 154 (1993)]. We rejected trials with incorrect responses, voltages exceeding ±100 μV, transients exceeding ±50 μV, electro-oculogram activity exceeding ±70 μV, or response times outside a 200-to 2500-ms interval. The remaining trials were averaged in synchrony with stimulus onset, digitally transformed to an average reference, band-pass filtered (0.5 to 20 Hz), and corrected for baseline over a 200-ms window before stimulus onset. Experimental conditions were compared within the first 400 ms by sample-by-sample t tests, with a criterion of P < 0.05 for five consecutive samples on at least eight electrodes simultaneously. Two-dimensional maps of scalp voltage were constructed by spherical spline interpolation [F. Perrin, J. Pernier, D. Bertrand, J. F. Echallier, Electroencephalogr. Clin. Neurophysiol. 72, 184 (1989)]. Dipole models were generated with BESA [ M. Scherg and P. Berg, BESA - Brain Electric Source Analysis Handbook (Max-Planck Institute for Psychiatry, Munich, 1990)]. Three fixed dipoles were placed at locations suggested by fMRI (left inferior frontal and bilateral parietal), and the program selected the dipole orientation and strength to match the exactapproximate ERP difference on a 216-to 280-ms time window, during which significant differences were found.
22. Participants were right-handed French students aged between 22 and 28 years (three men and four women in the fMRI version, five men and seven women in the ERP version). The project was approved by the regional ethical committee, and all subjects gave written informed consent. Stimuli were addition problems with addends ranging from 1 to 9 and sums ranging from 3 to 17. Ties such as 2 + 2 were excluded. For the exact task, the two candidate answers were the correct result and a result that was off by, at most, two units. In 90% of exact problems, the wrong result was of the same parity as the correct result, thus preventing the use of a short-cut based on parity checking [L. E. Krueger and E. W. Hallford, Mem. Cogn. 12, 17 (1984)]. For the approximation task, the two alternatives were a number off by one unit, and a number off by at least four units. A third control task of letter matching was also introduced, in which digits were replaced by the corresponding uppercase letter in the alphabet, and subjects depressed the button on the side on which a letter was repeated from the initial pair. Tasks were presented in runs of alternating blocks of trials with a 4-s intertrial interval, separated by resting periods of 24 s. Four such runs were presented in semi-random order, two runs alternating exact calculation (three blocks of 18 trials each) with letter matching (three blocks of nine trials each), and two similar runs alternating approximation with letter matching. For fMRI, we used a gradient-echo echo-planar imaging sequence sensitive to brain oxygen-level dependent contrast (30 contiguous axial slices, 5 mm thickness, TR = 4 s, TE = 40 ms, angle = 90°, field of view 192 mm by 256 mm, matrix = 64 by 64) on a 3-T whole-body system (Bruker, Germany). High-resolution anatomical images (three-dimensional gradient-echo inversion-recovery sequence, TI = 700 ms, TR = 1600 ms, FOV = 192 mm by 256 mm, matrix = 256 × 128 × 256, slice thickness = 1 mm) were also acquired. Analysis was performed with SPM96 software (www.fil.ion.ucl.ac.uk/spm). Images were corrected for subject motion, normalized to Talairach coordinates using a linear transform calculated on the anatomical images, smoothed (full width at half maximum = 15 mm), and averaged across subjects to yield an "average run" in each condition (the results were replicated when the same analysis was applied to individual data with 5-mm smoothing). The generalized linear mode was used to fit each voxel with a linear combination of two functions modeling early and late hemodynamic responses within each type of experimental block. Additional variables of noninterest modeled long-term signal variations with a high-pass filter set at 320 s. Because approximate and exact calculation blocks were acquired in different runs, the statistics we report used the interaction term (exact calculation - its letter control) - (approximate calculation - its letter control), with a voxelwise significance level of 0.001 corrected to P < 0.05 for multiple comparisons. In a separate session, ERPs were sampled at 125 Hz with a 128-electrode geodesic sensor net reference to the vertex [D. Tucker, Electroencephalogr. Clin. Neurophysiol. 87, 154 (1993)]. We rejected trials with incorrect responses, voltages exceeding ±100 μV, transients exceeding ±50 μV, electro-oculogram activity exceeding ±70 μV, or response times outside a 200-to 2500-ms interval. The remaining trials were averaged in synchrony with stimulus onset, digitally transformed to an average reference, band-pass filtered (0.5 to 20 Hz), and corrected for baseline over a 200-ms window before stimulus onset. Experimental conditions were compared within the first 400 ms by sample-by-sample t tests, with a criterion of P < 0.05 for five consecutive samples on at least eight electrodes simultaneously. Two-dimensional maps of scalp voltage were constructed by spherical spline interpolation [F. Perrin, J. Pernier, D. Bertrand, J. F. Echallier, Electroencephalogr. Clin. Neurophysiol. 72, 184 (1989)]. Dipole models were generated with BESA [ M. Scherg and P. Berg, BESA - Brain Electric Source Analysis Handbook (Max-Planck Institute for Psychiatry, Munich, 1990)]. Three fixed dipoles were placed at locations suggested by fMRI (left inferior frontal and bilateral parietal), and the program selected the dipole orientation and strength to match the exactapproximate ERP difference on a 216-to 280-ms time window, during which significant differences were found.
23. Participants were right-handed French students aged between 22 and 28 years (three men and four women in the fMRI version, five men and seven women in the ERP version). The project was approved by the regional ethical committee, and all subjects gave written informed consent. Stimuli were addition problems with addends ranging from 1 to 9 and sums ranging from 3 to 17. Ties such as 2 + 2 were excluded. For the exact task, the two candidate answers were the correct result and a result that was off by, at most, two units. In 90% of exact problems, the wrong result was of the same parity as the correct result, thus preventing the use of a short-cut based on parity checking [L. E. Krueger and E. W. Hallford, Mem. Cogn. 12, 17 (1984)]. For the approximation task, the two alternatives were a number off by one unit, and a number off by at least four units. A third control task of letter matching was also introduced, in which digits were replaced by the corresponding uppercase letter in the alphabet, and subjects depressed the button on the side on which a letter was repeated from the initial pair. Tasks were presented in runs of alternating blocks of trials with a 4-s intertrial interval, separated by resting periods of 24 s. Four such runs were presented in semi-random order, two runs alternating exact calculation (three blocks of 18 trials each) with letter matching (three blocks of nine trials each), and two similar runs alternating approximation with letter matching. For fMRI, we used a gradient-echo echo-planar imaging sequence sensitive to brain oxygen-level dependent contrast (30 contiguous axial slices, 5 mm thickness, TR = 4 s, TE = 40 ms, angle = 90°, field of view 192 mm by 256 mm, matrix = 64 by 64) on a 3-T whole-body system (Bruker, Germany). High-resolution anatomical images (three-dimensional gradient-echo inversion-recovery sequence, TI = 700 ms, TR = 1600 ms, FOV = 192 mm by 256 mm, matrix = 256 × 128 × 256, slice thickness = 1 mm) were also acquired. Analysis was performed with SPM96 software (www.fil.ion.ucl.ac.uk/spm). Images were corrected for subject motion, normalized to Talairach coordinates using a linear transform calculated on the anatomical images, smoothed (full width at half maximum = 15 mm), and averaged across subjects to yield an "average run" in each condition (the results were replicated when the same analysis was applied to individual data with 5-mm smoothing). The generalized linear mode was used to fit each voxel with a linear combination of two functions modeling early and late hemodynamic responses within each type of experimental block. Additional variables of noninterest modeled long-term signal variations with a high-pass filter set at 320 s. Because approximate and exact calculation blocks were acquired in different runs, the statistics we report used the interaction term (exact calculation - its letter control) - (approximate calculation - its letter control), with a voxelwise significance level of 0.001 corrected to P < 0.05 for multiple comparisons. In a separate session, ERPs were sampled at 125 Hz with a 128-electrode geodesic sensor net reference to the vertex [D. Tucker, Electroencephalogr. Clin. Neurophysiol. 87, 154 (1993)]. We rejected trials with incorrect responses, voltages exceeding ±100 μV, transients exceeding ±50 μV, electro-oculogram activity exceeding ±70 μV, or response times outside a 200-to 2500-ms interval. The remaining trials were averaged in synchrony with stimulus onset, digitally transformed to an average reference, band-pass filtered (0.5 to 20 Hz), and corrected for baseline over a 200-ms window before stimulus onset. Experimental conditions were compared within the first 400 ms by sample-by-sample t tests, with a criterion of P < 0.05 for five consecutive samples on at least eight electrodes simultaneously. Two-dimensional maps of scalp voltage were constructed by spherical spline interpolation [F. Perrin, J. Pernier, D. Bertrand, J. F. Echallier, Electroencephalogr. Clin. Neurophysiol. 72, 184 (1989)]. Dipole models were generated with BESA [ M. Scherg and P. Berg, BESA - Brain Electric Source Analysis Handbook (Max-Planck Institute for Psychiatry, Munich, 1990)]. Three fixed dipoles were placed at locations suggested by fMRI (left inferior frontal and bilateral parietal), and the program selected the dipole orientation and strength to match the exactapproximate ERP difference on a 216-to 280-ms time window, during which significant differences were found.
24. Participants were right-handed French students aged between 22 and 28 years (three men and four women in the fMRI version, five men and seven women in the ERP version). The project was approved by the regional ethical committee, and all subjects gave written informed consent. Stimuli were addition problems with addends ranging from 1 to 9 and sums ranging from 3 to 17. Ties such as 2 + 2 were excluded. For the exact task, the two candidate answers were the correct result and a result that was off by, at most, two units. In 90% of exact problems, the wrong result was of the same parity as the correct result, thus preventing the use of a short-cut based on parity checking [L. E. Krueger and E. W. Hallford, Mem. Cogn. 12, 17 (1984)]. For the approximation task, the two alternatives were a number off by one unit, and a number off by at least four units. A third control task of letter matching was also introduced, in which digits were replaced by the corresponding uppercase letter in the alphabet, and subjects depressed the button on the side on which a letter was repeated from the initial pair. Tasks were presented in runs of alternating blocks of trials with a 4-s intertrial interval, separated by resting periods of 24 s. Four such runs were presented in semi-random order, two runs alternating exact calculation (three blocks of 18 trials each) with letter matching (three blocks of nine trials each), and two similar runs alternating approximation with letter matching. For fMRI, we used a gradient-echo echo-planar imaging sequence sensitive to brain oxygen-level dependent contrast (30 contiguous axial slices, 5 mm thickness, TR = 4 s, TE = 40 ms, angle = 90°, field of view 192 mm by 256 mm, matrix = 64 by 64) on a 3-T whole-body system (Bruker, Germany). High-resolution anatomical images (three-dimensional gradient-echo inversion-recovery sequence, TI = 700 ms, TR = 1600 ms, FOV = 192 mm by 256 mm, matrix = 256 × 128 × 256, slice thickness = 1 mm) were also acquired. Analysis was performed with SPM96 software (www.fil.ion.ucl.ac.uk/spm). Images were corrected for subject motion, normalized to Talairach coordinates using a linear transform calculated on the anatomical images, smoothed (full width at half maximum = 15 mm), and averaged across subjects to yield an "average run" in each condition (the results were replicated when the same analysis was applied to individual data with 5-mm smoothing). The generalized linear mode was used to fit each voxel with a linear combination of two functions modeling early and late hemodynamic responses within each type of experimental block. Additional variables of noninterest modeled long-term signal variations with a high-pass filter set at 320 s. Because approximate and exact calculation blocks were acquired in different runs, the statistics we report used the interaction term (exact calculation - its letter control) - (approximate calculation - its letter control), with a voxelwise significance level of 0.001 corrected to P < 0.05 for multiple comparisons. In a separate session, ERPs were sampled at 125 Hz with a 128-electrode geodesic sensor net reference to the vertex [D. Tucker, Electroencephalogr. Clin. Neurophysiol. 87, 154 (1993)]. We rejected trials with incorrect responses, voltages exceeding ±100 μV, transients exceeding ±50 μV, electro-oculogram activity exceeding ±70 μV, or response times outside a 200-to 2500-ms interval. The remaining trials were averaged in synchrony with stimulus onset, digitally transformed to an average reference, band-pass filtered (0.5 to 20 Hz), and corrected for baseline over a 200-ms window before stimulus onset. Experimental conditions were compared within the first 400 ms by sample-by-sample t tests, with a criterion of P < 0.05 for five consecutive samples on at least eight electrodes simultaneously. Two-dimensional maps of scalp voltage were constructed by spherical spline interpolation [F. Perrin, J. Pernier, D. Bertrand, J. F. Echallier, Electroencephalogr. Clin. Neurophysiol. 72, 184 (1989)]. Dipole models were generated with BESA [ M. Scherg and P. Berg, BESA - Brain Electric Source Analysis Handbook (Max-Planck Institute for Psychiatry, Munich, 1990)]. Three fixed dipoles were placed at locations suggested by fMRI (left inferior frontal and bilateral parietal), and the program selected the dipole orientation and strength to match the exactapproximate ERP difference on a 216-to 280-ms time window, during which significant differences were found.
25. A. R. Damasio and N. Geschwind, Annu. Rev. Neurosci. 7, 127 (1984); C. Price, Trends Cogn. Sci. 2, 281 (1998).
26. A. R. Damasio and N. Geschwind, Annu. Rev. Neurosci. 7, 127 (1984); C. Price, Trends Cogn. Sci. 2, 281 (1998).
27. R. A. Andersen, Philos. Trans. R. Soc. London Ser. B 352, 1421 (1997); A. Berthoz, ibid., p. 1437; M. Jeannerod, The Cognitive Neuroscience of Action (Blackwell, New York, 1997); L. H. Snyder, K. L. Grieve, P. Brotchie, R. A. Andersen, Nature 394, 887 (1998).
28. R. A. Andersen, Philos. Trans. R. Soc. London Ser. B 352, 1421 (1997); A. Berthoz, ibid., p. 1437; M. Jeannerod, The Cognitive Neuroscience of Action (Blackwell, New York, 1997); L. H. Snyder, K. L. Grieve, P. Brotchie, R. A. Andersen, Nature 394, 887 (1998).
29. R. A. Andersen, Philos. Trans. R. Soc. London Ser. B 352, 1421 (1997); A. Berthoz, ibid., p. 1437; M. Jeannerod, The Cognitive Neuroscience of Action (Blackwell, New York, 1997); L. H. Snyder, K. L. Grieve, P. Brotchie, R. A. Andersen, Nature 394, 887 (1998).
30. R. A. Andersen, Philos. Trans. R. Soc. London Ser. B 352, 1421 (1997); A. Berthoz, ibid., p. 1437; M. Jeannerod, The Cognitive Neuroscience of Action (Blackwell, New York, 1997); L. H. Snyder, K. L. Grieve, P. Brotchie, R. A. Andersen, Nature 394, 887 (1998).
31. R. Kawashima et al., Neuroreport 7, 1253 (1996); H. Sakata and M. Taira, Curr. Biol. 4, 847 (1994).
32. R. Kawashima et al., Neuroreport 7, 1253 (1996); H. Sakata and M. Taira, Curr. Biol. 4, 847 (1994).
33. H. Kawamichi, Y. Kikuchi, H. Endo, T. Takeda, S. Yoshizawa, Neuroreport 9, 1127 (1998); S. M. Kosslyn, G. J. DiGirolamo, W. L. Thompson, N. M. Alpert, Psychophysiology 35, 151 (1998); W. Richter, K. Ugurbil, A. Georgopoulos, S. G. Kim, Neuroreport 8, 3697 (1997).
34. H. Kawamichi, Y. Kikuchi, H. Endo, T. Takeda, S. Yoshizawa, Neuroreport 9, 1127 (1998); S. M. Kosslyn, G. J. DiGirolamo, W. L. Thompson, N. M. Alpert, Psychophysiology 35, 151 (1998); W. Richter, K. Ugurbil, A. Georgopoulos, S. G. Kim, Neuroreport 8, 3697 (1997).
35. H. Kawamichi, Y. Kikuchi, H. Endo, T. Takeda, S. Yoshizawa, Neuroreport 9, 1127 (1998); S. M. Kosslyn, G. J. DiGirolamo, W. L. Thompson, N. M. Alpert, Psychophysiology 35, 151 (1998); W. Richter, K. Ugurbil, A. Georgopoulos, S. G. Kim, Neuroreport 8, 3697 (1997).
36. M. Corbetta, F. M. Miezin, G. L. Schulman, S. E. Petersen, J. Neurosci. 13, 1202 (1993); M. Corbetta, G. L. Shulman, F. M. Miezin, S. E. Petersen, Science 270, 802 (1995); A. C. Nobre et al., Brain 120, 515 (1997); M. I. Posner and S. Dehaene, Trends Neurosci. 17, 75 (1994).
37. M. Corbetta, F. M. Miezin, G. L. Schulman, S. E. Petersen, J. Neurosci. 13, 1202 (1993); M. Corbetta, G. L. Shulman, F. M. Miezin, S. E. Petersen, Science 270, 802 (1995); A. C. Nobre et al., Brain 120, 515 (1997); M. I. Posner and S. Dehaene, Trends Neurosci. 17, 75 (1994).
38. M. Corbetta, F. M. Miezin, G. L. Schulman, S. E. Petersen, J. Neurosci. 13, 1202 (1993); M. Corbetta, G. L. Shulman, F. M. Miezin, S. E. Petersen, Science 270, 802 (1995); A. C. Nobre et al., Brain 120, 515 (1997); M. I. Posner and S. Dehaene, Trends Neurosci. 17, 75 (1994).
39. M. Corbetta, F. M. Miezin, G. L. Schulman, S. E. Petersen, J. Neurosci. 13, 1202 (1993); M. Corbetta, G. L. Shulman, F. M. Miezin, S. E. Petersen, Science 270, 802 (1995); A. C. Nobre et al., Brain 120, 515 (1997); M. I. Posner and S. Dehaene, Trends Neurosci. 17, 75 (1994).
40. S. Dehaene, J. Cogn. Neurosci. 8, 47 (1996); S. Dehaene et al., Neuropsychologia 34, 1097 (1996); P. E. Roland and L. Friberg, J. Neurophysiol. 53, 1219 (1985); L. Rueckert et al., Neuroimage 3, 97 (1996).
41. S. Dehaene, J. Cogn. Neurosci. 8, 47 (1996); S. Dehaene et al., Neuropsychologia 34, 1097 (1996); P. E. Roland and L. Friberg, J. Neurophysiol. 53, 1219 (1985); L. Rueckert et al., Neuroimage 3, 97 (1996).
42. S. Dehaene, J. Cogn. Neurosci. 8, 47 (1996); S. Dehaene et al., Neuropsychologia 34, 1097 (1996); P. E. Roland and L. Friberg, J. Neurophysiol. 53, 1219 (1985); L. Rueckert et al., Neuroimage 3, 97 (1996).
43. S. Dehaene, J. Cogn. Neurosci. 8, 47 (1996); S. Dehaene et al., Neuropsychologia 34, 1097 (1996); P. E. Roland and L. Friberg, J. Neurophysiol. 53, 1219 (1985); L. Rueckert et al., Neuroimage 3, 97 (1996).
44. The exact and approximate tasks did not differ in mean response time (fMRI: 772 and 783 ms; ERPs: 913 and 946 ms, respectively) nor in error rate (fMRI: 4.4 and 5.3%; ERPs: 2.7 and 2.3%, respectively) (all Fs < 1).
45. -3, corrected), the finding of greater intraparietal activation during approximation than during exact addition was replicated in five of seven subjects, whereas significantly greater left inferior frontal activation during exact addition was observed in four of seven subjects.
46. S. E. Petersen, P. T. Fox, M. I. Posner, M. Mintun, M. E. Raichle, Nature 331, 585 (1988); M. E. Raichle et al., Cereb. Cortex 4, 8 (1994); R. Vandenberghe, C. Price, R. Wise, O. Josephs, R. S. J. Frackowiak, Nature 383, 254 (1996); A. D. Wagner et al., Science 281, 1188 (1998).
47. S. E. Petersen, P. T. Fox, M. I. Posner, M. Mintun, M. E. Raichle, Nature 331, 585 (1988); M. E. Raichle et al., Cereb. Cortex 4, 8 (1994); R. Vandenberghe, C. Price, R. Wise, O. Josephs, R. S. J. Frackowiak, Nature 383, 254 (1996); A. D. Wagner et al., Science 281, 1188 (1998).
48. S. E. Petersen, P. T. Fox, M. I. Posner, M. Mintun, M. E. Raichle, Nature 331, 585 (1988); M. E. Raichle et al., Cereb. Cortex 4, 8 (1994); R. Vandenberghe, C. Price, R. Wise, O. Josephs, R. S. J. Frackowiak, Nature 383, 254 (1996); A. D. Wagner et al., Science 281, 1188 (1998).
49. S. E. Petersen, P. T. Fox, M. I. Posner, M. Mintun, M. E. Raichle, Nature 331, 585 (1988); M. E. Raichle et al., Cereb. Cortex 4, 8 (1994); R. Vandenberghe, C. Price, R. Wise, O. Josephs, R. S. J. Frackowiak, Nature 383, 254 (1996); A. D. Wagner et al., Science 281, 1188 (1998).
50. Y. G. Abdullaev and N. P. Bechtereva, Int. J. Psychophysiol. 14, 167 (1993); Y. G. Abdullaev and M. I. Posner, Neuroimage 7, 1 (1998); A. Z. Snyder, Y. G. Abdullaev, M. I. Posner, M. E. Raichle, Proc. Natl. Acad. Sci. U.S.A. 92, 1689 (1995).
51. Y. G. Abdullaev and N. P. Bechtereva, Int. J. Psychophysiol. 14, 167 (1993); Y. G. Abdullaev and M. I. Posner, Neuroimage 7, 1 (1998); A. Z. Snyder, Y. G. Abdullaev, M. I. Posner, M. E. Raichle, Proc. Natl. Acad. Sci. U.S.A. 92, 1689 (1995).
52. Y. G. Abdullaev and N. P. Bechtereva, Int. J. Psychophysiol. 14, 167 (1993); Y. G. Abdullaev and M. I. Posner, Neuroimage 7, 1 (1998); A. Z. Snyder, Y. G. Abdullaev, M. I. Posner, M. E. Raichle, Proc. Natl. Acad. Sci. U.S.A. 92, 1689 (1995).
53. H. Hécaen, R. Angelergues, S. Houillier, Rev. Neurol. 105, 85 (1961); M. Rosselli and A. Ardila, Neuropsychologia 27, 607 (1989).
54. H. Hécaen, R. Angelergues, S. Houillier, Rev. Neurol. 105, 85 (1961); M. Rosselli and A. Ardila, Neuropsychologia 27, 607 (1989).
55. S. Dehaene and L. Cohen, Cortex 33, 219 (1997).
56. A. L. Benton, Arch. Neurol. 49, 445 (1992); Y. Takayama, M. Sugishita, I. Akiguchi, J. Kimura, ibid. 51, 286 (1994); M. Delazer and T. Benke, Cortex 33, 697 (1997). Note that in such left parietal cases, the preservation of exact arithmetic facts is clearest for very small multiplication and addition problems (23). As the numbers involved in an exact arithmetic problem get larger, subjects are more and more likely to rely on quantity-based strategies to supplement rote verbal retrieval [(7); see also J. A. LeFevre et al., J. Exp. Psychol. Gen. 125, 284 (1996); J. LeFevre, G. S. Sadesky, J. Bisanz, J. Exp. Psychol. Learn. Mem. Cogn. 22, 216 (1996)]. Indeed, supplementary fMRI analyses (available from S. Dehaene) showed that, within the present exact addition task, increasing problem sizes caused greater activation of the bilateral intraparietal circuit in regions identical to those active during approximation. This finding suggests that verbal and quantity representations of numbers are functionally integrated in the adult brain. Although only the verbal circuit is used for well-rehearsed exact arithmetic facts, both circuits are used when attempting to retrieve lesser-known facts.
57. A. L. Benton, Arch. Neurol. 49, 445 (1992); Y. Takayama, M. Sugishita, I. Akiguchi, J. Kimura, ibid. 51, 286 (1994); M. Delazer and T. Benke, Cortex 33, 697 (1997). Note that in such left parietal cases, the preservation of exact arithmetic facts is clearest for very small multiplication and addition problems (23). As the numbers involved in an exact arithmetic problem get larger, subjects are more and more likely to rely on quantity-based strategies to supplement rote verbal retrieval [(7); see also J. A. LeFevre et al., J. Exp. Psychol. Gen. 125, 284 (1996); J. LeFevre, G. S. Sadesky, J. Bisanz, J. Exp. Psychol. Learn. Mem. Cogn. 22, 216 (1996)]. Indeed, supplementary fMRI analyses (available from S. Dehaene) showed that, within the present exact addition task, increasing problem sizes caused greater activation of the bilateral intraparietal circuit in regions identical to those active during approximation. This finding suggests that verbal and quantity representations of numbers are functionally integrated in the adult brain. Although only the verbal circuit is used for well-rehearsed exact arithmetic facts, both circuits are used when attempting to retrieve lesser-known facts.
58. A. L. Benton, Arch. Neurol. 49, 445 (1992); Y. Takayama, M. Sugishita, I. Akiguchi, J. Kimura, ibid. 51, 286 (1994); M. Delazer and T. Benke, Cortex 33, 697 (1997). Note that in such left parietal cases, the preservation of exact arithmetic facts is clearest for very small multiplication and addition problems (23). As the numbers involved in an exact arithmetic problem get larger, subjects are more and more likely to rely on quantity-based strategies to supplement rote verbal retrieval [(7); see also J. A. LeFevre et al., J. Exp. Psychol. Gen. 125, 284 (1996); J. LeFevre, G. S. Sadesky, J. Bisanz, J. Exp. Psychol. Learn. Mem. Cogn. 22, 216 (1996)]. Indeed, supplementary fMRI analyses (available from S. Dehaene) showed that, within the present exact addition task, increasing problem sizes caused greater activation of the bilateral intraparietal circuit in regions identical to those active during approximation. This finding suggests that verbal and quantity representations of numbers are functionally integrated in the adult brain. Although only the verbal circuit is used for well-rehearsed exact arithmetic facts, both circuits are used when attempting to retrieve lesser-known facts.
59. A. L. Benton, Arch. Neurol. 49, 445 (1992); Y. Takayama, M. Sugishita, I. Akiguchi, J. Kimura, ibid. 51, 286 (1994); M. Delazer and T. Benke, Cortex 33, 697 (1997). Note that in such left parietal cases, the preservation of exact arithmetic facts is clearest for very small multiplication and addition problems (23). As the numbers involved in an exact arithmetic problem get larger, subjects are more and more likely to rely on quantity-based strategies to supplement rote verbal retrieval [(7); see also J. A. LeFevre et al., J. Exp. Psychol. Gen. 125, 284 (1996); J. LeFevre, G. S. Sadesky, J. Bisanz, J. Exp. Psychol. Learn. Mem. Cogn. 22, 216 (1996)]. Indeed, supplementary fMRI analyses (available from S. Dehaene) showed that, within the present exact addition task, increasing problem sizes caused greater activation of the bilateral intraparietal circuit in regions identical to those active during approximation. This finding suggests that verbal and quantity representations of numbers are functionally integrated in the adult brain. Although only the verbal circuit is used for well-rehearsed exact arithmetic facts, both circuits are used when attempting to retrieve lesser-known facts.
60. A. L. Benton, Arch. Neurol. 49, 445 (1992); Y. Takayama, M. Sugishita, I. Akiguchi, J. Kimura, ibid. 51, 286 (1994); M. Delazer and T. Benke, Cortex 33, 697 (1997). Note that in such left parietal cases, the preservation of exact arithmetic facts is clearest for very small multiplication and addition problems (23). As the numbers involved in an exact arithmetic problem get larger, subjects are more and more likely to rely on quantity-based strategies to supplement rote verbal retrieval [(7); see also J. A. LeFevre et al., J. Exp. Psychol. Gen. 125, 284 (1996); J. LeFevre, G. S. Sadesky, J. Bisanz, J. Exp. Psychol. Learn. Mem. Cogn. 22, 216 (1996)]. Indeed, supplementary fMRI analyses (available from S. Dehaene) showed that, within the present exact addition task, increasing problem sizes caused greater activation of the bilateral intraparietal circuit in regions identical to those active during approximation. This finding suggests that verbal and quantity representations of numbers are functionally integrated in the adult brain. Although only the verbal circuit is used for well-rehearsed exact arithmetic facts, both circuits are used when attempting to retrieve lesser-known facts.
61. S. Dehaene and L. Cohen, Neuropsychologia 29, 1045 (1991).
62. T. Dantzig, Number: The Language of Science (Free Press, New York, 1967); G. Ifrah, Histoire Universelle des Chiffres (Robert Laffont, Paris, 1994).
63. T. Dantzig, Number: The Language of Science (Free Press, New York, 1967); G. Ifrah, Histoire Universelle des Chiffres (Robert Laffont, Paris, 1994).
64. M. Kline, Mathematical Thought from Ancient to Modern Times (Oxford Univ. Press, New York, 1972).
65. S. T. Boysen and E. J. Capaldi, Eds., The Development of Numerical Competence: Animal and Human Models (Erlbaum, Hillsdale, NJ, 1993); I. M. Brannon and H. S. Terrace, Science 282, 746 (1998); S. Dehaene, G. Dehaene-Lambertz, L. Cohen, Trends Neurosci. 21, 355 (1998); C. R. Gallistel, Annu. Rev. Psychol. 40, 155 (1989).
66. S. T. Boysen and E. J. Capaldi, Eds., The Development of Numerical Competence: Animal and Human Models (Erlbaum, Hillsdale, NJ, 1993); I. M. Brannon and H. S. Terrace, Science 282, 746 (1998); S. Dehaene, G. Dehaene-Lambertz, L. Cohen, Trends Neurosci. 21, 355 (1998); C. R. Gallistel, Annu. Rev. Psychol. 40, 155 (1989).
67. S. T. Boysen and E. J. Capaldi, Eds., The Development of Numerical Competence: Animal and Human Models (Erlbaum, Hillsdale, NJ, 1993); I. M. Brannon and H. S. Terrace, Science 282, 746 (1998); S. Dehaene, G. Dehaene-Lambertz, L. Cohen, Trends Neurosci. 21, 355 (1998); C. R. Gallistel, Annu. Rev. Psychol. 40, 155 (1989).
68. S. T. Boysen and E. J. Capaldi, Eds., The Development of Numerical Competence: Animal and Human Models (Erlbaum, Hillsdale, NJ, 1993); I. M. Brannon and H. S. Terrace, Science 282, 746 (1998); S. Dehaene, G. Dehaene-Lambertz, L. Cohen, Trends Neurosci. 21, 355 (1998); C. R. Gallistel, Annu. Rev. Psychol. 40, 155 (1989).
69. K. Wynn, Trends Cogn. Sci. 2, 296 (1998).
70. S. Dehaene, The Number Sense (Oxford Univ. Press, New York, 1997).
71. Supported by NIH grant HD23103 (E.S.) and by the Fondation pour la Recherche Médicale (S.D.). We gratefully acknowledge discussions with L. Cohen, D. Le Bihan, J.-B. Poline, and N. Kanwisher.