Intelligence Intensification/Information Grasping

Vocabulary
Learn to be cognizant of words that you do not yet understand. Cognizant means to be consciously informed.

It is generally known that it is easier to learn something which you want to learn. When do we want to learn? One situation when we usually want to learn is when we have to in order to solve a concrete problem. This leads to the concept of information availability. If you have access to information when you need it, you will learn more. For instance, if you need to understand a word and have a handheld computer with a dictionary, you will probably look it up. If you haven't any dictionary available, and the need of knowing the word cease to exist, you never learn the word.

We also learn when we are curious. If you can access the information you need, again maybe using a handheld computer, you can learn while you are curious. Later in the day you might feel that other things are more important.

Metaphor
According to some cognitive scientists, notably George Lakoff, most abstract concepts are founded on metaphor. The number line represents quantity as distance, for example, and the Cartesian coordinate system generalizes that same metaphor to function in two dimensions. René Descartes did not invent this metaphor: people from many cultures say "the price went up" (or the equivalent), regardless of their native language. The metaphor "more IS up" (in Lakoff's notation) is quite old, but Descartes did formalize and extend it, allowing it to be applied to a new array of intricate concepts.

But a metaphor can only be extended by a certain amount, after which it loses its meaning. Much of the power of mathematics comes from systematic and well-documented limits to the application of its metaphors. As soon as a new metaphor is added to the field, mathematicians thoroughly test exactly where it does and does not apply, and record their results explicitly in theorems that guide future work.

The convention in other fields is to give metaphors much less explicit limits, forcing the reader or listener to find them out by trial and error. Returning to the "more IS up" example, some quantities tend to go up only with difficulty but go down quite easily, or vice-versa. In these cases, your intuitive understanding of gravity can be added to the metaphor, if you imagine that some quantities sink, while others float. If this doesn't apply, then the metaphorical "weight" of a quantity won't help your understanding.

Some teachers are cognizant of the role that metaphor plays in learning, but in many courses, the metaphors behind the concepts presented are only vaguely hinted at. Actively searching for them and thoroughly testing their limits as early as possible can be quite helpful.