Ramble, Part 34: Dysfunctional Notation
Oct. 13th, 2007 05:52 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowThe use of the notation y=f(x) to represent a functional relationship between two variables is deeply entrenched in modern mathematics, and the notation has proven extremely convenient for many purposes. Nonetheless, there are problems with it, and it is possible that some superior notation could be devised. More under the cut.
There is a longstanding tradition of using the notation f(x) ambiguously: on the one hand, it represents the function, the relationship between the two variables; on the other, it represents the value of the function - that is, the second of the two variables. Those of us who become mathematicians learn fairly soon to recognize which meaning is intended, but the fact of the ambiguity still niggles. There's been an increasing tendency to refer to the function as f and restrict f(x) to mean the value of the function, but it isn't universal, especially at lower levels.
Another problem arises from the very important operation of composition, whereby two functions f and g are applied in succession. Traditional notation places a small circle between the names of the functions, but in many contexts simple juxtaposition is used; that is, gf(x)=g(f(x)). This presents a bit of a psychological problem, in that functions compose right-to-left - it is the rightmost function that is applied first. This collides with the expectation of left-to-right reading, and students often stumble on it. There was a movement a few decades ago to write functions on the right - xf instead of f(x) - which would correct the problem, but it didn't gain much traction and seems to be defunct.
Finally, and most subtly: when one writes, say, f(x)=x2, is this an assertion regarding all possible xs, or only a specific x? That is, is x actually the name of a number (or whatever), or is it a dummy, standing in for every possible number (or whatever)? Let me clarify. In the symbol-processing language Mathematica, if you write f[x]=x2, then, whenever you write f[x], the program will interpret it as meaning x2, but if you write, say, f[y], it will not know how to interpret it: y is not x. To define the general squaring function, you must write f[x_]=x2; the underscore alerts the program that the x is not the name of some specific (though unidentified) number, but something generic. This sort of "slot variable" appears frequently in mathematics, and, again, most of us accustom ourselves to it, but students do struggle with it.
None of these, I think, are major problems, but in my experience they do cause difficulties for students. Good notation should make it easier to think, and should not cast up unnecessary roadblocks. Perhaps we mathematicians can take a cue from Mathematica, as may be happening with the equals sign, but I suspect it will take a long time to take hold, and longer still to percolate down to the lower-level classes where it might be most valuable.
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There is a longstanding tradition of using the notation f(x) ambiguously: on the one hand, it represents the function, the relationship between the two variables; on the other, it represents the value of the function - that is, the second of the two variables. Those of us who become mathematicians learn fairly soon to recognize which meaning is intended, but the fact of the ambiguity still niggles. There's been an increasing tendency to refer to the function as f and restrict f(x) to mean the value of the function, but it isn't universal, especially at lower levels.
Another problem arises from the very important operation of composition, whereby two functions f and g are applied in succession. Traditional notation places a small circle between the names of the functions, but in many contexts simple juxtaposition is used; that is, gf(x)=g(f(x)). This presents a bit of a psychological problem, in that functions compose right-to-left - it is the rightmost function that is applied first. This collides with the expectation of left-to-right reading, and students often stumble on it. There was a movement a few decades ago to write functions on the right - xf instead of f(x) - which would correct the problem, but it didn't gain much traction and seems to be defunct.
Finally, and most subtly: when one writes, say, f(x)=x2, is this an assertion regarding all possible xs, or only a specific x? That is, is x actually the name of a number (or whatever), or is it a dummy, standing in for every possible number (or whatever)? Let me clarify. In the symbol-processing language Mathematica, if you write f[x]=x2, then, whenever you write f[x], the program will interpret it as meaning x2, but if you write, say, f[y], it will not know how to interpret it: y is not x. To define the general squaring function, you must write f[x_]=x2; the underscore alerts the program that the x is not the name of some specific (though unidentified) number, but something generic. This sort of "slot variable" appears frequently in mathematics, and, again, most of us accustom ourselves to it, but students do struggle with it.
None of these, I think, are major problems, but in my experience they do cause difficulties for students. Good notation should make it easier to think, and should not cast up unnecessary roadblocks. Perhaps we mathematicians can take a cue from Mathematica, as may be happening with the equals sign, but I suspect it will take a long time to take hold, and longer still to percolate down to the lower-level classes where it might be most valuable.
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no subject
Date: 2007-10-14 12:10 am (UTC)Two things, a comment and a question:
1. As applications like Mathematica get used at successively lower levels, this ambiguity will become emphasized, and students will be forced to deal with it.
2. At lower levels, do you observe problems due to the ambiguity? Honest question, looking for answer based on your teaching experience.
no subject
Date: 2007-10-14 12:36 am (UTC)no subject
Date: 2007-11-13 08:53 pm (UTC)She told me two main things. First, the biggest problem algebra students have with functional notation is a tendency to interpret f(x) as the product of f and x. This is a different problem altogether, and one I hadn't thought of. The second is more complex. She says that if, for example, f(x) is defined to be x and g(x) to be 2x+1, students are apt to take (f+g)(x) as (x+2x+1)x. This seems to involve the first problem, but the things I mentioned in the post also seem to be involved. She agreed that dummy variables in general (and she pointed to dx in calculus) are hard for students to grasp. (This, too, I think, is related to the problems I mentioned.)
For what it's worth. The comprehension problems that students have in algebra and calculus can't all be handled by means of better notation, but flawed notation certainly won't help.
no subject
Date: 2007-11-15 03:33 am (UTC)f(x) -> multiplication. Ouch. I suppose we ought to strongly discourage the understood multiplication operator. [sigh] One wants notation that emphasizes the important relations and de-emphasizes the unimportant ones, but that tends to be context-dependent. And that is hard for novices to deal with. As is the extra complexity of non-context-dependent notation.
I am a physicist. My (limited) experience in trying to educate young physicists is that half (+/- a quarter) of their problem is algebra, or not being able to deal with simple symbolic relationships. That always came easily to me - so I have trouble recognizing students' difficulties. A good tutor is perhaps one who had difficulties with the concepts herself.
Thanks for the response. I owe you some on the recent Rambles. Which I have skimmed, but not yet digested. I really appreciate this series. Thank you.