A Misreading of Text by Zalman Usiskin


       Issue #24 (May, 2001) of Humanistic Mathematics Network Journal, an
occasional publication of Alvin White of Harvey Mudd College which one can
receive free of charge by writing to White at awhite@hmc.edu, contains a
reprint of a talk given by Zalman Usiskin at the 15th Annual UCSMP
Secondary Conference of November 6-7, 1999. ("UCSMP" is the University of
Chicago School Mathematics Project, which has created textbooks and much
else in the service of mathematics education in the schools, and of which
Professor Usiskin is the Director.)  His paper is called *Educating the
Public about School Mathematics*, and in an earlier version in print one
of its assertions was challenged by Hung-Hsi Wu as containing a
misrepresentation of some substance, which alas has not been repaired in
the present reprinting.

       On its first occurrence, i.e. when he gave the talk to the con-
ference mentioned, it was possible to attribute Usiskin's mistake to
carelessness, and maybe even a second time, in print, since perhaps he had
not been challenged concerning his misreading; but such an explanation
won't do after all this time and attention. Indeed, the error is hard to
attribute to carelessness at all.  However that may be, the Usiskin
misrepresentation of the California Mathematics Framework of 1999 must
once again be repaired, lest its incessant repetition persuade new readers
of its truth, and thereby also the truth of the title of his paper, or 
its other contents, much of which is in fact correct.

       Usiskin's paper as it appears in the Humanistic Mathematics
Network Journal begins with a summary view of the "New Math"
phenomenon of the 1960s, presenting it as a movement with damaging
consequences, ultimately discredited and followed by a reaction ("back to
basics") that was perhaps even more damaging, though short-lived. The
paper then goes on to compare the "New Math" in parallel time frames
with the progress of the present movement called "reform" or 
"Standards-based" mathematics as promulgated by NCTM in the period
1980-2000 and beyond.

        This second episode is not finished, of course, and while there is
already a reaction to the reforms of NCTM the outcome, as "Back to Basics"  
was the reaction to The New Math of the 1960s, is not yet clear, but it is
not Usiskin's purpose to complete the parallel; on the contrary, he favors
"reform" mathematics teaching, and wishes only to characterize the
attempted reaction now in progress here and there as ill-advised and
comparable to "back to basics" of the 1970s only in its potentially
deleterious effect.  Usiskin takes as a model, or prize example, of the
present reaction the California 1998 Mathematics Standards and the 1999
Framework (containing those standards, with amplification), which were
largely written by mathematicians rather than mathematics educators, whom
Usiskin sharply distinguishes from mathematicians.  And whereas the 1975
reaction to "the new math" was "back to basics", in his description, the
reaction to "reform" math is in this example from California, charac-
terized by him as nothing other than "the new math" redux.

	This too is curious, since the initial unfavorable reactions to
the California 1998 Standards had characterized it as a genuine "back to
basics" document, which would urge mindless memorization upon the
students, a misreading of the document in quite the opposite direction
from Usiskin's misreading.  "To a hammer, everything looks like a nail" is
an old saying, which in this context might be interpreted as meaning that
opponents of the California reaction to "reform" mathematics in the
schools, as defendeded by UCMSP and its allies in the National Council of
Teachers of Mathematics ("NCTM"), are tempted to see in the California
document what they are determined to see even before reading it, and even
if their interpretations are opposite to one another's.

       Usiskin early in the article draws the analogy, that
mathematicians are to mathematics educators ("teachers and those who
train them") as biological research scientists are to practicing physicians.
The analogy is not bad, but fails to imply what Usiskin thinks it does,
which is that mathematicians are not fitted to prescribe for mathematics
education.  Most biological research scientists have little interest in
clinical medicine, to be sure, nor do they pretend clinical competence,
and so most of them do not oversee or criticize the work of clinical
physicians; but there are those among them who do study and direct the
creation of new drugs and procedures, and their clinical trials, and who
understand their chemistry and practical problems well enough to write
the detailed accounts physicians rely on every day in their prescriptions. 
In the same way there are mathematicians whose oversight of school
mathematics education, were it more prevalent than it is now, could be
of analogous benefit to school mathematics teaching.  That is, Usiskin's
analogy itself does not support his condemnation of mathematicians'
interference in school mathematics policy at the content prescription
level.  It should, indeed, gladden the mathematicas educators that the 
creators of the science they would have correctly broadcast are, some of 
them, taking a hand in seeing that its more subtle points are correctly 
understood at the retail level.
 
       Following this analogy as if it did imply an ignorant or perverse
interference on the part of an unqualified group of ivory-tower
mathematicians, however, Usiskin goes on to give an account of what he
calls "Beliefs of the anti-reformers" to show (independently of any
analogy, actually, relevant or not) that they (the "anti-reformers") are
deficient in "knowledge of mathematics education and of students in
schools", and must not to be credited in this field despite the eminence
of some of them in mathematics itself.  "In one state of the Union they
have taken over," writes Usiskin, "and from this state we can obtain a
picture of the solution these mathematicians offer.  Their solution is
found in the Mathematics Framework for California Public Schools."

       So far we have opinion, which Usiskin is entitled to have and to
print, but when he gives what he takes to be a telling example from the
California Mathematics Framework, to show that the anti-reformers are
pressing a program that reflects the excesses rather than the lessons of
"The New Math," he has entered the domain of fact, which is another
matter.  He quotes from pages 154 and 155 of the California Framework,
where in the introductory section preceding the high school
recommendations, the Framework's authors explain what they would like high
school algebra to convey in the way of mathematical reasoning, and uses
the quotation to establish a point pleasing to his case, but false.

       Now, the Framework exhibits x - (1/4)(3x-1) = 2x - 5 as an example
of an equation to be solved, something that requires what could be called
clearing of parentheses, transposition and combination of certain terms,
with "x=3" the last line, and it describes briefly, using words of this
sort, how the usual high school (or middle school, these days) textbook
might word the solution.  We are all familiar with this sort of thing,
which generations of students have learned to perform, though not always
with much understanding of what they are performing.

       The California Framework authors are here at pains to point out
that this traditional wording, repeated without the teacher's
understanding its logical structure, can give students the idea that there
is no question of proof in such manipulations, and perhaps even that
"proof" is something found only in Euclidean geometry (which would be a
serious error), and that a more careful statement would make more sense
than a list of operations of such a mechanical nature as they have
outlined.

       At this point the Framework places in a distinguished boxed display
a very tedious double column proof showing (after noting that the
verification that 3 is a solution is the simpler part of the proof) that 3
is the only possible solution.  It has 16 steps, some of them very short,
as for example the deduction from "3=x" that "x=3", with a "reason" given
by quoting the general symmetric property of the equivalence relation
"equals". The entire display is intended to show that a statement in
algebra is required to be as carefully supported as a statement in
Euclidean geometry, in which the two-column format of proof was
traditional, in the days when Euclid was taught in the schools, as a 
device for exhibiting the logical sequence unambiguously.

	The text surrounding this display makes it clear that the authors
do not recommend such writing for students to imitate, and indeed that
teachers, too, should not be bound to write such things.  

	"In practice it would be impractical to demand such detail every
time a linear equation is to be solved," write the authors of the
California Framework, but they also explain that "Without the realization
that a mathematical proof is lurking behind the well-known formalism of
solving linear equations, a teacher would most likely emphasize the wrong
points in the presentation of beginning algebra."

       The crucial phrase in this last statement is "a teacher".  That is,
the California Framework, which is not a mere list of Standards but
includes more general guides for teachers and textbook writers, is here
telling the teacher *not* to imitate the tedium (and ultimate incomprehen-
sibility) that characterized the more pedantic writings of the "new math"
era, but to understand *themselves* the logical structure of what is being
done, in advance of presenting beginning algebra to students.  

	That teachers must know more than they expect their charges to
learn is a commonplace; in the present case the Framework authors give an
example of exactly what the teacher should be aware of, though not as
something to repeat it in that form for the student.  What the teacher
understands, in other words, does somehow carry over into the classroom
(and what the teacher misunderstands does the same) even if the words of
the classroom do not resemble those by which the teacher has learned the
correct understanding desired.  The California Framework here wishes to
provide an example to warn against the teacher's inadvertantly repeating
something logically invalid, as so often happens in textbooks we all know
and have in our time been afflicted with, but not that the teacher repeat
*all* the valid things that could, as some inadmissible cost in tedium, be
said in the classroom itself, for that was one of the errors of "The New
Math" of the 1960s.

       For generations, including the present, school algebra has been
too often presented as meaningless ritual by teachers and textbooks not 
warned by
such examples, and the California Framework writers, who were well
acquainted with the way the attempted cure in "New Math" days failed of
its intention, are here asking the high school teachers, at least, to
understand what the sequence of "steps" in the solution of a linear
equation are saying, before going on to explain the matter to students in
what might be called friendlier words, though just as accurate.  They take
explicit care, here, to warn the reader that they are not recommending the
teaching of this bit of "two-column algebraic proof" to students as a
method for students to follow, or even that teachers write such things for
themselves, except once in their lives perhaps, while learning their
trade, in order to observe that indeed "proof" is a feature of algebra as
much as of geometry.  They -- the authors of the Framework -- knew that
efforts to belabor this point to students in the 1960s was not productive,
but they could not, in their instructions to teachers and textbook
writers, permit the logical point to go unnoticed, lest the sort of ab-
breviation common in classrooms committed to an unthinking belief in
"basics" become a hindrance to understanding, when exhibited by a teacher
unaware of the logical structure of what solving an equation entails.

       [Footnote:  I have written a paper on this very point, the logical
structure of "the solution of an equation,"  to exhibit how when more than
linear equations are in question the understanding of the sequence of
statements has more meaning than one might at first imagine.  It was
published in two sections in the Humanistic Mathematics Network Journal,
Issues 17 and 18 (1998), and can be found reunited and reprinted at
.]

       Now Professor Usiskin quotes the 16 step proof verbatim, boxed
as in the original, but within the box, as a heading in boldface, as if it
were the title given for this display by the Framework, he prints
something that did not occur in the corresponding box in the Framework:

	"Recommended Proof of x - (1/4)(3x-1) = 2x - 5 
	by the writers of the Mathematics Framework" 

and at the bottom identifies the proof thus labeled as taken from the
Mathematics Framework for California Public Schools, p.155.  Just
before this display, commenting on the more customary description of
the process, with "transpose", "divide through" or words of that sort, as
had been given by the Framework with the comment that this sort of
thing is not a proof, Usiskin writes,

              "Not a proof? It looks very much like a proof to me, except
that I would emphasize doing the same things to both sides of the
equation, and avoid words like 'transposing' that suggest to students that
mathematics is a bag of tricks." -- In other words, Usiskin would improve
the brief, traditional version just a bit to make it satisfy his standards
of proof.  But of course this is what the authors of the Framework were
doing in suggesting that teachers -- teachers, mind you, not students --
go even further in their understanding of the details, though commenting
that "in practice it would be impractical to demand such detail" of
students' written work.  The only way the discussion differs from
Usiskin's commentary is that in the Framework the authors were instructing
teachers, and of course in more detail, deliberately and unto tedium, so
as to distinguish the depth of understanding a teacher must have from what
is to be expected of high school students.  But Usiskin goes on,

              "Their proof can be found at the top of the next page.  It
is, in my opinion, cruel and unusual punishment to inflict this kind of
pedantry onto young children."

       Goodness, yes. Who can argue with this?  Neither Usiskin nor the
authors of the California Framework intend "to inflict this kind of
pedantry on young children".  The Framework made this explicit, yet the
placing of this remark in Usiskin's paper is designed to imply that the
the Framework and Usiskin are of differing opinions on this matter.

       Armed with this misreading, what Usiskin says next is also correct:
"And in the process they have repeated one of the major excesses of the
new math era: the overemphasis on rigor." This would be true if, as in
some of the famous 1960's textboks by Beberman and Vaughan, or of Van
Engen and Hartung, such proofs were part of the student text, designed to
be imitated by them as little logicians.  (One might also remark that Max
Beberman, of "new math" fame, was a mathematics educator, not a
mathematician, and Van Engen too, despite his youthful PhD in
mathematics.)

       But Usiskin's misrepresentation, or misreading, of the California
Framework in this connection,  as evidenced by his gratuitous headline, 

	*Recommended Proof of x - (1/4)(3x-1) = 2x - 5 by the writers of
the Mathematics Framework*

is not yet done.  He writes, 

       "There is a significant marginal comment on this page.  'Without
the realization that a mathematical proof is lurking behind the
well-known formalism of solving linear equations, a teacher would most
likely emphasize the wrong points in the presentation of beginning
algebra.'"    

       Why would Usiskin call this "a *significant* marginal comment"?  
It does occur in the margin, to be sure, but as a repeat, a sidebar
created for emphasis, of words already in the text, a fundamental part of
that text, in fact, and not something smuggled in via the margins.  
Usiskin is within his rights to disagree about the advisability of telling
high school teachers things of this sort, but to present the comment as "a
marginal note", the authors caught out by Usiskin in the act of
recommending stupid rigor, so to speak, is to falsify the nature of that
sentence and its place in the Framework.

       Unless Usiskin has mistaken the sense of the Framework here, he is
implying that teachers ought *not* to be told to bear in mind the logical
structure of the procedure for solving an equation, before teaching their
students beginning algebra.  But of course this is not so;  Usiskin has
mistaken the purpose of the pedantically "new math" displayed proof, and
has leaped on the display to discredit the California framework as if it
were advocating logical tedium for children learning to solve a linear
equation for the first time.  This is emphatically not so.


	All this had been pointed out to Usiskin by Hung -Hsi Wu, the
actual author of that page in the Framework, in plenty of time for him to
amend his paper by the time of the appearance of the canard under
discussion here, or to announce his error after its appearance. Perhaps
someone may yet persuade him to acknowledge his error, and help set the
record straight on the intention -- and the words -- of the California
Framework.

             Ralph A. Raimi
             July 17, 2002