This is an open letter to the Friends School of Baltimore
community about the mathematics program. Please feel
free to copy it and distribute it. This letter, a
letter about my concerns about the lower school TERC
math program and the results of a quick survey of the
attitudes of college professors are all on my web site at
http://www.math.jhu.edu/~wsw/ED.
Dear Friends School:
First, I want to express my gratitude to the teachers
at Friends School for their dedication and devotion to
our children. Friends School is an extraordinary place.
The teachers are the foundation of Friends School.
They are the ones who touch our children on a daily
basis and they are the ones we are most indebted to in
our community. I thank them all for making Friends School
the wonderful place it is. I have faith in the teachers
at Friends School. I trust their competence to teach and
their desire to do what is right by our children.
I am writing because I believe there is a minor
misunderstanding which has serious consequences for our
children. I think the curriculum planners at Friends
School have been misinformed about the importance of
certain aspects of mathematics preparation. I do not
believe that the root of this misunderstanding is at
Friends School. On the contrary, there is a conflict going
on at the national level. The highest level mathematics
educators in the country have pushed hard for an emphasis
on ``understanding'' and problem solving, at the expense
of computational skills and substance. Any reasonable
institution would listen to this advice and act on it.
Friends has done so, with consummate skill it has
emphasized problem solving (at the expense of computational
skills and substance). I want to emphasize here that I am
well aware of the exceptional success Friends School has
with teaching conceptual understanding of mathematics.
Under no circumstances would I want Friends to back off
from this incredible achievement!
So, what is the problem? Where is the misunderstanding?
The mathematics educators educate mathematics educators.
They are not the people who think about mathematics, who
do mathematics, and who teach mathematics to our children
when our children get to college. They teach the people
who will teach children in K-12 how to teach mathematics.
There is a huge gap between these two groups of people:
people who teach mathematics education in college and
people who teach mathematics in college. It is natural
for someone who teaches mathematics in K-12 to look to
the mathematics educators for advice as to how to teach
mathematics and what to emphasize.
My premise is that the source of this misunderstanding
is the lack of communication between the mathematics
educators and the people who actually teach mathematics
at the college level. People who teach our children
mathematics at the college level are pretty unanimous
about the importance of computational skills. (See
http://www.math.jhu.edu/~wsw/list) I want to quickly
assert that they are not ``drill and kill'' advocates but
``thrill of skill'' advocates. They support the sort
of understanding which Friends School teachers know how
to impart to students but they want that understanding
to apply towards the traditional algorithms and basic
computational skills as well as to general problem solving
skills.
One can fairly ask: why do they want students to show
up in college understanding and proficient at basic
computational skills? We have calculators, surely if
they know what division means they don't actually have to
be able to do it quickly and accurately? When college
mathematics professors try to help students move on to
the next level of understanding in mathematics they have
found that a deep understanding of the number system,
an ease of manipulation with it, a comfort with all the
operations, is the most helpful prerequisite to moving
on and up. You cannot understand mathematics if you
cannot do it. Does this mean that we actually have them
doing long division on timed tests on a regular basis
in Freshmen mathematics courses? No, but those skills
make it most likely that the student can absorb the
next, more difficult, concepts we will throw at them.
[As an aside, students will be expected to know all of
the basic arithmetic operations (addition, subtraction,
multiplication, and division with fractions) of polynomials
(i.e. algebra) in order to move forward. A comfortable
working familiarity with this is impossible if a student
does not have similar skills with numbers.]
So, who should we believe, people who teach us how to teach
in K-12 or people who will teach our children in college?
The answer, of course, is both. Students should have
problem solving skills and computational skills.
I would like to put computational skills into a context
which would allow Friends school to teach them without
any change of philosophy whatsoever. Friends School is
fabulous at teaching students how to solve big complex
problem. Well, here is one of great importance. It took
civilization hundreds of years to solve this problem even
after the introduction of Arabic numerals. The problem
is: invent a system of hand computation which fits all
cases, one which does not require different strategies
for different numbers. Create algorithms which allow us
to compute arbitrary addition, subtraction, multiplication
and division problems without having to think through the
problem yet again. Make it work for decimals and fractions
and make sure it easily generalizes to polynomials.
The truth is that computational skills are the solution to
a significant problem: how to compute! Good computational
skills are the result of having solved a good problem.
Let me make yet another case for the importance of the
``traditional algorithms'' being taught and understood.
This time I will take the viewpoint of the pure
mathematician I am. The traditional algorithms which
solve the problem described in the previous paragraph
constitute a collection of THEOREMS of awesome power. The
Theorems are incredibly useful. It is true mathematics.
These results are probably the only examples of great
mathematics which you can show to and explain to a lower
school student. They are the culmination of years of
sustained study of numbers by generations of very smart
people. Don't dismiss them lightly. When a student has
understood and developed proficiency with these algorithms
then they can do any sort of calculation with ease and
understanding without having to rethink a strategy.
I have no objection to the constructivist approach of
letting a child develop their own strategy. At the end
of the day though, the child should be enlightened and
exposed to the traditional strategies which have been
distilled down to a truly elegant form of mathematics.
The point is that these strategies are fabulous, major
mathematical accomplishments and they are the only
such mathematical results which can be taught to small
children. This is their mathematical cultural heritage.
One should no more dismiss it than one should dismiss all
books written before 1990. Furthermore, viewed properly,
these traditional algorithms are incredibly exciting and
should be taught as such with great enthusiasm for the
very deep mathematics which they represent.
I would never presume to know when things should be taught
to our children. I trust the faculty to deal with concrete
problems like that. Perhaps the secret to long division
should be handed out with graduation diplomas? Frankly,
I think that awesome power should be shared earlier.
My guess would be a few minutes every day from grades
1 through 12 and anyone would become both knowledgeable
and proficient. Again, my guess is that the basics should
be taught in lower school and constantly reinforced in
middle school. Although the use of calculators should be
taught, dependency on them should be avoided at all costs
(i.e. they should be used rarely).
I have tried to be upbeat in my assessment of this
problem. However, it is time to emphasize the serious
consequences of this misunderstanding. The issue is
simple: our children will not do well in mathematics in
college without these fundamental computational skills,
and this will close off many occupational opportunities to
our children. When the word gets out, Friends School will
be known as a liberal arts school and our children will
have difficulty getting into certain types of colleges.
Actually, that is not the real future. What will really
happen is that parents will hire tutors for their children
when they fully understand what is happening. This will
prevent the dire description I have given from coming true.
However, Friends School will become (and is presently
rapidly heading there) a wonderful school to send your
child but where you must (and you will) get outside
tutoring in mathematics for your child so your child
can successfully deal with the post Friends School world.
Is that really what we want of our school? The one thing I
know, as a professional mathematics teacher at the college
level for 30 years: a student without these skills is
unlikely to do well in the Freshmen mathematics courses
taught by myself and my colleagues across the country.
While I'm not being upbeat, there is need to discuss
triage. A fair number of students have survived
mathematics at Friends School without learning these
important computational skills. For their sake, these
students must be tracked down and fixed, so to speak.
Sincerely,
W. Stephen Wilson
Parent of Saul
Wilson 4B
Postscript: On Monday, March 4, I spent most of the day
in Washington attending a debate between pro-arithmetic
activists and two presidents (past and present) of
the National Council of Teachers of Mathematics (NCTM).
The above letter was already written and I must retract my
description of the lack of communication between top level
educators and people who teach mathematics in college.
What I describe is not actually the case. The NCTM is
responsible for the standards which led to the Friends
School type programs. The present president of the NCTM
stated loud and clear that they were not interested in
the bright students (i.e. Friends School students) but
they were interested in the students who didn't use to
learn math. Friends School is teaching with programs
consciously developed for students who did not, in the
past, learn arithmetic. None of these students go to
Friends School. The reason why their standards support
the use of calculators and describe long division as
``obsolete'' is because their target students are not
going to go to college.
After the debate there was a diner attended by about 35
nationally known pro-arithmetic activists. Each of us
told the story of his or her involvement. It was actually
quite depressing. Aside from California there were few
success stories. It became clear that if our concerns
about the curriculum at Friends School are actually heard
and acted on it would be because Friends School is a
truly extraordinary place. I came away quite depressed.
It didn't help that during the debate, right on C-SPAN TV,
the head of the National Science Foundation (who was in
the audience two people down from me) stood up and said
she and her husband had tutored their kids in mathematics.
It became clear to me then: when confronted with a
curriculum like Friends School's curriculum in mathematics,
any parent who wants their child to go on to college
should get a tutor in mathematics and continue with the
tutor until the tutor says ``enough'' or until the child
says ``but mom, dad, we do this in school''. My first
response was to tell all parents this but I have a conflict
of interest. My son and I enjoy doing mathematics together
and if the school takes up teaching it then it could usurp
my roll with my son. Although I will persist at trying to
get Friends School to incorporate an understanding of the
basic operations of arithmetic and a good proficiency with
their use, I will do so with a light touch. Friends is a
very special place and the lack of an adequate mathematics
curriculum does not alter that fact.
Post-postscript: A 6-th grade parent recently forced
me to look at her kid's regular mathematics work in the
6-th grade. I was horrified! It is highly unlikely
that a school which approves of a program like that can
possibly be made to believe in the necessity to change to
an appropriate college preparatory program. Fortunately we
are a wealthy country and can continue to import properly
educated people from China and India and so, to some
extent, change is unnecessary at Friends School.