Slogans for Reform in Math Education
What does it mean to succeed in mathematics? If we can answer that, then we can evaluate what math education should accomplish. Compare the textbooks and curriculum in your school, to the slogans for real learning below.
The following slogans relate to the NCTM Standards for math education, the AAAS standards for science education, and the Expeditionary Design Principals for education. .
Mathematics is not memorization.
Mathematics is problem-solving 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 problem-solving, reasoning, or even natural learning processes.
Mathematics is not about knowing, its about not-knowing.
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. Problem-solving 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 problem-solving 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 self-awareness 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 cookie-cutter 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 cookie-cutter 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.
More on what math really means:
|See pages at any of these sections to see more about what it really means to use math to understand.|