Aristotle was a Greek philosopher. He was mentored by Plato and went on to become the mentor and tutor of Alexander the Great. His philosophical foundations draw upon a variety of sources including metaphysics, zoology, politics, ethics, and logic, just to name a few.
Aristotle's philosophical methods were empirical, based on analysis and observation, and critical.
The empirical aspect of his method employs observation and data collection, as well as categorization and classification. After making detailed observations and after all samples of data were collected, the student would proceed to make correlations that would provide categorization and classification of facts. This would provide the basis of knowledge. The student would then proceed to generalize a principle or establish a theory. Notice that these are the backbone steps of the scientific method.
The student, who is independently discovering facts aided by Aristotle, not instructed by him, will use inductive and deductive reasoning as critical thinking methods. Critical thinking through inductive and deductive reasoning is what the Aristotelian method is known for.
Syllogisms are the Aristotelian method of formulaic (critical) thinking. Syllogisms are formulaic because they use specific words to create a formula that will present a fact. A syllogism operates within established parameters of logic to verbalize a conclusion based on inductive and deductive critical thinking.
As a logical argument, the syllogism presents a (a) major premise, (b) minor premise, (c) conclusion. A syllogism is almost like a Haiku, if we were to compare it to poetry.
In the major premise, the general statement is expressed. The minor premise states an example or supporting detail for why the major premise is a fact or might be a fact. The conclusion should extend from the major and minor premises.
Major premise: A=C
Minor premise A=B and B=C
For the syllogism to be accepted as true, the first two premises must be facts and must be proven to be facts. Two negatives do not equal a positive in syllogisms.
Now, to the modern eye the Aristotelian method may look obvious. We all gather and collect data, then analyze it before reaching a conclusion. However, we do these things now because of the Aristotelian method.
If we compare Aristotle's method to Plato's, the significance of Aristotle's method of teaching is that Plato attempted to define the abstract and the ideal, while Aristotle attempted to explain the concrete and the real. Both philosophers were correct in their applications, so much that they were immortalized in Raphael's "School of Athens" (1509-1511) in which Plato and Aristotle take center stage as they discuss their philosophical views. Here, Plato points at the sky, to the abstract. Aristotle points down and center, to the concrete. Such is the importance of Aristotle's method of teaching.