Re: [Phys-l] What are your answers for this teacher?

["John Clement" <clement@hal-pc.org>]

The triangle method is used because the students can't handle 3 variable
equations.  Essentially you are giving them a method to get through the
tests.

But in the process, the reason why the students can't handle 3 variable
equation is the lack of proportional and compensation reasoning.  The
triangle gets around the problem, but does not treat it.  Indeed, it just
puts off the day of reckoning.  Once the students have the necessary
reasoning, the triangle is not needed, and students will even wonder why
then might need it when the reasoning is so obvious.  JD is right that it is
a crutch which allows them to get by without confronting the real problem.

Speed problems are easily recast in an understandable way by using motion
maps.  The motion maps make the reasoning visible.  Then students have to
learn the meaning of 2 m/s.  They will say it is "the car travesl 2 meters
per second", and your reply is, "no you just read it out loud but didn't
give the meaning.  If they say it means the "velocity is two", you reply,
that is just the name of the quantity, but not its meaning.  They have to be
able to say "the car traveled 2 meters every second".  They use the word per
as a stand in for the division sign without thinking of what it means.  At
that point they can reason out how far something travels in 5 seconds.
Other problems again can be reworded.  For example finding the velocity is
finding how far the car travels each second, given its distance and the
time.

There are many things that are done to avoid the necessity of thinking
proportionally, but the students need to be confronted with tasks where this
type of reaoning is necessary.  The triangle method avoids this.  It is
another example of where students are taught to do something by rote which
prevents them from using reasoning.  As a result 75% of US seniors do not
have proportional reasoning ability.

So your choice is clear.  If your job depends on their getting higher
scores, you might have to teach the triangle.  But if you can improve their
reasoning, they will do much better in the end.  There is too much
expediency in the US school system.  If you want to improve reasoning, look
at the publication of Shayer & Adey.  9th graders are at an age where
proportional reasoning should be possible (age 10+), but 4th graders are
not.  Standard methods of teaching do not do a good job of improving
proportional reasoning, but IE methods combined with making the students
aware of metacognition does work.  In 4 weeks I improved proportional
reasoning ability from about 40% to 60% in community college students.

John M. Clement
Houston, TX

> My 9th grade physics students were taught a method for 
> solving speed problems in middle school that does not require 
> algebra. The variables are separated in a diagram of a 
> triangle (or a circle). When you cover up the variable you 
> are solving for, the diagram shows you the recipe on how to 
> solve the problem. If you're not familiar with what I'm 
> talking about, here's a link to an example using Ohm's Law: 
> http://www.electronics-tutorials.ws/dccircuits/dcp_2.html
>
> I have avoided use of this device for a variety of reasons. 
> First of all, it has zero meaning. I feel that it is 
> important to reinforce my students' algebra skills, which 
> this method completely skips. I also prefer Hewitt's method 
> on using equations as "guides to thinking" which these 
> triangles/circles avoid. Finally, I have found in the past 
> that various memory devices in math (e.g. FOIL, cross 
> multiplication, etc.) have limited function and students tend 
> to use them incorrectly much of the time.
>
> However, I am wondering if I should reconsider my stubborn 
> position. I have had many students approach me this year 
> showing me the triangle and claiming "here's an easier way to 
> solve the problems!" In addition, at the NSTA conference I 
> was introduced to a text entitled "Active Physics" by Arthur 
> Eisenkraft which uses the circle for every equation given in the book.
>
> Again, these are 9th graders I am teaching, so maybe I should 
> not be such a stickler on going through all the algebra. This 
> device is primarily used for "plug and chug" problems anyway, 
> which have little meaning themselves.
>
> Do any of you use this method with your students? Have you 
> found it useful? Do you have any tips to make it more 
> meaningful and universal? Do you see it as a hindrance?
>
> Thanks in advance for your collective knowledge and wisdom!
>
> bc thinks the picture is not a model.