On Monday, October 21, 2002, I was on a panel at the annual
conference of the Association of Independent Maryland
Schools. There were 130 math teachers from private schools
in the audience. Here is a list of the questions for the
panel and my personal responses. Feel free to do anything
you want with them. They are also posted on my web page.
http://www.math.jhu.edu/~wsw/ED
The format of the panel was: two minutes of opening
remarks, each panelist had a special question (I took on
the first one by choice) for which they had 4 minutes,
each could talk for 2 minutes on the other questions.
There was a very short question and answer period.
There is no record of what others said but I would like to
mention one thing. One of the other panelist is the
principal of a major boy's private school in Baltimore.
Before being the Principal of the Upper School he was the
Academic Dean and before that Head of the Mathematics
Department. He said something which I found truly
astonishing. He thinks so highly of technology (i.e. he has
made his students so dependent on calculators) that he tells
his students that when they visit colleges they should
interview the chair of the mathematics department and find
out if they allow the use of calculators. If they don't then
they should consider the college to not be a good fit.
Steve
W. Stephen Wilson (410) 516-7413
Department of Mathematics FAX (410) 516-5549
Johns Hopkins University
Baltimore, MD 21218 wsw@math.jhu.edu
http://www.math.jhu.edu/~wsw/
Opening remarks:
I am very pleased to be here. It gives me the opportunity
to thank you all for the job you do. We entrust our
children to you and you have the awesome responsibility of
educating them. I consider your job the most important in
the country. However, I suspect that this doesn't show up
in your pay checks. I want to express my personal profound
gratitude and appreciation for what you do. Thank you.
Unfortunately, I also want to explain that your job is
harder than you are perhaps aware. When you are done
with your students you send them to me. I regularly
teach freshman calculus in college. Regrettably though,
I teach it to about 200 at a time. I can convey a love of
mathematics, an excitement about the Calculus and a real
desire that they learn the material; but I do not get to
know their names or how they think, and I cannot really
claim to teach them anything. I run the material by them
at high speed in a large lecture and then I examine and
grade them. If they survive, it is because you prepared
them properly, not because of anything I do. I apologize
if I add to your burden, but it is just the way things are.
1.To what extent does the autonomy of each division and
its particular mission get in the way of a coherent math
curriculum across the grades? Who should be determining
the curriculum?
I cannot address this issue for all of K-12. However,
I can deal with it at the college level. Almost all
students in college who are taking mathematics courses are
doing so because their major requires it. Their majors not
only specify the math course which they must take but the
material covered in the course. Mathematics departments
in college are service departments, we teach for other
departments. Even when we teach for mathematics majors,
the material is well defined and not optional.
If we fail to teach our students what is required then we
hear about it and lots of pressure comes to bear on the
department for us to get back in line. If, for example
our department (or our University) decided that premedical
students should learn statistics instead of calculus,
then none of our students would get into medical school
(which would be a disaster for Johns Hopkins).
The bottom line is: we, in college, have NO autonomy.
The end user determines the curriculum.
Because my 200 Calculus students have just come from
high school I can say something about the autonomy of
high school programs. Mathematics professors are in fair
agreement about what we need students to know when they
come to college. Because Johns Hopkins University is a
fairly selective school, if a student is not prepared
then they just don't get in. So, high school's have,
from my point of view as a teacher at Johns Hopkins, only
two choices: prepare the student, or they won't get in.
For those colleges which are not so selective, most will
give placement tests to entering students. Last fall,
in all 4 year colleges and universities in the US, there
were 676,000 students taking courses before calculus,
including arithmetic. Only 440,000 were in calculus.
These numbers do not include community colleges where the
ratio is even higher. Very few of the students in the
first group will ever complete calculus.
It was extraordinarily difficult to get numbers to
demonstrate that fact. I finally found one large state
university which had them. Of its over 5,000 recent
students who had placed into college algebra (material
they could have learned in high school) only 7.5% of
them ever successfully completed 1 semester of calculus.
The lesson is that if you don't prepare a student for
college level mathematics before they go to college,
then it is very unlikely that they will ever successfully
take college level mathematics, thus excluding from their
life-options many many careers.
So again, the high school autonomy really isn't there
either. There are only two options: prepare the student
for college mathematics, or they will most likely never
take college mathematics.
In an attempt to answer this question for everyone, it
seems to me that curricula should be pretty much determined
by the requirements of the next stage where you are sending
students. That's certainly the way it works in college.
2.What set of skills and mathematical dispositions
should students have as they leave high school and head
to college?
First: Entering freshmen need to know how to add,
subtract, multiply and divide. They need to be able to
do this quickly, easily, and thoughtlessly for numbers,
fractions, decimals and polynomials. There is no better
preparation for the next step in mathematics than an
intimate familiarity with our number system. If a student
cannot add or divide fractions and do long division quickly
and easily then they do not have the necessary depth to
move on to the next stage.
Second: Students need to be able to solve complex
multi-step word problems. I'll put a good lower school
example from a French text up on the screen in just
a minute.
Lily and Vincent divided a cake proportionally to their
ages. Lily was six years old, and Vincent was four. Lily
ate one third of her share for lunch, and three fourths of
the remainder at dinner. She gave the rest to their cat.
Vincent ate three eights of his share at lunch, and five
sixths of the remainder at dinner. He gave the rest to
their dog. Which had more of the cake, the dog or the cat?
Third: And this is harder to grasp because it doesn't
affect admissions or placement, but is extremely important.
I wish my students were capable of picking up a textbook
and learning the material on their own. After all,
that's what they have to do in most college math courses.
I feel I was very lucky to have grown up in Kansas where
the nearest person who knew calculus was 25 miles away.
I had to teach myself calculus from a text, without help.
When I went to college, I was ready.
3.Many schools report conflict in the transitions between
divisions over computational skills? To what extent is
the mastery of computational skills, specifically the
traditional algorithms, critical to future success in
mathematics?
The traditional algorithms are the only collection of
serious mathematical theorems which can be taught to
lower school students. These theorems solve the age old
problem of how to do basic computations without having to
use different strategies for different numbers. If taught
with enthusiasm and admiration, a student should find them
exciting and appreciate the awesome power they give.
I have already stressed the necessity for proficiency
with basic calculations. This can be achieved with any
reasonable algorithm, so why insist on the traditional
algorithm? My interest in K-12 education has made me
pay much more attention to my college teaching and I find
that I am constantly using the traditional long division
algorithm to solve problems for my students from freshmen
calculus to senior math major courses.
It is true I am using the long division algorithm not
usually with numbers but with polynomials or even in
more abstract settings. The traditional long division
algorithm easily generalizes to many different situations.
If a student doesn't use the traditional algorithm then
I can't talk to them and they can't talk to their peers.
If you don't teach them the traditional algorithms, then
when they get to college they will be isolated from the
human interaction which is so very important to learning.
5.Many middle schools are beginning to use graphing
calculators as early as sixth grade and yet high school
teachers are hearing from returning college students that
many college professors are banning graphing calculators.
Although the SAT permits their use, the GRE does not.
Given the uncertainty of calculator use, what advice do
you have for K-12 teachers?
Calculators are an important tool and students should
certainly learn how to use them. My advice is, however,
that a student should always do a calculation by hand
if possible. Every bit of practice matters.
I asked a cross section of mathematicians whether they
agree or disagree with the following statement:
In order to succeed at freshmen mathematics at
my college/university, it is important to have
knowledge of and facility with basic arithmetic
algorithms, e.g. multiplication, division,
fractions, decimals, and algebra, (without having
to rely on a calculator).
I got 93 positive responses and no one disagreed. This is
particularly remarkable because if you ask this same
bunch what mathematics is, then no two of them will agree.
There are at least a couple of dozen who would normally
disagree just to be disagreeable. Personally, I don't
know anyone who allows the use of calculators on exams
in college.
4.Some critics have argued that in attempt to engage
students the new curricula have sacrificed mathematical
rigor. They feel that the repetition once relied upon to
internalize procedures is being ignored in the interest
of conceptual development. Comment on the relative merits
of traditional instruction, constructivism, and any other
approaches to teaching mathematics.
I am in no position to tell you how you should teach.
If I have anything to offer on this panel it is to let you
know what a prepared college bound student is and also to
let you know what happens to unprepared college students.
Unprepared college students don't go on to careers in
medicine, they don't go to business school, they don't
study science or engineering or computers. They have to
do something else. They don't have a choice. They are
not prepared.
All of us who worry about what to teach and how to teach
it are caught on a bit of a treadmill. I have to produce
students with certain skills and knowledge. In order
to do that I have to get students with certain skills
and knowledge. All of us are in this same situation.
The curriculum is built, to a large extent, from the
top down. To be honest, I don't know where it starts up
there, I just know my little part on the treadmill.
Do whatever works for you and your students. Just don't
try to redefine mathematics and teach skills and knowledge
to a student which will not allow the student to succeed
at the next level.
W. Stephen Wilson (410) 516-7413
Department of Mathematics FAX (410) 516-5549
Johns Hopkins University
Baltimore, MD 21218 wsw@math.jhu.edu
http://www.math.jhu.edu/~wsw/