A mathematical misunderstanding
Picture the scene….
A math instructor and her student are working through a problem together and on the first line it says
f(x) = 2(x+3)
“So” says the instructor “What do we have here?”
The student thinks for a second before replying, “f of x equals 2 of x+3”
The instructor gives a long sigh and puts her head in her hands.
“Have I taught you nothing?” she asks desparingly as the student looks on in bewilderment. Who has the conceptual misunderstanding of this situation; the teacher or the student?
I would take the student’s wording to mean “I have two (x+3) or (x+3)+(x+3)” but I guess the professor is taking it to mean “I have function called 2 which is (x+3).” I don’t think there is enough information to make a judgment on who isn’t parsing the language or concepts correctly.
One error is that of language.
The other is that of the concept of functions and mathematical notation.
I’d want to ask the student to explain what they meant by their statement, or to solve the function for f(2).
I’d ask the professor what they believe the student meant by that statement, such as “do you believe they thought 2 was a function, with x+3 as the variable or what?”
The instructor is being sloppy in their notation and relying on the implied ‘*’ between 2 and (x+3), the student is being sloppy by not defining ‘2’.
A valid answer could be:
‘f of x is 2 of x + 3, where 2=@(x) x*2’ (using MATLAB syntax for lambda functions), or ‘f of x is the function taking x as input and returning the function ‘2’ evaluated at the result of plus(x,3), where ‘2’ = … and ‘plus’ = …’
(I may have been reading about compilers too much recently)
The teacher seems to have a student willing to assume not only natural, rational, irrational, and imaginary numbers, but “functional” numbers as well. Seems like the makings of a great mathematician to me.
I would ask the student: “Assuming only the need for consistency, where would you go next with your functional numbers? That is, what other equations could you write? Maybe: 2(x+3) ==== 3(y+2)? How few new conventions can you get away with?”
And I get the feeling this student might reply: “Just as you have shown that there is only ONE imaginary number, i, there is only ONE functional number, 2. But unlike the imaginary number i, which can be used in equations with ANY number of the other types of numbers, my functional number 2 can only be used in this ONE equation.”
This wouldn’t be a problem if we all used Mathematica notation: square brackets enclose function arguments, parentheses indicate grouping.
When I first learned Mathematica I thought the square brackets were really strange. Now I think they were brilliant. A slight change to standard notation made it possible for Mathematica to handle numerous situations more elegantly.
The teacher was using standard mathematical syntax, which has an ambiguous grammar. ?(x) means apply the function ? to x when ? is a letter. It means multiply ? times x when ? is a number. The students need to be taught that explicitly. That is not common practice.
I realize that even after being taught some of the rules of mathematical syntax, students will still be fooled from time to time. So when that happens, remind them. But if the teacher doesn’t tell them in the first place, it’s the teacher’s fault.
@Drake – Absolutely right! There is not enough information in the scenario to work out who is in the wrong.
@Matt – Yep, everyone is being sloppy here. Traditional mathematical notation is about as sloppy as it gets sometimes :) MATLAB and Mathematica notation is much more precise IMO.
@John – Couldn’t agree more. In fact, you have stolen my thunder because that was going to be (part of) the conclusion when I posted a part 2 to this post.
@Charles – It’s even more ambiguous than describe. Without me giving you any context how would you parse a(b+1) for example?
I wonder if anyone can find any real exam questions or textbook exercises where the questioner has not given enough information about the symbols used and so assume that the student will just guess correctly.
what is the problem? no need to put your head in your hands, “two of three” is six, and “two of x plus 3” is 2*(x+3).
I think you are all reading way too much into this. It seems to me to be just an ambiguity about the order of operations … “2 of x plus 3” means (2*x) + 3 so the teacher is simply lamenting the student’s algebra.
Neither!
The student is not understanding – not her fault! – the standard function notation f(x), and then overgeneralizes, thinking the parentheses in 2(x+3) mean the same thing. It has nothing to do with “sloppiness” or “error”. Mathematica, e.g., gets around this by writing f[x] for functions.
Hardy in his book “Pure Mathematics” has an example, as I recall (many moons since I read it) in which “A” occurs twice in a formula. Hardy has a footnote explaining the second occurrence of “A” is to be interpreted differently from the first occurrence. He could have used “B” but chose not to. Sloppy? Not really – but less precise than computer science e.g. Wrong? No. Convention? Yes.
As Bill Thurston has written, the language of mathematics can be (quite) confusing.
Nice example, nice discussion. Congrats!
I read “2 of x + 3” as meaning “2 times x+3” (if you get my meaning re spoken math expressions).
I’d be interested to know how others would say “2(x+3)” aloud.
Just wait until probability theory then…
@Rock Hyrax: I’d most likely say “two times x+3” in normal conversation, but I believe the ‘proper’ way would be to say something to the effect of “two times QUANTITY x+3” to denote x+3 is a grouped thing. :D