I haven't posted anything here in a week, so, why not. (Note: I really doubt that most of the entries in the file will ever get written. There's simply too many and the motivation to write them is lacking.)
One thing that often happens when I'm helping students in the MathLab is that -- well, they'll suggest something that makes no sense, often phrased in terms of manipulating the symbols on the page, and I'll respond with something to the effect of "Let's talk about the numbers, not the symbols on the page".
Of course, in a very real sense we are talking about symbols on a page! What the student did wrong was to suggest something that doesn't respect the tree sructure of those symbols. We represent mathematical expressions as strings, but really, they are trees. Well, OK, we don't quite just represent them as strings -- and some things, like superscripts, subscripts, radical signs, fraction bars, etc., help to display the tree structure a bit more directly.
My attempts to correct these errors often involve saying -- well, suppose there was a sin(x+3), and they talked about sin(x); I would say "There's no sin(x) here. There's an x+3, and a sin(x+3), but no sin(x)." (Yes, you could expand it out with the angle addition formula. That is not the point.) This seems to work pretty well for individual instances, but does it really help with the underlying error? That's not really something I can see!
Now anyone who works with mathematical expressions regularly understands that expressions are trees, but apparently a lot of people don't, and it makes me wonder -- is this something we should be teaching more explicitly, and earlier? Related is the confusion about "order of operations", and how lots of people think this is a rule of mathematics, rather than a convention about how to write things (how to represent trees as strings).
Of course, by all accounts, lots of teachers don't understand this stuff either, but hey, we can try...
-Harry
One thing that often happens when I'm helping students in the MathLab is that -- well, they'll suggest something that makes no sense, often phrased in terms of manipulating the symbols on the page, and I'll respond with something to the effect of "Let's talk about the numbers, not the symbols on the page".
Of course, in a very real sense we are talking about symbols on a page! What the student did wrong was to suggest something that doesn't respect the tree sructure of those symbols. We represent mathematical expressions as strings, but really, they are trees. Well, OK, we don't quite just represent them as strings -- and some things, like superscripts, subscripts, radical signs, fraction bars, etc., help to display the tree structure a bit more directly.
My attempts to correct these errors often involve saying -- well, suppose there was a sin(x+3), and they talked about sin(x); I would say "There's no sin(x) here. There's an x+3, and a sin(x+3), but no sin(x)." (Yes, you could expand it out with the angle addition formula. That is not the point.) This seems to work pretty well for individual instances, but does it really help with the underlying error? That's not really something I can see!
Now anyone who works with mathematical expressions regularly understands that expressions are trees, but apparently a lot of people don't, and it makes me wonder -- is this something we should be teaching more explicitly, and earlier? Related is the confusion about "order of operations", and how lots of people think this is a rule of mathematics, rather than a convention about how to write things (how to represent trees as strings).
Of course, by all accounts, lots of teachers don't understand this stuff either, but hey, we can try...
-Harry
