**Notes on the Brookings Institution Brown
Center Symposium **

on

__Algebraic Reasoning: Developmental,
Cognitive, and__

__Disciplinary Foundations for Instruction__

__Washington, D.C., Sept 14 and
15, 2005. __

Tom Loveless, director of the
Brown Center, a subsidiary of Brookings devoted to education, invited some
mathematicians (Milgram, Howe and Bass), some math educators, some
psychologists and some cognitive scientists to make formal presentations, seven
of them (the Milgram-Howe-Bass presentation was the seventh), each followed by
comments and questions from the invited guests, who sat at the big circle of
conference tables along with the speakers, except that the current speaker
generally chose to speak from the front, often with a Power Point or slide
show.

Among the invited guests
present were Cathy Seeley (current President of NCTM), Jerome Dancis, Madge
Goldman, Richard Askey and me. Several
people from the government were there, big types in charge of various things,
and I did talk to some of them over coffee or lunch at our seats there, but I'd
have to look up their cards, which I have filed somewhere, to remember who they
were. It's nice in Washington; you feel
close to God, though rather distant from human beings as such.

Daniel Berch
("NICHS/NIH") gave the first presentation, "Aims and Objectives
of the Meeting". It was short, and
I took no notes except to say he posed the question: should children's learning
of algebra be prepared via "pre-algebra" education down to the early
grades, or should the schools be silent on such things until a full-scale
Algebra I in Grade 8 or 9? And he
probably asked a couple of others, likewise ignored in the sequel.

Martha Alibali, "a
cognitive and developmental psychologist who studies children's knowledge and
communication about mathematical concepts", was next. According to the notes passed out to all of
us, Alibali "earned her PhD in Psychology at the University of Chicago,
and she is currently Professor of Psychology and Educational psychology at the
University of Wisconsin-Madison. Her
research focuses on mechanisms of knowledge change in cognitive development and
learning ...."

My notes here are extensive,
since she made me angry. Her lead-off
statement, which I had heard before, somewhere, was something like "Prior
knowledge can be a stumbling-block."
That is, some children have trouble getting things straight under
algebra instruction because they already know something they'd have been better
off without. My notes say:

"Prior knowledge"? Prior __mis__information!."

(En passant: Once we know
what her research focuses on, how on earth can we now get even more
particular? The word "focus"
is a favorite in the education world, and some papers focus on six things at
once.)

Professor Alibali repeated some
of the things stated by Catherine Kieran, a well-known writer on such things,
in a paper in JRME a few years back.
There, Kieran attributes a detailed analysis of the various meanings of
"variable" and "equals" to Usiskin, among others, and notes
that even Usiskin wasn't alone, or first, in analyzing the problem. (Every key
statement in an education-research paper is followed by a reference, without
naming a page number, to about seven books.)
I don't remember if Alibali named Kieran in her talk, but I happen to
have been a subscriber to JRME in those days and was reminded most strongly of
that Kieran paper as Alibali spoke. Thorndike, Van Engen, Matz and Firth also
had been mentioned in Kieran's paper, in connection with the uses of "variable".

(See
<http://www.math.rochester.edu/people/faculty/rarm/algebra.html>.)

What are these things being
repeated, maybe focused on, by Thorndike, Kieran and, latterly, Alibali? Previous knowledge as a stumbling
block. Example: Children interpret the symbol "="
as a command to do something (this is previous knowledge) and are therefore
confused when they see "4 + 5 + 8 = 4 + □".

Now, in school math such
sentences as "4 + 5 + 8 = 4 + □" never come with capital letters or periods,
let alone quantifiers or other explanations, or requests followed by question
marks. Why can their teachers and textbooks not see in this incomplete syntax
the total explanation of the alleged stumbling-block? The difficulty has nothing to do with mathematics and nothing to
do with the alleged multiple meanings of the symbol "=". The prior "knowledge" Alibali
refers to is really only a prior reflex, that children have been taught goes by
the name of mathematics. To analyze
their mind-set as if it were a natural phenomenon requiring cognitive science
to unravel is overkill. Worse, it leads
to wasteful and false teachings.

What is needed here is English
lessons, not research. Well, maybe
research helped in this case. After
presenting the problem, Professor Alibali did show that when a linear equation
was presented to students in Grades 6, 7, and 8, in the form "For what m
is 3m + 5 = 23 true?", she got quite a few correct answers. Eureka.

Then she went on to the
"meaning of variables". In a
situated problem (I didn't take very good notes here) there was a question as
to whether "d" stood for "dime", "the value of a
dime", or one of some other things suggested by her subjects. She said it could stand for any of the
suggested things, but allowed that others claimed it had to stand for
[something in particular, I forget what].
My note here was, "Why not __say__ what it stands
for?" Is the meaning of
"d" to be uncovered by a deep understanding of algebra? In other words, what this psychologist was
doing was asking meaningless questions and making much of the kinds of answers
the kids would give. My own reaction
was to think that this sort of experimentation on human subjects should be
forbidden by law.

And then I see this mysterious line
in my own handwriting: "Test
question: Which is larger, 3n or
n+6?" I wish I could take notes at
the speed they talk. This is clearly
apropos of the meaning of the variable 'n'.
Best I can say. For further
details of how psychologists can mystify the retention of false or incoherent
prior conditioning, and think it is inherent in the subject-matter or (worse)
in the brain, go back to the URL (five or six paragraphs above) pointing to my
discussion of the Kieran paper in JRME (Journal of Research in Mathematics
Education).

Next came David Carraher,
listed as a member of TERC, the federally sponsored educational think tank that
wrote one of the famous elementary school math programs that is stirring up
parental dissatisfaction around the country.
His paper, __Lessons From Early Algebra Research,__ recommends
"deepening" the existing curriculum.
If the kids have to argue about the value of 7 + 4, or even spend a few seconds figuring it out,
that "gets in the way" of his conversations with them which he was
intending to lead to algebra. He
strongly recommends the learning of the elementary number facts before
attempting the transition to algebra.
(Hear, hear!)

Carraher's trials are with 5th
grade students, seeing if they can make sense of open situations which at first
they interpret "definitely".
(This needs some explanation, I think, which I can supply from the
Kieran paper earlier referred to. A
researcher, seeking to find out how much algebra they know, asks some children
the following question: "Let x be
any number. If we add three to it, how
much is the result?" One of the
answers he gets is "9"; why?
Well, the kid thinks, if x can be any number, then it can be 6, and 6+3
is 9, isn't it? This is an example of a
child "interpreting definitely" the open expression
"x+3". (And it is wrong!)

The researcher, Carraher
continues, is thus faced with the problem of getting the kid to understand the
phrase "any number" in his opening gambit, "Let x be any
number", or so he thinks. For the
moment, he believes that kid to be ignorant of algebra, and is conducting
research on the problem represented by this "incorrect"
interpretation of "any" as allowing any definite substitution. To him it seems that the kid doesn't
understand the phrase "x+3", or "any number, plus
three". I confess I don't,
either. If Mr. Carraher would please
ask me a mathematical question I might do better.

Carraher now gives an example
of the sort of lesson he hopes will lead to algebra once such
misinterpretations are straightened out.
He takes a class of 5th graders and tells them,

"John has a wallet with some money in it. Mary's wallet has three times as much."

Here he stops. He expects some response to this; he
silently watches for their reaction. I
guess they immediately ask how much John has, since they are used to such
problems (not that a problem has been stated here). Somehow, Carraher
believes, they have to get used to __not__ having such information; the
"answer" they are supposed to discover in this case is something like
"M = 3J". A linear graph to
that effect would also be a good answer.

But I am seriously annoyed
here. It is bad enough to teach
children to regard the equals symbol as a command to do something, by giving
them "5 + □ =
11" and calling it a problem without even putting a period at the end, let
alone writing out a question. Now the
man says something (not much) about the wealth of Mary and John, thinking it a
sort of "open-ended" problem, and imagining the consequent general
discussion of Mary and John will lead the children into an understanding of
algebraic formulations. But it is not
yet a problem at all. The whole thing
reminded me of the famous "(a+bn)/n = x; donc Dieu existe. Répondez!"

To my mind this is bad
pedagogy, and I recognize it, from having read parts of TERC's Investigations
in Number, Data and Space, as typical of that organization's approach to
elementary school education. Some
children, with educated parents or good teachers, might learn to write and
understand, via open-ended conversation, how an equation such as M = 3J, with
proper definitions, models the Mary and John monetary relationship, but in the
typical schoolroom, without direct explanations, some will gradually be
chivvied into saying "Mary is three times John", while others will
simply give up mathematics. Mathematics
is not mind-reading, and should not be presented as such. Ask a silly question and get a silly answer,
the saying goes; ask a silly question and expect learning to result is
self-deception.

The TERC theory behind such
"open-ended" lessons is this:
Children must be taught to behave like mathematicians, like real users
of real mathematics. Not every situation
in which math is applicable comes with fully defined parameters; they can also
come with unnecessary information, or incomplete information. Children have to get used to this, and when
confronted with incomplete problems they should be able to identify the missing
data, name them, and write what is known about them with algebraic symbols.

But in this case where was the
problem? Every mathematician I know
deals in problems. He may be faced with
incomplete data, or superfluous, but he must have a question relative to which
he recognizes what is incomplete and what is superfluous. One doesn't tell a
mathematician, "Let E be a locally convex real topological vector
space", and watch his resulting behavior.
A real mathematician, after a few seconds, would reply,
"Yes?" His response to the
information that Mary's wallet has three times the money that John's has will
be the same, were a response demanded.
If the researcher says that the children have to learn that the amounts
of money in each of the two wallets are missing data, what then? Any idiot cans see that. On the other hand, why would they even want
to know those amounts? The world is
full of missing information. After
all, the name of the captain is also
missing from this account. Without a
problem, who is to say what is missing and what superfluous?

How much time a teacher of
algebra can save, and how much misery, by not being so open-ended! It is a perfectly good lesson to make the
statement about Mary and John and their wallets and teach the kids to "model
the situation" by calling Mary's fortune P and John's fortune Q (in
dollars), and then exhibit the equation (P=3Q) with some of its values
tabulated, and its graph. The
"problem" here is to draw a good graph, expressing the information in
hand. I would have the class do a dozen
such modelings, and go on in the
following days to problems, stated as problems, for which such expressions have
some value, and then solving the equations that arise. Yes, this is the way teachers have been
doing it for a hundred years, and usually without much success, but I was not
persuaded that a day of floundering, which the children are bound to take as
evidence of their own incompetence, makes a good lesson. The agony of having such a Mary and John
piece of information and having to talk it over, without a question or demand,
all in pursuit of a theory of learning, is more than children should bear. (I do hope Carraher didn't help out, when
questioned, by telling the children that Mary's wallet has "any amount of
money" in it, in explanation of the original "some money".)

Joan Moss, of the Ontario
Institute for Studies in Education, was next, and described her work with
Robbie(?) Case, using patterns to approach algebra. She described, or referred to, "Ferrini-Mundy's Problem"
having to do with tiling a swimming-pool.
It didn't seem to me that she gave us any reason to believe this kind of
instruction is worth the time.

Why is it that people want to
teach algebra without teaching algebra, but by means of analogies more persuasive
to grown-ups who already know some algebra than to the children? Years ago, some mathematician in an
unguarded moment defined mathematics for a popular audience as "the
science of patterns"; and in a certain poetic way it is. Furthermore, the patterns visible in daily
life, the ones that exhibit symmetries of some sort, or order, as a
checkerboard or the sequence of multiples of three, are indeed examples that
can help illustrate certain mathematical ideas, but they are not the patterns
the poetic mathematician really had in mind.

At the 1970 International
Congress of Mathematicians, held in Nice, France, the algebraist John Thompson
won a Fields Medal, a great honor in mathematics, and he was immediately asked
by a local newspaper reporter to explain what he had done. Since his work had been in group theory, he
explained what he could to the reporter, whose article, printed the next day,
seriously represented Thompson as a designer of wallpapers. Every analogy has its dangers; one can
suppose that Thompson did not use wallpaper the next time he explained groups
to reporters.

Unfortunately, the school math
experts tend to take this "patterns" dictum literally, as a prelude
to school algebra. Jean Moss and the
Ferrini-Mundy problem reminded me of the Cuisinaire blocks used by my daughter
Diana in her early grades. They were
rods, actually, blue, green, red, purple, etc, in lengths of 1, 2, …, 10
centimeters, with unit square cross-sections, and the kids learned to put them
end-to-end to produce addition, such as the lesson "A blue and a green
matches a purple and a yellow", where blue is 4 and green 5, while purple
is 7 and yellow is 2. After a few weeks
of this the kids get quite well used to the equivalent combinations of the
rods, but no numbers are mentioned, not even as lengths of the rods. The rods also model multiplication, as areas
occupied by several rods of the same color set side by side to form rectangles
measured by a rod of a suitable color in the perpendicular direction.

My daughter got good grades in
school, and I didn't worry much about the rods. She was good at some arithmetic
even before she went to school, in fact, for we sometimes had mental arithmetic
discussions at dinner, with her and her older sister. In later years Diana
became a lawyer, and one time, when she visited Rochester, the Cuisinaire rods
came up in conversation. She told me
she was a grown woman before she realized they had been trying to teach her
arithmetic with them. She laughed at
the recollection, but it was no joke. I
say, why not teach arithmetic, rather than try to sweeten it, creeping up on it
in the dark? The same should be true
for algebra.

Kenneth Koedlinger, of
Carnegie-Mellon, was next, and the title of his presentation was, __What makes
algebra hard for learners?__ My notes
say, "Words, words, words."
You can always tell a pointless essay by the number of times you see the
word "learners" in it.

Next, as "discussant"
to Koedlinger's paper, was David Geary, of the University of Missouri. He is
the author of a book called __Children's Mathematical Development__, and, according to Daniel Berch (who had
introduced the whole session) is "an evolutionary scientist". Geary posed the questions, "How do we
mold minds formed 100,000 years ago, to impart Algebra? Why do we have schools?" These are excellent questions. Geary then said we have somehow to connect
the artificial to the natural. (Well,
yes, isn't that what he had just said earlier?) He considers anything but hunting and gathering artificial, it
seems. My notes say, "How does he
explain children's games?" But
this is really by the way; his point, that algebra was not as natural to me, in
my childhood, as pulling my hand out of the fire, is obviously valid.

Geary's Power Point presentation, by the way, was in
darkish purple against a slightly lighter purple-gray background. A young man in good health could read it if
he really paid attention. Geary talked
mainly about China, which he had visited, and the style of teaching there,
which to us would seem 19th Century. He
described it as disciplined, the class giving answers in chorus, and the exam
grades posted on the bulletin boards for everyone to see. By Grade 5 they are familiar with notations,
and arithmetic facts expressed in algebraic terms, as that "6x=42 if
x=7". These displays seem to me as
unnatural, as unlike real life 100,000 years ago as I can imagine. I'm waiting to see what Geary will advise,
to overcome the gap.

The word "equals" is
not problematic for the Chinese, he says, and they use it as the verb it is,
even when they are children. (He
implies that it wouldn't be problematic for us, either, despite Kieran's
mystifications – which he didn't mention – if we used it properly in daily
life, too.) In short, what they learn
in China in the early grades is arithmetic, not "algebra-prep"
by analogies. Geary thinks this works, and so do I, but it was not clear to me
what evolutionary biology had to do with what he was actually studying.

That was the first day, apart
from discussion and questions.

On September 15 the sessions
took up only the morning hours, and there were only two formal presentations,
though preceded by a few historical notes given by Loveless, our host. He asked, why did public education in the
19th Century begin in Prussia and the U.S., only later followed by other
countries? I don't remember the answer,
or whether it really had any bearing on algebra instruction. He gave a few other sociological
observations, but not much.

The first presentation was by
John Anderson, a cognitive scientist, introduced to us as a leader in the
field. He reported on an experiment he
had conducted expressly for the present conference, and he was able to exhibit
on overhead displays his x-ray brain photographs showing "brain
activity" at certain points in the brain, serving as if lights turned on,
in the course of a child's solving the equation "3x + 2 = 17". That brain activity was quantified
(intensity of the display, I guess), and plotted against time, as the child
went "through the steps" of solving it. Anderson exhibited its graph, which began at 0, rose with
increasing steepness for a while, leveled off to a maximum, and then descended
with ever diminishing slope, like e^{-x}. The whole graph looked like y = Kxe^{-x} for some
constant K>0.

And K turns out to be
important, because Anderson repeated the measurements over a period of three, I
think, days. Three trials, anyhow, were
graphed. What he found was that the
activity was greatest on the first try, when the kid was first learning how to
do the problem. Given the same problem
a day (or more?) later, the subject solved it with brain activity of similar
sort, but less intense. That is, K for
the second day was smaller than for the first.
And the third was less than the second, though again in the same form.

Activity in those brain areas
are, he said, the signals of "conflict". My notes don't indicate how germane this is; I suppose it meant
that conflict and trying to solve a problem are similar brain states. His experimental equipment was impressive,
what with brain scans in real time and computer read-outs. The result, that practice makes perfect,
seems to me something I have heard before.

Last on the agenda was: __Panel Presentation: Analysis of NAEP
Items Classified Under the Algebra and Functions Content Strand__, given in three parts by the mathematicians
Hyman Bass (Michigan), James Milgram (Stanford), and Roger Howe (Yale), and was
an analysis, from three points of view, of the current NEAP math exam questions
on algebra, or at any rate those released by NAEP for either public scrutiny or
just for the use of this panel, I didn't know which.

Bass said the sample he looked
at failed to address certain important elementary algebraic skills, such as
simplifying an expression containing something like 6√(54). All told, he said, the algebra questions
fell far short of covering the ground.
However, he and the others had no way of knowing how fully
representative the questions they looked at actually were, and the NAEP
officials did not permit them to look at the whole spectrum of examination
questions as administered. No amount of
assurance of confidentiality would persuade them to relent.

Howe pointed out that the "rules
of arithmetic" are part of anything called "algebra", and in
fact the first and most essential part.
He decried the madness for "patterns", including the NAEP
pattern questions, now sweeping the country under the rubric of
"algebra". In addition, the
NAEP problems, at all levels, lacked depth, being usually one or two step
calculations, or numerically so trivial as to lend themselves to guessing.

Milgram, the third speaker,
said that 20 percent of the problems were mathematically incorrect. I suppose
that in most cases these errors are a matter of incomplete or incoherent
statements, though statements ( I should think) everyone actually knows the
intent of. (That people “know what you mean” is of course no defense in such
cases. Even if they don’t affect the
scores on the exam, such questions help corrupt the public understanding of
mathematics, and of rational discourse more generally.) Milgram added that he
saw no prospect for improvement for the next five years, i.e., that NAEP was
fixed in its principles until a new governing board could be elected (or
appointed?).

Tom Loveless then returned to
lead a general discussion. He suggested
that educational research concerning school algebra should concern:

(a) The place of patterns;

(b) How to "intervene",
to motivate students;

(c) Efficiency in instruction,
to shorten time in the classroom.

Hyman Bass returned to remark
that improving examinations will not solve the problem of teaching algebra in
the schools. His view is that primary
emphasis should be on the education of teachers. A spokesman for NAEP was in the audience, and hotly disputed
Milgram's allegation of errors in NAEP questions. He outlined the elaborate process by which NAEP questions were
formed and tested in advance of being used, as if that answered the allegation
of errors. Milgram did not retreat, and
the company finally dispersed at lunch time.

Ralph A. Raimi

20 October 2005