WE STILL NEED LONG DIVISION, MORE HIGH TECH MATH NEEDED
\doc\web\99\09\must.txt 1999
http://www.csun.edu/~hcbio027/standards/conference.html/
Conference on Standards-Based K-12
Education California State University Northridge Transcript of R.
James Milgram (Mathematics Professor, Stanford)
A realistic long term is maybe 15 years. If we are lucky, in 15 years the
average student may get near the standards if everything goes just right.
In a shorter time than that, it is almost inconceivable to believe that
this will happen.
[3rd grade long division is needed for polynomials and calculus later on]
... the scope of things in mathematics is so long that an ordinary
second, third, fourth grade teacher is not equipped to make a
judgment about whether a subject is needed or not needed.
[Quote: "If you want to learn mathematics, you must learn it precisely.
Mathematics is precision and one of the first objectives in teaching K -
12 mathematics is for students to learn precise habits of thought."]
This transcript should be dissminated to ALL state BOE members, school
board members, elementary school teachers, and HS math and science
teachers!!!!!
1999 Conference on Standards-Based K-12 Education
California State University Northridge
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---- Transcript of R. James Milgram (Mathematics Professor, Stanford) W(edited by the speaker) Biography:
Mr. Milgram: I would like to start by again thanking David Klein and Cal
State Northridge for arranging and organizing this wonderful opportunity
to get together and compare ideas on the incredibly challenging times
ahead of us. Professor Wu brought up a number of critical points in his
discussion and one of them that he mentioned -- that this is a long term
challenge -- is particularly important.
I'd like to fill in somewhat what the problem is here. First of all, "long
term" has generally been understood to be in the order of perhaps three
years, and there seem to be real expectations of being able to meet the
standards in that time frame. But this is very unrealistic!
A realistic long term is maybe 15 years. If we are lucky, in 15 years the
average student may get near the standards if everything goes just right.
In a shorter time than that, it is almost inconceivable to believe that
this will happen. California today ranks just about at the bottom in the
United States, in terms of the level of mathematical achievements of
students in K-12. The United States ranks near the bottom among all the
developed countries in the world in terms of math achievements of
students. We have an incredibly long way to go because you have to
remember that the new California Mathematics Standards were written to
match the levels of the standards of the top achieving countries in the
world. Meeting these standards is a daunting challenge and we had better
take it seriously.
We now look at the reasons we clearly needed new standards in mathematics.
They can be subsumed in three main areas. REASONS FOR NEW STANDARDS
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---- The increasing failure of the present system to produce enough
technically skilled graduates to meet national needs
Curricular problems which leave more and more students without the
prerequisites needed for their majors, particularly in technical areas
Lack of a clear understanding - on the part of teachers and math educators
- of the major goals of the mathematics component of K-12 education
The next three slides explain a little bit about how we see some of this
so we cannot escape from these issues. The facts quoted in these slides
come from recent newspaper articles for the most part.
INDICATIONS OF FAILURES
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---- >From 1990 to 1996 there has been a 5% decline in high-tech degrees
-- engineering, math, physics, computer science -- in this country and the
trend is continuing.
Of the decreasing number of high-tech degrees awarded a significant and
growing proportion go to foreign nationals.
At the doctorate level 45% of high-tech degrees were granted to non-U.S.
Citizens
>From 1990-96, there's been a 5% decline in high-tech degrees overall in
>this
country. And the trend is continuing -- in fact, the trend is
accelerating. Even though the number of high-tech degrees is decreasing,
it is vital to note that an ever increasing portion go to foreign
nationals. At the doctorate level, for example, 45% of high-tech degrees
are granted to non-U.S. Citizens and at Stanford, in the mathematics
department, two thirds of our graduate students are foreign-born. Even 10
years ago, less than half were.
As a result of this situation it has been impossible to fill all our
technical jobs with United States citizens. This is particularly true in
Silicon Valley. To find qualified people to fill these positions Congress
was intensely lobbied by Silicon Valley, and Congress was forced, much
against their will, to provide 142,500 more visas for foreign nationals to
fill jobs in Silicon Valley.
Currently it is estimated that the number of foreign-born residents of
Silicon Valley is about 25% of the population.
Among all the states as I said in the beginning, California colleges
showed the greatest decline in high tech degrees.
INDICATIONS OF FAILURES - II
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---- Last year Congress was forced to provide 142,500 more visas for
foreign nationals with high-tech skills
Currently it is estimated that the number of foreign born residents of
Silicon Valley is about 25% of the population
Among all states, California's colleges showed the greatest decline in
high-tech degrees awarded.
So the first point is that the system today is simply failing to produce
enough technically qualified graduates to meet national needs. The
foremost problems and most dramatic declines are here in California.
Curricular problems are overwhelming here and leave more and more students
without prerequisites needed for developing and learning technical skills
in college. When they come to us, even at Stanford, more and more of them
are just not able to become engineers and scientists, even though this is
their original intent. They just don't have the background any more. It is
a dramatic change.
Finally, and sadly, because I have the utmost respect, and I think we all
do, for the practicing teachers, the level of understanding on the part of
teachers and above all of math educators -- that is members of the
educational schools throughout the country -- that is required for
teaching mathematics in K-12 is just not there any more.
Look at the effect of this lack of understanding on our students.
INDICATIONS OF FAILURES - III
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---- The percentage of entering students in the California State
University System who are place into remedial mathematics courses after
taking the ELM placement exam is about 88%
Overall, well over 50% of entering students are placed into remedial
mathematics courses.
The average level of the questions on the recent version of the ELM is
about grade level 6.9 according to the new California Standards.
This 88% is a statistic that astounded me. And it is correct, differing
from the failure rates commonly reported (which are bad enough). The
percentage of entering students in the California State system who are
placed into remedial mathematics courses after taking the ELM placement
exam is 88%. Let me emphasize this: 88% of those students taking the exam
fail it. Some of you may know a statistic of about 55% for the failure
rate. Unfortunately, this is calculated by counting the 40% of the
entering students who are not required to take the exam as having passed
it.
These 40% are counted as passing it probably so the statistic will look
reasonable. I reiterate that the actual statistic is 88% taking the ELM
fail it, and it is not that hard an exam overall. In any case, well over
50% of entering students in the California State University system are
placed into remedial math courses.
Those are some of the reasons for our current problems. They stare at us.
We can't avoid or deny them.
Now, I would like to give you an idea of the real complexity of the
problem and the consequent difficulty with trying to fix it.
On our first slide the second problem with mathematics that I indicated is
the lack of understanding of curricular development on the part of math
educators.
Curricular development is a very complicated issue. As an illustration,
I'm going to look at one topic, long division, now. Long division is
something that a lot of professional math educators want to take out of
curriculum. So let's just look at why it is in the curriculum.
CURRICULAR PROBLEMS
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---- The recent fashion of not teaching material like long division and
factoring polynomials is based on claims that such skills are no longer
useful.
This reflects a deep lack of understanding of the role of mathematics in
fields like science, engineering and economics.
In mathematics many skills must be developed for many years before they
can be used effectively or before applications become available.
First of all, I claim that taking -- even asking to take it out of the
curriculum -- shows a profound ignorance of the subject of mathematics.
The point is, in mathematics, many, many skills develop over an extended
period of time and are not really fully exploited until perhaps 10, 12, or
even 15 years after they've been introduced. Some skills begin to develop
in the first or second grade and they do not come to fruition or see their
major applications until maybe the second year of college. This happens a
lot in mathematics and long division is one of the key examples.
SOME SKILLS DIRECTLY
ASSOCIATED WITH LONG
DIVISION
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---- Students cannot understand why rational numbers are either
terminating or (ultimately) repeating decimals without understanding long
division.
Long division is essential in learning to manipulate and factor
polynomials.
Polynomial manipulation and factoring are skills critical in calculus and
linear algebra: partial fractions and canonical forms.
So just to start, understanding that decimals represent rational numbers
if and only if they are terminating or ultimately repeating -- a skill
that was requested be put into the standards by math educators -- cannot
be understood without long division. It is only in understanding of the
process of taking the remainder in long division that you see the
periodicity or termination happen.
I regard the repeating decimal standard as relatively minor, but some
people seem to think it is important. The next topic is critical and
almost everyone thinks it's minor (Laughter). Long division is essential
to learning to manipulate polynomials. Without it, you simply cannot
factor polynomials.
So what, you ask? Again, this is a question that doesn't come up until
the third year in college. At this point the skills that have come from
long division through handling polynomials become essential to things like
partial fraction decomposition which is important in calculus but finds
its main applications in the study of systems of linear differential
equations, particularly in using Laplace transforms, which is the critical
construction in control theory. It is also essential in linear algebra
for understanding eigenvalues, eigenvectors, and ultimately, all of
canonical form theory -- the chief underpinning of optimization and design
in engineering, economics, and other areas.
The previous slide indicated what I call the static applications of long
division. The next slide illustrates some of the "dynamic" applications.
DYNAMIC SKILLS ASSOCIATED
TO LONG DIVISION
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---- The process of long division is one of successive approximation, with
the accuracy of the answer increasing by an order of magnitude at each
step.
The skills associated with this process become more and more fundamental
as students advance.
They include all infinite convergence processes, hence all of calculus, as
well as much of statistics and probability, to say nothing of differential
equations.
Long division is the main application of the previously learned skills of
approximation.
Long division is the only process in the K - 12 mathematics curriculum in
which approximation is really essential. The process of long division is a
process of repeatedly approximating and improving your estimates by an
order of magnitude at each step. There is no other point in K - 12
mathematics where estimation comes in as clearly and precisely as this.
But notice that long division is also a continuous process of
approximation, the answer keeps getting more and more accurate and when
the students learn how to do long division with decimals they learn to
carry the process to many decimal places. This leads naturally -- in a
well conceived curriculum -- to students understanding continuous
processes, and ultimately even continuous functions and power series. The
development of these skills are all contingent on a reasonable development
of long division. I don't know of any other or any better preparation for
them.
What happens when you take long division out of the curriculum?
Unfortunately, from personal and recent experience at Stanford, I can tell
you exactly what happens. What I'm referring to here is the experience of
my students in a differential equations class in the fall of 1998. The
students in that course were the last students at Stanford taught using
the Harvard calculus. And I had a very difficult time teaching them the
usual content of the differential equations course because they could not
handle basic polynomial manipulations. Consequently, it was impossible
for us to get to the depth needed in both the subjects of Laplace
transforms and eigenvalue methods required and expected by the engineering
school.
But what made things worse was that the students knew full well what had
happened to them and why, and in a sense they were desperate. They were
off schedule in 4th and 3rd years, taking differential equations because
they were having severe difficulties in their engineering courses. It was
a disaster. Moreover, it was very difficult for them to fill in the gaps
in their knowledge. It seems to take a considerable amount of time for
the requisite skills to develop.
APPLICATION OF THE SKILLS
ASSOCIATED TO LONG DIVISION
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---- The combination of these skills is used critically in economics,
engineering and the basic sciences via Laplace transforms and Fourier
Series.
Without a thorough grounding in these topics it is impossible to do more
than routine work in most areas of engineering, the most active current
areas of economics and generally, any area involving optimization.
So you see the problem. The problem is that the scope of things in
mathematics is so long that an ordinary second, third, fourth grade
teacher is not equipped to make a judgment about whether a subject is
needed or not needed.
SCOPE IN THE MATHEMATICS
CURRICULUM
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---- The long division story illustrates one of the chief problems with
curricular development in mathematics. The period needed before a learned
skill can be fully utilized can be as long as eight to ten years.
It takes real knowledge of mathematics as well as how it is applied to
make judgements regarding curricular content.
I think the long division problem illustrates the problem described on the
slide above very well. And I put that dragon up there advisedly.
EDUCATORS TELL US OF THE
NEED FOR CONCEPTUAL
UNDERSTANDING AND MATH
REASONING SKILLS IN OUR
STUDENTS
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---- These skills ARE critical in todays technological society.
What many math educators tell us represent examples and exercises for
developing these skills are NOT relevant and/or NOT correct.
The first slide mentioned a third aspect of the problem, which was the
lack of knowledge of the subject on the part of math educators. To make it
clear, I'm talking about math educators and not teachers. Teachers learn
what they are told in the education schools and just hope that this
background prepares them sufficiently. They do the best they can and have
the most demanding job that I know of. As a group I believe they are the
most dedicated people I know of. But if you do not provide teachers with
the proper tools, they can't do a proper job.
MATH EDUCATORS OFTEN
HAVE LIMITED KNOWLEDGE
OF MATHEMATICS
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---- For example, three of the 14 problems originally proposed by the
presidential commission on the eighth grade national mathematics text
and/or the "solutions" they gave were INCORRECT. This commission included
many of the best known math education experts in the country.
The next slides discuss one of these problems.
I just want to spend a few minutes, now, looking at some of the problems
that we have seen in the last few years when we -- as professional
mathematicians -- have looked at some of the things that math educators
are trying to tell the world is mathematics. I will concentrate on
problems that these people suggest for testing mathematical knowledge.
A PROBLEM FROM THE
NATIONAL EIGHTH GRADE
EXAM
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---- We are given the following pattern of dots:
At each step more dots are added than were added at the last step.
How many dots are there at the twentieth step?
This is a problem from the original proposed 8th grade national exam,
produced by a presidential commission including most of the best known
math educators in the country. The problem appears to be simple and every
person I've asked, who I haven't warned to think hard and carefully about
it, has answered immediately, "Oh, it's of the form n times n plus 1, so
you are looking at the 20th stage, therefore the answer is 20 times 21."
But that's not right. The words need to be read carefully.
The point is, the words tell you the only thing you are actually given --
namely, that there are more dots added at each stage than the previous
stage. That's all you are given, and the picture is just a picture.
ANALYSIS OF THE PROBLEM
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---- The answer given by the Presidential Commission on the National
Eighth Grade Exam was 20 X 21 = 420
This is incorrect! The correct answer is that any number of dots is
possible as long as there are at least 267.
As was pointed out, the Presidential Commission that proposed this problem
included many of the best known math educators in the country.
ANALYSIS OF THE PROBLEM - II
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---- This can be seen by considering that you must add at least seven dots
to get to the fourth stage, eight to get to the fifth, nine to get to the
sixth, and so on, but, of course, you can always add more.
So the formula for the number of dots at the nth stage with n>2 becomes:
any number at least as big as 6+(6+7+8+...+(n+3)) which equals
any number at least as big as
(n+3)(n+4)/2 - 9 = 267
Hmm? Actually that problem was about as complicated as any problem I've
seen at this level, and it was proposed for the 8th grade national exam!
When you read it carefully, it is a problem a 12th grade senior would have
trouble solving.
So what is the moral here?
If you want to learn mathematics, you must learn it precisely. Mathematics
is precision and one of the first objectives in teaching K - 12
mathematics is for students to learn precise habits of thought.
The next slide presents a problem that Wu is very fond of (Laughter). It
can be found in many sources, but in particular it was included as part of
the original Mathematics Standards Commission's proposed California
Mathematics Standards.
A PROBLEM FROM THE
ORIGINAL STANDARDS COMMISSION
STANDARDS
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----
You have a friend in another third grade class and want to determine which
of your classrooms is bigger. How do you do it?
This problem is often proposed as an example which shows that "there is no
single correct answer" since you could use perimeter or volume or area to
measure size.
Of course, this is incorrect!
The trouble is that bigger is not precisely defined. And if every term is
not precisely defined, your problem is not well posed. So technically this
is not a well-posed problem. Of course, we realize that is a little
technical. We have an idea that bigger has certain connotations -- but
unfortunately, a lot of them: perimeter, area, volume, and maybe even
combinations of the three such as 3A + 2.4P + 7V.
ANALYSIS OF THE
COMMISSION PROBLEM
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---- The difficulty here is that bigger is not precisely defined, and to
do mathematics you generally have to know exactly what each term means.
However, mathematics does provide for the situation where terms can have
different meanings. There is still a single "correct" answer. It consists
of the set of all answers.
But since bigger can mean anything, the set of answers is uncountably
infinite, and this problem is totally inappropriate for any but the most
advanced high school students.
You see, when you put in a linear combination of the three, you get an
uncountable number of possible definitions of bigger. That's all right.
Mathematics allows for this, as long as you can make some sense of the
problem. Mathematics says the correct answer to the problem is all
possible answers to the problem (Laughter). If you are going to take that
problem at face value, you have to give me an uncountable number of
answers.
MORE DETAIL ON SOLUTIONS
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---- Here is an example which illustrates the point that the "answer" is a
collection of "all solutions".
Consider the system of two equations in three unknowns:
2x + y + z = 1
x + 2y + z = 0
A solution is x = 1, y = 0, z = -1. The answer is
x = 1 + y
z = -1 - 3y
So, what is the point? One of the most important things, as I indicated,
that students should learn in doing mathematics is precise habits of
thought. Suppose we start with a "real world problem", given, as is
typical for such problems, very imprecisely. We want students to be able
to break the problem apart into smaller problems, make sense of them, and
solve them or recognize that it is not possible to solve them with the
information given. One of the first things that mathematics should
prepare student for is making the best possible (rational) decisions when
faced with real problems.
SUMMARY - I
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----
One of the most important things that students should learn from studying
mathematics is precise thinking.
They should understand how to recognize when a problem is well-posed.
They should be able to decompose a possibly ill-posed problem into pieces
which can be made well-posed, and solve the individual sub-problems.
Now, I don't for a minute want to minimize the fact that students have to
learn basic number skills, certainly they have to do that too. And they
have to learn things like statistics, I mean, this is critical in our
world today, and it is a wonderful thing that it is commonly taught today.
It helps prepare students to defend themselves from tricky claims and
fake uses of statistics. Students also have to learn how to survive in
the monetary world. So a key part of our request for changes when the
State Board of Education asked some of us at Stanford to help revise the
California Math Standards was that compound interest be put back into the
7th grade standards.
SUMMARY - II
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---- They should also learn the basic mathematical skills needed to
survive in today's society.
These include basic number-sense
They also include skills needed to defend themselves from sharp practices,
such as being able to determine the real costs of borrowing on credit
cards.
Additionally, they include being able to recognize illegitimate uses of
statistics.
I think everybody has the idea now. I have many more problems here, all of
which are incorrect and all of which are due to some of the top math
educators in the country. But I think you all get the idea of what the
level is here and what we are trying to deal with, so I think we can skip
most of them. But there is one more example that is worth noting
(Laughter).
A PROBLEM FROM THE NEW
NCTM STANDARDS
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---- The following is proposed as a Kindergarten problem:
How big is 100?
This suffers from exactly the same difficulty. I asked one of our best
graduating seniors this problem (he has a fellowship to study in Germany
for next year and the year afterwards will continue his graduate work at
Harvard).
This is from the current new proposed version of the NCTM standards. "How
big is 100?" It suffers from every one of the flaws I mentioned before.
But I loved the response from the student above.
A PROBLEM FROM THE NEW
NCTM STANDARDS II
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---- Without even a moment's hesitation he answered:
Oh, about as big as 100!
Indeed, any other answer would involve elements of perception and
psychology, not mathematics.
Okay. I think probably I'll finish up now and say again that it's a long
process ahead. It is a serious, serious thing we are trying to do. But I
think it is something that we can do. It's just something we cannot treat
lightly and cannot treat in any way as a casual enterprise. For example
if you hear someone say something to the effect that "Oh, we're going to
give the teachers the Standards. We are going to say, now teach -- and
it's over -- no problem," be very suspicious.
IMPLEMENTING THE MATH
STANDARDS
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---- Problems California students rank at or near the bottom among all the
states in average mathematics competency Generally teachers in grades K-4
have little competence in mathematics above their grade levels
Expectations We cannot solve these problems all at once Time is needed,
and skills and competencies should be introduced gradually. The new
California Math Framework shows the most important skills that must be
learned first.
It is a huge process -- of re-education on everyone's part, it is a
process we all have to contribute to and work on with full attention. But
I think there are grounds to hope that we can actually do it. And the one
thing that has the potential to help with this process is the Framework.
The Framework is something that Wu and I worked on with Janet and the
Curriculum Commission, and with many of the best people in many aspects of
education throughout the country. The Framework has been designed to ease
our way into the teaching to the Standards. It's something that I think we
have to focus on a lot more in the next few months as we try to figure out
how to reach the levels needed.
I would like to just say one word about one of the ways in which the new
Framework can help.
IMPLEMENTING THE STANDARDS
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---- In first grade there are only five emphasis topics in the Framework
out of 30 total topics: Count, read and write whole numbers to 100 Compare
and order whole numbers to 100 using symbols for less than, greater than
or equal to Know the addition facts and corresponding subtraction facts
(sums to 20) and commit to memory Show the meaning of addition and
subtraction Explain ways to get the next element in a repeating pattern
The critical thing about this is that the Standards for first grade have
about 30 basic topics. Well, those topics are, for the most part, quite
difficult at the first grade level and will take a great deal of time and
effort to teach properly. Fortunately, it turns out that only 5 or so of
them are essential. The Framework identifies the essential standards and
makes your jobs as teachers and your jobs as curriculum developers much
easier because the textbooks in the next textbook adoption will be focused
on the emphasized topics, rather than the entire 30 topics in the
Standards. So this will allow us to focus on just a few pieces and make
your job of reaching the levels needed a little simpler.
I think this is where I'll stop (Applause).