Mathematics is not memorization.
Mathematics is problemsolving and reasoning. Anything
that is memorized is not a problem to be solved. Things that are merely
memorized are not reasoned about.
People who succeed in mathematics do not memorize the "rules." Instead,
they find ways to organize ideas so that concepts may be easily discovered.
Educational research has borne out that students that perform high at
mathematics break the "rules." Those that perform poorly try to memorize
the list of "rules."
Test your school's curriculum and math texts against this knowledge.
Do they follow the traditional approach of asking students to memorize
the "rules." If so, they are not promoting problemsolving, reasoning,
or even natural learning processes.
Mathematics is not about knowing, its about notknowing.
It is easy to fall into the trap of believing that learning means knowing.
More learning means more knowing. But mathematics is the thinking we do
when we don't know the answer or the method. Problemsolving requires
a situation where neither the answer nor the method is known up front.
Once we know the the answer and the method we are no longer doing the
problemsolving and reasoning that make up mathematics.
This creates a challenge for math educators. To promote math education
requires learning that is not based in knowledge (Bloom's lowest skill.)
So what then, if not knowledge, are we teaching? This is the very confusion
that creates resistance to real mathematics curriculum reform. If not
knowledge, then what, in fact, does it mean to learn? We must really learn
about our own thinking.
Test your school's textbooks against this? Do they drill on knowledge
of "math facts," or have they risen to promoting thinking skills?
Mathematics is a vast web
Most school math curriculums treat mathematics as a ladder. The faster
a student climbs to the top, which traditionally is has been calculus,
the higher the student is believed to achieve. But what have they missed
on the way?
Another reformer has compared mathematics to a tree. Without the diverging
branches mathematics has no life or beauty! To race students up the middle
is to alienate them from its very nature. The ladder approach cuts off
both the life and beauty of mathematics.
Most accurately, mathematics is a web, like the Internet. A learner can
get onto the web at any entrance point, and find his way to any other
point in the web, using one of many possible paths. Starting with arithmetic
and proceeding through algebra and geometry to calculus is merely an historical
and cultural bias. No natural basis for this approach exists. This is
what NCTM refers to when it talks about mathematical connections. Let
students explore in any direction in the web from the point they are at.
Again, compare your textbooks to this concept. Do they direct the student
in only one "right" direction. Or do they open you up to the possibilities
that exist in all directions?
Mathematics exists within your own mind
To learn mathematics is to ask, "what am I capable of understanding?"
"What am I capable of thinking?" "What can I figure out on my own with
the available information?" "What are the potentials and limits to my
ability to discover new information?"
As such, mathematics looks inward, not outward. Mathematics means, "Know
thyself!" If mathematics is inward looking then it is not made up of facts
in books. Books can only be used to support the introspection.
Evaluate your textbooks by this standard. Do they build selfawareness
that leads to mastery of individual thinking, or do they make the book,
instead of the student, the authority which has mastery over the material?
Arithmetic is to mathematics as spelling is to writing
Arithmetic is a collection of specific procedures and facts, much as
spelling is a collection of language rules or facts. The real substance
of of writing has nothing to do with spelling. Similarly arithmetic contains
none of the substance of mathematics.
Imagine limiting the first six years of our language education to spelling.
Do we believe that students would be able to write better? Do we think
they would show any interest at all in writing? Nonetheless, we spend
six years teaching them nothing but arithmetic. And we wonder why they
can't perform in mathematics! We wonder why they are unprepared to do
any creative mathematical thinking when the reach algebra. Six years of
arithmetic has crushed their spirits just the same way six years of spelling
would crush their desire to write.
Evaluate your school's elementary math textbooks. Do they start with
real thinking and end with arithmetic as a tool to support that thinking?
Or do they start with arithmetic, and fit in some mundane cookiecutter
problems to create the illusion of thinking?
People hate mathematics because we taught them to hate mathematics
Don't think, memorize! Don't develop ideas, follow the procedures! Don't
examine real problems, do a set of uninspiring cookiecutter formula problems
 memorize the steps first. Don't ask why, follow the rules! Disregard
interesting asides, race for the top! How could anyone maintain their
interests and intellect when treated this way.
We have them race to the top  of what? Who defined calculus as the top?
Who created all theses rules anyway? Why do we have to learn them? The
students are right to ask!
If you are involved in educational reform use the slogans above to help
you evaluate your goals in changing your math program.
