In an email from Hoover Liddell, San Francisco 1/29/01:
I just finished writing a book with the title, 'Journey from Kilimanjaro'. It is autobiographical about my life in Africa and San Francisco as a traveler and a teacher. One chapter has the title, 'A Mathematical Expedition', and I refer to you in the chapter. The book is available at Amazon.com and Barnes and Noble (bn.com). Below is this chapter:
'We struggle for high school
students to learn algebra when elementary school students are capable of
learning calculus. Why is it not
our mission that the most highly skilled and understanding teaching force be
available in the classroom?
I sometimes find life in
elementary school classrooms profoundly interesting.
My wife is teaching summer school again, and I am the math teacher.
Several years ago I visited a public housing resident (Theresa Coleman)
and I found a fascinating book in her overcrowded dwelling.
It is about teaching calculus to kindergartners.
How surprised I was that it was there.
I ask to borrow the book and reluctantly she agrees.
I give her my word that I shall return it in a week.
The concept is powerful and
engaging. I am glad to read and to
study it. When my wife invites me
to teach the math class, the idea is still in me about teaching calculus to
young children. I search the
information highway and I find material on teaching calculus to the young.
I communicate with Don Cohen who wrote the material for 'Calculus By
and For Young People' and had been using it for seventeen years.
He talks with me, and I am impressed with his materials and his ideas on
learning. I start with a lesson on
the sum and convergence of an infinite sequence on the first day. It is a fascinating adventure that involves all of us.
The class each day is intense and demanding.
The students work hard. They
learn to use calculators and computers in new ways and to diligently study the
partial sums as they endlessly merge, crowding toward a point.
One day in class a black girl who
is struggling with infinite repeating decimals asks why we are doing this
college work in elementary school. She
is the only one who voices such a concern.
The others work on. What I
tell the class is that we are working problems that are sometimes difficult to
grasp and understand. It is
important to know that playing, studying, laughing, being good, and doing
difficult things are all a part of life. If
we do not work on hard things, read books, or see the beauty in a leaf, we are
not really living. I tell them to
take school and education seriously and to learn, new, difficult, and demanding
things as we go through life. It is
important that they do.
The students are nine and
ten-years-old and will be going into the fourth and fifth grades next school
year. The class is mostly Black and
Chinese. There are two White
students and one Latino. I think it
is quite possible for four or five students to master such concepts of calculus.
The deep goal is to have all of the students in the class participating
and learning. There are thirty
students in the class, and my mission is to have every soul here moving deeper
into mathematics than they ever thought possible.
One student is from China and
cannot speak English. She grasps
the math with much more certainty than the English. She does all the work that everyone else does and does it as
well. Others help her with her
English, which is significantly improving, sometimes daily. She is a new immigrant with a deep personal mission.
There are two special education students who started slowly but stay with
the problems and learning process and gain strength in the subject and in this
world we all form. One student,
Lawrence, can see deeper than others into problems and can comprehend abstract
and novel situations. It is good
that he is here because there seems to be no ideas beyond his understanding.
It is often his insight and understanding that ignites the class and
excites their thinking. Some things make us question our capabilities.
Other things deeply transform and move us.
There are also many students who struggle with multiplication,
subtraction, and division. The
division algorithm is too complicated for many to master so we view division in
other ways such as subtracting quantities that are contained within other
quantities and sometimes discover new ways to understand our results.
Fractions, decimals, addition, and division are not isolated exercises
but are used in determining the limit of an infinite set of partial sums.
It is important that we learn what we can now, no matter how we got to
where we are.
Calculus is a collection of
thoughts, objects, and symbols closed in a system of descriptions.
New methods and streams of descriptions come into our consciousness,
bringing a hint of the real world, revealing it through symbolic form.
One day I would like for us to drop and propel objects and to understand
their displacement in space. This
can show us that this mathematics does work in the real world.
Some students do not become deeply
involved in learning because of social and academic inequality from tracking,
segregations, a diminished curriculum, and negative categorizing.
This mathematics class where we all worked hard, profoundly challenged
and changed us all. We find learning, discovery, and understanding, which
connects each of us to a larger world. We
all live diminished lives when the environments that we exist in do not
challenge and deepen our capabilities'.
The above is from the book 'JOURNEY FROM KILIMANJARO' by Hoover Liddell; published by Writer's Showcase; ' 2000. Used by permission of the author.
To
order Don's materials