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 misinformation!."

(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 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