## Evidence from Brain Damaged Patients

First, consider patients with major left-hemisphere lesions. As one would expect given that language faculties reside in the left hemisphere, such damage is highly associated with severe problems in understanding and speaking language. But patients with this kind of damage can also be acalculic, showing profound deficits in their ability to deal with numbers: an inability to understand number words or relate them to each other and an inability to perform relatively simple arithmetical calculations such as adding 9 + 7. However, in the face of this extreme impairment, nonverbal digit recognition, magnitude judgments, and some arithmetical operation abilities may still be functioning relatively normally. For example, one patient, NAU, who had suffered a major lesion of the posterior left hemisphere, had great difficulty reading number words and moreover was unable to solve even such simple arithmetic problems as 2 + 2! However, when presented with digits, he could indicate where they should fall on the number line by pointing to roughly the correct location, showing that his digit recognition was preserved and that he was able to access the approximate magnitude of the number represented by the digit. Moreover, and most interestingly, whereas he was unable to give the correct answer to a problem such as 2+2 and unable to state accurately the truth or falsity of arithmetical statements such as 2 + 2 = 5 or 2 + 2 = 3, he was able to state the falsity of statements such as 2 + 2 = 9! Moreover, when presented with two digits, he could immediately say which of the two was larger. He had also apparently retained the approximate, but not the exact, magnitude values of numbers, stating (for example) that there are about 6 or 10 eggs in a dozen, about 350 days in a year, and about 50min in 1 hr. In sum, NAU was able to identify and operate over the approximate magnitudes of the digits and obtain the approximate magnitudes of the solutions to simple arithmetic problems. In contrast to such effects of damage to the left hemisphere, damage to the right hemisphere has little or no effect on number processing.

Consider now lesions to the inferior parietal cortex. According to the model, such patients can be expected to have a preserved ability to call up memorized arithmetic facts and to correctly read and recognize number words and Arabic digits, but to have no sense of the meanings of these numbers and no understanding of numerical magnitudes and magnitudinal relationships. Again, numerous cases fit this description.

Mr. M., who had a lesion to the inferior parietal cortex, was unable to subtract (saying, for example, that 3—2 = 2) and had great difficulty in saying which of two digits was larger. He was also unable to generate the solution to what number lies midway between two given numbers; not only were his responses wrong, they were frequently ridiculous, for example, he replied that the number that lies midway between 10 and 20 is 30 or 25. In essense, Mr. M. had lost all sense of the meanings of numbers. However, he was well able to read number words and digits and to generate answers to addition and multiplication problems learned by rote in school, such as that 3 x 9 = 27.

Finally, split-brain patients present an interesting test of the model. Because all three numerical codes are present in the left hemisphere, numerical tasks presented to the left hemisphere should show normal performance. However, because the right hemisphere lacks the verbal number system, numerical tasks presented selectively to the right hemisphere should show the same pattern of results as that for patients with left-hemisphere lesions. Indeed, this is the case: larger-smaller comparisons are performed by split-brain patients equally well (and on a par with the performance of normal controls) whether presented to the left or right hemisphere. In contrast, performance on mental calculation tasks equals performance by control subjects when the tasks are presented to the left hemisphere but is severely impaired when the tasks are presented to the right hemisphere.

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