Re: apples and oranges

["L. R. Cartwright (Larry)" <physics@scnc.cps.k12.mi.us>]

"David W. Steyert" wrote:

> First, in an earlier post, Larry had written "Years ago I acquired a
> math major and certification, and was taught that math teachers 
> (except at the most elementary levels) shouldn't be thinking of apples 
> and oranges;  they should be thinking about numbers in an abstract and
> generalized manner."  I think Larry put his finger on a crucial 
> problem that science educators face.  Students are good at abstract 
> problems (5+6 = ___ or 4x + 3= 27) but cringe at "story problems."  
> Because they aren't thinking about apples and orang es, they wilt in 
> the face of problems like "Bob has three more oranges than Sue, and 
> Sue has ..." 

David,I think you are misinterpeting what I meant by "apples and oranges".
The thread was about the difficulty of taking simplistic visualizations of
concepts ("multiplication is repeated addition" and "you can't add apples
and oranges") and trying to use them to explain a more sophisticated
concept (multiplication of physical measurements).  I was taught to avoid
doing that, and to develop the more sophisticated concept on its own
theoretical and logical foundation.  You are interpreting "apples and
oranges" to mean applied mathematics, the use of mathematical concepts for
"practical" problem solving, which certainly should be in the repertoire
of every high school math teacher. 

That said, I think you have raised an important issue, one that seems to
perplex all physics educators up to and into college level.  Problem
solving always requires a considerably higher order of thinking and
maturity than does mastering the theoretical background.  But I have to
give you the sole credit for pointing it out, as it wasn't what I was
thinking of when I wrote previously.

> I'd like to learn from the group how we get students across that 
> divide: from concrete science problem to the equation they need to 
> solve... and back to the scientific interpretation.

My take on this problem is that students who screw up problem solving
usually do so because they haven't taken the time to set the stage before
attempting the solution.  They don't understand the problem they are
trying to solve but want to come up with "thuh anser" in 60 seconds or
less, and hence fail to solve it correctly.  Sometimes it almost seems
like they've been intentionally trained to value speed more than success. 

My biggest challenge is to get them to slow down and spend some time
becoming familiar with the situation posed by the problem, and to *plan*
its solution before *executing* its solution.  Unless the problem is so
simple as to be transparent, the only strategy that can be counted on over
the long haul is something like: 

(1) Sketch a picture or diagram of the situation
(2) Identify what you know: any data that you have been 
    given or know or can determine (including units)
(3) Identify what you are supposed to find (including units)
(4) Identify any relationships that you might use to tie together
    what you know and what you are trying to find out
(5) Plan a computation or series of computations that should
    produce the result you are supposed to find, expressed as
    equations that can be used with your calculator
(6) Execute a solution, including units
(7) Examine the solution; does it make sense?

It doesn't take long to get them to understand how to do this, but it
takes forever to get them patterned to actually do it.  At any and every
opportunity, they will slip back into what I call HS&BS mode: 
high speed and bull ... stuff! 

Best wishes,

Larry

------------------------------------------------------------
Larry Cartwright
Physics, Physical Science, Internet Teacher
Charlotte High School, 378 State Street, Charlotte MI 48813
<physics@scnc.cps.k12.mi.us> or <science@scnc.cps.k12.mi.us>
------------------------------------------------------------