One pillar of our education enhancement effort is the concept of “pre-understanding” which argues that there usually is a step that has been skipped in education and that is the overview or guidance or “lay of the land” step that comes before courses become efficacious. To tackle a 900-page text-book seems soul-crushing in the absence of “pre-understanding” (i.e., where are we and why are we doing this) other than the coercive power of schools (grades, scholarships, recommendations, grad school admissions, etc.)?
Can we give students a “pre-understanding” that opens a backdoor or side window into the field, where such doors and windows were never seen or noticed?
Immediate concerns are of course price, flexibility of ticket, safety reputation of different airlines, schedules, weather forecasts, routes, etc.
A person might argue: Flight A stops in Tokyo and I can make use of that because my friend who lives in the area will put me up for a weekend, whereby we can do the town and sights, talk about old times, re-connect, etc. There’s also some other task or chore there I could do and so the Tokyo interruption is to my liking. There’s some risks associated with this (i.e., my fiancée who’s traveling with me might find it boring). I’m not sure (uncertainty).
You don’t realize that you’re making subtle decisional calculations where risks and uncertainties that cannot be quantified, are somehow being weighted and weighed and quantified by you, implicitly and the decision calculus is quite complicated.
Suppose you were now given to understand that economics is about economizing (i.e., budgeting your costs, benefits, risks and uncertainties, some of which are qualitative and subjective) but you find a way to assign some kind of numbers and weighting factors (i.e., importance to you) in your actual but more likely, intuitive calculations.
Goaded and prompted by this “pre-understanding” you might then pick up a standard guide to actual cost-benefit analysis (such as Mishan’s classic book) and go through this previously unseen “door” into the field without being crushed by the feeling that it’s all so tiresome in its appearance.
Similarly, if you take a math concept like the square root of minus one, think of it as an imaginary “unicorn” of the mind, then how is it that it appears constantly in all science and math such as Euler’s equation, Schrödinger’s equation, electrical engineering textbooks, etc.
How can something so elusive be so useful?
This “pre-understanding” quest or detour or episode could give you, the student, a deep nudge through a hidden window or door into “math world.”
Without this “trampoline of pre-understanding,” an “ocean of math intricacy” seems to loom before you.