Learning by Logical Induction vs. Logical Deduction : Induction is the defining characteristic of highly effective pedagogy, and a critical learning experience skilled teachers contribute to successful classrooms.
Unambiguous clarity demands that textbooks and computer programs be written using DEDUCTIVE logic, where the general rules of a system are precisely defined, and thereafter each input results in a correct output when the rules are correctly applied. If the governing rules of a system are correctly defined, then the result of any input will always be correct output under logical deduction. As an example of deductive logic, consider the rule that the area of a rectangle is always base multiplied by height. A textbook would be correct in presenting this formulation, and once programmed with this rule, a computer will always correctly return the area of a rectangle given base and height.
Like computers, humans can learn by deduction. An important contrast to computers, humans can also learn naturally by INDUCTIVE logic, where specific individual observations of a system are repeatedly made by the student until a reasonable guess at the general rules of the system can by hypothesized which accounts for observations. By this method NEW knowledge can be created, even in systems where the governing rules are not explicitly given. As an example of inductive logic, picture any rectangle drawn on graph paper, and consider the foundational definition that the area is the number of little square units inside that rectangle. A student can find the area by counting little square units. After repeated exposure to examples of rectangles within this system, as the tedium of the task spurs innovation, students will begin to formulate that a shortcut might exist in skip-counting, or multiplying. Such organic hypothesizing of general rules governing a system derived from repeated observation defines inductive logic.
When presented with a new shape of unknown area, like a triangle, a computer reliant on deductive logic would be unable to compute area until programmed with new rules. The opposite is true for the student equipped with inductive logic, which once flexed in the discovery of rules governing rectangles, will often transition readily to the organic generation of rules for other polygons without requiring further instructions.
Another way to think about the difference between Inductive logic and Deductive logic is the way people learn language. All people learn their primary language by 1nduction, which means they pick it up naturally from repeated exposure to examples of the language spoken daily until the rules of the system become deeply confirmed understandings. The opposite logical process of Deduction is at work when people try to learn a second language from a rule-set, like a textbook. While this rule-based deductive approach could flawlessly teach a computer new languages, humans seem to experience better results and retention learning languages by Induction.
A personal thought experiment: would you rather learn to speak a foreign language from six years of classroom study, or six months of unstructured travel in a place where that language is exclusively spoken? This defines the difference between six years of exercising your brain by deductive logic, versus six months of exercising your brain by inductive logic.
While our ever-expanding body of scholarly knowledge must always be chronicled in clearly deductive logic according to provable rules, for humans learning is experiential. The high art of skilled teachers shines in the transformation of what textbooks document deductively, into what students experience and discover inductively within a customized jungle-gym of learning. For teachers, the power to tailor deep learning experiences increases with the development such learning jungle-gyms, where interactive presentation allow the rules at work within a system to be isolated, illustrated, animated and then fluidly exposed in multiple representations (e.g.: equation, table, graph).