e:\doc\web\98\04\fordham.txt
Date sent: Wed, 11 Mar 1998 21:36:53 -0900
From: family
To: Alaska Ed LOOP
Subject: Fordham Foundation - State Mathematics Standards - Alaska "C"
It has been brought to my attention that the Alaska DOE has sent out the
proposed Alaska Math Standards for an independent review by the Council
for Basic Education (CBE).
There is some question about the basis of rigor that the CBE uses in
it's evaluation. I will send that opinion under a separate heading.
Every evaluation has something of value and the DOE should be given
credit for trying to get an outside opinion. The following is an
independent review of State Math Standards recently published by the
Fordham Foundation that includes Alaska.
IMPORTANT: Please PRINT and READ the excerpts that I have compiled. Then
call and order the full report, or download it from their website. We
must be clear and concise on where "Alaska Math Education" is heading.
This should have as much, if not more, importance as the time spent
reviewing the Ed Week Quality Counts report.
Dr William Pfeifer
-----------------------
Fordham Foundation Home Page
http://www.edexcellence.net/
FORDHAM REPORT:
Volume 2, Number 3
March 1998
"State Mathematics Standards"
by Ralph A. Raimi and Lawrence S. Braden
http://www.edexcellence.net/standards/math.html
The Thomas B. Fordham Foundation commissioned evalutations of state
mathematics standards. The evaluations can be found in the .PDF file
available at the address above.
In all, the standards from 46 states plus the District of Columbia and
Japan
were examined. The grades were
A 13-16 3 states (& Japan)
B 10-12.9 9 states
C 7-9.9 7 states (Alaska)
D 4-6.9 12 states (DC)
F 0-3.9 16 states
N not evaluated
This report is particularly notable in that the newly adopted California
Mathematics Standards received the highest, and highest possible,
rating:
"If teachers and textbooks can be found to carry it through properly,
this
Standards outlines a program that is intellectually coherent and as
practical
for non-scientific citizens as for the future engineer. Whatever of
'real
world' application school mathematics can have, is found here, set upon
a
solid basis of necessary understanding and skill. " (p 26, col 1)
Alaska's Math Standards scored at a 7.5, just edging in to get a grade
of "C". Alaska Math Standards received a .7 out of 4 points for CLARITY.
I have cut and pasted Alaska and California information below.
You can contact the foundation directly below to get a copy of their
report, or you can download it from their website. Single copies are
free by calling 1-888-TBF-7474.
The Thomas B. Fordham Foundation
1015 18th Street, NW
Suite 300
Washington, D.C. 20036
(202) 223-5452
(202) 223-9226 (fax)
(888) TBF-7474 (publications line)
http://www.edexcellence.net
-------------------------------------------------------
ALASKA page 23
The Framework (1) is extremely vague,
offering examples of lack of clarity, and
also of inflation: “By strategically applying
different types of logics, students will learn
to recognize which type of logic is being
used in different situations and respond
accordingly.” (“Different types of logic” is,
we believe, a reference to induction and
deduction, about which much is made
these days, without much effect. And
“respond accordingly” couldn’t be vaguer.)
On page 4-12, under “benchmarks” for
Math Content Standard A (Content of
Math), at the 16-18 year-old level, the fol-lowing
appears: “A student would be able
to . . . explore linear equations, nonlinear
equations, inequalities, absolute values,
vectors and matrices.” “Exploration” is
not a standard, and “absolute values” is
curiously misplaced in this list. Clarity has
suffered here, and more than clarity.
At the grade 3-5 level, under “Problem
Solving,” readers are told, “Evaluate the
role of various criteria in determining the
optimal solution to a problem.” This is not
only unclear, but is Inflation. For high
school: “Recognize how mathematics
changes in response to changing societal
needs.” (We believe mathematics is eternal
and unchanging. There certainly are many
things that vary in response to social pres-sures,
but the document puts it badly.) The
Standards (2) partly makes up for the defi-ciencies
in the Framework (1), especially in
its avoidance of inflated language, but it
outlines a program lacking in sufficient
content, especially at the secondary level.
S T A T E R E P O R T C A R D
Alaska
I. CLARITY 0.7
II. CONTENT 2.3
III. REASON 2.0
IV. NEGATIVE QUALITIES 2.5
TOTAL SCORE (out of 16) 7.5
GRADE C
--------------------------------------------------
CALIFORNIA page 25
The Standards is a scant 37 pages in
length, and is classified by grade level from
K-7 and then by subject headings (the sub-jects
are strangely called “disciplines”),
from Algebra I (in Grade 8) through cours-es
which prepare the student for AP
Calculus and AP Statistics. The writing is
always terse and to the point.
At the start of each grade (K-7), the
expectations for that grade are summarized
in a hundred words or so, permitting a
rapid and accurate overview of the whole.
The details follow by rubric, the same for
all grade levels:
i Number Sense
ii Algebra and Functions
iii Measurement and Geometry
iv Statistics, Data Analysis, and
Probability
v Mathematical Reasoning
Although naming a rubric “Algebra and
Functions” is stretching things at the lower
levels, there is in general no undue multi-plicity
of rubrics, e.g. a strand labeled
“Calculus” which at least one other state
mysteriously included in its framework
right down to the first grade.
“Mathematical
reasoning” is the
only one here whose
presence might be
questioned, for its
demands are of a
rather general
nature, and some
could be considered
“inflation” by the
authors of this report
if they were not
thoroughly exempli-fied
in the content
standards of the
other rubrics.
Each rubric is headed by one or more
general admonitions which would also
tend to be labeled not “Definite” or
“Testable” were it not that their subhead-ings
explain exactly what is meant. Under
“Algebra and Functions,” grade 3, for
example, students are to “...represent sim-ple
functional relationships” (which is
vague to say the least), but then the
provincial teacher imagined in the Criteria
is immediately told what that means in
terms of content, e.g. “solve simple
problems involving a functional relation-ship
between two quantities (e.g., find
the total cost of multiple items given
the per unit cost).”
Again, under “Mathematical
Reasoning,” grade 3, we find that “Students
use strategies, skills and concepts in finding
solutions.” So often admonitions of this
sort are simply left hanging as empty
exhortations, but here follow six specifica-tions,
e.g., “express the solution clearly and
logically using appropriate mathematical
notation and terms and clear language, and
support solutions with evidence, in both
verbal and symbolic work.” A tall order,
perhaps, but conveying (in passing) anoth-er
important point: In speaking of “the”
solution, the phrasing insists that mathe-matical
problems have a single solution.
Here the “reform” philosophy of what its
opponents sometimes have called “fuzzy
math” is firmly rejected.
In grade 4, instead of reading
“Investigate the relation between the area
and the perimeter of a rectangle” (a popu-lar,
though confusing, entry in many state
standards, and in any case “investigate” is
not a content standard), we find (1.2) “rec-ognize
that the rectangles having the same
area can have different perimeters,” and
(1.3) “understand that the same number
can be the perimeter of different rectan-gles,
each having a
different area.” It is also
refreshing to observe, in
grade 4 “Number Sense”
and in the Glossary, that
mathematical educators
in California know what
prime numbers are and
tell us carefully. (cf.
Pennsylvania, below.)
Scientific calculators
are mandated for the first
time in grade 6, but only
for good reason, and after
the essential properties
of the real number sys-tem
have been assimilated through hand
calculations. (Japan introduces these elec-tronic
aids in grade 5, while most
American states demand their use from the
beginning.) By the end of grade 5,
California students are to be able to do
long division with multiple-digit divisors
and to represent negative integers, deci-mals,
fractions, and mixed numbers on a
number line. By the end of grade 7 they
graph functions, use the Pythagorean
theorem, evaluate algebraic expressions,
organize statistical data, and in general are
prepared for high school algebra and
geometry.
The years 8-12 are described by subject:
Algebra I and II, Geometry, Probability
and Statistics, Trigonometry, Linear
Algebra, Mathematical Analysis, and as
the “Advanced Placement” subjects of
statistics and calculus. The Standards is not
prescriptive in its pedagogy. How the
teacher, or the textbook, goes about the
job is left to the discretion of the teacher
or school district. A table is provided sug-gesting
placement of this material by year,
so that integrated curricula are possible by
the same standards as the subjects would
demand when taken in the form of courses.
It is clear that a course-by-course program
would place Algebra I in the 8th grade,
Geometry in the 9th, and Algebra II in the
10th, completing the state-mandated cur-riculum
for graduation.
Algebra I names 25 items, some neces-sarily
mechanical and some with welcome
attention to logical structure, including
knowing the quadratic formula and its proof.
“Practical” rate problems, work problems,
and percent mixture problems are to be
studied as well as the (impractical?)
Galilean formulas for the motion of a parti-cle
under the force of gravity.
In Geometry, students must be able to
prove the Pythagorean theorem and much
else, including proofs by contradiction, and
classical Euclidean theorems on circles,
chords, and inscribed angles. “Geometry”
also includes the basic trigonometry of
solving triangles, and the properties of rigid
motions in the plane and space.
Mathematical Induction is introduced
in Algebra II, which material is apparently
intended for the 10th grade or earlier.
Experience will have to show if this and
certain other ambitious demands are really
possible, or should rather be left for the fol-lowing
years and a volunteer audience.
S T A T E R E P O R T C A R D
California
I. CLARITY 4.0
II. CONTENT 4.0
III. REASON 4.0
IV. NEGATIVE QUALITIES 4.0
TOTAL SCORE (out of 16) 16
GRADE A
Here the “reform”
philosophy of what its
opponents sometimes
have called “fuzzy math”
is firmly rejected.
----------------------------------
CALIFORNIA (cont) page 36
Other topics are either preparation for col-lege
work or are (these days) college-level
work themselves (de Moivre’s Theorem,
the Binomial Theorem, Conic Sections
with foci and eccentricities, etc.). We are
not told how much of this program must be
taken by all California students, or whether
the courses named Trigonometry and
Statistics are intended as options.
There is no mention of “shop math,”
“finite math,” or “business math” (etc.),
directed at the non-college bound student,
or remedial courses for older students who
have failed earlier. Perhaps the
Mathematics Framework (see below),
required by California law to be revised for
1999, will address these issues. The authors
of this report did not downgrade California
(or any other States) for this sort of omis-sion,
when the core curriculum is adequate
and well stated. We do, however, expect
that California will in due course find it
advisable to add something appropriate to
its “elective” curriculum at the high school
level, as does, for example, Tennessee,
which offers a branching of alternate tracks.
If teachers and textbooks can be found
to carry it through properly, this Standards
outlines a program that is intellectually
coherent and as practical for the non-sci-entific
citizen as for the future engineer.
Whatever of “real-world” application
school mathematics can have, is found
here, set upon a solid basis of necessary
understanding and skill. Initial reaction to
the adoption of this document included a
widespread apprehension that this “return
to basics” represented an anti-intellectual
stance: rote memorization of pointless rou-tines
instead of true understanding of the
concepts of mathematics. The opposite is
true. One can no more use mathematical
“concepts” without a grounding in fact and
experience, and indeed memorization and
drill, than one can play a Beethoven sonata
without exercise in scales and arpeggios.
There is always a danger that intellectu-ally
challenging material, be it in music,
literature, or mathematics, will in the
hands of ignorant teachers, or bowdlerized
textbooks, become reduced to pointless
drills. The history of American school
mathematics in the 20th century has large-ly
been a chronicle of conceding defeat in
advance, teaching too little on the grounds
that trying for more will fail. California is
to be commended for taking up the chal-lenge
head-on, and announcing its
intention in the clearest terms in its
Content Standards.
It is the more curious, then, that the
adoption of this Standards has been attend-
ed by an extraordinarily
bitter public debate
centering on their char-acterization
as a
reactionary document
discouraging, rather
than demanding, the
“real understanding” of
mathematics. An earli-er
version of this
Standards had been
composed by a special
Commission on
Standards which
worked through most of
1997 on standards for
English and mathemat-ics,
to be approved by
the State Board of
Education. The
appointment of the
Commission was itself
extraordinary, and the consequence of pub-lic
dissatisfaction with current teaching of
core academic subjects.
California by law publishes a
Framework for mathematics instruction
every seven years. Past Frameworks includ-ed
standards of the sort under review here,
along with pedagogical and administrative
information, and the most recent was pub-lished
in 1992 in the midst of enthusiasm,
on the part of the school administrators, for
the point of view represented nationally by
the NCTM Standards of 1989, and often
called “reform.” (The “reform” trend had
been visible in California even earlier.)
Two foci of opposition soon appeared:
“HOLD” in Palo Alto and
“Mathematically Correct” in San Diego,
both citizens groups (including mathemati-cians
and engineers) publishing web pages
designed to persuade readers that the
“reform” represented by the 1992
Framework and its progeny, should be dis-carded
in favor of something usually
(though simplistically) called “traditional.”
In particular, they pointed to what they
said were deteriorating scores of California
children on national tests.
In a word, the anti-reform camp
claimed “Johnny can’t add,” but instead
spends his school time measuring play-grounds
and talking it over with his
classmates; and he uses a calculator when
asked to multiply 17 by 10. Thus the new,
ad hoc Commission on Standards was
appointed, quite apart from the legally
mandated Framework committee, to—in
effect—adjudicate this controversy. The
Commission, a citizens’ commission not
intended to be expert in mathematics (new
standards for other
core subjects were
also part of its
charge), took advice
from experts of its
own choosing and
ended sharply divid-ed.
It voted by a large
majority in favor of a
document that it sub-mitted
to the State
Board of Education
on October 1.
The Board, which
has final say, heard
much public testimo-ny,
including
opposition expressed
by mathematicians,
and rejected the draft.
The Board used that
draft, however, as a
beginning for the very substantial revision
it ultimately approved in December of
1997. That revision, which is the Standards
reviewed here, was mainly prepared by a
group of mathematicians at Stanford
University, and its publication has generat-ed
more public controversy than anything
seen earlier. Apart from segments of the
public, two groups of professionals are now
in contention: the mathematics education
community, or a vocal part of it, against
the mathematicians’ community, or a vocal
part of that.
(In the meantime, the Board has the
task of reconciling the mathematics stan-dards
implicit in the new Framework,
expected to be published in 1998, with the
revised Standards under review here.)
Newspaper reports of the controversy
make it apparent that those opposed to the
Board’s revisions, and who wish the Board
to return to the document submitted to
them by the Commission in October,
include the California Superintendent of
Public Instruction and at least one high-ranking
official of the National Science
Foundation. This party portrays this revised
Standards as a return to the failures of past
years, a document devoted to the mindless,
pointless manipulation of outdated algo-rithms.
The authors of this report believe
such a characterization is mistaken, and
that the mathematicians who participated
in the final revision had no such intention,
and their product no such result—except as
poor teaching might make it so. It is to the
better mathematical education of teachers
that California (and the rest of us) must
look for improvement of result.
One can no more use
mathematical “concepts”
without a grounding in
fact and experience, and
indeed memorization and
drill, than one can play
a Beethoven sonata
without exercise in scales
and arpeggios.