Which Came First...

When I'm sitting in class, I like to think that everyone else values their time as much as I value my own.  Apparently that is not the case sometimes.  As I was sitting in class today, it was interesting seeing all the different approaches we thought of when trying to solve the Great Gumdrop Mystery as I am going to call it from here on out.  About 30 minutes into class, I'm scratching my head, wondering if there is something I have missed.  And then it happened...we got to the point of no return...the discussion turned into a matter of semantics.  Here's the thing: guessing is guessing no matter if you guess 2 or 2 trillion.  If you pulled the number out of your you-know-what, it's guessing.  Now to argue that one form of guessing is different than another is null and void; when the discussion comes to guessing methods, you're wasting my "valuable" time.   BUT, alas, my callused heart and rational mind have come to a truce!  Now, arguing whether the sky is robin's egg or baby blue IS a just an argument rooted in semantics (it's freaking light blue!  just walk away!) but what if you ask a student what 12x4 is and their reply is, "I don't know, I haven't learned my 12's yet" and then you ask them what 4x12 is and they say (with great confidence) "48"?  Math, it turns out, is all a matter of semantics!  You see, 12x4 is 4x12 but many students don't see it that way because they are taught facts and not ideas!  We can argue that 12x4 and 4x12 are both different and the same...that would be a matter of semantics...we need to teach students that whether you have 4 groups of 12 or 12 groups of 4, you have the same amount of whatever...let's just say gumdrops for today :)  It's all goes back to the whys...not the hows...if I ever start a math club, I'm going to call us The Whys!