CA MATH STANDARDS WARS 12/97
Date sent: Sat, 20 Dec 1997 11:09:44 -0600 (CST)
To: LScheffers
From: mkr@merle.acns.nwu.edu (Marilyn Keller Rittmeyer)
Subject: Please Forward to Lists and Anyone Else You Desire
From: "Ralph A. Raimi"
Subject: Luther furioso
To: "Paul R. Gross"
Message-id:
Content-type: text/plain; charset=US-ASCII
Content-transfer-encoding: 7bit
You may quote all or part of the following to anyone.
Dear Paul:
Let me begin (and maybe end) by taking the easy way out. Mr.
Williams, whose name I do not find in the combined directory of the mem-
bership of AMS, MAA, and SIAM, the three major mathematics professional
societies, has argued from a false premise. It is not necessary to
determine if there is an agreed body of research information concerning
the relative value of "back to basics" math education as allegedly
practiced in the 1970s and "reform" math education as allegedly fostered
by the National Council of Teachers of Mathematics (NCTM) 1989 *Standards*
and its progeny in the California experimental programs Luther Williams
refers to so threateningly. One must begin with the charge that the
revision of the California Standards currently under examination by the
California Board of Education is in fact representative of "back to
basics" as characterized by Mr. Williams. If it is not, Mr. Williams is
attacking a straw person, and the fairness, intelligence, and basis in
research of his attack have no bearing on the validity of his conclusions.
The recent history of these proposed Standards is this. California
appointed a Commission on Standards to write a document of the kind now
common in all other states (save Iowa), designed to tell local school
districts and even individual teachers and textbook-selection committees,
what children should know and be able to do by each stage in their
mathematical education, K-12.
(In California the resulting *Standards* was must be regarded as a
complement to a more comprehensive "Framework", simultaneously written by
yet another committee, which includes, along with its own "benchmark"
expectations about curriculum, much other information on the philosophy,
pedagogy, and administration of mathematics education for the state.
Its draft was submitted last September, and I am not certain of its
present legal standing. I mention this here only to warn the reader not
to be confused if the public debate comes to mention the Framework, too,
as in the future it might.)
The Standards Commission was a citizen's commission, containing no
mathematicians or mathematics educators; but its members took advice, both
open testimony and privately. Its report, after acrimonious debate
involving the writing of several drafts, each closer to the style
favored by professional mathematics educators than its predecessor, was
approved on October 1, with two negative votes and two abstentions. The
abstentions were from members who had been presented with the final draft
on the morning of the vote. That final draft
(see http://www.ca.gov/goldstandards)
was more distant from its immediate predecessor than any predecessor had
been from its own predecessor, and was a surprise to those who abstained.
The leader of the opposition, who has posted a Standards of his own, which
he presented to the Commission and the public as an alternative:
(http://www.rahul.net/dehnbase/hold/platinum-standards),
was Bill Evers, a political scientist at the Hoover Institution at
Stanford. Evers took his mathematical advice from a cross-country
collection of referees to whom he had the Commission mail successive
drafts for comment. I was among these referees and proofreaders, but I
*believe* his principal advisors were some of the mathematicians from
nearby Stanford University. The Commission majority also had advisors, of
course, and indeed the final text itself had to have been written by some
of them, as the Commission did not profess competence in mathematics or
mathematics teaching.
The October 1 document was studied by the California Board of
Education, which took public (and nation-wide email) testimony, some of
it from mathematicians, among then Henry Alder, former president of MAA
(Mathematical Association of America, which is explicitly concerned with
mathematical education, albeit primarily at the college level). Alder, a
mathematics professor at U Cal Davis, and a former member of the
California Board of Education, testified on October 20 that "the content
is not stated in sufficient detail to give a clear picture to the teachers
as to the depth with which to cover the content." Also, "Most of the
sample tasks given are the simplest and most routine ones imaginable for
each of the topics listed." And in the end he advised that "...to publish
these Standards without all these badly needed changes would be far worse
than not to publish any Standards at all."
Any mathematician would be apalled to sample the Glossary alone, for
that matter, which was pieced together (it has a bibliography of its own)
by people who didn't know mathematics. It is filled with errors and
ignorance.
The Board of Education rejected the October 1 document and asked a
group of mathematicians from Stanford University to prepare a revision.
That group avoided publicity, but did get mention in the op-ed pages and
was spoken for by one of its members, Professor Ralph Cohen. Not only the
four from Stanford, but several of their friends from other universities,
were involved in the rewrite, which was drastic. The professed content of
the original document was retained, and much of its phrasing, but a good
half of the original was simply dropped. That half was the "Clarification
and Examples" sections, mentioned above by Henry Alder as being largely
trivial.
More than trivial, mathematicians would say: often time-wasting
and indefinite. Page 73, at the Grade 8 level, under the rubric
"Problem-solving and Mathematical Reasoning", has
"2.3 Solve for unknown or undecided quantities using algebra,
graphing, sound reasoning and other strategies
"2.4 Make and test conjectures using both inductive and deductive
reasoning";
and in the right-hand columns, under the associated "Clarifications and
examples", is written
"Sample tasks...
" * Research geometric proofs of the Pythagorean Theorem.
Design an interactive visual display to demonstrate one of the proofs.
Attempt to apply the proof to non-right triangles and demonstrate its
failure."
2.3 and 2.4 are models of indefiniteness, and an 'interactive
visual display' can well fill a week's time. The verb 'research'
(imperative mode) is perhaps overdrawn at the 8th grade level, and there
is no telling how much time its consequence can be made to fill.
The mathematicians not only removed all the examples and
clarifications, but reorganized the text to omit entirely the "Problem
Solving" part of the strand named "Problem Solving and Mathematical
Reasoning", which appeared in the original at every grade level. They
retained much of the text (of all kinds) where it said something valuable,
however, though often under other rubrics. For example, again at the 8th
grade level, this time in the strand "Number Sense", the original said,
"Use properties of numbers to construct simple valid arguments of, or
formulate counter examples to, claimed assertions." (p63)
The corresponding standard in the revision reads, under "Algebra 1",
"1.1 Students use properties of numbers to demonstrate that assertions
are true or false."
This is the same instruction, but characteristically avoids the
original's grammatical construction, so popular among educators, that
lists several prepositions in sequence (though only two in this case)
before trumping them with a single object.
The rationale for removing the strand "Problem-solving and
mathematical reasoning" is obvious to any mathematician. Prob-
lem-solving, as construed at the State Education Departments, means
setting up a physical or financial (or other "real-life") situation in
(usually) algebraic terms, thence to obtain a real-life solution; while
*mathematical reasoning* is the deductive process by which one
mathematical statement entrains another. There is no reason to join the
two into a single strand, and good reason not to, since the second part is
generally slighted without notice in most State Standards, and the famous
1989 NCTM Standards, while the phrase "and mathematical reasoning" remains
prominent. Where "problems" concerned algebraic methods, then, the
Stanford mathematicians put them under "algebra", and so with other
exercises of the sort that used to be called "story problems", and only
where reasoning in the *mathematical* sense was at issue did the items
remain in this strand, now renamed "Mathematical Reasoning".
As the Stanford mathematicians wrote in the Introduction to their
revision of December 10, "Mathematical reasoning and conceptual
understanding is [sic] not separable from content." They might even have
intended that singular verb. Most mathematicians, describing a model
curriculum, would therefore not even retain a separate strand for
"mathematical reasoning", and I'm not sure why the Cohen group did;
perhaps to emphasize the centrality of reason in the use of mathematics.
(There is of course a formal study of reason, called "logic", but nobody
wishes to get into logic, except incidentally, at the K-12 level.)
The revised draft of the Standards is posted by the Board of
Education at
www.cde.ca.gov/board/ [with separate links to K-7 and post-7 levels]
and it is these two drafts that have generated the present controversy.
The K-7 draft is, by the way, very hard to read as posted, as it consists
of the print-out of the Commission October 1 draft with the Examples, etc.
removed and the modifications of the rest presented interlineraly or in
brackets indicating substitutions and the like; while the Post-7 section
is totally new, organized by the names of courses (Algebra I, Geometry,
Algebra II, etc.) rather than by strand. It is perhaps the Post-7 section
that has generated the most heat among mathematics educators in
California, for it explicitly rejects the original draft's call for an
"integrated curriculum", and its extreme brevity gives credence (in the
mind of persons unaquainted with the ways of mathematics and
mathematicians) to the idea that it recommends nothing but memorization
and drill concerning each of its headings.
It is ludicrous to suppose that the Stanford mathematicians have the
intention of reducing mathematics education to the kind of drill in
mindless algorithms their opponents paint them as advocating. Above all
else, mathematicians want the schools to teach that which generates
understanding and not memorization of routines. Above all persons,
mathematicians understand the implications of lessons that affect to do
the one or the other.
One important topic of controversy concerns the algorithms called
"long multiplication" and "long division". These teachings are a
touchstone of the battle at the elementary school level. The Stanford
mathematicians wish all students to learn these routines, as well as the
means of calculating with fractions. The "reformers", as their opponents
in the NCTM and the State education departments call themselves, consider
these to be mere "skills", now outmoded by the calculator. The reformers
cannot be made to believe that learning how to find the quotient 685/22.4
as a decimal expression, done in columns with due attention to trial
divisors, subtractions and "carrying" (or whatever) is anything but a
waste of time. Mathematicians, cognizant of the nature of the decimal
system and what it takes to use it effectively, including mental
estimation, tend to believe that use of the calculator in childhood
lessons of this sort will not only inhibit this understanding, but will
fail to prepare for more sophisticated divisions (for example) concerning
polynomials, something the typical 5th grade teacher, compelled to teach
"long division" to an unhappy class, knows little of. It is so much more
*interesting* to have that 5th grade class work on a project: to
"research" the dimensions of the playground and design paths and positions
of tables and swings in order to meet a certain budget, using a calculator
when numerical divisions are needed. Is this not real-life behavior?
Thus it has been for several years now, that anyone who opposes the
tenets of the reformers (read "NCTM") tends to be regarded by them as
advocating the return to the (idealized) grey classroom of Mr. Gradgrind.
And if mathematicians want people to learn "long division" it must be
because as mathematicians they see their expertise in long division being
eclipsed by the computer, and fear for their honor and emoluments. There
are those who believe that to be a mathematician at the university level
must mean an ability to perform long divisions of great length very
rapidly. They can easily understand when the State eduation department
derides exercises in long division as needless, when a calculator is
available.
Mr. Luther Williams is quite wrong to write that the revision
"vacates any serious commitment to elevating problem-solving and critical
thinking skills to K-7 mathematics standards."
and
"The wistful or nostalgic "back-to-basics" approach that
characterizes the Board standards overlook the fact that the approach has
chronically and dismally failed. It has excluded youngsters from engaging
in genuine mathematical thinking and therefore true mathematical learning,
and has proposed a disproportionate mathematically illiterate citizenry."
and
"...school systems that embark on a course that substitutes
computational proficiencies for a commitment to deep, balanced,
mathematical learning."
The new Standards drafts being considered by the Board of Education
have been posted since they were written, and are not final. Indeed,
there already have been two versions of the 8-12 Standards ("post_7"), the
second a great deal less austere than the first, pubic input having
already had an effect on the amplitude of the prose if not its essence,
and it was that first one, as I recall, that generated articles and
letters-to-the-editor in the New York Times and elsewhere, some of them
misrepresenting the document as badly as Mr. Luther Williams.
I will not defend the new Standards in detail, and indeed I don't
know what it will look like when done. It might turn out to be a poor
document from a practical point of view, or even in principle; but I would
like the interested world to know that it is flat-out false that the
revision has as its purpose the "return to [rote-learning] basics." This
is only its characterization by mathematically illiterate people, such as
those who wrote parts of the October 1 draft and its Glossary, and perhaps
by other persons deceived by them.
Your question on the research basis underlying the more general
controversy about the amount of "basics" needed in a program that
inculcates mathematical understanding and facility (*mathematical*
facility, not the ability to multiply long numbers without a machine) is
an important one even if it is largely irrelevant to the California
controversy, where the document being prepared by the Board of Education
envisions more (non-basic?) mathemtical content than is common in any
State today. Harold Stevenson, professor of psychology at the University
of Michigan, can lead you to references on the matter. The literature is
enormous, perhaps of the order of the amount devoted to psychoanalysis, a
science similar to "education" in many ways; and a scientifically inclined
reader needs a steady guide.
Ralph A. Raimi Tel. 716 275 4429, or (home) 716 244 9368
University of Rochester Fax. 716 244 3361, or (home) 716 442 3339
Rochester, NY 14627 Homepage: http://www.math.rochester.edu/u/rarm