My 13-year-old sister has a problem which, given the way math is currently taught, I doubt is anything but all too common. She has a low grade in her math course and only ever attempts to memorize formulas and tricks, but never actually learn any of the reasoning behind the math. Cross multiplication is the perfect example.
She knows that from
.. she can "cross multiply" to get
$30 = 5x$
.. and from there get $x=6$. She has absolutely no idea what any of this means, however. She's simply memorized a pattern and is applying that pattern to a recognizable arrangement of numbers.
Given the incremental nature of math, her performance has gotten worse as her lack of understanding has compounded. She'll occasionally ask me for help, but is always upset that I won't simply give her the answer to the problem at hand or the "formula" for what she's trying to do. As I ask questions to test her understanding of something, she begins to randomly guess numbers either out of thin air or numbers that I'd mentioned in my explanations, but she doesn't appear to be actually thinking about the problem and considering the answer. After about an hour, she begins to claim she's tired, can no longer focus, and that we're spending too much time on a single problem and that she has more to do.
The inspiration for my finally posting this question and reaching out to the mathematics community came from the homework she had today. She wanted to know how to find the circumference of a circle. After a few questions I had determined that she had no idea what the radius, diameter, or circumference of a circle even were. She even attempted to guess "area" at one point. After relating circumference to the circumnavigation she had learned about, radius to the rays of a sun, and diameter to meaning two (even though this isn't the proper etymology of diameter), she was at least able to label the parts of a circle. Instead of giving her the $c=\pi d$ formula she wanted so badly, I wanted her to understand that $\pi$ represented the amount of times the diameter "fits" into the circumference and that this is the relation between the parts of the circle. I measured as accurately as possible the perimeter and diameter of the mouth of a cup I had and showed her that dividing the numbers produced approximately pi. This unfortunately didn't provide the "ohhh" response I was looking for, which signified that she didn't intuitively understand division. So I tried with a much simpler example. Our conversation went something like:
"The circumference of the glass divided by the diameter gave me pi, what does that mean?"
"Er... I don't know?"
"Well, if I divide 10 by 2, what do I get?"
".. and what does that mean? How many twos are in ten"
"five twos go into ten?"
"Right, so if I divide the circumference by the diameter and get pi, how many diameters are in the circumference?"
"WHAT?!? Why seven?"
"because diameter means two?"
and so on ad infinitum.
She doesn't have any learning disabilities or mental handicaps, so it irritates me to no end that she won't put any effort into learning things that are essential to her understanding and that she could easily grasp.
How do you teach someone to understand math when they are capable but unwilling to do so?