The secret to “thinking outside the box” is to never have your thinking put in a box in the first place. Thanks, Mom. This would not have happened to me without you making it happen, and I only just now figured this out.
Since these two problems are really the exact same problem, in two different forms, why not just use “x” to teach it, from the beginning, in elementary school, instead of using the little box? The two symbols have the exact same meaning!
To the possible answer, “We use an ‘x’ for multiplication, instead, so doing this would be confusing,” I have a response: why? Using “x” for multiplication is a bad idea, because then students have to unlearn it later. In algebra, it’s better to write (7)(5) = 35, instead of 7×5 = 35, for obvious reasons — we use “x” as a variable, instead, almost constantly. This wouldn’t be as much trouble for students taking algebra if they had never been taught, in the first place, that “x” means “multiply.” It’s already a letter of the alphabet and a variable, plus it marks spots. It doesn’t need to also mean “multiply.”
Why are we doing things in a way that causes more confusion than is necessary? Should we, as teachers, not try to minimize confusion? We certainly shouldn’t create it, without a good reason for doing so, and these current practices do create it.
These things may not be mysteries to others, but they certainly are to me.
[Note: for those who do not already know, I am a teacher of mathematics. However, I do not have any experience teaching anything at the elementary level. For this particular post, that’s certainly relevant information.]
Software used may be tried here: http://www.software3d.com/stella.php.
Software used: see http://www.software3d.com/stella.php.