[BUG] Trusting "use what you know" vs. an algorithm

[Wangberg, Aaron]

> Am I correctly understanding that it's the method of Lagrange
> multipliers which does "not make much sense", rather than any direct use
> of drvec?  This wouldn't surprise me; my experience is that students
> view Lagrange multipliers as a mysterious algorithm to be followed
> blindly, even when I've tried to motivate it geometrically.

In the past (i.e. not using geometric reasoning), I've found that students who work with Lagrange multipliers have found the method confusing.  In particular, they seem to miss the logic behind viewing the constraint as a level curve of a function.  They also seem to miss the geometric reason for using lambda to set grad f = lambda grad g, and instead blindly just follow the algorithm - they look at the equality of the components of the two vectors but are not sure why they are doing that.
 
I'd hoped that using dr vector would help students understand the method of Lagrange multipliers, and it caused mixed results.  They understood how to find the maximum or minimum of a function along a constraint using dr vector, and this method of solution seemed to be a by-product of their understanding of the geometry of the problem.  On the other hand, for the method of Lagrange multipliers, they still had a difficult time seeing the geometric reasoning behind using the gradient vector associated with the constraint.  Viewing a constraint as a level curve of a function g, and then finding the gradient vector of g, and then going back to restrict solutions of grad f = lambda grad g to those just existing on the level curve does not seem as easy as "using what you know".  It is nice that this method combines many different ideas into one problem.

> > I find it interesting that students followed someone else's algorithm
> > (Lagrange multipliers), even as they were confused by it, over a solution
> > method they had previously used comfortably.
> > 
> > (In all fairness, from my students perspective, though, I'm sure they found it
> > confusing that we even discussed the method of Lagrange multipliers in the
> > first place, when we had previously just "used what we knew" about the
> > constraint, dr, the master formula, and df = 0.)

> Perhaps the answer is not to teach these as disjoint topics.  Surely
> drvec can be part of the argument used to justify the method of Lagrange
> multipliers?  I'm not sure I've ever actually tried this, and wonder if
> it would work better than the traditional pictures of level curves
> tangent to the given curve.
 
The nice thing about using drvec and grad f is that both live in the domain.  I think that the traditional approach to Lagrange multipliers could be more successful if I could put more emphasis on the relationship between a gradient vector and the level curves very early in the course.  For instance, I can see that it might be useful to give students the gradient vector field of a function and show them or have them draw the level curves of the function.  Doing this early may help students realize the connection between gradient vectors and level curves.  On the other hand, this might be too much to learn about the gradient vector when I first introduce it.

Aaron

________________________________

From: Tevian Dray [mailto:tevian at math.oregonstate.edu]
Sent: Mon 10/8/2007 11:39 PM
To: Wangberg, Aaron
Subject: Re: [BUG] Trusting "use what you know" vs. an algorithm 



I'm going to respond first just to you since I'm not sure I fully
understand your concerns yet.

> I believe that it will be very helpful to use dr vector when covering
> double and triple integrals.  However, I've also come across a few
> situations where doing things without dr vec seems to "not make much
> sense" to the students.  For example:

My first reaction was that you meant *with* drvec...  After reading the
rest of your message, I now believe you really do mean "without", but
then "However" seems out of place.  Am I correctly understanding that
it's the method of Lagrange multipliers which does "not make much
sense", rather than any direct use of drvec?

> I find it interesting that students followed someone else's algorithm
> (Lagrange multipliers), even as they were confused by it, over a solution
> method they had previously used comfortably.
> 
> (In all fairness, from my students perspective, though, I'm sure they found it
> confusing that we even discussed the method of Lagrange multipliers in the
> first place, when we had previously just "used what we knew" about the
> constraint, dr, the master formula, and df = 0.)

Perhaps the answer is not to teach these as disjoint topics.  Surely
drvec can be part of the argument used to justify the method of Lagrange
multipliers?  But let's move this part of the discussion to the BUG list
as soon as I'm sure I've understood your first point.

Best wishes,
        Tevian



________________________________

From: bug-bounces at science.oregonstate.edu on behalf of Tevian Dray
Sent: Wed 10/10/2007 12:21 AM
To: Bridge Users Group
Subject: Re: [BUG] Trusting "use what you know" vs. an algorithm 




> However, I've also come across a few situations where doing things
> without dr vec seems to "not make much sense" to the students.

Am I correctly understanding that it's the method of Lagrange
multipliers which does "not make much sense", rather than any direct use
of drvec?  This wouldn't surprise me; my experience is that students
view Lagrange multipliers as a mysterious algorithm to be followed
blindly, even when I've tried to motivate it geometrically.

> I find it interesting that students followed someone else's algorithm
> (Lagrange multipliers), even as they were confused by it, over a solution
> method they had previously used comfortably.
> 
> (In all fairness, from my students perspective, though, I'm sure they found it
> confusing that we even discussed the method of Lagrange multipliers in the
> first place, when we had previously just "used what we knew" about the
> constraint, dr, the master formula, and df = 0.)

Perhaps the answer is not to teach these as disjoint topics.  Surely
drvec can be part of the argument used to justify the method of Lagrange
multipliers?  I'm not sure I've ever actually tried this, and wonder if
it would work better than the traditional pictures of level curves
tangent to the given curve.

Best wishes,
        Tevian
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