TITLE: What are some conceptualizations that work in mathematics but are not strictly true?
QUESTION [87 upvotes]: I'm having an argument with someone who thinks it's never justified to teach something that's not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is to iteratively learn and unlearn along the way. 
I'm looking for examples in mathematics (and possibly physics) where students are commonly taught something that's not strictly true, but works, at least in some restricted manner, and is a good way to understand a concept until one gets to a more advanced stage.

REPLY [6 votes]: Geometry, and differential geometry, are full of these kinds of shortcuts, for the simple reason that geometry is a very visual, accessible, and venerable topic, resting on a deceptively deep body of theory required to make it rigorous. You are taught concepts in high school that will require many years of topology, linear algebra, measure theory, real and functional analysis, etc. before you have the full story.
Some examples:

You are taught to calculate the volume and surface areas of solids by two kinds of slicing. Rarely is it explained why surface area requires approximating a surface with slices that are generalized cones, while generalized cylinders are good enough for volume, and I've never seen discussed why and for which kinds of surfaces these techniques don't work at all. More generally, it is often assumed that if you want to measure something about a smooth surface, you can do so by looking at a limiting sequence of discrete surfaces that converge to the smooth surface under "reasonable" refinement. Defining "reasonable" requires great care (beyond the scope of high school geometry) to avoid counterexamples like the Schwarz lantern.
Concepts like arc length, surface area, curvature, etc. are not defined in a complete or rigorous way. On the one hand you have some formulas that clearly require a certainly level of differentiability, and on the other you have no problem calculating e.g. the surface area of a polyhedron, and probably don't bat an eye at the fact that polyhedra satisfy a meaningful version of Gauss-Bonnet, despite ostensible missing two degrees of differentiability! In addition to regularity, subtleties about orientability, compactness, etc. are likely not discussed. 
Many theorems in plane geometry require objects to be in "general position" (usually left vaguely defined) and the corner cases are ignored. Similarly, pathological counterexamples to the formula for the Euler characteristics of polyhedra (e.g. polyhedra with odd Euler characteristic) are usually ignored.
Finally, while not the same as teaching students incorrect facts, many geometric arguments you see in early classes implicitly rely on unproven "obvious" results like the Jordan curve theorem, invariance of domain, Sard's lemma, existence of triangulations, etc. that are far from obvious.