[BUG] The Valley

[Martin Jackson]

Stuart's question and Tevian's response about the inclusion of C in 
the line integral notation has prompted me to organize some of my 
thinking on notation.

I think issues of notation deserve careful thought.  We need to ask 
ourselves "How do students at this level read notation?"  I think 
some (perhaps many) students at this level read notation as an 
instruction set.  A particular group of symbols indicates a procedure 
that is to be carried out.  For example, the usual notation for 
definite integral signifies the procedure "find an antiderivative and 
evaluate the difference at the endpoints" while the usual notation 
for indefinite integral signifies the procedure "find an 
antiderivative and be sure to put + C at the end."  These students do 
not see the group of symbols as denoting a particular object (a 
number in the case of a definite integral and a function (or family 
of functions) in the case of an indefinite integral).

Our choice of notation often involves balancing the level of 
simplicity with the  level of explicitness.  The issue is forest vs. 
trees.  As students get more mathematically sophisticated, they can 
use simpler notation (showing the forest) and keep track of the trees 
in their head.  For example, when first doing implicit 
differentiation in Calc I, many students need to start out by 
literally substituting f(x) for y in order to see how to compute the 
relevant derivatives correctly.  Many students can then train 
themselves to compute correctly without this substitution.  (It would 
be interesting to explore how students come to think about this.  I'm 
sure some think along the lines of "When I see y^2, I must write 2y 
dy/dx"  while others "When I see y^2, I must remember that we are 
thinking of y as a function of x in this context because we are asked 
for dy/dx and not dx/dy.")  However, the least sophisticated students 
are not able to move away from substituting f(x) at least in the time 
period we allow for thinking about implicit differentiation.

Another issue for students is being able to distinguish similar 
groups of symbols.  At a trivial level, some students do not 
distinguish between upper and lower cases of the same letter.  In a 
more relevant example,  consider denoting two line integrals with the 
same vector field F but different curves, say C_1 and C_2.  I'm sure 
that some students would not see the difference that is buried in a 
subscript on a thing hanging out below an integral symbol.

Now that I've finished these thoughts, I'm not so sure how they 
relate to the question at hand.  Some of the tension here is the 
level of simplicity.  Do we name C in Problem 1 so that we can refer 
to it in Problem 2?  Tevian is correct in pointing out that the 
wording of the problem does specify the curve and so a name is not 
necessary.  However, some students will go straight to the symbols 
(the instruction set for the procedure to be carried out) without 
carefully reading the sentence in which the symbols are embedded.  I 
think the context of this activity is so clear that no confusion 
about what curve to use will result.   In other contexts, the choice 
of notation may be more critical.  I tend to err on the side of more 
detailed notation although I am becoming more conscious of the need 
to help students learn how to be flexible with notation as they 
mature.

This is a long post so here's some specific questions that might 
generate a discussion:

1.  Am I right in thinking that some students at this level read 
notation as an instruction set?  Some, many, or most?

2.  Is it a good strategy to start with explicit, detailed notation 
and then work toward simplified notation?  Is it better to start with 
simple notation and
have students learn to resolve possible ambiguity by considering the context?

3.  Is notation a bigger issue in multivariate and vector calculus 
because there is more information to take care of?  (For example, a 
directional derivative involves a function, an input, and a 
direction.)  Does this make notation a qualitatively different issue 
in multivariate/vector calculus?

Martin



>  Just a minor point:  In problem number 2 students are asked to choose a
>  > curve.  In problem 3 they are asked to evaluate a line integral along
>>  the curve C.  However, C was never defined!
>>
>>  If this lab is to be done as a way to introduce line integrals (as I'm
>>  planning), it should probably be made explicit that the curve chosen in
>>  problem 2 is in fact C. 
>
>Good point.  This raises an interesting question of notation: Why not omit
>the C altogether?  The integral in problem 3 is clearly a line integral, and
>the problem specifies the location in words.  So why does it feel wrong to
>leave the limits out?  Especially when each group will choose a different
>curve, all of which are being called C.
>
>Tevian
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