Why are students good at math? I have had this idea for along time that any person that can visualize text is able to process the content better. What do I mean by this? Simple. I am sure you have talked to someone and realize that you have read the same book. Usually this strikes a conversation about the book. You may converse about pieces of the plot or character development. Anyway, as you read you develop an image of what something looks like. Some images are really clear and other images are not as clear. So let's look at an example of what I mean. Say you are at a really good part in a book. One scene that comes to my mind in a Grisham book is when the main character and, I think their cohort and suspect, are searching for a body. They are walking from the road into the woods to where the supposed body is. Grisham's words become so clear in my mind that I can actaully hear the leaves and branches breaking under the character's feet. I think you get the picture.
So let me take this idea and parallel it with mathematics. First, let's look at a real simple concept like one-to-one correspondence. When I say the number "two" what comes to your mind? For me, two bananas comes to mind. 'I really like bananas.' You may picture something completely different. Moving towards something more complex try to answer this question, "how many steps does it take to walk to your mailbox?" If you don't have a mailbox think of something else that you could walk to that is outside. Now, as you are imagining your lovely stroll with a hot morning drink try to picture yourself taking steps towards the mailbox. This imagining is a simple skill that many of us take for granted. I would assume a great number of us could visualize ourselves walking. Maybe some can't. However, let's throw into the equation trying to figure out the number of steps.
Here's another assumption. Some people may get stuck here and others would naturally ask more questions. Such as, "I can't quite comprehend the amount of steps to the mailbox but I could assume there are _____ steps to the tree, and if there are _____ steps to the tree and I and I am halfway there then I'll just double the amount to the tree." Simple, right. Wrong.
Attending to detail here we could analyze the breakdown of skills involved here but we would stray off topic.
So, back to my initial idea of the movie playing in our mind. From my experience talking to students about their inadequacies which impede these skills we take for granted I strongly believe that if students or anyone for that matter are able to play a movie in their head of the math concept at hand then that person will have success at deciphering tougher math problems.
In a way, isn't this what we ask student all the time, "can you visualize the blah blah and then see how the blah blah changes?"