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Forum: Making it all add up

How to teach mathematics? In the debate between 'back to basics' and 'higher order thinking,' Richard Wertheimer looks for the best of all possible worlds

Sunday, November 10, 2002

What does it mean to be "mathematically correct"?

  Richard Wertheimer, a mathematics teacher, is education manager of City Charter High School (wertheimer@cityhigh.org). 

As you've been reading in the paper recently, the Pittsburgh school board is contracting with four mathematics "experts" to provide them (and us) with insight into what mathematics should be taught in our schools. Unfortunately, as we've already seen, discussions of mathematics curriculum often become polarized into a fight between right and left, conservative and liberal, "back to the basics" vs. "higher order thinking."

A distinction should be made before we enter the fray. There is a difference between mathematics and mathematics education.

Mathematics, the science studied and practiced by mathematicians, is a language that quantifies the world around us. In its applied form, it is used by workers in most walks of life. Unfortunately, most people see mathematics as cold, abstract, difficult and beyond their reach.

Mathematics education, the science studied and practiced by teachers, cognitive scientists and educational researchers, pertains to how best to teach mathematics. This is an important distinction, because the group that opposes the current reform effort is often populated by mathematicians, while the group advocating for change is mostly mathematics educators.

A second distinction pertains to why we should learn mathematics. On the one hand, the study of mathematics can lead to a career as a mathematician. Although this happens in very rare cases, it is an important issue to address.

Our country benefits greatly through the work of mathematicians. The education necessary to develop a mathematician is deductive, conceptual and rigorous. It is not an easy road to take and, as most of us are aware, the road is littered with people who veer off along the way.

On the other hand, skill with mathematics can empower an individual to succeed in any number of careers including finance, architecture, engineering, computer science, business, medicine, bookkeeping or many trades. Application of mathematics in common careers is different in nature. It requires use of computational fluency, technology, problem solving, inductive thinking and the ability to communicate mathematically with others.

So who's right in the debate between "back to basics" and "higher order thinking"?

First, an observation pertaining to the common experience people have learning mathematics. Over the last century, most of us experienced mathematics through the basics. Think about your education and your feelings about mathematics. You spent an enormous amount of time drilling on computation -- addition, subtraction and multiplication and division facts. You did many problems and worksheets on fractions, decimals and percents. You may have taken a traditional algebra or geometry course in high school -- pages and pages of equations, quadratics, factors, variables and proofs. And lots of comments like "when are we ever going to use this?"

How did you like it? Did it empower you? What are your memories of your mathematics classes? Are you someone who feels that mathematics has helped you? Or, are you someone who feels that because you don't understand mathematics, you were held back regarding many career choices?

The refrain "I'm no good at math" is the product of a century-long basic math agenda. Critics of the reform movement would suggest that it is the reform movement that is to blame. There is not enough space here to address the foolishness of that statement. I would simply say that you should use your own experience to judge how successful your mathematics education was.

Second, a number of personal experiences with the Pittsburgh program may shed some light on the current debate.

I was a mathematics teacher in the district from 1979 to 1988 and a mathematics supervisor from 1988 to 1993. My two daughters attended Pittsburgh schools for their K-12 education. They both experienced the "Everyday Mathematics" and "Connected Mathematics" programs with great success. Finally, my doctoral thesis, completed last year, was a study of the Pittsburgh fourth-grade mathematics program. Specifically, my research focused on teachers who implemented the "Everyday Mathematics" curriculum correctly and how they balanced teaching basic skills while using calculators.

My research found that "Everyday Mathematics" teaches students how to add, subtract, multiply and divide numbers in their head and on paper. Students are tested on their ability.

"Everyday Mathematics" also uses calculators for problem solving, difficult computations and learning new concepts. I found that the present curriculum is balanced with respect to basic skills, use of calculators and higher order thinking. I also found that there are a number of talented teachers in the district who understand that balance and are providing an excellent mathematics education for our children.

However, finding that balance is not easy; it takes a great deal of content knowledge and pedagogical skill on the teacher's part. That is why Diane Briars, the district's director of mathematics, has obtained $10 million in National Science Foundation grants to support the Pittsburgh mathematics effort. These grants are used to help train teachers on the intricacies of teaching a standards-based curriculum while maintaining academic integrity. Briars and Lauren Resnick (from the University of Pittsburgh's Learning Research and Development Center) recently published a research paper (www.cse.ucla.edu/CRESST/Reports/TECH528.pdf) that showed striking evidence of achievement when the current program is implemented correctly.

My research suggests that the "Everyday Mathematics" curriculum, the Pittsburgh professional development program called PRIME and the district's use of the "New Standards" assessment program are integral to helping teachers raise student achievement.

Mathematics education is an extremely difficult and complex task. If you don't believe so, look around you. How many of us freely admit we are terrible at math? As you listen to the dialogue pertaining to the mathematics program, keep one thing in mind. The mathematicians are among the few survivors of the traditional mathematics program. They are trying to apply what they know to the entire population. I commend their desire to engage in the dialogue.

But I have great reservations about their ability to understand the complexity of educational methodology that achieves results beyond a select few. Mathematics education needs to be successful for everyone.

As test results and research are beginning to demonstrate, the Pittsburgh mathematics program is a shining light in a dark time in our district's history. Proceed with care.

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