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?
cups 1 4
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:
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