Formulas

Math is riddled with formulas that provide short cuts to solutions.  They are simplifications of long hours put in by great mathematicians to help us find a quick answer.  These mathematicians probably assume that we understand all of the "behind the scenes" work and intuition of the formulas, but that's rarely the case.  In my opinion, this is very unfortunate because by explaining how we get these formulas, we could deepen students' understanding and not only help them remember formulas, but know why and how they work.

I saw an example of misunderstanding in my classroom observations.  The algebra students were given two points, (3,4),(-5,2) and asked to find the slope.  Some students graphed the two points and found the slope by counting and dividing rise over run.  Others attempted to use the formula Y2 -Y1/X2-X1.  As I walked around the room, looking at students' work on their whiteboards, I noticed the numerators all seemed to be 4-4.  I was so confused as to how they were getting the second 4.  This point was clarified for me when the teacher asked one student to explain his work to the class.  The student started reading off the generic formula, but instead of saying Y2, as in the second Y, he said Y squared.  Ah, they were squaring 2 to get the second 4.  It was a simple problem of not reading the formula correctly.  However, this could have been avoided if the students' had understood intuitively what the slope formula was saying.  Clearly, in the graphs they understood rise over run, as the difference between the Y coordinates divided by the difference in the X coordinates, which is exactly what the slope formula says, just with some subscript to clarify the coordinates' order.  But the students missed the connection between that explanation and the creation of the slope formula.  If this point had been clarified, I believe that they would not have been confused about the subscript notation because they would have known Y squared has no use in rise over run.