Date: Sun, 25 May 1997 21:39:07 -0400 (EDT) 
From: "Ralph A. Raimi" 
To: [a certain few friends]
Subject: NCTM Rochester Meeting 

   I attended the regional NCTM meeting just held here in Roc-
hester, Thursday morning through noon Saturday, May 24. 

   This meeting is the first NCTM I've been to, and did not much
resemble AMS meetings, though Convention Centers with attached
hotels by skyway are much the same from Atlanta to Rochester.
There was no crowded message center and very few small groupings
of folks with pencils and backs-of-envelopes; and very many of
the announced presentations were cancelled at the last minute or
later without even announcement. The Program was hard to read,
for me at any rate. The listings were color-coded by a small dot
indicating K-6, 5-8, or 9-12 or combinations thereof, but in a
rather disorganized way; and there were no plenary sessions. I
saw no great crowds anywhere. 

      Most of the talks were practical: demonstrations of class-
room devices, "how-to" almost everything instructional, and
sometimes pitches for particular programs or manipulatives or
calculators, not necessarily by the authors or publishers but by
happy customers.

   Many of the events took place in rooms with four to six round
tables for the audience-participants, plus a blackboard for the
speaker; I imagine many school classes are held this way these
days. One session I attended featured the use of a TI-83 (I
think), a box of which a sad-looking TI salesman sat guarding
near the doorway, passing them out to us as we came in and
collecting them as we left. Most of us couldn't figure out how
to use them, since the speaker was using and illustrating a
TI-82 (I think) up front, and we didn't have time to translate
what she was doing to our superior (I think) machine. 

   But it was audience participation, even though the calcula-
tors were really irrelevant: the talk was called "From patterns
to functions: Patterns, Algebraic Formulas & Function Graphs."
The speaker, Irene Jovell, explained that what we were doing
followed the recommendations of the Standards, which demand
calculators for data analysis. Her data on the present occasion
was, she said, of the sort called "sequences" by seniors, but
"patterns" by younger children, and this is the way she illus-
trated one of them: 
     days   1  2  3  4  5  6 
     built  6 11 16 21 

      The question is, how many bicycles will be built by the
12th day? It could be done in various ways. She illustrated by a
linear graph as one way, pointed out that constant first dif-
ferences should be shown to the kids as indicative of linearity,
getting to a formula for f(n) for more advanced lessons. This
problem, as I understood her to say, is part of a BOCES pilot
for something going on now at Syracuse University, "Math A" I
think she said. That's a New York designation that I didn't get
an explanation for. How the (small) factory took no time at all
to produce the first bicycle was not explained. For illustrative
purposes, I suppose, one must not *always* have b=0 in a linear
function. Real life. 

   She showed a sequence with constant second differences and
stated it was quadratic, and showed how to explain this sort of
thing (no proof), and to get the formula in a case she exhibi-
ted, by postulating f(n) = an^2+bn+c and plugging in three
values to get three equations for a,b,c. That was for high
schools. Guessing patterns seemed to amount to recognizing
arithmetic sequences associated with things very like bicycle
shops. How you recognize a quadratic pattern by eye, without in-
struction about second differences, she did not explain. She
spoke some about curve-fitting and experiment, and described
using the "Slinky" to illustrate Hooke's Law. (Because it is
extremely responsive to changes in weights it shows the lineari-
ty very well.)

   All told, what she did was pretty good school stuff, for a
while, though it represented a lot of 'real-life' time for the
slender amount of mathematics introduced. Was Slinky the kind of
thing Hooke had in mind? In fact, it would be very hard to
deduce a Young's Modulus for a Slinky by merely knowing the
Young's Modulus for the steel in it, along with its curious
spring-like shape. Has anyone ever wondered why a real spring's
linear response to stretching forces follows from a linear
response to stretching (i.e. constant elasticity) shown by the
material of which the spring is made? I myself (not expert in
physics or engineering) doubt that it does, except to the degree
that almost everything in nature is locally linear. Even the
exponential function seems so -- for a while, as when young
people experience inflation as "only 3%". 

   She ended with a summary, part of which I took down:

        Take risks;
        Be creative;
        Use technology; 

and the *leitmotif* was: Follow the Standards and don't be as
dull as things used to be. But we in the audience had no need of
the technology provided, and it was not easy to see what the
risks were. Her lesson was well described, intelligently presen-
ted and utterly unconnected with any reform, unless close atten-
tion to student response is something new in teaching. Since it
was not clear to me what grades were intended for what parts of
her lesson, I could not say whether this represented a dilution
of curriculum or an acceleration, compared with, say, 1980.

   The same theme was handled with much more wit and brio by
Steven Leinwand, a math ed supervisor of some note in Connec-
ticut and (as I discovered the next day) one of the 15 authors
of the Scott-Foresman-Addison-Wesley *Middle School Mathematics
Course 3*, and probably others in the series, though I didn't
look. Hearing Leinwand was an unexpected bonus and the high
point of my experience at the Meeting. He was called at the last
minute to replace Gail Burrill, who had been suddenly called to
Washington (D.C.) for a meeting concerned with the National 8th
Grade Math Test. Much of the audience that had gathered to hear
Burrill (on a different topic, concerned with the future, I
think) drifted away after that announcement, but about thirty of
us remained, and it was well worth it. 

   The man Leinwand is a showman, mesmerizing, fast-talking,
prepared and fast-thinking both. He came down into the audience
three or four times to make a point or talk directly to someone
who had raised a hand, and each time he called that person by
name (we wore name tags) and remembered the name later when back
on the podium. 

   He began with a Calvin and Hobbes cartoon implying that math
was the most boring class in school (and school the most boring
part of the world, for that matter), and he proposed to cure
that by making math relevant and fun. But without *our* enthusi-
asm and participation and devotion, he explained, nothing would
come of it. Why *bother*?, he said repeatedly, if the folks back
home pay no attention? Why bother to do group work, why bother
to use technology, to use the Standards, to alienate colleagues
and make things harder for them? For his part, he said,
   "I've bought the Standards... so we can go back and face the

   He gave several sample lessons taken from real-life situa-
tions, that would engage the students. Example: a breathalyzer
test taken 20 minutes after the arrest shows .13, and a blood
test at the station 40 minutes later shows .15 (a real story, he
said): Was the judge right to let the guy off on the grounds
that at time zero he showed so close to the legal .1 as made the
case impossible to prove? The story was livened with detail,
commentary on error possibilities in each measurement, questions
about the concavity of the function "blood level of alcohol
against time" and so on. 

   Example. The "traditional" text asks: Let F=4(S-65)+10 (S >
65); find F when S=81." How deadly dull. His shoulders drooped
with the dullness of it all. He shows us a page from some book
where problems like this fill the page with hardly a word or
picture. (The book had to be pre-1960, in my experience, to
contain such a page.) 

      Then, leaving the formula on the blackboard, he tells the
real-life story of being caught on a Interstate 91 in southern
Vermont doing 81 miles an hour. The fine for speeding in VT is
ten dollars plus four dollars more for every ... The audience
gets the point, and is especially appreciative of his throwaway
line at the end: "Actually, I was going more like 90!"
      Then he goes on to compare the formulas (both linear) used
by CT and VT for speeding fines, and asks the students which
State they would rather be in if they were going 70, 75, 80, 85?
Isn't that more interesting than a page full of problems like

        y=19x+79  ? 

   Who *cares* what (x,y) solves such a pair, in this, the
second Ninetndo generation? Who *cares* about a couple of graphs
on a piece of paper? *Think* about the present generation with
their computers and statistical information. How can you inte-
rest them in such stuff?

   But the two equations just now quoted were actually telephone
rates, comparing MCI rates "collect" against using cards. *Now*
the kids will show an interest. When they go away to college,
which phone service will their parents want them to have? What
will it depend on? 

   We must give kids ACCESS, he said, via


and even then, this is only the easy part. He goes on with the
prescriptions of the Standards: 


[there were a couple more but he was too fast for me]

   "So here we are in the 6th grade, Chapter 10, problem on page
367 [he reads]: 

   "Sandra is buying party favors for the friends she is in-
viting to her birthday party. The price of the straws is 12
cents for 20 straws. The storekeeper is willing to split a
bundle of straws... She wants 35 straws... How much... ?" 

   Leinwand's mockery reaches a peak with this problem. *San-
dra*? Where's this guy coming from? Who is this Sandra? What
about [here he gives several popular current names, mocking the
very idea of anyone being interested in a goody-goody named
*Sandra*]? And "party favors"? And splitting a package of
*straws*? **Straws**? Etc. You gotta go to where the kids are
coming from, he explains. 

   His talk contained many summaries, slogans, THINGS TO REMEMBER. 

Obstacles: Isolation, beliefs, ignorance, fear of change, ab-
sence of support, fear of failure, lack of confidence. 

Antidotes: Sharing [as in Japan, where videos of both classroom
lessons and common room discussions among teachers show adhe-
rence to his present prescriptions], risk-taking, support.
   Leinwand wrote an article in The Mathematics Teacher a couple
of years ago, either called or containing "Four Teacher-Friendly
Postulates," he said, which goes into some of these things. Two
of them, or maybe two of something else, are the following: 

   # The traditional curriculum was designed for the people and
the problems of another era, not our own. 

   # We cannot expect reform to change anything by more than 10%
per year. 

   He gave examples of things no longer educationally relevant,
but cautioned against trying to throw too many out at once. The
dinosaurs would howl (my phrase, this time; Mr. Leinwand didn't
mention dinosaurs at the end, but made it clear why 10% is the
upper limit.)

   All told, Steve Leinwand made a good case for not using phony
story problems from old textbooks, and for finding interpreta-
tions for linear equations that made sense when teaching the
elements of algebra. But his most striking illustrations touched
on only a small bit of the potential school curriculum. Some of
that curriculum he damned in passing, but without much argument
apart from his stricures on dull *teaching*. One of those yearly
10% deletions he recommended included "simplification of radi-
cals", for example, but I couldn't catch the other sample he
gave at that point. He didn't mention quadratic anything. He had
also showed a page of arithmetic problems, without telling us
the year (again, it had to be pre-1960), a real beauty, the page
downright gray with problems, each one looking like

and he urged us to get rid of them.
   But while he stirred the enthusiasm of the audience, I could
not see that he made any case at all for calculators or other
technology, except to say we were in a Nintendo generaton. Nor
did I see that he had made a case for "collaborative learning",
which he mentioned but did not illustrate, except in his tales
of collaboration (?) between inspiring teacher and students who,
seeing the light, would say "Oh, wow" as a group, instead of
grimacing with the solitary boredom of Calvin (or Hobbes). 

   I spoke with Leinwand just briefly after his presentation (he
was in a hurry, but polite). First, I made the mistake of tel-
ling him that he should be selling real estate in Florida, but
before I could go on he told me he didn't take that kindly. I
apologized, saying it was intended as a humorous comment on his
persuasiveness, but he said he had heard such comments from
those who were implying he was selling snake oil. I disavowed
all that, and, having thus used up half what time of his I could
decently ask, went on with

      1. When these teachers go back and find the same deadly
textbooks that were there last week, how can they *use* the
enthusiastic things he was advocating? I can't remember enough
of his answer to summarize it, except that I remember it as
sensible enough.

      2. Why doesn't the NCTM print reviews of books, to let
these people know what will permit them to be interesting and
informative at the same time? His answer to this was that this
question was "very controversial", but he added that the books
I might want *are* "out there", and he waved in the direction of
the publishers' booths. 

   I did not then know that he was part-author of any of them.
The following day I looked at a few books. In Saxon I discovered
that "vectors in polar form can be added if and only if their
angles are the same or differ by 180 degrees". As I saw it this
was typical, Saxon's being a recipe book not concerned with
mathematical statements as such. I had a hard time trying to
read Saxon because the salesman was badgering me with infor-
mation and questions about my health and biography, and I had to

      In the Addison-Wesley-etc. booth I mused to the salesman
that the thing had been written by 15 people! How could such a
thing be done, I asked. I was told proudly that Leinwand was the
chief author -- this without my having mentioned that name --
and that they passed manuscript back and forth a lot before
publication. So Leinwand is of some importance. What I could see
of the book before the salesman's importunities drove me off did
not show much difference from most others being printed today,
and in particular did not show the kind of enthusiasm and imagi-
nation Leinwald had exhibited in his lecture.  Actually this is
probably not fair, as no book can do that.

   In Leinwand's book there was indeed a picture (p.515) of a
black girl named Tanisha, not Sandra; but I think I could make
merry with a page of Addison-Wesley-Scott-Foresman problems like
this one on page 365: 
      "Use the square-root key X on your calculator to deter-
mine whether each of the numbers is a perfect square:
(a) 222
(b) 169
(c) 351.56"

and deride the textbooks of 1997 as easily as Leinwand did the
textbooks of his own illustrative, unnamed years. The main
difference is that one does not print a whole gray page of such
things any more, but the colors, pictures and words do not, as I
see them, accomplish the exciting things his talk pointed
towards; and there is a question in my mind whether anything
can. Life has its rewards, and excitement and so on do turn up
here and there, but inspirational speakers imply more than is
possible, even granting the truth and interest of what they have
in mind. I see TV commercials for Pepsi that do the same thing,
portraying all the world as populated with beautiful young men
and women on a perpetual rollicking beach, laughing and drink-

   Also on display in one of the publishers' booths were a
number of audio tapes and disks containing music written or
arranged by one Gary Lamb, to be used as directed on the handout
sheets, from which I quote:     "Music ... trains the brain.
When children exercise cortical neurons by listening to clas-
sical music, they are also strengthening circuits used for
mathematics. Music, say the UC team, 'excites the inherent brain
patterns and enhances their use in complex reasoning
tasks.'...[from *Your Child's Brain*, Newsweek, 2/19/96]" 

   "Teachers throughout the country are realizing the valuable
benefits of using instrumental music in the classroom. From
silent reading to creative writing and yes, even math testing,
this music is a wonderful way to help students to focus and
reduce stress and will enhance concentratrion during the lear-
ning process. Highly recommended for ADD/DHD students." 

   A disk player was playing this music all day. When I stopped
at the booth, the woman pointed out the disk then being played.
It was called, "The Language of Love" (solo piano), and clas-
sified as "New Classical (60 beats per minute)". That 60 beats
thing is supposed to calm people down by resembling the proper
rate for a calm and healthy heart, and is intended to be a
constant background in school. A few months ago I was complain-
ing (to either mathed or math-teach) of the omnipresence of such
music, and sarcastically suggesting it be used to make our
mathematics lectures and classes more understandable, if it
could do that for TV commercials. Well, life imitates art. My
own heart beats at about 75 most of the time, sometimes 70. 

   Later that afternoon I went to the presentation by Mark Saul
and Eva Browder, which concerned the GOPM (Gelfand program).
There was an audience consisting of two mathematician friends of
the speakers ( Sam Gitler, a topologist of the U of R, and me),
and two people I didn't know, but who sounded interesting. There
was no music, and very little excitement; but then there was
very little audience. Maybe we're in a losing game. 

   I saw only a small part of what was going on at the NCTM
meeting, of course, but very little of what I saw, or what was
printed in description of things I didn't see, implied that
learning more mathematics was required by our present or next
American cadre of schoolteachers. The idea was always that what
teachers knew (about mathematics itself) was sufficient, and
what was to be got across (to the students) was already settled
as well; and that the principal problem was that of motivating
students, with secondarily the problem of making explanations
more lucid, or understanding more secure, by using technology,
social interaction devices, and the imaginations of the students

      The Standards are regarded as synonymous with reform, but
are called on only as defining methods of teaching, not as
defining of what is to be taught and when. No speaker I heard,
or person chatting at a lunch table or in a publishers booth,
could conceive of anyone's being opposed to some part of the
Standards unless that person had an interest in maintaining
things as they were in a rather idealized version of what the
math classroom looked and sounded like in 1940. That is, that
only a dinosaur could quarrel with the Standards. (Not that the
possibility of a quarrel ever came up within my hearing.)

   Any reform in the Standards themselves will therefore have to
advertise itself as an improvement of an already sacred docu-
ment, something like the 'bringing up to date' of the King James
Bible that has taken place recently in (at least) the Anglican
church. It will have to be a *revised* "Revised Standard Ver-
sion". To write a new *sort* of Standards might not be hard, and
I hope it happens; but to present it as a genuine departure
would violate the sensibilities of very many people. Unlike the
"Back to basics" cry of 1975, which was presaged by widespread
dissatisfaction with both the tenets and the practice of the
"new math", whatever cry there might be (and such cries are hard
to find at NCTM meetings and publications) against the current
reforms is not rooted in a comparable dissatisfaction within the
profession itself. 

Ralph A. Raimi 
University of Rochester  
Rochester, NY 14627