__Let's
Make it Happen: __

__What Exemplary
Curricula and Professional__

__Development Can
Contribute to School Reform__

__Review of Speech by
Glenda Lappan__

__at the University of
Rochester, October 23, 2001__

Each year the Warner Graduate School of Education and Human Development at the University of Rochester presents an annual George Eastman Medal of Excellence to some educator, who comes here to deliver a corresponding lecture in exchange. This year's honored recipient of the Medal was Glenda Lappan. The audience was made up mostly of graduate students of education and in some cases practicing teachers, though there were also some younger students and some professors. Among the latter was Phil Wexler, the recently resigned Dean of the Warner School, who is on leave in Jerusalem but returned to attend the coincidental Inauguration of his successor, Raffaella Borasi. Wexler is a sociologist but will return as professor of education next year.

I
don't know why he resigned. Borasi is a
specialist in math education, which Wexler was not, and she is an enthusiastic
follower of current NCTM doctrine.

Glenda
Lappan is professor of mathematics education at Michigan State University and
former President of NCTM. During her
lecture she referred offhandedly to her husband's being a mathematician,
something I hadn't known, and indeed there is a Peter Lappan, also a professor
at Michigan State, listed in the AMS directory. Both of them are members of MAA and AMS. Glenda Lappan is known
today mostly as the co‑director of the __Connected Mathematics Program__,
a middle school program financed by the NSF and widely used in America, though
strongly objected to by such organizations as Mathematically Correct and a good
number of parents' groups in school districts across the country which have
adopted CMP. CMP was the only middle
school program designated as Exemplary by the U.S. Department of Education in
1999, in a list it was required by law to create at stated periods. An objection to this designation, and the
other nine as well, was raised in a widely publicized open letter to Secretary
of Education Riley, signed by about two hundred mathematicians and scientists,
and generating a political storm of more complexity than I can describe here.
Lappan's CMP is avowedly in line with NCTM doctrine, while the opposition to
CMP, quite bitter in many cases, comes from persons not usually found inside
colleges of education. This opposition
usually cites the failure of CMP to include instruction in such things as the
algorithms of arithmetic and the theorems of Euclid, while celebrating the use
of calculators and inductive reasoning, small groups discussing a problem and
writing accounts of their discoveries.

The
audience last night, counting me and Wexler, contained 56 women and 12
men. They were not all, indeed they
were mostly not, teachers of mathematics in particular. Lappan's talk therefore went easy on the
math. She began by putting up, via
Microsoft's Power Point, a "problem":

45 ¸ 7

There
were no words or other punctuation in this display, but she called it a
"problem", and the audience accepted it as such. She warned the audience that she was not
trying to stretch their mathematical competence, that they mustn't worry or
expect an exam. There was friendly
laughter, then and later, at her suggestions that the audience was not to be
frightened by some of the things she would be saying, as she explained that her
examples were for pedagogical illustrative purposes, mathematical in nature
since mathematics happened to be her field; but that the lessons she would draw
(not quite her words here) were applicable to education more generally.

Having
exhibited the problem, she asked the audience if they had at least "a
ballpark notion of the answer". Apparently
yes, they did [laughter]. She then said
that in olden days kids would get a page of fifty problems like this, but that
recent reforms are putting a stop to such stultifying waste of time. She then exhibited three or four questions
printed (in Powerpoint) in English,
e.g.,

1.
Suppose each ticket to the senior ball cost $7; how many can you buy for $45?

2.
Jamie worked seven hours for $45. How
much did she get paid per hour?

3.
45 children are to be taken in mini‑vans on a field trip. Each mini‑van holds seven
children. How many mini‑vans are
needed?

As
they were being exhibited, Lappan said, including us all in her understanding,
"Get the idea?" This was
real, she explained, interesting to children, not mere formulas; and the

answers are not all the same, either. She then also exhibited 6.4285714 as another
answer to the problem, given by her calculator, but then pointed to the mini‑van
problem and (with

the audience) laughed at the inapplicability of
such accuracy. In her day, she explained, children in classrooms were instead
presented with hundreds of dull problems, things like “45 ¸ 7",
where today we consider “the whole complexity” of mathematics. Skill comes with interpretation. What is reform ** about**,
then? What she studied in school in her
day, she said, was "insufficient for today".

I
would comment here that it is easy to see in this dichotomy the subtext from
PSSM and the 1989 Standards, that real mathematical problems, even “45 ¸ 7", have more than one answer. The acceptance of "45 ¸ 7" as a "problem" at all,
however, has made for some lack of communication between NCTM and the
mathematicians, who tend not to regard 45 ¸ 7 to be a problem at all. It is only the name of a certain number, I
would say, a number also named 45/7 and 6‑3/7. But in schoolhouse parlance for many years it has been considered
a full statement of a problem (with 6‑3/7 its answer), and lists of such
things did in fact fill the Exercises pages of typical schoolbooks of
1935. Sometimes "6‑3/7"
would be the problem and "45 ¸ 7" the answer. All this is to be
deplored, and certainly the "New Math" reformers of the 1960s,
including a majority opinion from NCTM, did deplore it and tried to do
something about it, but most of the outcome was more complicated than can be
summarized here.

It
is unfortunate that NCTM today, instead of saying the expression is not a
problem ‑‑ it is not even a sentence, after all ‑‑
shows with some tedium that "it" has many interpretations and
answers. An entire mystique of
mathematical meaning has been built up around such simple misstatements, followed by psychological research into the
difficulties children have in reifying this and conceptualizing that. Our (mathematicians')
objection to the notion that problems may have several answers, all correct,
thus rests on a premise the world of education doesn't recognize. Worse is that we are seen as narrow‑minded
in our adherence to such doctrines as the importance of fully stated
mathematical sentences. Not that
this linguistic curiosity is a
particularly important issue in the general debate -- it is only one item --
but when we say something like " ‘45¸7' is not a problem," many current
educational researchers will say, and I have heard it, in tones of superior
understanding, "But that's not the way the children see it."

Well,
of course the children will see " 45¸7'
" as a problem, or a command to do something, if that's the way teachers
and books have told them to see it, just as they will later come to see a
"variable" as a shifty sort of not‑quite‑truthful number,
wriggling in a way "constants" do not, though somehow mysteriously
subject to the distributive law and so on when the chips are down. Upon which the educational researchers
conduct lengthy enquiries concerning "the child's understanding of
'variable'", or "the difficulty children have in working with '3+x'
as a number rather than a command to do something." Such researches, which fill journals such as
NCTM's __Journal for Research in Mathematics Education__, are very often the
study of the psychological artifacts set up by confused previous teaching
rather than of innate psychological phenomena.
Mathematicians' observations that properly paced instruction, taking advantage
of the past few thousand years of human experience, would obviate the
artificially created problem at the outset, are considered naive, excusable to
some degree because of their inexperience with children. (Parents are more of a political problem,
and must be treated in a different way.)

Lappan's
presentation did not directly address such subtleties, however, and she seemed
secure in the conviction that the reformers (whom I might as well call NCTM)
have now achieved a new insight into mathematics, that their predecessors did
not have; and that those who object to the curricula, such as CMP, that are
based on such insights, or, more accurately, based on the notion that these
insights are the essence of mathematics teaching, are partisans of ritual
recitation of answers to gray pages of isomorphs of "45¸7' ".

Lappan
interspersed such examples (there were several other problems she adduced as
she went along) with inspirational words about how teachers influence the
future, touch the generations, and so on.
She put up this list of the three important things in education:

1. An exemplary curriculum;

2. Good instruction;

3. Good assessment.

These
are my own abbreviations (I hadn't time for full transcription) of longer
phrases. In particular, assessment was emphasized as something to be used for
the student's benefit, not ours.
(Subtext: standardized exams teach nothing to the students, and ought to
be abandoned as unauthentic besides.
But she didn't say this.) The
first point, about exemplary curricula, speaks for itself, CMP being well‑known
to the mathematics teacher part of her audience as anointedly exemplary.

Lappan's
voice sinks to a whisper when she is being particularly sincere, and my ears couldn't
catch some of what followed here, but she did also list three Goals (for
something apparently different from that addressed by the list of three
Important Things given just above). I'm
sorry the three phrases below are not parallel parts of speech, but they are
key phrases in what she said and wrote concerning Goals:

1.
Developmentally responsive;

2.
Academic excellence;

3.
Socially equitable.

The
first point took some explaining; she was anxious that "developmentally
appropriate," not be construed, as it often is, as an excuse for delay in
the introduction of difficult things. I

didn't get much of what she was saying, however,
and so cannot praise her unequivocally for this mention. It was, however, about as close as she got
all evening to stating that adults were

proper judges of what students ought to be told
to do or know. "Development"
requires challenge, she said, and children who are not challenged don't learn
much The third point rather softens the
bite of the first, however, "equitable" having a rather special
meaning these days. As she described it, equity in practice implies that the
teacher or program has to be different for each child. My view is that this is something rather
expensive to achieve. Expense was not
on the agenda, however.

Next,
"What do teachers need?"

1.
Deeper knowledge;

2. Skill
in adapting ... ;

3.
Problem‑solving orientation.

Again,
these are key phrases, hurriedly transcribed from longer Power Point displays
that didn't give me much time for writing.
Lappan also spoke of classroom morale, and she told an anecdote to
illustrate the statement that "We have to open up our classrooms in ways
that permit students to show their talent."

A
classroom or school or some other entity she knew of, she told us, which was
being measured for research purposes, and using an exemplary program I suppose,
decided to use as a test of its success an examination from outside the
project, one which had been much used across the country, much studied, normed
and refined. It was known to be quite
trustworthy. It had 32 questions, she
said, and of the entire population of a hundred or more (I didn't get that
exactly) in the tested group there were only two students who got a perfect
score, 32 correct, on the examination,
“AND THOSE TWO STUDENTS (her voice dropped to a whisper) were __learning
disabled__!” The lesson speaks for
itself: Never give up, she
explained. I believe this anecdote
illustrates point (2) above, but am not sure.

Next,
she returned to an example, which in retrospect I take to have been put there
to show the value of technology in learning mathematics, and the value of an
interactive method of teaching, which she was now going to illustrate by
speaking to us as if we were the class she were teaching. It had to do with ladybugs, which are
prevalent in Michigan at about this time and so beloved of children that it is
natural to want to protect them. On the
computer‑game screen were a depicted a ladybug and a leaf. The problem for the ladybug was to find the
leaf and get under it to hide from predators.
The game provided several buttons to push: a step up, a step down, a
step to the right, a step to the left, a quarter‑turn clockwise (without
other motion), and the same but counter‑clockwise. If the display were interpreted as the first
quadrant of the coordinate plane, which it was not, one might say the initial
positions had the ladybug at (2,6) and the leaf at (5,1), measured in step‑lengths
delivered by pushing the buttons.
Lappan asked the audience which buttons they would push, to get the
ladybug to the safety of the leaf.

Someone
hesitantly named the "forward" (i.e. one horizontal step)
button. Lappan created a right‑pointing
arrow of the unit length in the space at the bottom of the screen, where one
could picture the steps in sequence without actually using them, and asked how
many would be needed. As an experienced
consensus‑maker, she quickly got a consensus of 5, which was obviously
(to me) an overshooting to the right, but which would have been correct if the
bug had begun on the y‑axis.
Lappan sort of encouraged that incorrect (or inefficient) response,
attractive perhaps because 5 was the x‑coordinate of the leaf, and rushed
on to what to do next after the horizontal motion. It would have been indecent
to make her back up. There was more hesitation here, but a quarter‑turn
clockwise was suggested by someone, and Lappan noted that one down. And then some vertical steps, sure.

Pushing another button then printed the
path dictated by the successive commands, and moves the ladybug accordingly,
ending at the approximate vertical position of the leaf but two steps too far
to the right. The audience agrees to
take another clockwise quarter‑turn
plus two steps to the left to finish the problem.

See
how the computer aids in geometric intuition?
And this can be generalized, she said, by asking, for example, for a
minimal solution. (Minimal for ladybug
safety, or for predator's pleasure, she did not say.) Or, she went on, one could generalize to learning about other
motions, that are not only steps and turns.
"We have so little precious time", she said, "Don't bring
a thing into the classroom unless you know what it will accomplish." And she again derided the fifty identical
arithmetic problems characteristic of school mathematics instruction before
NCTM.

It
seemed to me that today's average ten‑year olds (my grandchildren a few
years ago were examples) play computer games much more complicated than this
one as a matter of course, and are much more alert to combinations of such
geometric moves dictated by joysticks and pictured buttons than the people in
Lappan's audience, who were therefore learning how to teach children exactly
what today's children don't need, while neglecting such things as fractions and
long division as stultifying and obsolete.
Had coordinate descriptions of the moves and their compositions been a
goal of the exercise there might have been some novelty in it, or mathematical
progress. Nobody needs professional
tutoring in how to cross a room without bumping into the furniture.

"How
do teachers learn what they need to know and be able to do?" Her answer was to use the "exemplary
curricula." Teachers learn from
experience. The ultimate goal is
student learning, but professional development is an ongoing process.... "Create learning
environments..." She referred
several times to the 1991 NCTM Professional Standards for the Teaching of
Mathematics, whose ideals "we" are still seeking to attain. Constantly ask yourself, she explained,
"What skill, algorithm, and higher order thinking will this task
provide?" "How will I reach
all students with this task?"

This
nod to "algorithm" ‑‑ she actually used the word ‑‑
would not, I believe, have been made before PSSM, the Year 2000 revision of the
1989 Standards. Despite many confident words, NCTM spokesmen have in fact been
sensitive to complaints from the public, that their prescriptions did not
include arithmetic skills. The 1989
NCTM Standards regarded systematic arithmetic skills as being no more than
something once practical for grocers before the invention of adding machines,
but not in the modern age. It did not admit their place in really conceptual
mathematics instruction. But since the
public had not in the following five years or more shown signs of understanding
that point, NCTM has been, at least since about 1998, political enough to try
to reassure them without giving the lie to what they were really doing.

To
show the kind of language that can accomplish this feat, let me interrupt this
account with a brief quotation from a paper Glenda Lappan presented in a 1997
symposium, a paper entitled

** Lessons from the Sputnik Era in
Mathematics Education**:

".......
The way in which basic skills are attended to in innovative curricula must be
clearly spelled out so that parents and administrators are satisfied that
students will not be harmed. Each project must gather evidence that students
are performing at an acceptable level on basic skills. The tradition that
arithmetic has a place in the curricula because of the need to develop basic
skills for trade and managing one's affairs is very strong. Reform is in peril
when parents and administrators are not satisfied that basic skills are a part
of reform curricula. The NCTM Standards documents have been greatly
misinterpreted in this area. By trying to move toward balance among conceptual
development, problem solving, and skill development, the documents open
themselves up to the interpretation that kids do not need to learn their
"facts." The real message is that estimation and mental arithmetic
are more important than ever! In a
technology environment a sense of the size of a number that is expected as a
result of a computation is essential to monitor the reasonableness of
results."

It
is clear from this quotation, which is not a bit out of context, or abbreviated
to make sound worse than it is, that Lappan here is evading the question of
what "basic skills" are; she says that parents (and administrators,
the dinosaurs!) are to be mollified with the
knowledge that we are now teaching "estimation and mental
arithmetic", as if she really believed these are what the public objections to programs like CMP are all
about. The objections of mathematicians and scientists are not even mentioned,
as it would be embarrassing to place them in the same category as the
objections of the ignorant, i.e., the parents and supervisors. In another forum, a television interview as I
recall, it was either Lappan or Gail Burrill (another NCTM President) who
insisted that NCTM, misunderstood, has always maintained that children should
learn the multiplication tables ‑‑ as if that were sufficient as a
characterization of the skills in question.

To return now to last night's event at
Rochester, Lappan concluded her talk with a brief mention of the problem of the
reform‑resistent teacher, old‑fashioned, perhaps only a year or two
from retirement, who bristles at something like CMP and wants to go her own
way, the way she learned in school when she herself was a child, the way she
thinks is the right way. This sometimes
can't be entirely remedied, but, says Glenda Lappan, we do have remedies. It is
helpful to create an environment in the
school that in effect mandates change by making the old way impossible to
pursue. Discussion with other teachers
who take on themselves the role of helpful advocates for reform, the school‑mandated
use of an exemplary book or program ‑‑ such things make it impossible
to avoid the new methods, and can accomplish by indirection what direct
confrontation can not.

A little before the speech ended,
Glenda Lappan gave an example of a classroom assignment illustrating
cooperative learning, and how students can learn to talk and write about the
mathematics they were doing, as well as create methods of their own for
problems they had not been taught as a routine. She exhibited the actual papers turned in at the end of a certain
classroom exercise, which was attacked by the children in groups of four, with
one of them assigned to write the consensus solution and explanation. The problem had to do with what she called
"proportional reasoning."

The data comprised several choices of
mixtures of orange‑juice concentrate with plain water. Which of the following pairings, when mixed,
will be "more orangy" in flavor?

Concentrate Water

cups
1 4

4 8

2 3

3 5

She
let us contemplate the possibilities for a bit, then flashed copies of
students' work on the screen. The first
written explanation (in students' handwriting) explained that the group first
made all the cups of concentrate "equal to 1" and changed the water
part, so:

1 4

1 2

1 1½

1 1⅔

Solution:
that the third combination is best, as having the least water. The explanation was not luminous, but
adequate for a child.

I
won't here review the other solutions Lappan showed us. They were filled with words, some more
sensible than others, and generally ended with the correct choice. One of them involved adding the rows and
getting percentages of concentrate in the mixture, and another treated each
pair of numbers as defining a fraction, i.e. 1/4, 4/8, 2/3, 3/5, the ordering
of which seemed to give an answer. I
suppose this is what NCTM means by treating fractions as representing ratios,
but one wonders just what such "fractions" mean as numbers.

My
main thought was this: The first
solution, exhibited above, shows the writer well acquainted with fractional
parts, and able to use the mixed numeral for "one and two‑thirds"
freely as a name for 5/3. He also
recognized that 5/3 represented the water‑to‑concentrate ratio as
one way to assess concentration. To such a person a mixture problem should be
transparent when formulated in a much less convoluted manner, and in the manner
that scientists do when they want to keep their thoughts straight, that is, to
express the concentrate part of each mixture as a fraction, and order the
fractions. This was actually done by
one of the children, or groups of children, in this example, though for some
reason that child converted all the fractions to percentages before making the
comparison. Maybe because he couldn't
order the fractions without a calculator.
(Lappan was silent on whether calculators were being used here.)

I would conclude such a lesson, which doesn't have to be accomplished in a day of course, by defining "concentration" as used in chemistry, and showing how data on mixtures can be converted into statements of concentration, a systematization of the problem Lappan gave. We would arrive at a method, a theorem if you like, from which one can go on. But Lappan gave no hint as to what the teacher was supposed to do with these papers, except to announce that they were all praiseworthy. Of course there was a right answer, but Lappan's presentation wasn't concerned with that, and indeed didn't emphasize anything concerning the ease with which this or that method worked, or its clarity, or even whether the answer was right on this paper of that. I had the feeling that the class, like Lappan's explanation of what was happening, rather trailed off, and didn't quite know what it had learned that day.

It
is curious that such problems are presented as open‑ended and worthy of
doing in many difficult ways, with sedulous avoidance of giving the students a
few definitions that might

make later work of this sort transparent and a
stepping‑stone to future complications, while the corresponding problems
in statistics are introduced, even by NCTM prescription, by as rigid a device
as anything found in an arithmetic book of 1935. Here is an exercise I have never heard mentioned as a subject for
exploration before exercises in the definitions of mean, median, quartile,
outlier and the like are prescribed:

Given two sets of exam scores, one from Classroom A and the other from
Classroom B, we ask: “Which class did better
on the test?” I can imagine children
floundering for days on this, if

they weren't first coached in means and medians,
and if older, in standard deviations and modes. Even so, they can't answer the question, which of course depends
on having some definition of "better" as applied to a class. Yet the avoidance of a definition for
"concentration" was counted a virtue, even after the exploration,
according to the Lappan example.

That
is, in today's K-8 instruction the words "mean", "median",
"quartile" and so on are __given__ to the students, and only then
are questions asked, such as "Which class did better?" Why isn't the question asked first, and the
children permitted to spend a while discovering for themselves, in small
groups, such measures as "mean" and "median" by which to
analyze the question? Some of them
might even come up with "standard deviation", for children are
amazingly inventive.

Providing
children with adult-generated, canned definitions for "median" and the
like would be entirely analogous to providing them with the definition of
"concentration", by which to analyze the intensity of orange
flavor. But the latter definition is
concealed, in Glenda Lappan's ideal classroom exercise, evidently for some deep
pedagogical reason, while in another part of the curriculum the statistical
words are carefully dinned into them without question. There appears to be some inconsistency here
in the constructivist doctrine. I
wonder why.

*October 24, 2001*