Putting The Pieces Together Is Tough For Some

I've seen a lot of examples in recent months of students who can learn Math skills in one (often rather rigid) context but then be quite lousy at figuring out how to use them in other situations. To understand what I mean, let's look at an example.

Suppose you knew that the formula for the area of a rectangle was length times width, and had known that for years. Also assume that you learned in Grade 9 how to multiply binomials and since then have even acquired the ability to solve simple quadratic equations. All of that knowledge would seem to position you perfectly to answer the following question:

"A rectangle is x + 6 units long and x - 4 units wide. If its area is 56 square units, what are the dimensions of this rectangle?"

Someone who's comfortable with Math at a high school level or higher will quickly realize that they need to solve the equation:

(x + 6)(x - 4) = 56

which is to say, solve:

x**2 + 2x - 24 = 56, or x**2 + 2x - 80 = 0

Factoring that quadratic yields:

(x + 10)(x - 8) = 0

meaning that only x = -10 and x = 8 will solve it. Since a value of -10 gives us negative values for the length (-4) and width (-14), we can safely discard it, leaving x = 8 as the only reasonable answer.

Sure enough, if we use 8 for x, we get a rectangle with a length of 14 and a width of 4, good for an area of 56 square units. Problem solved!

Every step in this solution involved Math of a sort that a typical high school student in Gr 10 or higher would know how to do. But because it starts off as a word problem (Strike 1!), has dimensions that involve variables as well as numbers (Strike 2!!) and requires the solving of a quadratic equation "out of the blue" (Strike 3!!!), it's just a full stop for many of them. I think part of the problem is that some students aren't willing to just try stuff in order to see where it leads them, in addition to a general but deep-rooted aversion to applying what's been learned for one type of problem to questions that don't initially seem similar.

I'm always shocked and dismayed when one of my Math tutoring pupils has all of the tools necessary to solve something like the example used here and yet can't even get going on it, without someone like me to push them in the right direction. I can't help but feel that some essential ingredient has been missed in their education up to this point... I'm just not sure yet what exactly that omitted component is.