The Mathematical Proof Thread

[RozKek]

My kid is studying some stuff in her 8th grade advanced honors algebra class. I'll have to run this by her to see if she understands. Once you start throwing letters into a math equation, I get lost. Letters don't belong with numbers for us dumb folk.

In high school and in college, I was actually not interested in math at all.  However, two years ago, I became interested in it and have taught myself (no classes or formal education; I'm simply an independent learner, nothing more) high school algebra, pre-calc, trig, calc I,II,III (I absolutely loved the u substitution), differential equations (odes w/ a brief intro to pdes), elementary linear algebra, and discrete math (I'm currently learning this). My point in making this recollection is that with the exception of basic linear algebra and discrete math, I was definitely more focused on the numerical and computational aspects of subjects, rather than gaining a true appreciation for the underlying theory. Hence, I was too grounded in computational thinking, and I can tell you that it is entirely normal to be thrown off by letters, as they represent a shift in thinking (from the specific to the general) which takes time to properly cultivate.
 
As a result, when studying proofs (at least in the beginning) it may be helpful to work out a few numerical examples just to see the concept being concretely illustrated. For example 2 is even because, 2=2*1 (one is an integer).  Likewise, 4 and 6 are even because they can be written as 4=2*2 and 6=2*3, 2 and 3 are integers.  Hence, this process can be extended to a more general understanding of even numbers: an integer n is even if n=2a for some integer a (the definition of an even number). Hence, all the letters do is acknowledge what we already know to be true in individual cases: it extends those known facts and connects them to a broader general theory.  Once you get more acclimated with proofs, then you can often do what we just did above in reverse: prove a theorem in more general terms via definitions, lemmas, or other theorems and then reinforce that general understanding with specific examples and cases in order to gain a complete understanding of the mathematical concepts.

You taught yourself so much in two years? That's quite awesome, damn.