Those of you who
have been with me for an appreciable chunk of the dozen years I have been
newslettering will be pleased to know that this one is back to normal form: I
am happily writing about a pair of events I enjoyed a lot and the excellent
impact they have had.

The
events in question featured Ruth Parker, a passionate advocate of the kind of
teaching for understanding that I so much believe in. She has developed an
expertise in an area where we have a dire need: communicating to parents and community members what is
at the core of what has been called Standards-based teaching or Reform
teaching. Since both of those terms have become loaded, I prefer to call it
teaching for understanding, at least until that one, too, takes on strange
undertones and overtones.

Ruth's
presentation was one from which anyone who went actually wanting to know what
this kind of approach is about could learn a lot. One of her fundamental tenets
is that to learn effectively one must be engaged, so she engaged the whole 400
or so of us in the Roosevelt High School auditorium in some mental math. First
came a very elementary one: 43 - 18. We worked in silence until enough of us
had indicated (not by raising our hands, but by putting a fist, thumbs-up,
under our chins -- much less distracting to other thinkers!) that we had an
answer we were convinced about. Then she took answers -- or rather, the answer,
since no one had an alternative one. Then came the meat of it: she asked people
how they got that answer. I lost track of how many ways there were, partly
because I was so much enjoying the exclamations from around me as each tactic
turned up. Adding two to both numbers or subtracting three from them were early
entries, and it went on from there. Next she gave a similar problem and checked
it more briefly, since the major question was "How many of you used one of
the methods you just heard from someone else?" Subsequent examples
involved multiplication, fractions, area and turkey sandwiches, and had various
specific sub-messages to point up, but the dominant message, which came through
loud and clear, was that if children have a good understanding of numbers
themselves, in particular of place value,
they can use that to build up a confident, competent handling of all the
operations, and will have an
understanding that will serve as a foundation for building further mathematics.
Algebra, in particular, is a completely natural development out of numerical
operations, provided the operations are understood. So why aren't they? Ruth's pretty
convincing argument is that most of the blame lies with too-early introduction
of the standard algorithms, by which, she feels, we routinely un-teach place
value. Consider, for instance, multiplying 68 by 7: "7 times 8 is 56, so
we write the 6 and carry the 5. 7 times 6 is 42, add the 5 and we get 47, so
the answer is 476." As a shorthand, it's just fine, but not before the
longhand is established. You aren't really multiplying the 7 by 6, you're
multiplying it by 60, and getting 420, and since what you carried wasn't really
a 5 but actually 50, they add up to 470. All of which children can grasp and
handle just fine if the don't have a magic formula blocking the way. And once
they are entangled in that formula, and have accepted that math isn't supposed
to make sense, it is much harder for them to have the confidence to think, and
from there you get teachers feeling that "Kids can't do that sort of
thing". That last comment was me embroidering on what Ruth said, but I
maintain it was implicit! She gave a lot of examples of tactics her fifth
graders had developed, many quite stunning. In a conversation since then she
described the turn-around of one skeptic who happened to drop in on a third
grade while they were checking the hypothesis that if you're multiplying it's
OK to double one side and take half of the other. They opted to check it out on
7 x 7, undisturbed by the fact that they had had no instruction whatsoever in
fraction multiplication. They understood halves very well, and successfully
arrived at 14 x 3 1/2 = 49. Whereupon one of the girls piped up with "If
this works for all numbers it ought to work if we do it again!", and by
golly, they figured out half of 3 1/2 and multiplied that by 28 and got 49
again (to triumphant celebration!) Ruth did point out that not every session is
quite so inspiring, but it was clear that even without the triumphs
"Number Talk", as she calls it, is never boring.

The
next morning Ruth ran a follow-up session at the Stanford Center -- Seattle
Public Schools' central building. This one was an invitational breakfast
meeting designed to help the many stake-holders in school math think about how
to collaborate. There were teachers and parents and school board members and a
legislator and business leaders and a bunch of us from UW and a number of
people from the Seattle school administration. In fact, the meeting was called
by Superintendent Raj Manhas, whom I had not met before. A very intelligent and
alert guy, which made his resignation less than a week later a real blow. It
seemed to me the meeting went well. Details have blurred (7:30 meetings are not
my forte) but the need for building up mathematical strength of those who teach
mathematics in K-12 was definitely on the table. I think it really needs to be
not merely on the table, but jumping up and down and stomping its foot, but I
claim no originality for that thought.

The
evening talk was well attended, and I have had the good fortune to have access
to the responses of a number of the attendees. One bunch of responses came from
my Math 170 class. Going was one of their options for a project, and around 3/4
of the class took that option. Judging by the reports, a large majority
expected to ho-hum their way through and then cudgel their brains for things to
fill two pages, single-spaced. Great was their shock when they found themselves
completely engaged for two full hours -- Ruth picked up a lot of converts on
the spot! Other responses came from my colleagues, a fact which has led us into
a terrific Brown Bag development: after 12 years of one-shot topics from widely
ranging areas of teaching and learning, we have a real seminar going. For me it
is tremendously exciting. I love being jarred out of some of my long-held
positions and having a whole bunch of reading to do to fill in unsuspected gaps
in my knowledge (not to mention some more-than-suspected ones!) So far the
central challenge has been algorithms. We've all moved a bit in our positions,
and the big question is how close we will end up. We have no need to be in the
same spot at the end, but much need to be able to make sense of each other's
stances. Details are definitely not yet available, but tune in again in a few
weeks or months -- there's no telling where we may have gotten to!