One of the most comprehensive studies of mathematical giftedness was conducted in the Soviet Union between 1955 and 1966, by a team of researchers headed by psychologist Vadim Krutetskii. In describing the results of this research, Krutetskii noted innate differences in the cognitive capacities of children. While carefully sidestepping the Soviet prohibition on linking talent to inheritance, he reported on the individual differences in the ability to generalize relationships and associate related concepts:*
The difference between capable, average, and incapable pupils, as our research permits us to conclude, comes down to the following. In able pupils these associations can be formed “on the spot”; in this sense they are “born,” if one can so express it, already generalized, with a minimal number of exercises. In average pupils these associations are established and reinforced gradually, as a result of a whole series of exercises. They form isolated, concrete associations, related only to a given problem, “on the spot.” Through single-type exercises these associations are gradually transformed into generalized associations. In incapable pupils, even the isolated, concrete associations are formed with difficulty, their generalizations are still more difficult, and sometimes such generalizations do not occur at all.
In his attempts to determine how the mathematically gifted differ from others, Krutetskii and his research team presented a series of mathematics problems to students in grades 6 through 8 (ages 11 to 13) and observed their thinking processes. One such problem, designed to compare their capabilities in spatial visualization is shown below. You might want to try this one before reading the solution.
Problem:
Each face of a cube is marked with a letter. The cube is shown in 3 different positions.
Determine the letter on the face opposite:
• the face marked X. • the face marked A. • the face marked C.
(Check your answer.)
The problem above tests visualization skills. These require strong short-term memory as well as the capacity for abstraction, that is required in imagining how the cube will look in various orientations. This capacity for abstract thinking plays a key role in understanding metaphor in literature, relationships in mathematics, and making connections between things that the average person would regard as unconnected. For example, in the 1860’s, Father Gregor Mendel “mated” pure tall pea plants with pure short pea plants, and observed that the offspring were always tall. But when the tall plants of the second (hybrid) generation were mated, he discovered that about one in four of these second-generation plants were short. He synthesized these independent observations into a conclusion that laid the foundations of genetics and heredity. So many great intellectual discoveries, including the development of calculus by Newton and Leibniz, the theories of relativity of Einstein, as well as quantum physics and the revolutionary medical and technological breakthroughs have come from a gifted person’s ability to problem solve.
Modern neuroscience, using brain scans during cerebral activity, is currently attempting to discover what structural differences in some brains make them faster and more efficient in problem solving and dealing with abstraction. Based on their PET scans, Richard Haier and his research team hypothesize that brains capable of rapid absorption of concepts have a more efficient neural structure and consume less glucose in problem solving activities than the average brain.** However, these findings are somewhat tentative and neural functions are so complex, that we still know very little of the mechanisms that underpin intelligence. What we do know from observations, such as those made by Krutetskii and his team, is that people differ significantly in the speed at which they acquire an understanding of concepts and in their capacity to understand abstract ideas.
When we observe a gifted child, we usually see someone who absorbs information like a sponge and derives great joy from deep focus and engagement. As a precocious toddler of 3, Jeff insisted that he should have a bed instead of a crib, but his mother, Jackie, denied his request. A short time later, his mother discovered him with screwdriver in hand, dismantling his crib and transforming it into a real bed. Jeff attended a Montessori pre-school where he became so engrossed in each project that he had to be picked up–chair and all–and moved to the next activity. Three decades later, Jeff Bezos founded Amazon.com.
*Krutetskii’s description of the difference between capable and non-capable students is taken from Krutetskii, Vadim A. 1976. The Psychology of Mathematical Abilities in Schoolchildren. Chicago: Chicago University Press. p. 262.
**Haier, Richard. (2016) The Neuroscience of Intelligence. Cambridge University Press.
