Interview
with Mathematics Professor James Milgram
Dr.
R. James Milgram is a professor of mathematics
at Stanford University. He is on the board
of directors for the National Board for Education
Sciences, the NASA Advisory Council, and
is a member of the Achieve Mathematics Advisory
Panel. He is currently one of the directors
of the new National Comprehensive Center
for Instruction.
From
2002 to 2005, Professor Milgram headed a
project funded by the U.S. Department of
Education that identified and described the
key mathematics that K-8 teachers need to
know. He also helped to direct a project
partially funded by the Thomas B. Fordham
Foundation that evaluated state mathematics
assessments.
He
is one of the four main authors of the California
Mathematics Standards, as well as one of
the two main authors of the California Mathematics
Framework. He is also one of the main authors
of the new Michigan and new Georgia mathematics
standards. Among other honors, he has held
the Gauss Professorship at the University
of Goettingen and the Regent’s Professorship
at the University of New Mexico, and has
published over 100 research papers in mathematics
and four books, as well as serving as an
editor of many others.
He
currently works on questions in robotics
and protein folding. He received his undergraduate
and master’s
degrees in mathematics from the University
of Chicago, and a Ph.D. in mathematics from
the University of Minnesota.
Why
has the California Math Framework been so controversial?
There is a 7 year cycle on standards and frameworks
in California. The 1992 framework and standards
were so controversial and the results were
so poor that enormous pressure had been put
on the State to speed up that cycle.
In 1996 enabling legislation was passed
to start this process. Then the Department
of Education setup a standards commission in
collaboration with the legislature and the
state board of education. A framework commission
was also setup. The two commissions worked
relatively independently of each other.
About five of us at Stanford’s Department
of Mathematics were very much involved. We
were watching the process and tentatively trying
in indirect ways to get advice to people. Ralph
Cohen was appointed to the framework committee,
but none of us were involved in the Standards
committee.
After the Standards Committee had produced
a set of standards, they were given to the
State Board of Education for final approval.
We had been talking to and advising a number
of members of the State Board, so they brought
the standards to us to look at and we were
just appalled.
It was counted up later that there were over
100 major mathematical errors in that document
- complete misunderstandings of what was going
on.
As a result, a hearing was held by the State
Board. Ralph Cohen, Gunnar Carlsson, mathematicians
at Stanford, Dick Askey from the University
of Wisconsin at Madison, and I testified. After
that the Board asked us to revise the Standards,
acting as consultants for the Board.
What were some of the weaknesses of the previous
California Math Frameworks?
I should start out by saying that mathematics
isn’t a continuum. There’s mathematics
and there’s non-mathematics and there’s
absolutely nothing in between. I think we were
all implicitly making the assumption that what
was being taught in the schools was actual
mathematics.
I came to learn and I think we all came to
understand later that much of what was being
taught in the schools was not mathematics.
And then the question becomes how does that
happen? The answer is that it happens when
people who don’t know the subject but
have other missions and motivations take it
over.
What do you mean by missions and motivations?
I was in school in the late 1940s through
the mid-1960s. At that time mathematics was
only thought to be important in specialized
areas. It wasn’t a major focus. What
was a major focus in instruction was that all
kids were able to learn to a very solid level
how to work with numbers – the basic
operations of arithmetic.
The reason was that in commerce and day to
day living they needed that kind of skill continuously.
Since they didn’t have readily available
technology at that time, they had to actually
be able to do calculations by hand and do them
fairly accurately. That was what was taught,
but mathematics itself was not regarded as
an essential subject. Only for the very small
minority of people doing things like the hard
sciences or engineering.
Over time mathematics became ever more central
to what people had to be doing in their day
to day lives and in the workforce. Not just
arithmetic, but actual mathematics. These days
if you look at a production line, it’s
almost inevitably a robotic production line.
The skills involved in working with a robotic
production line are far different than they
are if you are manually carrying an item from
one place to another.
Such a line is characterized by short production
runs and constantly reconfiguring and changing
the line. That means reprogramming robotic
mechanisms. In order to do that you have to
have a really solid background in algebra,
because that algebraic type of thinking is
what’s involved in programming.
That’s just one example. If you look
at the information technology that really has
taken over our age you should realize that
all of this is just a physical realization
of mathematics. It’s simply become absolutely
essential that students have qualifications
in mathematics to get anywhere in the workforce.
Just to survive at more than a subsistence
level.
But this means that mathematics has become
a social “good.” As such,
there are people who believe that what is important
is simply a piece of paper that says a student
has successfully sat through a math class. They
do not necessarily think that there is actually
any content that needs to be learned.
You mentioned calculators. Why is the mathematics
community so negative about calculator usage
in the early grades?
The answer is that first of all, we’re
not negative about calculator usage per se.
In fact, we don’t have any problem with
using technology to do calculations. Where
we have a problem is more basic. Mathematicians
universally agree that mastery of early arithmetic
is the essential foundation for developing
mathematical competency. It is certainly true
that mathematics at the highest levels does
not involve arithmetic, but arithmetic is the
foundation for what we do.
Without that mastery, which can only be developed
through thoroughly understanding what is going
on, you will never get anywhere. So what we
try to say and what we strongly believe is
that in the early grades calculator usage interferes
with the development of that mastery.
We are
not against calculators, but we are strongly
against them in the early grades and more than
that, our reaction is, if somebody is a teacher
or somebody is standing out there telling me
that students can use these calculators to
do their calculations and substitute that for
the development of numerical skills, then my
reaction is very simple, these people literally
have no idea what they are talking about.
Some states use tests that are not
aligned with the California Framework.
Schools or districts that want to use textbooks
that are aligned with the Framework still
need to be held accountable to state tests.
What would you recommend for these schools?
You’re kind of stuck between a rock
and a hard place, aren’t you? I evaluated
one of the Maryland state exams and found that
it was one of the exams with the usual 20-25%
incorrect problems. So I don’t know what
you’re testing in Maryland; it’s
certainly not mathematics.
Why do the philosophies of mathematicians
and educators seem to vary so widely?
The people holding the power in the education
community today hold the belief that the major
function of the public schools is to keep children
out of the workforce.
The recollection is the horrors of child labor
from the 19th century. The objective was to
keep them out of the workforce as children,
but that was it. They also believe that kids
should have a good time in school because implicit
in their belief is the conviction that kids
will not have a good time as adults.
That’s shocking to me. I’ve read
about the Math Wars, but I’ve never hear
that viewpoint expressed.
The debate in the Math Wars was between math
educators and mathematicians. Somehow the people
in the education schools proper stayed out
of it. But when you come right down to it,
you have to deal with the people in the education
schools.
Ultimately and what really was remarkable
to me when I got to know a number of these
math educators is they were consistently telling
me of their feelings of powerlessness. We were
assuming they were the ones that are responsible.
They don’t necessarily agree with us
100 percent, but they agree with us a lot more
than you would expect.
This type of agreement reminds me of the paper
you co-authored, “Reaching for Common
Ground in K-12 Mathematics Education.” Did
the “Common Ground” discussions
bring the mathematics and math education communities
closer together?
Yes and no. There is common ground with most
of the serious math educators in the country.
The math disagreements with them are much smaller
than you would expect. But the direct outcomes
that were hoped for by math educators were
not produced. They thought that when mathematicians
and math educators finally got to talk with
each other that the mathematicians could help
the math educators.
That has not materialized because it turns
out that we have a common enemy: the educations
schools proper. One might say they have their
hearts in the right place and there are many
competent people there, but the overall philosophy
of the education schools is incompatible with
what either the math educators or the mathematicians
want.
Is this due to the philosophy that school’s
main purpose is to keep children out of the
workforce and keep them entertained and happy?
Yes, I believe that.
Despite the resistance of the education schools,
has the increased communication between the
communities had any concrete results?
Things are very much in a state of flux. What
we need is a more educated workforce. We have
to have that in order to remain competitive.
And the real question is how do we develop
an educated workforce given the realities of
the world as we see it now.
Common Ground has really facilitated communication
between the math education community and the
mathematicians. We are talking to each
other now in a way we haven’t done for
probably 50 to 100 years.
For example,
next month National Council of
Teachers of Mathematics (NCTM) will roll out
something they call the Focus Topic, direct
recommendations from NCTM of what instruction
should look like in grades pre-K through
8. It looks nothing like what you might expect.
Typically if you look at state standards in
third and fourth grade you’ll see 65
or even 85 separate topics listed in each grade.
NCTM says that 60-80% of instruction time in
each grade in mathematics should be devoted
to three topics, which will differ from grade
to grade.
This is almost exactly what the math community
has been saying for years and years. Indeed,
the very able mathematician Sybilla Beckman
at the University of Georgia was a key author
of the Focus Topic. Roger Howe at Yale
and I were outside evaluators.
Previous to Common Ground, the NCTM had not
dealt with mathematicians at our level and
not dealt with them as the outside evaluators
for any project in a very long time. So I would
say that Common Ground has had a huge effect
within this community.
The
Math Wars
Written by Phil Folkemer, BCP SES Program
Director
The Cold
War, more than any other conflict before
it, was at heart a war of minds and ideas. When the Soviets launched the
world’s first man-made satellite into
space in 1957, policy makers and citizens alike
began to fear that the U.S. was falling behind
the Soviet Union in science and technological
advances. The U.S. was terrified of the
new technological threat and decided that,
in order to close the supposed Soviet scientific
advantage, our early education system needed
a drastic overhaul. Enter: New Math.
Created in the early 1960s, New Math was an
attempt to keep up with the increasing educational
demands of a changing world. Emphasizing
mathematical structures over specific skills,
it did away with rote learning and scaled back
instruction in arithmetic and other traditional
basics. It hoped to expose students
to a wider range of mathematical ideas, readily
sacrificing a few right answers for what proponents
hoped would be a more holistic understanding
of abstract mathematical concepts.1
Unfortunately, the new system didn’t
work. Teachers barely understood it,
students failed at it, and a backlash began
to grow against it. Opponents openly
derided New Math as “Fuzzy Math.” By
the early 1980s, a new battle over mathematics
education was beginning.
One of the most important battlegrounds was
in the nation’s largest school system:
California. Knowing public opinion was
beginning to turn against them, California’s
New Math advocates (at this time somewhat ironically
called Progressives) published three new influential
works on early mathematics education. The
California Department of Education published
a Framework for California Schools in 1985
and a revised version in 1992, and the National
Council of Teachers of Mathematics (NCTM) produced
the Curriculum and Evaluation Standards for
School Mathematics in 1989.
These
three documents further solidified the new
math movement. They advocated
a further scaling-down of traditional pencil-and-paper
arithmetic and, for the first time, promoted
a greater use of calculators in primary and
secondary schools.2
In response, California’s traditionalists
(as those opposed to New Math were called)
spread their own ideas, publishing influential
works in newspapers and academic journals around
the country. They advocated a return
to a more traditional approach to mathematics
education: a rigorous study of arithmetic and
precise mathematical tasks, specific and defined
curricula, and the solving of mathematical
questions without the aid of calculators. From
1995-1999, the two sides would battle it out
in the schools, the state legislature, and
the court of public opinion.
During that time, California’s universities
began to notice the negative effects of years
of New Math in their entering freshmen classes. In
a mere eight years, the California State College
System saw a leap from 22% to 52% of remedial
math among incoming students.3
It was around this time that I was finishing
my undergraduate study in mathematics at the
University of Maryland, College Park. There
I was teaching a class of Calculus I and a
class of Calculus II students as part of a
mathematics scholarship. I found that
many of my students, despite being brilliant
and dedicated, were struggling in rather unexpected
areas. Students would successfully complete
the integration of a complex system, only to
get stuck trying to simplify the resulting
algebraic equation: “x2 + 2x +1.” I
witnessed students using TI-82 calculators
to solve arithmetic problems such as “10
x 10” or even “7 + 1.” This
lack of basic algebra skills and reliance on
calculators left them with something less than
a complete understanding of the subject matter.
There are no shortcuts to mathematics education. We
must build a foundation: securely, concretely,
and patiently. Without this, nothing
else will stand. Students need to learn
arithmetic without calculators: not just because
it has always been done that way, but because
it is essential for understanding number relationships
on a more sophisticated level later on.
This was the heart of California’s traditionalist
movement. To teach fewer things, but
to teach them better. To not move on
just because most of the students understand
most of the time. To build mastery, not
simply familiarity. To create a structure
that will not crack but grow as more weight
is piled upon it.
Eventually the traditionalists won their battle,
and the guidelines for mathematics education
developed by a panel of university level mathematicians
and scientists as well as educators and policy
makers became the state-wide norm—the
1999 California Math Framework. The 1999
Framework provided, among other things, an
approved list of textbooks supporting the traditionalist
approach. Included on that list is Connecting
Math Concepts, the texts used in the Baltimore
Curriculum Project’s charter schools.
In California, the positive effects of these
changes were evident almost immediately. In
1998, the final year before the 1999 Framework
became the norm, California second-graders
averaged only the 43rd percentile (outperforming,
on average, 42 of 100 nationwide second-graders)
on the SAT-9, a standardized mathematics exam. By
2002, some three years after the 1999 Framework
was implemented, that score has risen 20 points
to the 63rd percentile. During that same
time period third-graders improved from 42nd
to 64th (22 points), fourth-graders 39th to
58th (19 points) and fifth-graders jumped from
41st to 58th (17 points).4
There is much work still to be done to build
the nation’s mathematics programs to
where we dreamed they could be fifty years
ago. Many states, including Maryland,
are still using curricula and standardized
tests based on the New Math ideas of the 1960s. Fortunately,
however, some of those states have begun to
reevaluate those standards. In Maryland,
the Governor has appointed an Advisory Committee
on Mathematics, Science and Technology Education
to look at the state math standards and curriculum. Hopefully
states like Maryland will build on the lessons
learned from California. Only when the
state makes a commitment to the traditionalist
approach—both in curriculum and standardized
tests—will it be possible to utilize
the 1999 California Math Framework effectively. Until
then the ghosts of New Math will continue to
haunt our schools.
- Wikipedia: “New
Math”
- Anthony
Ralston, Notices of the American Mathematical
Society, November 2003
- Jerome
Dancis, Professor at the University of
Maryland, interview with The Baltimore
Curriculum Project, Feb/Mar 2006 Newsletter
- Wayne
Bishop, Dept of Mathematics, Cal State
Univ., LA, “Four Years of California
Mathematics Progress”
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