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Topic: [ME] More "Math Wars" in Phi Delta Kappan
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Jerry P. Becker

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Registered: 12/3/04
[ME] More "Math Wars" in Phi Delta Kappan
Posted: Oct 5, 2000 12:52 PM
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BILL JACOB and JERRY BECKER had an article on the politics of
California school mathematics in the March, 2000, Phi Delta Kappan:
Becker, J.P. and Jacob, B. (2000 / March) The politics of California
school mathematics: The anti-reform of 1997-99, Phi Delta Kappan,
Vol. 81, No. 7, pp. 529-537. The article can be viewed at . Here is an
ABSTRACT -- The authors tell the story of a powerful group of
parents and mathematicians in California who manipulated information
and played off of the public's perception of our "failing schools" to
acquire political clout. Through this telling, they hope that other
states will be able to adopt a more rational course as they
reconsider their policies.
A response to the article, written by Professors Deborah Tepper Haimo
and R. James Milgram (mathematicians) appears in the October, 2000
Phi Delta Kappan, along with our reply to them. Both their response
and our reply are given below with the complete citations -- JPB and


CITATION: Haimo, D.T. and Milgram, R.J. (2000 / October).
Professional Mathematicians Comment on School Mathematics in
California, Phi Delta Kappan, Volume 82, Number 2, pp. 145-46.

Professional Mathematicians Comment on School Mathematics in California

At Odds/Mathematics Standards

The authors respond to the March article on California school
mathematics, by Jerry Becker and Bill Jacob - pointing out only a few
of their most serious concerns.

By Deborah Tepper Haimo and R. James Milgram

WE WISH to respond to the article by Jerry Becker and Bill Jacob on
the recent changes in school mathematics in California ("The Politics
of California School Mathematics: The Anti-Reform of 1997-99," March
2000). Although we would have no trouble providing a more complete
treatment and going through the article line by line pointing out
difficulties, we will confine our comments to a few of our most
serious concerns.

Right at the start is a most unfortunate assertion that cannot be
ignored. The subhead of the article, chosen by the editors from the
authors' second paragraph, reads: "The authors tell the story of a
powerful group of parents in California who manipulated information
and played off of the public's perception of our 'failing schools' to
acquire political clout."

There is no evidence whatsoever for this statement. Do the authors
really want to claim that professional mathematicians in California
(Jacob included) who have devoted a substantial portion of their time
to K-12 mathematics education over the past five or more years have
done so because they wanted "to acquire political clout"? All our
observations indicate that these mathematicians have engaged in this
activity because of their strong commitment to excellence in
mathematics education for California's students. They are deeply
concerned about the sharp drop in the performance in mathematics of
California students compared to students in almost any other state
and, above all, compared to students in other countries.

However, we have even more serious difficulties with the body of the
article. We find, for example, that the point of view presented is
excessively biased in a direction that tends to widen the gap between
the vast majority of professional mathematicians and K-12 teachers.

In the current reality, upwards of 75% of high school graduates in
this country attempt higher education. Thus, regarding matters of
content, K-12 teachers must seriously heed the concerns of
professional mathematicians who teach college mathematics courses,
just as elementary school teachers must take account of the concerns
of high school teachers. It is no longer acceptable to argue, as
Becker and Jacob do, that the reason for dismissing the concerns of
professional mathematicians is that, to the authors' knowledge, "none
of these mathematicians [who were involved in the development of
California's new school mathematics policies] ever taught in K-12

The authors' persistence in characterizing the current California
mathematics standards as a return to the past or a curriculum of the
past is strange. Why not call the curriculum internationally
benchmarked mathematics? Or, as we prefer to call it, real
mathematics? It is amusing that they refer to the mathematics of the
previous California standards - standards that have been around for
over a decade - as "reform math."

We conclude with two additional items that must be mentioned. Becker
and Jacob constantly pretend that suggestions given to teachers in
the California framework are meant instead for students. They do this
even though the foreword to the framework clearly states that it
"provides instructional guidance to teachers to enable them to raise
their benchmarks for achievement and mastery in realistic ways."

As an example of this type of distortion, Becker and Jacob consider
one of the key examples in the framework.1 They criticize the
discussion as set too high when the authors of the framework attempt
to inform teachers of the simple way in which the problem can be
transformed to be mathematical. Instead of clarifying the
difficulties, Becker and Jacob seem to believe that these authors
merely confuse. Their own solution is to evade mathematics being
introduced. The example in question is the following: "The students
are given a picture that shows in succession a rectangle, triangle,
square, rectangle, triangle, square, blank, triangle, square. The
students are asked to fill in the blank."

This problem is not well posed, as is pointed out by the authors of
the framework. At this point, however, there is little explanation
about the mathematical way of stating similar examples correctly.
Becker and Jacob pounce on this, stating that "we must remember that
this is a discussion about teaching mathematics in kindergarten!"
(Assuredly, everyone is keenly aware of this.) Furthermore, they
assert that these authors wish to introduce formal mathematical
language and rules for reasoning in kindergarten.

In fact, Becker and Jacob have taken this completely out of context.
In the previous paragraph, the problem that was discussed in detail
was the following: "A picture of three objects, a basketball, a bus,
and a tennis ball, is shown to the students, and they are asked to
tell which one does not belong." This is similarly not well posed.
Here, however, there is substantial discussion of ways of stating the
problem correctly and at the appropriate level. For example, with
this picture in hand, students might be asked the following: "We want
to collect balls. Which of these objects should we select?" A number
of other examples of correct and grade-appropriate variations are
suggested before the problem criticized by Becker and Jacob is
introduced in the next paragraph.

There is some question as to why the authors of the framework did not
show, in equal detail, how to replace the second problem by one that
was both mathematically correct and grade appropriate. The reason, of
course, was that it was assumed that the audience of teachers would
be able to do so for themselves, using the example of the first
problem as a model. Was this unreasonable?

The message conveyed by the framework is precisely that elementary
school teachers must become aware of the fact that there are ways of
presenting mathematics both accurately and at the appropriate grade
level. We find it ironic that Becker and Jacob attempt to show the
complete opposite.

In exactly the same way and with the intent of again seeking to
reverse the actual content, Becker and Jacob criticize the
introduction to the discussion of the grades 8-12 standards,2 saying,
"Students are expected to provide a 16-step, two-column proof of such
algebra facts as. . . ." First, it is clearly and repeatedly stated
in the surrounding discussion in the framework that this section is
for teachers. Furthermore, the objective is to make sure that
teachers are aware that problem solving and proof are essentially the
same thing. To emphasize this point even further, immediately after
the 16-step proof that Becker and Jacob object to, we find these
clarifying remarks:

In practice it would be impractical to demand such detail each time a
linear equation is solved. Nevertheless, 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.

This is one of the points where the California framework most
explicitly rejects the approach of the "new math" of the 1960s. Thus
we find it interesting that Becker and Jacob - by omitting the
surrounding text - attempt to portray this as a "new math"
prescription of deadly logical formalism for children.

In another vein, Becker and Jacob are critical of the use of the term
"hidden agenda," which appears on page 110 of the California
framework (body of discussion and sidebar). Perhaps this phrase might
better have been replaced by "mysterious element" or, as Becker and
Jacob suggest, a more formal term, such as "missing assumptions."

Jacob took full advantage of the formal procedures for public input
on the final draft, giving substantial public testimony. Also, he
never commented on this phrase at the time nor during the long period
afterward when public input as well as corrections were sought.
Apparently, he chose instead to save that particular objection for a
more favorable occasion. The phrase could have been changed easily in
proof, and it would have been, if it had been pointed out. That Jacob
ignored this phrase until he could use it later for purposes of
attack appears to have been more than an oversight.

We urge interested readers to check everything out for themselves as
the published version of the California framework is available on the
Web at

1. Mathematics Framework for California Public Schools, Kindergarten
Through Grade 12 (Sacramento: California Department of Education,
1999), pp. 109-11.
2. Ibid., p. 155.



CITATION: Becker, J.P. and Jacob, B. (2000 / October). Look at the
Details: A Reply to Deborah Haimo and James Milgram, Phi Delta
Kappan, Volume 82, Number 2, pp. 147-48.

Look at the Details: A Reply to Deborah Haimo And James Milgram

At Odds/Mathematics Standards

We need to be vigilant and careful and not be fooled by the seemingly
impressive credentials and writing of critics, Mr. Becker and Mr.
Jacob respond. Content knowledge is no substitute for knowledge of
how students' understanding develops and can be nurtured.

By Jerry P. Becker and Bill Jacob

DEBORAH Haimo and James Milgram question our use of the phrase
"political clout" and assert there is no evidence to support it. We
ask Kappan readers to consider the following events and judge for

In 1997, four mathematicians (Milgram included) substantially revised
the draft California mathematics standards. Their changes were
accepted by the state board of education without seeking public input
or the involvement of K-12 teachers. (The fact that four people,
acting as advisors to the board, met in private to discuss the
revision of a public document appears to violate the state's public
meeting act.) In 1998, three mathematicians wrote sample problems for
the state mathematics framework, and two mathematicians and a
cognitive psychologist wrote significant portions of it. None of them
discussed this work in public as part of an open process. Some of the
work was presented to the Curriculum Commission, which had little or
no time to work with it.

But most of the sections cited in our March 2000 article were not
contained in the last draft made available for public comment (on 8
October 1998); they were inserted in November just prior to the state
board's vote. During 1999, the state board adopted a new policy
requiring panelists to have a Ph.D. in mathematics as a prerequisite
to serving on a Content Review Panel (CRP) for the California K-8
mathematics adoption - a doctorate in education was not allowed.
Although members of the Instructional Materials Advisory Panel
included teachers and did review materials, in the end it was the CRP
members who determined the state board's decisions. And in the case
of Everyday Mathematics, a CRP report was rewritten by a
mathematician two months after the panels had disbanded. Also during
1999, two mathematics professors reviewed and rewrote AB 1331
professional development materials, and the state board accepted
their judgments without allowing any further review (and subsequently
appropriated $43 million for AB 1331 programs during the 2000-01
school year).

So in California, the mathematics standards, framework, instructional
materials, and professional development have all been very tightly
controlled by a small group of university mathematicians. High-stakes
tests and accountability measures that have profound effects on the
lives of teachers and students are linked to all of these policies.
Yet the voices of the teachers who know their students best have been
omitted from the process. In our view, being allowed to circumvent
the public process and get a single vision of policy uniformly
imposed on a large state like California is ample evidence of
"political clout."

The other main criticism leveled by Haimo and Milgram is that we
distort the California framework discussions by confusing comments
intended for teachers with expectations for students. The discussions
in question all come from "Grade Level Considerations," chapter 3 of
the Mathematics Framework for California Public Schools, Kindergarten
Through Grade 12, not from the "Instructional Strategies" or
"Professional Development" chapters, where such information for
teachers might have been appropriate. The focus of chapter 3 is on
how mathematics is to be presented to students.

In fact, the question of whether the passages are written "for
teachers" or "for students" completely misses the point. Of course
teachers will read the framework, not students. The point of our
examples was to demonstrate that the formal thinking of
mathematicians about mathematical content is now driving California
policy with respect to how children are first to encounter
mathematical ideas. This is true whether discussions focus on how
teachers might view the mathematics being presented or on how
students should receive the mathematics. Readers who review our March
2000 article after reading the commentary by Haimo and Milgram will
see this point clearly. The issues of well-posed problems or formal
proof are concerns of mathematicians. They are certainly relevant,
but there are many other issues of far greater educational
significance. We do not agree with the statement of Haimo and Milgram
that "problem solving and proof are essentially the same thing."
Proofs are of enormous importance in higher mathematics, but they
come at the end of a long developmental process, and they are not the
same as children's problem solving. Yet such mathematical formalism
drives the approaches in the California framework, and, in our view,
this is not appropriate.

Haimo and Milgram comment that "Jacob took full advantage of the
formal procedures for public input on the final draft, giving
substantial public testimony." The fact is that, although Jacob did
write some letters to the Curriculum Commission during 1998, he never
once gave public testimony that year (largely because of travel time
and expense). So their comment is inaccurate. Regarding the phrase
"hidden agenda" and why Jacob "never commented on this phrase at the
time," there is a simple response. The entire Preface to Kindergarten
Through Grade 7, in which it appears, was not included in the final
draft of the framework that was available for public comment on 8
October 1998. So it would have been impossible for any person outside
of the state board's select authoring group to read this section
prior to state board approval of the framework. The fact that this
particular phase was singled out by the authors of the framework
prior to publication to be highlighted as the sidebar on page110
suggests that they felt strongly about its message. Now that Haimo
and Milgram feel uncomfortable about it, they wish to place the blame
on somebody who was not part of the authoring or proofreading team.

After the last paragraph, many readers may be thinking that
Californians must be crazy to be arguing over such points. If so, we
agree. Such wording will probably have little or no effect on
classroom practice. But the existence of these arguments and the
tenor of the discussions about them are revealing and point to real
problems in California. The state has failed to forge a true
collaboration among parents, teachers, mathematics educators,
mathematicians, and others, and instead a small collection of
professors of mathematics have autocratically set state policy
relating to mathematics curriculum and instruction. The widening
"gap" between professional mathematicians and K-12 teachers, noted by
Haimo and Milgram, began in California when the state's mathematics
standards were forced on K-12 teachers - many of whom do not
subscribe to them. The continuing practice of the state board in
relying almost exclusively on the same small groups of content
experts only widens the gap.

On the other hand, we never dismissed the interest and concerns of
either K-12 or higher education. Indeed, as professors of mathematics
and mathematics education, we both work closely with teachers and
bring our concerns to them. In doing so we have developed the highest
respect for the understandings of our K-12 colleagues and believe
that those who have devoted their professional careers to working
with our children should play a leading role in setting policy.
Understanding mathematical content as an adult is not the same as
understanding how children's thinking develops. In our article we
showed how California's politicians and some professors have failed
to understand this. In our view, the situation is a travesty. It is
time to set a national agenda - paying highest respect to the
professionalism and knowledge of our nation's dedicated K-12 teachers.

Finally, as we mentioned in our article, the National Council of
Teachers of Mathematics (NCTM) has now released the final version of
its Standards 2000 document, Principles and Standards for School
Mathematics. It outlines a balanced view of teaching for
understanding and also addresses the issues of both skills and
problem solving. We hope that in response to the Principles and
Standards we do not again hear about lack of mathematical precision,
lack of skills (with emphasis on "standard algorithms"), mathematical
errors, inappropriate calculator use, low standards, and the
"research" that supports the critics' views.

As we mentioned in our article, we need to be vigilant and careful and not be fooled by the seemingly impressive credentials and writing of critics. We need to look carefully at the details. While we recognize that there is always room for improvement in any endeavor, these critics - instead of joining forces with teachers and contributing to the process - may well keep up their efforts to interfere with those who seek to reach more students in a constructive manner. They are likely to continue to work to secure their vision of 13 years of precalculus symbol manipulation. Content knowledge is no substitute for knowledge of how students' understanding develops and can be nurtured, but this point seems lost on these critics. Again, we ask readers to examine the NCTM document. We are convinced that they will see the same merit in it as we do.
Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
Carbondale, IL 62901-4610 USA
Phone: (618) 453-4241 [O]
(618) 457-8903 [H]
Fax: (618) 453-4244

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