As opposed to the most harmful heuristics, what are the most useful heuristics which
are hand-waving,
are conducive to proper mathematical education, and
you have seen taught or taught yourself?
In this context:
Hand-waving means imprecise, intuitive, ambiguous, with a purpose of impressing or convincing.
Proper mathematical education means that a person can understand, use, discuss, and derive the learnt mathematical claims after finishing the education process to the levels (a) advertised by goals of the education process and at the same time (b) having, up to some allowed degree of ambiguity, the same, widely accepted meaning in the community. Example: “Real Calculus” could mean “basics of differentiation and integration over the functions $\mathbb{R}\to\mathbb{R}$”.
Seen taught means you closely observed or participated as a learner in the educational process.
Taught yourself means you were a lecturer or an author of used educational material.
I like Richard Feynman’s heuristic of understanding a generality via a simple (but nuanced enough), well-understood example. The start of an inductive proof is sort of an application of this: Convince myself that the statement is true for some simple cases, and see where there might be a general pattern for every such case.
I like testing probability deductions with real life experiments. Especially dice problems are really illustrative for people who are just starting out.
Teaching and using differentials in elementary calculus. They help with linear approximations, the product rule, the chain rule, arclength, Cavalieri’s principle, applications of integration. In each case the handwaving can be made rigorous, but the effort at rigor obscures the underlying idea.
Picking up on Tao’s comment there, one of the most useful heuristics is thinking of exponentiation as iterating an infinite number of infinitesimal multiplications. This is a useful heuristic not merely in Lie groups but any time one is dealing with the infinitesimal generator of a flow. In fact the flow can be thought of as the shadow of a walk by infinitesimal steps (of course infinitely many of them).
At a more elementary level, thinking of $\frac{dy}{dx}$ as a ratio and ignoring the boos from the audience 🙂
A picture is worth a thousand words.
For young students, it is helpful to introduce the concept of multiplication by arranging objects into groups having an equal number of objects in each group, or by arranging the objects into a rectangular array of the desired number of rows and columns.
For algebra students, we can illustrate that $(a+b)^2 = a^2 + 2ab + b^2$, by drawing a square that is $a+b$ on each side, and dividing it with a horizontal and vertical line into four regions: squares that are $a^2$ and $b^2$ on each side and two rectangles that are $a$ by $b$ in size.
For calculus students, when introducing the concepts of derivatives or integrals, it is helpful to illustrate the problem we are trying to solve using a graph, then approximate the solution using finite methods and consider how we might converge to the desired solution using limits.
Thus, we might plot a line tangent to a curve at a specific point and ask “How can we determine the slope of the line?” Then, introduce a finite approximation, such as the secant method, and observe that we obtain a better approximation as the two points are moved closer together.
A similar strategy can be used when introducing definite integrals, by asking how to determine the area under a continuous curve over a closed interval. Introduce the midpoint method as a way to approximate the area and consider how the approximation is improved as we reduce the width of the rectangles.
When introducing Fourier Series, plot examples such as $sin(x)+sin(3x)/3$, then $sin(x)+sin(3x)/3+sin(5x)/5$, etc. to show how the sum more nearly approaches a square wave as the number of terms increases. This also provides an opportunity to discuss topics such as overshoot and ringing, or how a low-pass filter might affect such signals.