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PDE AS A UNIFIED SUBJECT

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  • cours - matière potentielle : to the articial distinctions
GAFA Geom funct anal Special Volume GAFA X c Birkhauser Verlag Basel GAFA Geometric And Functional Analysis PDE AS A UNIFIED SUBJECT Sergiu Klainerman Introduction Given that one of the goals of the conference is to address the issue of the unity of Mathematics I feel emboldened to talk about a question which has kept bothering me all through my scientic career Is there really a unied subject of Mathematics which one can call PDE At rst glance this seems easy we may dene PDE as the subject which is concerned with all partial dierential equations According to this view the goal of the subject is to nd a general theory of all or very general classes of PDEs This natural
  • linear theory
  • range of applications of specic pdes
  • general framework
  • complex analysis
  • nonlinear evolution equations
  • pdes
  • solutions
  • equations
  • subject
  • problem

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Learning to Think Mathematically
Walk into any classroom or lab bra (such as moving quantities A Shift in Priorities
with computers, and you are like- across the equals sign). Drill-and- In the past several decades,
ly to see students working with practice programs basically research has transformed how
j mechanize the familiar flash mathematics software. And more we think about learning mathe-
cards we all used in first grade. than any other type of math pro- matics. In the past, the teacher's
I gram, you will see drill-and-prac- Not only do drill-and-practice focus was primarily on getting
I tice routines. These present stu- programs turn the computer into 1 children to give the "right
I dents with problem after prob- an over-qualified babysitter, but answers" to mathematical prob- I
lem, grilling them for the "basic" most of them leave out funda- lems. We now know that to think i
math facts of arithmetic (such as mental aspects of mathematics flexibly for problem solving in I
the multiplication table) or alge- learning. Even though math facts
- - - @b~z=u-s+{
-Imporrant, the central goator i- hug& ri5Foi.11k9ke answer, or
mathematics education should pravduct, but the thinking, or
be to support students in learning process involved. This thinking R3's CORNER
to think mathematically. The involves many compon-
National Council for Teachers in ent skills. In the last software letter, I
Mathematics suggests that what is The mathematician George described the major categories of
"basic" includes being able to Polya divided the problemsalv- educational software and sug-
call up knowledge and strategies ing process into four major gested criteria
flexibly and efficiently to solve for evaluating phases: defining and understand-
new problems. software within Continued an next page
each category.
Specific evalua-
tion criteria are
indeed hel~ful.
But can we also I define general
guidelines that apply to all
educational software? I think so.
In many ways, choosing soft-
ware for kids is like a
good book for them to read. The
first quality that parents should
look for in choosing any software
is ease of use. The way the child
communicates with the program
(often called the user-interface),
should be essentially transparent
so that the child can use the pro-
gram very rapidly gnd not spend
Continwed on back page
Now that's what I call user friendly! ing the problem, making a plan
for how to solve it, trying out the
plan or alternative plans until
some success is met, and consoli-
dating what one has learned so
that this problem-solving experi-
ence may be applied to similar "It combines mathematical
problems.
principles with linguistics We can see from this list that
the "right answer" is only a small
and child development
part of a problem-solving
approach. Getting there is much theory and applies them
more than half the fun. Just as
to Candyland." most of us write and revise many
times to arrive at a good essay, so
we may need to approach an
equation many different ways to
arrive at a solution.
Problem-Solving attributed to the availability of example, in "Rescue Mission"
in the Real World computers to aid mathematical (Holt, Rinehart and Winston) stu-
Of course, even drill-and-practice thin king. Computer programs dents must navigate a boat in
software that drills students on (such as "TK! Solver," an order to save a whale trapped in
single-digit subtraction presents equation solver) can free the a fishing net. Various arithmetical
"problems" that the student ecologist and the city planner to and geometrical skills are needed
must solve. But the problem-solv- define their problems and set up to use the naval instruments
ing involved in most of mathe- equations, rather than get necessary for navigation. Similar-
matics is much more interesting bogged down in the tedious cal- ly, Sunburst has created simula-
and complex. culations necessary to work tion programs in which students
In the world outside school, through those equations. With use math skills as aids to planning
problems do not come defined in some creative work on the teach- and problem-solving in real- -
electronic workbooks, and there er's part, these same programs world situations, like running a
are no answer sheets to turn to. can free up the student as well. business ("The Whatsit Corpora-
Say an ecologist needs to know iton") and trip planning, con- This problem-solving approach
the effect of a larger beaver struction, and best-buy shopping to learning math focuses on
population on the deer that ("Survival Math"). Programs like projects rather than exercises. It
populate a certain ecosystem, or these can motivate mathematical does away with the drudgery of
a city planner is predicting the thinking in otherwise ambivalent remembering and practicing
change in the flow of traffic a students. mathematical mechanics, like
proposed new building would Another way to motivate long division (once the principle
mathematical thinking cause: most likely no one has run is to pro- of is thoroughly under-
vide a social context for math across exactly these problems stood). And it brings math back
before. Each of people problem solving. Computer pro- from the textbook into the realm
must define the problem and fig- grams can help create.an envi- of practical reality, where the
ure out ways to attack it. And ronment in which students can motivations for mathematical
working on the problem, the per- discuss, reflect upon, and col- thinking arose in the first place.
son may come to realize that a laborate on the math necessary In this way, math can become for
new variable must be added to to solve a problem. The "Bank the student "my math" rather
the problem definition. Psycholo- Street Laboratory" (Holt, Rine- than someone else's, with
gists have discovered that solving hart and Winston), for example, unknown origins and purposes.
a problem is a back-and-forth is composed of hardware devices
(recursive) process, not a linear which plug into the computer Motivating Math
one. What students need to and can be used to measure and Programs like "TK! Solver" are
know is not just answers, but graph changes in light, tempera- designed for professionals as well
ture, and sound over time. Sup- how to define and solve prob- as students. But how can good
lems that require this recursive plementary teacher materials sug- math software be specifically
mathematical thinking. gest activities where students tailored for students?
work together as a team to apply
One way is to present a situa-
Computers in Math mathematical thinking in making
tion in which the math to be
scientific discoveries. The recent shift in the focus of learned is necessary to deal effec-
math education can also be tively with the problem. For Continued on back page Mathematically. . . in the shape of the graph. They ing entry into empirical geometry
Continued from page two discover equations for ellipses, (induction), and it can be used to
lines, hyperbolas, and parabolas. complement classroom work on
Computer-Aided Intuition proofs (deduction). These are exciting times for
learning and teaching math. The A second function of educational A second important mental
rigid attitude that the mind is a software is to help students activity central to mathematical
muscle to be exercised through understand and use the different thinking is understanding the
mechanical repetition is giving mental activities involved in math mapping relations between dif-
ferent ways of representing a way to a richer view of the crea- thinking. Using "The Geometric
mathematical problem. "Green tive, exploring mind that comput- Supposer" (Sunburst), for exam-
ple, students make conjectures Globs" (in Graphing Equations by er tools can aid and enhance. By
about different mathematical Conduit) and "Algebra Arcade" infusing life into the learning tools
objects, like medians, angles, and (Brooks-Cole) do this for alge- for mathematics, children can
bisectors, in geometrical con- braic equations and graphs. A realize how mathematics offers
random distribution of little fig- personal power. In this way, a structions. In this way students
ures appears on a Cartesian x-y child learning is can discover theorems on their
own. The program is an electron- coordinate grid. Students must learning but one more way to
think, and to fully develop his or ic straight-edge and compass. It create algebraic equations, which
her personal voice in the world. comes with "building" tools for the computer graphs, to hit the
- Roy D. Pea defining and labeling construc- figures and score points. Students
tion parts (like the side of a tri- become facile at knowing how
angle or an angle), and measure- changes in the values of equa- Roy Pea is Associate Director
ment tools for assessing length of tions, such as adding constants or and senior research scientist wirh
the Bank Stref Center for Chil- lines and degrees of angles. Stu- changing factors (x to 3x, for
dents find this program an excit- example), correspond to changes dren and Technology.
R3's Corner . . .
Continued from first page down fighters, the child will functions, and have no parallel in
the world of books. For example: quickly lose interest. too much time trying to figure
Can the child print out what she's A program with some degree of out how to make it work.
done? Does the software make depth or complexity, however,
This is analogous to choosing good use of things like color and invites the child to return time the right reading level. As a par- sound where they are appropri- and again, as might a good book. ent, you choose books that you ate? Is the program interactive? And as her skill improves, she'll
know your child can read with a (Can the child exercise discretion be able to take more advantage
minimum of frustration. If the and judgment?) of what the program offers. You'll
reading level is too high, the
One way to find these things find that this is particularly true of child will never be able to get to
out is to ask a knowledgeable some of the better simulations the content.
person. If there's a teacher or and adventure games.
The second quality to look for parent at school who has a repu-
is depth. If a child uses the Good software doesn't "talk tation for knowing about soft-
program once or a few times, down" to kids or trivialize what ware, then it's worth a call.
does she lose interest? This is they're doing. It has a direct link Books and magazines about
often true of game software. You to an activity that's valuable in computers are also available. Our
may be able to shoot down ten the adult world. A good music own Bank Street Family Comput-
fighters or one hundred fighters, processor, for example, lets the
er Book was, in part, an attempt
but if all you can do is shoot child compose music by ear, then
to fill this knowledge gap. Use
displays the notes, prints them on
these sources to find out more
paper and plays the song in dif- specifics about good software The Bank Street Software Letter is ferent ways according to the and which publishers can be published by the Media Croup/ child's direction. The product counted on to produce high- Bank Street College/610 W. 112th that the child creates is musically quality materials. Then you can StreetINew York, NY 10025. (212) correct, and the activity has a decide which of the good soft- 663-7200. Richard R. Ruopp, direct link to a very important ware is right for your child. President, Bank Street; James A. adult creative activity.
- Richard R. Ruopp Levine, President, Mdia Croup;
Of course, there are some President of Fay Wheeler, Editor-in-Chief;
questions that apply to computer Bank Street College Jaan Auclair, Assistant Editor. Computer Maintenance Myths
check luggage, and metal detec- that disk drive doors should be
PART II
left closed when not in use. This tors that check people. X-rays will
not harm disks: the microwaves advice is not completely un-
In the last issue we shattered used by metal detectors will. In- founded, since closing the door
myths that have evolved around formation on the disk is stored prevents dust from getting into
the maintenance of computer magnetically, and microwaves the drive. But leaving it closed
hardware. This month we'll tackle also applies constant pressure to can change or destroy some of
another class of computer myths the magnetic structure. That is the readlwrite head of the drive
- those surrounding the care of and to the spindle that keeps why you are often warned to
disks and disk drives. a disk in place during use, and keep disks away from magnets,
this pressure could gradually library security systems, televi- One superstition making the
cause the disk drive to become sions, telephones, and other elec- rounds is that if you're going to
unaligned. So all other things trical appliances. throw a disk down on a table
being equal, it's best to leave the Since most airports have X-ray top, without its protective enve-
door open. If you are storing the machines right next to the metal lope, it's all right as along as the
computer for a long time, over detectors in~the areas where they label is facing up, because the
the summer, for example, cover check passengers and carry-on actual data won't come into con
the computer system with a cloth luggage, it is probably still not tact with the dust, spilled soda,
to protect it from dust. safe to let your disks go through and sharp objects cluttering up
the X-ray machine. It is better to your computer table. In fact, on
have the guard hand-check your single-sided disks the data is all
disks and walk them around both stored on the "bottom" side (the
machines, a few feet away. side without the label). It's not a
If you have disks in a suitcase good idea to throw your disks
around at all, especially with- you're not carrying on board
out their little jackets. But if with you, then they will only be
X-rayed, and are safe-except you must abuse them, remember
that temperatures ir: the storage that it is their backsides that The disk drive is the part ot
compartments of airplanes can are most sensitive your computer that is most likely
get very low. Acceptable tem- Another debate about disks is to need repair. When disks will
peratures for disks are 50 to 125 whether they can be harmed by not boot, or when they run er-
degrees Fahrenheit. Blank disks, taking them through airport en- ratically, it is often because the
which have not been formatted trances. In fact, there has been readlwrite head in the disk drive
and contain no data or informa- no acce~ted wisdom on this. and is out of alignment or needs to
tion, can be X-rayed or micro- it took ;any phone calls to get be cleaned, or the disk drive is
waved, since there is nothing to some. The answer is, well, yes not timed properly. All of these
be lost. and no. Two kinds of machines repairs can be taken care of at
are used at airport gate en- A ruling that often gets handed home, but specific procedures
trances: X-ray machines that down from the computer gurus is vary among different kinds of
computers. Check with your
local computer users' group or
relevant magazines.
Remember that your home
computer is a household appli-
ance, and basically a reliable
machine which, with proper
care, can last many years. You
can apply the same common
sense principles to its mainte-
nance that you can to your car,
bike, or home stereo.
- Kathy Rehfield
Kathy Rehfield is a Field Advisor
for Computer Outreach Services at
Bank Street College oi Educat~on.
What's a 2?