[Edu-sig] True or False? - A translation is a function.

michel paul mpaul213 at gmail.com
Fri Nov 28 18:26:34 CET 2008

We were talking about translations in math class, and I was going to show
the students how the translation notation in their text could TRIVIALLY be
turned into Python code, and it occurred to me that maybe I should first
find out if they understood that translations already WERE already functions
or if functions and translations were two different kinds of things.

The results were very interesting.  Lots of hesitation and disagreement
among the students.  Some had no problem at all saying that yes, a
translation is a function.  Others got stuck on the idea of a vertical
translation somehow violating the 'vertical line test', so they thought
vertical translations would not be functions.  That was what I was
suspecting - our curriculum creates a really boxed in sound-byte
understanding of what functions are all about.  The kids get trained to
respond "is a function" when they see a 'U' shaped parabola and to respond
"is not a function" when they see a 'C' shaped parabola.  However - the
important issue is not a vertical line test - the important issue is clearly
designating our input and output.  A horizontally opening parabola is simply
a function in the form x = f(y).  They also tend to develop the boxed in
notion that functions are things that operate on and produce single scalar
values.  That's the effect the current curriculum has.  The idea that a
function can take in ordered pairs or lists seems strange to most students
who haven't been exposed to programming.

I found that a useful discussion to have, and I thought it might be fun to
see what the teachers in the math department believed.   So, I went around
and asked.  I just wanted to see what their honest in the moment response
would be.

Amazing - I got the same responses as from the students.  Some had no
problem with the idea.  Others did get it right, but with much hesitance.
And, several did not get it right.  The reasoning on the part of the
teachers who were either unsure or incorrect was the same as the students'.

It really does seem that our curriculum fosters among both students and
teachers an understanding of functions as things that operate on and produce
single numbers and as graphs that pass the 'vertical line test'.  All the
more argument for weaving in a computational thread in the curriculum.

- Michel

P.S.  A couple of weeks ago I gave my first presentation at a math
conference.  I had a small group in my session, but they really seemed to
get it.  The title of my presentation was *Fractions are Objects, not
Unfinished Division Problems*.  It was the first time any of them had seen
Python.  They loved it.
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