AFTER EDIT: Thanks to all who have replied for the interesting comments. I discovered this link while digesting replies I received on three different email lists to a request to name experts on mathematically precocious young people. (That was for work.) Tanya Khovanova, the author of the blog post submitted here, was one name suggested to me as an expert on precocious mathematics learners. When I saw her personal website,

http://www.tanyakhovanova.com/

I remembered that I had seen her blog post "Should You Date a Mathematician?"

http://blog.tanyakhovanova.com/?p=319

posted to Hacker News (and other sites I read) before. I'll read more of her more purely mathematical blog posts over the next few days. I see one I can use right away in the local classes I teach to elementary-age learners.

On the substance of the post, I'm seeing several comments that equate "genius" to "person with a high IQ score." That was indeed the old-fashioned way that Lewis Terman (1877 to 1956) labeled a person with a high IQ score as he developed the Stanford-Binet IQ test. But as Terman gained more experience, especially with the subjects in his own longitudinal study of Americans identified in childhood by high IQ scores, he didn't equate high IQ to genius, and he became more aware of the shortcomings of IQ tests. Terman and his co-author Maude Merrill wrote in 1937,

"There are, however, certain characteristics of age scores with which the reader should be familiar. For one thing, it is necessary to bear in mind that the true mental age as we have used it refers to the mental age on a particular intelligence test. A subject's mental age in this sense may not coincide with the age score he would make in tests of musical ability, mechanical ability, social adjustment, etc. A subject has, strictly speaking, a number of mental ages; we are here concerned only with that which depends on the abilities tested by the new Stanford-Binet scales."

Terman, Lewis & Merrill, Maude (1937). Measuring Intelligence: A Guide to the Administration of the New Revised Stanford-Binet Tests of Intelligence. Boston: Houghton Mifflin. p. 25. That is why the later authors Kenneth Hopkins and Julian Stanley (founder of the Study of Exceptional Talent) suggested that is better to regard IQ tests as tests of "scholastic aptitude" rather than of intelligence. They wrote

"Most authorities feel that current intelligence tests are more aptly described as 'scholastic aptitude' tests because they are so highly related to academic performance, although current use suggests that the term intelligence test is going to be with us for some time. This reservation is based not on the opinion that intelligence tests do not reflect intelligence but on the belief that there are other kinds of intelligence that are not reflected in current tests; the term intelligence is too inclusive."

Hopkins, Kenneth D. & Stanley, Julian C. (1981). Educational and Psychological Measurement and Evaluation. Englewood Cliffs, NJ: Prentice Hall. p. 364.

So on the one hand there is the acknowledged issue among experts on IQ testing that IQ scores don't tell the whole story of a test subject's mental ability. A less well known issue is the degree to which error in estimation increases in IQ scores as IQ scores are found to be above the norming sample mean. Terman and Merrill wrote,

"The reader should not lose sight of the fact that a test with even a high reliability yields scores which have an appreciable probable error. The probable error in terms of mental age is of course larger with older than with young children because of the increasing spread of mental age as we go from younger to older groups. For this reason it has been customary to express the P.E. [probable error] of a Binet score in terms of I.Q., since the spread of Binet I.Q.'s is fairly constant from age to age. However, when our correlation arrays [between Form L and Form M] were plotted for separate age groups they were all discovered to be distinctly fan-shaped. Figure 3 is typical of the arrays at every age level.

"From Figure 3 [not shown here on HN, alas] it becomes clear that the probable error of an I.Q. score is not a constant amount, but a variable which increases as I.Q. increases. It has frequently been noted in the literature that gifted subjects show greater I.Q. fluctuation than do clinical cases with low I.Q.'s . . . . we now see that this trend is inherent in the I.Q. technique itself, and might have been predicted on logical grounds."

Terman, Lewis & Merrill, Maude (1937). Measuring Intelligence: A Guide to the Administration of the New Revised Stanford-Binet Tests of Intelligence. Boston: Houghton Mifflin. p. 44

Readers of this thread who would like to follow the current scientific literature on genius (as it is now defined by mainstream psychologists) may enjoy reading the works of Dean Keith Simonton,

http://www.amazon.com/Dean-Keith-Simonton/e/B001ITRL1I/

the world's leading researcher on genius and its development. Readers curious about what IQ tests miss may enjoy reading the book What Intelligence Tests Miss: The Psychology of Rational Thought

http://www.amazon.com/What-Intelligence-Tests-Miss-Psycholog...

by Keith R. Stanovich and some of Stanovich's other recent books.

Readers who would like to read a whole lot about current research on human intelligence and related issues can find a lot of curated reading suggestions at a Wikipedia user bibliography

http://en.wikipedia.org/wiki/User:WeijiBaikeBianji/Intellige...

occasionally used for the slow, pains-taking process of updating the many Wikipedia articles on related subjects (most of which are plagued by edit-warring and badly in need of more editing).