by Glenda Lappan, NCTM President 1998-2000

*NCTM News Bulletin*, March 1999

Michael Jordan. He's got a gift for basketball, most people agree. Whereas our society accepts that all students can and should be involved in sports and other physical activity, we also accept that some, like Jordan, because of their talent, interest, and commitment, reach levels of excellence in sports that others do not. The same is true of mathematics. For mathematics educators, the dilemma is how to make available in mathematics opportunities that encourage such advanced pursuits of excellence without creating programs that deny other students access to high-quality mathematics.

Our primary goal must be mathematical power for all students. We speak often about providing rich opportunities for disadvantaged students. But among the students we have in our mathematics programs are some that have either high abilities or high interest, or both. Our programs must include opportunities for these students as well. These students are likely to become significant users of mathematics as our future scientists, mathematicians, statisticians, engineers, technologists, and researchers. They deserve programmatic attention just as students with other kinds of special needs do.

We underserve high-ability and highly motivated students when we let them breeze through our programs without ever learning to grow mathematically. If students have excellent memories and the ability to quickly mimic the teacher, they might pass through school thinking that mathematics, as well as other subjects, is easy. They might never learn to think about and work on a mathematics problem for hours, days, or even weeks. They might never learn to seek needed information or to collaborate in a meaningful way with others. Because they haven't been encouraged to stretch their thinking, they might give up on themselves and quit the study of mathematics when they reach the limit of what they can do with minimal effort.

All too often, we cope with these students by giving them more of the same. If the less able students do 10 problems, the more able students do 25. This does little to encourage deeper mathematical thinking and might even turn capable students against mathematics. We also sometimes cope by pushing the more able students through the standard material, only faster. This has always seemed to me to be such a wasted opportunity to expose students to beautiful mathematics that they might not otherwise get to see. Instead of more of the same, why not add more depth in the form of both in-class and extracurricular opportunities for able, motivated students to engage with some mathematics that is not in the program?

Students can be given resources to read and study on their own. Semester courses or seminars that look at challenging and beautiful mathematics can be offered. Mathematics clubs can provide extracurricular opportunities to explore problems, to read interesting books about mathematics or its history, or to study graph theory, approximation techniques, mathematical modeling, or many other topics that teachers will find interesting as well.

No matter what direction is taken, we need resources. NCTM publications can help fill these needs at all the grade levels. The Mathematical Association of America, the American Mathematics Society, the American Statistical Association, and others also publish resources that are appropriate for high school and even middle school students.

One excellent resource available to teachers and students is **Quantum** magazine, published six times a year by the National Science Teachers Association (NSTA) in conjunction with NCTM and the American Association of Physics Teachers. **Quantum** is directed at high school students but has articles and problems that are also accessible to middle school students. The magazine is mathematically sophisticated, including many situations where students are asked to follow or provide a proof of a conjecture using precise mathematical language. For students who have never seen a mathematical proof, this can be an exciting and very worthwhile experience.

In the November/December 1998 **Quantum**, an article on polygonal patterns, entitled "Lattices and Brillouin Zones," shows beautiful geometry in vivid color. The article can be examined at different levels. Some high school students will enjoy puzzling through the conjectures, lemmas, and proofs that are given or suggested. Other students will enjoy studying the pictures to find for themselves relationships among the layers of the polygons. The same issue of **Quantum** appeals to middle school students with brainteasers like this one: "Each of three successive months has exactly four Sundays. Prove that one of the months is February."

We talk a lot about applications of mathematics to science. Since **Quantum** is published by science and mathematics organizations, it includes many articles on results of importance to both science and mathematics and is a source of many examples where science and mathematics come together in meaningful ways.

We can and we must challenge our high-achieving, highly motivated students. They need opportunities to struggle with mathematics that is challenging to them, just as all students do. We want all our students to be encouraged to reach for their mathematical stars. That's a true vision of "mathematics for all."

For more information about **Quantum** ($17.70 for students, $25 for personal use, and $45 for libraries/institutions; six issues a year), see the NSTA Web site, __www.nsta.org/quantum/__, or call (800) 777-4643.