For several years, I've read John Denker's posts about the problems with
sig figs. His arguments convinced me reasonably early on that sig figs
give students a poor and misleading concept of uncertainty, and I found
that students coming into my physics classes had learned how to apply
sig figs from their chemistry classes, but had no idea what it meant or
why they were doing it.
My concern wasn't so much about whether it would be better for students
to teach them about estimating, specifying, and propagating uncertainty,
but about how much class time it would take, and whether that was better
use of their time than the content it would replace.
What ended up convincing me was the realization that one of the big
holes in most kids' K-12 science education is that they don't learn
anything about error analysis; anything I could teach them about it
would have benefits far beyond simply giving them a better tool than sig
figs. So this year I decided to teach them a unit on uncertainty and
error analysis as part of the math & measurement topic.
I based what I teach them on an error analysis tutorial from Columbia
University, at <http://phys.columbia.edu/~tutorial/>. I taught them how
to estimate uncertainty and how to propagate uncertainties through their
calculations. This took about 3 class days, including a hands-on
measuring/estimating activity. (Most of my students have not taken a
statistics course, so I opted not to spend an extra couple of class days
getting them to understand what standard deviation is. The kids who
have taken statistics already know; for the others, it's a function on
their calculator and if they want to understand it, I'm happy to teach
them after school.)
Because most of the formulas we use involve only multiplication,
division and exponents, most of the time their error analysis will
consist of estimating uncertainties in measurements, calculating the
relative error of each, adding the relative errors, and converting the
relative error of their final result back to an absolute error. A
typical high school lab has only two or three measured quantities and
only a handful of data points, so this doesn't add a huge burden of
additional work for the students. The fact that I can already see that
they have a sense of what their uncertainties mean (whereas they had no
idea after a year of chemistry with sig figs) is evidence that this was
class time well spent.
As an amusing aside, honors students tend to cling more tenaciously than
average students to ideas they were taught by their previous teachers.
My honors students vastly preferred the idea of specifying uncertainty,
but they didn't want to let go of sig figs. In order to help illustrate
the problem with sig figs, I showed them one of John Denker's graphs as
a way of explaining the limitation. The graph was a Gaussian
distribution of 3.8675309 +/- 0.1 using all of the digits vs. rounding
to tenths vs. rounding to hundredths. (I looked for it on av8n.com, but
it looks like John has since edited the page it appeared on, and the
graph is no longer there.) One of my students looked at the number and
blurted out, "Oh my God! Jenny!"
My question for the list is: what else would it be useful (and
practical) for kids to learn about error analysis in high school?
--
Jeff Bigler
Lynn English HS; Lynn, MA, USA
"Magic" is what we call Science before we understand it.