All is Fundamental
Remember the infamous Fundamental Theorem of Arithmetic...?
What the student remembers (or hears): Each number can be factored in exactly one way, ignoring order.
So, that ever so clever student replies: "Not true!
14 = (2)(7)
14 = (1/3)(10)(0.2)(21)
14 = [4+sqrt(2)][4-sqrt(2)]
14 = [8+5sqrt(2)][8-5sqrt(2)]
14 = (4.5-2.5i)(4.5+2.5i)
How do you respond?
It perhaps would not help to restate the theorem's exact wording, as students will hear the same thing already in their heads.
Moreso...any idea how the Fundamental Theorem of Arithmetic is proven?
Can you state the Fundamental Theorem of Algebra? Geometry? Probability? Statistics? Calculus? Perhaps such do not always exist...
Who decides what makes a theorem "fundamental"? That is, what are the necessary criteria...and can their be more than one for a content area?
Source: Adapted from G. Merriman's To Discover Mathematics, 1942, p. 51