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Topic: PAPER I: Becker-Jacob Paper on the CA 'Math War' Situation
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Jerry P. Becker

Posts: 13,035
Registered: 12/3/04
PAPER I: Becker-Jacob Paper on the CA 'Math War' Situation
Posted: Dec 22, 1998 2:10 PM
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[Note: Several months ago we wrote a paper "'Math War' Developments in the
United States (California)" that was published in the ICMI Bulletin (No.
44, June, 1998, pp. 16-25.) ... we are sending the article in this note.

Subsequently, Henry Alder, professor of mathematics at the University of
Califorinia at Davis, submitted "Another Perspective" in reply to our
article, to the ICMI Bulletin. Shortly thereafter, we received a copy of
Alder's "Another Perspective" from the Editor along with an invitation to
submit a response. This we did - "Response to Alder." Both Alder's
"Annother Perspective" and our "Response to Alder" are being published this
month in the ICMI Bulletin (No. 45, December, 1998, pp. 9-13 (Alder) and
pp. 14-16 (Becker-Jacob)). We are sending these two as a second note,
following shortly after this one ... J. Becker and B. Jacob.]
**********************************************************

'Math War' Developments in the United States (California)

Jerry P. Becker, Southern Illinois University at Carbondale

Bill Jacob, University of California, Santa Barbara

INTRODUCTION

If there was ever a time in the United States when no one cared about
mathematics education, it certainly has not been the past couple years.
Mathematics education has been written about in the local, regional, and
most important national newspapers and magazines. Reports have also
appeared on radio and national television. The focus of attention has been
the so-called "Math Wars" that center on reform in the school mathematics
curriculum and its teaching. In particular, a "backlash" against the
California Mathematics Framework and the National Council of Teachers of
Mathematics' (NCTM) Standards in California has been prominent in the news.

California is the largest among the 50 states. By virtue of its
size, it has a very significant influence on school textbook and test
publishers. Controversies in education - philosophical, political or social
- that come up in California can presage similar controversies and trends
in other states. Thus, there is great interest throughout the United
States in what transpires in California. As a consequence, and because the
1992 California Mathematics Framework is consistent with or fits closely
with the NCTM's Standards, the NCTM has a great interest in these
developments in California; in fact, its Standards (currently undergoing
revision) have been the focus of much expression of unrest.

The so-called battlefields for the California math war extend beyond
concerned parents meeting with teachers and school boards. Involved are
state education agencies, their advisory panels, and ultimately the state
legislature. As one might expect, newspaper accounts capture only a small
part of a rather complex story, so we shall try to provide a few more
details here.

BACKGROUND

Like other states, California has a State Board of Education (SBE).
Its members are appointed by the Governor of the state, subject to the
approval of the state legislature. There is also a Superintendent of
Public Instruction (SPI). This person is elected by popular vote and is
the head of the California Department of Education (CDE). [Presently the
Governor and the SPI are from different political parties.] The function
of the CDE is to provide administrative support for various groups and
agencies, implement the policies set by the SBE, prepare and disseminate
documents, and provide information to citizens, school districts, and the
public.

Due to recent changes in California, however, the curriculum
standards, frameworks, and adoptions of instructional materials for the
schools now fall under the purview of the SBE, not the SPI. The SBE
appoints a Curriculum Commission (CC) whose job is to supervise the
drafting of frameworks and instructional materials criteria, recommend
instructional materials for state-wide adoption, and make policy
recommendations to the SBE. Its members are mainly K - 12 teachers, with a
few administrators or representatives from higher education. The SBE also
appoints (using CC recommendations) members for framework committees and
instructional resources evaluation panels (IREP). The mathematics
framework committee spends about a year drafting the curricular framework
and materials criteria, which is revised by the CC and then submitted to
the the SBE for approval. The IREP spends four months reviewing
instructional materials submitted for adoption, comparing them against SBE
adopted criteria. For details about the materials adoption process and
further background, see Jacob (1998).

Standards are new to California. Prior to the SBE approval of the
new mathematics standards last December, the state only had curriculum
frameworks. Standards that are established at the state level for the
schools are voluntary. However, California has established a new testing
program for the entire state that must be aligned with the new state
standards in 1999. So in reality, the tests will induce districts to
adhere to and follow the standards. The standards are drafted by the
Standards Commission (SC, which is unrelated to the SBE, CDE, or the CC),
whose members are appointed directly by the Governor, the SPI, and the
legislature, and what they develop must be approved by SBE.

THE PERIOD 1985 - 1997

The California SBE adopted a new Mathematics Framework in 1985. Many
ideas in the new framework were new to teachers, as well as to publishers
of school textbooks, for in the next year the CC rejected all commercially
published materials that were submitted for consideration for adoption in
California's schools as none of them met new criteria established by the
SBE. In 1987, the CDE published the Mathematics Model Curriculum Guide
(CDE, 1987) which included 88 pages devoted to "teaching for understanding"
with classroom examples. This document clarified many themes from the
Framework and historically proved to be quite influencial, both for
teachers interested in change and textbook developers. Also, "state
replacement units" were made available to teachers so they could try out
some of the new approaches and so textbook companies would have models to
consider. However, until 1995, because of the failed adoption, the
curricular materials used by nearly all California K-8 teachers remained
almost identical to those sold during the early 1980's. In 1992 a new
Mathematics Framework and criteria were approved by the SBE. This new
policy document is still in place and is a central ingredient in the the
raging controversy that exists at this time.

California developed its own state-wide testing program during the
early 1990's (the California Learning Assessment System/CLAS) and this
performance-based exam was field tested in 1993 and administered state-wide
in 1994. But some of the free response questions on CLAS were considered
controversial and the Governor vetoed funding for the program, resulting in
a three-year gap in the state's testing program. California's test results
in 1993 and 1994 were poor and they were confirmed by the results of the
National Assessment of Educational Progress, in which California ranked
41st out of 43 states. This let to an attack on the 1992 California
Framework. Well-organized groups claimed that emphasis on cooperative
learning, problem solving and applications of mathematics diminished the
importance of individual accountability and the importance of mastering
basic computational skills. (NCTM, 1998). The concern about the "failed
reform" led the state legislature to enact a new law (A.B. 170) in 1995
requiring the SBE to adopt instructional materials that are "based on the
fundamental skills, ... including basic computation skills." By the summer
of 1995 the verdict was clear, the 1992 Mathematics Framework had failed.
The timing was amazing. In September 1995 [1] for the first time in over a
decade, new instructional materials in alignment with the Framework became
available for state-wide use in grades K-8.

During 1995 and 1996, the SBE became quite active in mathematics,
producing a "Program Advisory" in July 1996 that called for a "balance of
basic skills, conceptual understanding, and problem solving" in mathematics
education. And in November of 1996 the SBE appointed a new framework
committee - it rejected the majority of the CC's nominees to that
committee and added fourteen others recommended by a recent appointee to
the SBE, a person who holds views of mathematics teaching and learning
very much at odds with the recommendations in the 1992 Framework. In 1997
the framework committee met, developed and sent a draft document to the CC
with all eight CC-recommended framework members voting against it. Central
features of the draft were a listing of topics required at each grade level
and an elimination of all discussion of pedagogy.

In a supplemental adoption (September 1997), the SBE rejected two
programs highly recommended by the CC (both were NSF-funded curriculum
development projects) citing mathematical errors and other problems.
Examples of mathematical errors noted by the SBE included writing "ratios
instead of fractions" and a number theory mistake that "30 divides the
product 36x45" which the SBE explains in their written report is an error
because "30 is not a factor of either 36 or 45". A middle school program
that included a "pizza pirate" in a story problem was cited as violating
the states patriotism and morality code. (For details see Nicholas (1997)
or Jacob (1998)).

The Standards Commission (SC) approved the school mathematics (K -
12) standards in September, 1997, after a year of deliberation and
considering public reactions to the document. In fact, the Commission's
standards were substantially revised during July and August as a result of
public comment, and the rapidity of these last-minute changes resulted in
some glitches that both sides of the debate critcized and worked to correct
during the Fall. The SBE adopted a substantially revised document in
December, relying almost solely on the work of four Stanford University
mathematics professors. They claimed the revisions were necessary to
increase the "mathematical precision" in the statements of the standards
and to remove the "Clarifications and Examples" whose purpose was to
illustrate what the standards mean in the classroom, but were interpreted
as "prescribing pedagogy" by the SBE. The process by which this document
was approved, and the content of it, are major factors in the controversy
that presently rages.

IMPRESSIONS GIVEN BY THE MASS MEDIA

While the media has tried to make sense of the debate that surrounded
the controversial developments, it seems that they would not or could not
get to the heart of each side's position. Typical news reports stated that
the controversy was over "the best way to teach math" and that the
arguments over teaching were about such issues as "uses of real-world
problems vs. skills" or "integrated vs. traditional curriculum". While on
the surface such statements are not incorrect, they miss the central
issues. The Framework committee agreed not to "prescribe pedagogy" in its
document, but could not come to agreement, or even agree on a format to
discuss, how to balance skills with problem solving. Even with pedagogy
off the table, the committee could not agree on content. In their written
"homework" (and therefore, public documents) for the committee on "How to
balance K-6 Mathematics," two northern California mathematics professors
offered contrasting views: "the curriculum should include extended projects
or capstone problems that require the student to synthesize and integrate
concepts and calculational techniques", and "I suggest that our goals and
expectations of elementary school children should be pretty much limited to
arithmetic". (CFIR, 1997) How mathematical questions could be posed in the
document was particularly problematic. For example, describing an area
problem on a geoboard was rejected by the committee majority for two
reasons: it "prescribed manipulative pedagogy" and "the appropriate tools
for geometry are the straightedge and compass." Disagreement arose over
appropriate use of calculators and technology. The media has by-and-large
bypassed these central controversies.

The press reports on the Standards debate (see Wu, 1998) usually
reduce the question to high standards vs. low standards. SPI Delaine
Eastin's quote that the SBE "dumbed down" the CS's standards received
prominent coverage, as did the SBE's statements that it is "removing
pedagogy from the document" and striving for "greater accuracy". But here
again, while publicizing each side's favorite "one-liners", the press has
failed to dig out the basic differences over content. Central issues
included the SBE's consistent removal of such phrases as "estimate" or
"explain" and replacing them with "calculate," the removal of the study of
patterns from the Algebra and Functions strand in elementary school, or the
complete removal of all exemplars that were designed to help K-12 educators
(and text designers) understand how topics are to be approached at a given
grade level. The press seemingly never examines why both sides claim their
views represent "high standards." Does moving mastery of computational
skills to a lower grade level raise standards? Or, does adding an
expectation that students explain what a number procedure means
geometrically raise standards? However one views the situation, these
actions by the SBE have seemingly established California as the center of
the opposition to the NCTM's Standards-based reform. Due to many factors,
the media a prominent one, many parents, policy makers and some teachers
and professors came to the view that the 1992 Framework had failed.
Following this thinking, therefore the NCTM Standards were also wrong and
could be regarded as the culprit. (Ross, 1998)

RESEARCH

An important factor in the California debate is the requirement that
state-adopted instructional materials "incorporate principles of
instruction reflective of current and confirmed research" (CA Education
Code 60200c-3). The SBE invited Prof. E. D. Hirsch to speak on this issue
in April 1997. In the written version of his comments, he described
"mainstream educational research" as found in "journals such as the
Educational Researcher" explicitly stating "This is a situation that is
reminiscent of what happened to biology in the Soviet Union under the
domination of Lysenkoism, which is a theory that bears similarities to
constructivism." (Hirsch, 1997, p. 3) After some explanation, Hirsch
continues: "I shall briefly outline the conflicts between educational
Lysenkoism and mainstream science in testing, math, and early education
...," and citing math education experts Anderson, Geary, and Siegler about
what research shows math students need, "They would tell you that only
through intelligently directed and repeated practice, leading to fast,
automatic
recall of math facts, and facility in computation and algebraic manipulation
can one do well at real-world problem solving." (Hirsch, 1997, p. 6) Hirsch
received a standing ovation from the SBE, and indications are that it is
proceeding, based on his recommendations.

In spite of the SBE instructions to base the Framework on research,
the Framework committee never discussed any research articles. In July
1997 the SBE awarded a contract to Prof. Douglas Carnine to provide a
review of high quality mathematics research upon which the Framework's
instructional strategies would be based. In this document, presented to
the SBE in March (see Dixon, (1998)), we find "From a total of 8,727
published studies of mathematics in elementary and secondary schools ...
only 110 passed the multi-level evaluation criteria we developed to
identify high quality studies." All studies are experimental, most
consider interventions over very short time intervals, many deal with
learning disabled students, and many use "instructional booklets" in order
to eliminate teacher-pupil or pupil-pupil interaction (which were
considered "confounding variables".)

The American Educational Research Association's Special Interest
Group for Research in Mathematics Education has written a public letter to
the SBE (signed by 73 researchers) protesting the poor design of the
Carnine report. But in spite of numerous errors (for example incorrect
reporting of grade levels, content, or experimental design), the SBE will
prepare new statements on "math instructional strategies" for inclusion in
the Framework this summer, based upon the Carnine document. Due to the
lack of time, the CC will not be able to participate in this part of the
process. Various observers anticipate an endorsement of direct instruction
followed by repeated practice, opposition to having students try to develop
ideas through problem solving, and quite possibly (based upon Carnine's
work) discontinuation of the use of manipulatives in elementary schools
(since they interfere with automaticity in fast recall).

SOME OBSERVATIONS ABOUT THE TWO SETS OF STANDARDS

In California, as noted above, there were two sets of standards -
those developed by the SC and the revision of that set under the
supervision of the SBE. To observe that these two standards were the focii
of unabated interest, across the land and for many months, is an
understatement! For example, the NCTM devoted the front page of one issue
of its News Bulletin to unflattering comments about the SBE's revised
standards. (NCTM, 1998) Wu (1998) reported that "The reaction to the
revision was swift and violent." (p. 1)

To illustrate the changes between these two documents we consider the
original third grade Number Sense section where one has "[2.2] build up
multiplication table from 0x0 to 10x10 and commit to memory," which has in
its corresponding Clarification and Example column: "Students see that
understanding properties and relationships within the multiplication table
can assist them in memorizing facts (e.g., 4 x 7 = 7 x 4; 7 x 6 is easy if
you know 7 x 3 and know that you can double it to get 7 x 6, etc.)" In
its revision, the SBE replaces these with "[2.2] Memorize to automaticity
the multiplication table for numbers between 1 and 10." In Grade 7, where
preparation for grade 8 algebra has became a central Standards and
Framework objective, we find the SC's Algebra and Functions standard
including "[3.2] generalize numerical and geometric patterns using
algebra, and relate the equation, graph and table of values resulting from
the generalization." The Board revised this standard to read "[3.2] plot
the values from the volumes of a 3-D shape for various values of its edge
lengths." As with the Framework, one finds that discussion of students'
learning of mathematical relationships (like the use of commutativity to
facilitate learning basic facts) was eliminated either because they
"prescribed pedagogy" or "lacked precision."

University of California, Berkeley mathematics Professor H. Wu
prepared a paper based on a lecture he gave in California describing his
assessment of the two standards. (Wu, 1998) He does this from both his
mathematical and educational perspective. Since he is known for his
critiquing of the current reform, his thinking is worthy of examination.
He regards the SC's standards as a thoughtful document, into which a lot of
care was used in setting forth its goals (p. 2). But overall, Wu focuses
on the importance of "getting the mathematics right" in his article. He
felt there are many errors that need to be corrected (for details see pp.
3-10), topics are omitted, and there is an ambiguous mixture of pedagogical
statements with content statements. For example, he cites the exclusion of
the division algorithm in the elementary grades and the Fundamental Theorem
of Algebra in the higher grades.

Wu strongly objected to a grade 4 Geometry Standard which reads:
"Students understand and use the relationship between the concepts of
perimeter and area, and relate these to their respective formulas"
mentioning that the trouble "is that there is no relationship whatever
between perimeter and area, or between volume and surface area, unless it
be the isoperimetric inequality. However, the latter would be quite
inappropriate for students at this level." (p. 4) About his perceived
errors, Wu has strong language: "I very much regret to say that this kind
of mathematics standards would guarantee the deterioration of mathematics
education for a very long time." (p. 4) While this standard may constitute
an error in the eyes of one research mathematician, a fourth grade teacher
explained to us how she interprets it: "We want students to understand at
their level that perimeter 'goes around' and area 'covers,' and then to be
able to explain (for example) in the case of a rectangle why 2xl + 2xw
can be understood as measuring the 'going around' while l x w counts
covering (say by square tiles)." We think a teacher can learn more about
this from the Clarification and Examples in the SC's Standards that were
eliminated by the SBE. So, we find in the standards debate a serious
breakdown of communication between elements of the mathematics community
that values precise abstract constructions upon which they conduct their
work, and members of the K-12 educational community who have learned to
interpret the informal presentations of ideas that children use as they
encounter mathematical concepts. Finally, we note that Wu supports the
position in the SBE's standards that calculators should not be used in
state-wide testing for grades K-6.

WHAT IS YET TO COME

In California, the governor's staff has announced that $250 million
($60 per student) will be spent in the next year on new math textbooks
(Morain, Los Angeles Times, May 13, 1998). The legislature is revising the
process by which curricular materials are adopted (proposed AB 2517), so
that the new criteria to be written by the SBE this summer will take effect
in 90 days. This means that schools may be limited to using materials that
follow the "three-phase" approach outlined as "best" in the Executive
Summary of the Carnine Report: (1) Direct instruction in skills ("other
regulation"), (2) followed by a "help phase", concluding with (3) "self
regulated" drill and practice. In recent (public) discussions with the CC,
SBE members have stated that the "terrible" manipulative-based elementary
math programs must be stopped immediately. Exactly how all this will play
out depends upon fast-track legislation that will be considered during the
coming months.

Nationally, a Standards forum was held on the program of the
Baltimore AMS-MAA mathematics meetings in January, 1998. The panel was
comprised of representatives of six ARGs. [An ARG is an "Association Review
Group" from which NCTM requested assistance and input in revising its
Standards - e.g., the Mathematical Association of America (see for example,
Reports 1,2, and 3 of the MAA Task Force on the NCTM Standards at
http://www.maa.org/ ); the American Mathematical Society; the Association
of Symbolic Logic, etc.]. Many mathematicians shared their views on the
NCTM Standards. In particular, NCTM was credited for tackling important
issues in mathematics education, for addressing the needs of all students
in mathematics and for drawing people inside and outside mathematics to
discuss the issues. (Ross, 1998) At the same time, various concerns were
expressed; for example, that the Standards need to be made less vague and
less subject to misinterpretations, since a multitude of things are being
done in the name of the Standards. Also, it needs to be made clear that
mathematics is not always fun, not always easy, and that it is a myth that
only some people can do mathematics. (Ross, 1998, p. 4) Other commentators
expressed views consonant with Wu's views; for example, that the Standards
need to be made shorter and crisper and more specific, and that more
attention needs to be paid to logic and reasoning in mathematics. The
over-arching theme of problem solving was supported in the discussions.
But the ARGS only provide suggestions for the NCTM to consider. The NCTM
revision of its Standards (called Standards 2000) is scheduled for
completion in 2000.

As U.S. mathematics educators deal with the "backlash," there are
other important issues that are being raised. Among them, the manner in
which precise mathematical language and logical arguments (from informal
reasoning to proof) are developed - in particular, what are our
expectations across grade spans and how do educators help students develop
these understandings - and communicating how "real-world" problems can
help enhance mathematical understanding and eliminating a possible
over-emphasis on them where the distractions of the context obscure the
mathematics. Beyond the curricular issues, there still remain those of
teacher preparation. There is insufficient support for continuing teacher
education and there is a great need to revamp preservice teacher
preparation programs. Using the new curricula requires greater teacher
understanding of both the mathematics and the approaches different students
will take in learning. Finally, there are those who feel a need to further
examine under what circumstances cooperative learning is effective and when
it may not be, and more generally, the issue of how constructivist thinking
is influencing, or should influence, approaches to teaching (cf.,
Kilpatrick, 1997).

U.S. Secretary of Education Richard W. Riley has concerns about the
deep divisiveness of the debate over mathematics education reform. In a
major talk at the Baltimore AMS-MAA meeting, he mentioned in a forceful
manner that "This leads me back to the need to bring an end to the
shortsighted, politicized and harmful bickering over the teaching and
learning of mathematics. I will tell you that if we continue down this
road of infighting, we will only negate the gains we have already made -
and the real losers will be the students of America." (Riley, 1998)
Referring to the California "math wars" he continues, "Let me say right now
that this is a very disturbing trend, and it is very wrong for anyone
addressing education to be attacking another in ways that are neither
constructive nor productive. It is perfectly appropriate to disagree on
teaching methodologies and curriculum content. But what we need is a civil
and constructive discourse." Perhaps we can see that the California "math
wars" have, in the final analysis, served a useful purpose. Overall they
have served as a lesson on how not to behave in the future, in rethinking
and reconstructing school mathematics education for the benefit of our
students.

Footnote [1]: State law requires 30 months between setting criteria and an
adoption. The materials approved in October 1994 by the SBE were aligned
with the 1992 Framework criteria, and state funding for purchase of these
materials was available for the 1995-96 academic year.

References

California Department of Education (1985) Mathematics framework for
public schools, kindergarten through grade twelve, Sacramento, California.
[http://www.cde.ca.gov/index.html]

California Department of Education (1987) Mathematics model curriculum
guide, kindergarten through grade eight, Sacramento, California.

California Department of Education (1992) Mathematics framework for
California public schools, kindergarten through grade twelve, Sacramento,
California.

California Department of Education (1997) Mathematics framework for
California public schools, kindergarten through grade 12 (field review
draft).

California State Board of Education (February 5, 1998) The California
mathematics academic content standards [http://www.cde.ca.gov/board.html].

Curriculum Frameworks and Instructional Resources Office (CFIR) (1997)
California Department of Education 1997, Mathematics Framework Committee
Homework (public documents available at each meeting).

Dixon, R., Carnine, D. Lee, D., and Chard, D. (1998, March) Report to the
California State Board of Education and addendum to principal report -
review of high quality experimental research, mimeographed - 99 pp.

Hirsch, E. D. (1997) "Address to Cal State Board, April 1997,"
mimeographed 9 pages, California State Board of Education (public document).

Jacob, B. (1998) Instructional materials for K-8 mathematics classrooms,
the California adoption, 1997, Issues in Contemporary Mathematics, edited
by E. Gavosto, S. Krantz, and W. McCallum, Cambridge, England: Cambridge
University Press (to appear).

Japan Society of Mathematical Education (1990, January) Mathematics program
in Japan (kindergarten to upper secondary school), Tokyo, Japan.

Kilpartick, J. (1997) Confronting reform. The American Mathematical
Monthly, (104, 10) [December], 955-962.

Morain, D. (1998) Wilson Proposes Major Education Funding Hike, Los
Angeles Times, May 13, 1998, p. 1.

NCTM (1989) Curriculum and evaluation standards for school mathematics,
Reston, VA: The Council. [http://www.enc.org/online/NCTM/280dtoc1.html]

NCTM (1998) New California standards disappoint many. National Council of
Teachers of Mathematics, News Bulletin, (34, 7), 1 and 5.

NCTM (1998) The California mathematics academic content standards for
grades K - 12. Report submitted to the National Council of Teachers of
Mathematics Board of Directors, March 25, 1998, 4 pp.

Nicholas, J. (1997) "To: State Board Members, Subject: Math Adoption:
Dale Seymour Series," California State Board of Education (public
document), September 9, 1997.

Raimi, R. and Braden, L. (1998) State mathematics standards, Fordham
report of the Thomas B. Fordham Foundation, Washington, D.C., (2, 3).
[http://www.edexcellence.net/standards/math.html]

Riley, R. (1998) The state of mathematics education: Building a strong
foundation for the 21st century, speech at Conference of the American
Mathematical Society and the
Mathematical Association of America, Baltimore, Maryland, January 7, 1998.
Text available in Notices of the American Mathematical Society, (45),
487-490 (1998).
[http://www.ams.org/notices/199804/riley.pdf]

Ross, K. (1998) Reality check: At Baltimore standards forum, all quiet
along the "math wars" front, Mathematical Association of American, Focus
(Newsletter), (18, 5) [May/June], 1 and 5. [http://www.maa.org/]

Wu H. (1996) The mathematician and the mathematics education reform.
Notices of the American Mathematical Society, (43, 12), 1531-1537.

Wu H. (1997) The mathematics education reform: Why you should be
concerned and what you can do. Mathematical Association of America, The
Monthly, (104, 10) [December], 946-954.

Wu H. (1998) Some observations on the 1997 battle of the two standards
in the California math war. mimeographed, 20 pp.
_____________

Jerry P. Becker is Professor of Mathematics Education, Department of
Curriculum and Instruction, Southern Illinois University, Carbondale, IL
62901-4610, USA [jbecker@siu.edu]

Bill Jacob is Professor of Mathematics, Department of Mathematics,
University of California, Santa Barbara, CA 93106-0001 USA
[jacob@math.ucsb.edu]
************************************************************

THE SECOND NOTE WILL FOLLOW SHORTLY.

************************************************************
*
*
Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
Carbondale, IL 62901-4610 USA
Fax: (618)453-4244
Phone: (618)453-4241 (office)
E-mail: JBECKER@SIU.EDU





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