To me, this is the million dollar question, the one that ultimately motivates all the other questions I ask here. This is my first gesture towards an answer.

Imagine five students in a mainstream high school math class: Alice, Beth, Claire, Dana, and Eve. All five of them are highly motivated, hard-working students, but they differ in their abilities and their grades in math.

Alice could be considered a "math genius." When she was little, her stroller had a string of beads on it. She remembers organizing the beads into groups of one, two, three, and so on, to see how many groups she could make out of the beads. She eventually deduced the idea that the number of beads left over would always be smaller than the number of beads in the group--if she arranged the beads in groups of three, for example, there could be one or two left over, but there could not be more. Given that she was in the stroller at the time, she could not have been older than about three when she figured out how remainders worked.* Throughout her life, she has continued to make discoveries about math on her own and without prompting. Not surprisingly, she has found she can earn A's in class with minimal attention in class and minimal effort on the homework. She has already learned the basic principles on her own; the classes merely give her the vocabulary for what she has learned.

Beth does not teach herself math on her own in the same way Alice does. However, she has been able to deduce some implications of what she was taught without being told. For example, when she was about four, her parents told her how the number line worked, drawing the numbers starting from zero on it. She guessed that there would have to be negative numbers and explained the concept to her parents, although she did not know the term "negative numbers." Beth has to pay more attention in class than Alice does, because unlike Alice, she is actually learning the material in class. However, once she understands the concepts presented in class, she does not need them repeated, and finds the homework easy because it seems to follow logically from what she has learned. Not surprisingly, she also earns A's.

Claire has always been able to learn the concepts presented in class. However, she needs the teachers to provide examples of how to apply them, or else she applies formulas to the wrong problems. Claire finds straightforward calculation-based math problems the easiest, while word problems and homework that require her to use principles not explicitly taught in class are harder for her. Nonetheless, like Alice and Beth, she is an A student and perceives herself as good at math.

Dana is much like Claire, only she generally earns B's and C's rather than A's. She is one of the average students in her class.

Eve, though currently in a mainstream class, has a math learning disability. Although she works incredibly hard to understand the concepts presented in class, remember them, and apply them to her homework, she neither understands them well nor calculates accurately. Never mind the unfamiliar problems that Claire sometimes struggles with, Eve has difficulty doing the basic problems. She works with a tutor and spends hours struggling with her math homework, but is still failing the class.

(This is not an exhaustive list of types of math students, of course. There are also those like Fay, who find that "the complex is simple and the simple complex"--she took three years to learn the multiplication tables and looked like either Dana or Eve until she was first introduced to algebra, when she earned A's and felt like she understood math for the first time. At this point, it became clear that she had a keen logical mind, much like Beth's, and could understand math concepts at an abstract level. However, her difficulties with calculation and working memory prevented her from applying her understanding. Like Claire, her best work earned her B and C grades, and she saw herself as rather bad at math. Fay could be considered to combine Alice's and Beth's math talents with Eve's math disabilities).

Students like Fay aside, one can see a pattern: talented students achieve more with less expenditure of effort than average students, and average students achieve more with less effort than disabled students.

One could roughly generalize that talent is basically "learning or achievement per unit of effort."

Of course, people have been theorizing about talent and disability forever. The real question is: would this theory stand up if tested in the lab?

At the moment, I'm not sure, because it's not clear how one could test this hypothesis. Achievement and amount of learning are fairly easy to measure. Teachers, cognitive psychologists, psychometricians and many others have come up with numerous ways to measure these things. But how does one measure effort? Is it time spent? The subjective feeling of "working hard?" Engagement of the brain's attentional network? Time it takes to lose focus or feel fatigued?

How would you measure the effort a learner expends? And how would you define talent and disability?

*Alice's feat--and all other details mentioned here--come from people I know (genders changed, in some cases).