Imagine what would happen if a first-grade teacher, instead of teaching her students addition and subtraction, attempted first to teach them algebra–or, even worse, in the name of intellectual rigor and in an effort to offer a program of exceptional quality–decided to dive right into calculus. Obviously, the students would learn nothing. Being unable to grasp the basic operations of mathematics with specific numbers, they would be completely unable to grasp the idea of a variable that abstracts away from any particular number–let alone advanced equations involving complex algebraic calculations.
Fortunately, few schools would attempt to teach math in this way. But they commit the same error, in less extreme and far less noticeable form, throughout all subjects–that is, the error of teaching certain content before students have the context necessary to grasp it.
All abstract knowledge depends, for its meaning and validity, on other knowledge that sets the context for it. For example, algebra depends on addition, and calculus depends on algebra. The more complex the knowledge, the more extensive the knowledge that must precede it.
One major aspect of the fact that knowledge depends on other knowledge–the aspect most relevant to and most violated in education–is that more abstract knowledge depends on less abstract knowledge. This is the principle of the hierarchy of knowledge.
All abstract knowledge of reality begins with our only direct point of contact with reality: perceptual observation. For example, the first concepts formed by a child are those formed directly from perceptual data; they are what philosopher Ayn Rand, the first philosopher to fully identify the hierarchical nature of knowledge, called “first-level concepts.” A child looks out at the world, perceives entities, and integrates his perceptions into these first-level concepts–concepts such as “cat,” “dog,” and “horse.”
Using these first-level concepts, a child is able to form simple generalizations such as “cats meow” or “dogs bark.” From these early concepts and generalizations, and with further observation, he is then able to form more abstract concepts (concepts further removed from the perceptual level) and to make more abstract generalizations. For example, noting the essential similarities between cats, dogs, and horses, he is able to form the concept “animal.” He does not look at the world and see “animals”; he looks and sees cats and dogs and the like–from which observations he is able to form the corresponding first-level concepts. Then, having formed these abstractions, he is able to form the broader abstraction of “animal.” Eventually, having made countless observations, and having grasped a complex range of concepts and relationships on various levels of abstraction, he is able to form generalizations such as: “All animals are mortal.”
A concept or generalization is more or less abstract according to its cognitive distance from the perceptual level. Concepts and generalization exist in a hierarchy, from the perceptual level to the highest level abstractions.
Highly abstract concepts presuppose a very long chain of prior conceptualizations. This is why, for example, so much knowledge must be gained for students to learn calculus.
Knowingly or not, parents encounter the issue of hierarchy all the time. When my daughter Lana was 2
