Comparing and contrasting is always an interesting pursuit. At the
moment I am on the Olympic Peninsula, having an excellent opportunity
to learn a bit about the lives of teachers in rural schools -- really
eye-opening, after years of working with teachers from urban schools.
Some things are very familiar, both the amusing (what to do when a
paper airplane sails past your nose) and the painful (how to help a
child whose family is non-existent or worse). Others are much less so.
Rural teachers are highly visible members of a tightly knit community,
with very little option of anonymity in almost anything they do.
Support from that community can greatly strengthen their teaching
options, but opposition can be catastrophic. One thing that I had
not previously run into is a report of a school that has been closed
down under the No Child Left Behind Act. The report came from a teacher
at the closest school, 30 miles away, to which the children are
theoretically being bussed. Some ten or fifteen of the sixty emerge
each day from the bus -- not always the same ten or fifteen. The impact
on WASL scores has left the recipient school anxious about the
possibility of being closed down. As I understand it, the nearest
school for them is 50 miles away. Not a good scene.
Some things are universal, though, among the teachers who participate
in an Institute of the sort I am currently helping to run. The most
notable is an awe-inspiring energy and enthusiasm for anything that the
teachers feel will help them to do a better job of teaching. It's a joy
to work with.
Before I launch into the AWM column that jogged me into sending this
out, I have a swift follow-up on a previous topic. In Newsletters 117
(http://www.math.washington.edu/~warfield/news/news117.html) and 118
(http://www.math.washington.edu/~warfield/news/news118.html) I reported
on a newly fledged Math Fair. There have been several since then. For
the most recent, I had to leave town after all the preparations, but
before the Math Fair itself. One of the graduate students who helped
out kindly made a report, complete with pictures. I liked it so well
that I got his permission to point you towards it: it is at
And now for the AWM educations column to which, as I recall, I gave the title of "Metaphor".
For the past few weeks I have been playing with a metaphor. It keeps
evolving, and each evolution produces a new bit of insight.
Possibilities for further evolution still glimmer enticingly, but
sundry deadlines are staring at me, so I think the time has come to
renounce ownership and turn the metaphor over to you all for further
Actually, before renouncing ownership, I should admit to not having
been the owner in the first place. The metaphor originated with my
friend and colleague Guy Brousseau. He and I have been working on a
series of papers about the teaching of rational numbers and decimals.
In the seventies Guy led a team of researchers in Didactique in
producing a sequence of Situations covering all of the material on
rational numbers and decimals that the French National Program
required, and his wife Nadine led a team of fourth grade teachers in
helping to develop the Situations and teaching with them. The
Situations on whose description we are currently working induced the
students to develop decimal numbers by taking the whole class through a
collection of games for which non-decimal rational numbers could be
used but were extremely cumbersome. The end results were highly
successful — most of the students finished with a solid
understanding of decimal numbers — but the process
presented a heavy challenge for both the class and the teacher.
Inevitably, this leads to the question: why put them through so much?
What are they learning during the long stage in the middle of the
process when the whole class is struggling together and not even the
brightest of them can see where they are going? To respond to this, Guy
came up with a metaphor involving a child learning to play rugby.
Guy's central image is that of the learning done in a neighborhood
pick-up game. All you need to know to join in is that you have to help
get the ball to a particular end of the field, and that you should
neither cry nor sock somebody if they get it away from you. What you
then have is a bunch of kids working together towards the joint
achievement of getting the ball into the goal. Some of the team
members, by talent or prior experience, play a larger part than others
in that achievement, but the triumph and the score belong to the whole
team, and all of the members are learning from each other as they go.
By the same token, students who are put in a Situation where they work
as a whole class to achieve a mathematical goal can all make some
contribution, share in the excitement of achieving the goal and learn
from each other as they do so. The only difference is that one set is
doing rugby and the other is doing math.
For my own purposes, I converted the metaphor to soccer. It’s the
only sport in which I can claim some expertise (I know, for instance,
that the most stressful position on the field is mother of the goalie).
I then tossed in a bit of coaching at the youth soccer level, and found
some new parallels. One was in the matter of skill drill. Clearly, any
player can benefit from practicing the art of controlling the ball
while running down the field, or trapping a pass, or shooting on goal.
Equally clearly, if every minute of soccer time turns into running
figure eights around a pair of cones the level of team enthusiasm is
going to dwindle pretty rapidly. There needs to be a balance between
time spent drilling and time spent playing, having fun and discovering
where the results of the drills can be used. The coach that can find
that balance will have a team that progresses rapidly and keeps wanting
to progress more. Does that sound familiar?
Moving on from there, I came to yet another of our tough issues:
assessment. I refuse even to think about imposing grading on a youth
soccer coach, but suppose for instance that some players are being
sought for a select team. Certainly the selection process should
include some evaluation of skills. On the other hand, if the assessment
is made entirely on the basis of ability to control the ball, then
the select team is going to miss out entirely on little Johnny,
who is not too swift on his feet but somehow always manages to be in
the right position to block a breakaway run by the striker, or Suzy,
whose dribbling still needs some work, but who always seems to know who
is open for a pass when she needs to get rid of the ball. Again it is a
question of balance, this time complicated by the fact that it is a lot
easier to test for skills than it is to assess the far vaguer issue of
understanding of the game itself.
There are some other metaphorizable aspects in there that I haven't
managed to follow up on yet. For instance, there is the question of how
to keep players passing to everyone on the team, and not just to the
player they've all decided is Superman or Superwoman. There is also the
matter of keeping the weaker players from getting discouraged and
quitting, by finding ways to encourage them that don't involve false
Cogitating on generalizations of that last aspect led me into thinking
about how much I have learned about teaching from the wonderful
director of my Medieval Women's Choir. But that's another metaphor