[BUG] Bridge activities in a multivariable course

[Martin Jackson]

My multivariable class did The Valley today so I am prompted to 
report on my experiences for the semester.  I intended to report more 
frequently but....

The context here is a semester multivariable course at a small, 
undergraduate school.  This is the final semester of our 
three-semester calculus sequence.  There is a lot packed into the 
course.  On Monday, I arrived at vector analysis proper with two and 
a half weeks to go.  This is clearly not enough time to do the topic 
justice.  On the first day, we discussed vector fields and line 
integrals.  I gave students the assignment of using Matthias's Vector 
Field Analyzer to plot some vector fields. (and encouraged them to 
play around with some other features.)  You can read my specific 
instructions to them on the course web page at
http://www.math.ups.edu/~martinj/courses/spring2004/m221/m221.html
under the "Daily Note" of Monday, April 19.  In class, we did one 
example of computing a line integral.

With this one day of preparation, my students were ready for The 
Valley.  As might be expected, some groups struggled with getting 
drvector for their curve.  All groups got through this either with 
geometric reasoning or by computing a differential and substituting. 
(I encouraged groups who initially did it one way to think through 
the other way.)  All groups finished part 3 with good understanding 
and several groups got through part 4.  We did not have time for much 
wrap up.  I will do this in class tomorrow.  My sense is that most 
students gained understanding with the meaning of line integral and 
comfort in computing line integrals.

So far, I have used four activities: Which Way is North?, 
Acceleration, The Hill, and The Valley.  These and the Bridge 
materials overall have impacted my course in the following ways:

1.  I am much more careful in working with geometry first and then 
coordinates.  I have consistently delivered the message: plane or 
space first, geometric objects next, coordinate system and components 
last (if needed).  Which Way is North sets this up beautifully and 
Acceleration reinforces the message nicely.  A specific example of 
how this approach benefits students comes early in working with the 
equation of a plane.  Following most texts, I used to write something 
like {N dot (r-r0)=0} where N is a normal vector, r is a position 
vector for a general point and r0 is a position vector for a point 
known to be on the plane.  Now I write {N dot vector(PQ) =0} where P 
is a general point and Q is a point known to be on the plane.  The 
difference here is that the first expression uses position vectors 
which requires a defined origin at the very least.  To draw r and r0, 
you need an origin (and usually draw a full set of cartesian 
coordinate axes).  For example, see the figure on page 861 of the 5th 
edition of Stewart.  The coordinate axes only clutter and confuse the 
geometry in this figure.    I was fortunate here in that the current 
edition of our text (Strauss, Bradley, and Smith, 3rd ed) has gone to 
a {N dot \vector(PQ)=0 }style.  I feel that my students had much 
better understanding of this form for the equation of a plane.

2.  On a related note, I now express all vectors in terms of unit 
basis vectors (ihat, jhat, khat for cartesian coordinates) rather 
than as an ordered triple (perhaps with special brackets).

3.  I have used differentials regularly.  This aspect has infiltrated 
my Calc I classes as well.  I'm now not sure why I had such an 
objection to differentials in the past.  (Part of this is learning 
the general lesson that rigor is not equivalent to good pedagogy.) 
In Calc I, I explain differentials in the following way:  Let Delta y 
and Delta x represent rise and run along the graph of y=f(x).  Let dy 
and dx represent rise and run along a line tangent to the graph of f. 
Since this is a line, we know dy=(slope)dx=f'(x)dx  For equal runs, 
Delta x=dx, we have Delta y not equal to dy in general but Delta y is 
approximately equal to dy so we can use
Delta y approx dy=f'(x)dx.
I then do examples where it is useful to trade equality for the 
simplicity of a linear relation.  I like the propagation of error 
from measured quantity to computed quantity for this.

4.  In doing vector analysis, my plan is to do line integrals and 
surface integrals before divergence and curl.  Although I am only two 
days into this, it feels right so far.  I am also only covering line 
integrals for vector fields and not line integral for scalar fields. 
(This is something I have done for a while because of the time crunch 
I face and the relative importance of vector fields over scalar 
fields here.)

This is becoming a rather long post, but I do have an issue on which 
I am seeking advice: The beginning of the multivariable course is 
traditionally about vectors.  I find that this takes more time than 
the topic seems to merit.  I am willing to wager that most students 
in a multivariable course look back at the introductory material on 
vectors and wonder why it seemed difficult in the beginning. 
Students gain a lot of familiarity with vectors during the course. 
Is there a faster way to get them started?  When I poll my students, 
about half have seen vectors in a high school or college physics 
course and about half have not.  There are a lot of things going on 
in the vector material, including a new way of thinking and the issue 
of working in 3 dimensions.  Does anyone have suggestions for how to 
move students through the introductory ideas on vectors so that there 
is more time for the substantial ideas of vector analysis at the end 
of the course?

Martin