I am often surprised by the diverse opinions on two questions:

- How well are U.S. students doing in mathematics?
- How do students come to understand mathematics?

**How well are U.S. students doing in mathematics?**

There are several standards that can be applied to answer this question:

- An international standard - are we world class?
- An historical standard - are we better or worse than at some earlier time?
- An empirical standard - are we as good as we need to be?

In an international comparison based on the National Assessment of Educational Progress (NAEP) the United States ranked thirteenth out of fifteen nations in the percentage of 13-year-old students rated proficient (Pashley and Phillips, 1993). But on a state-by-state basis, half of the participating states scored at or above the international median. Are we world class?

Historical comparisons are also slippery. The decline of SAT scores has been well documented. The question is, what does this say about mathematics achievement over the decades? Does the decline of SAT scores document a decline in average levels of achievement? According to the NAEP, average mathematics performance has not declined since 1973, and has significantly increased for 9- and 13-year-olds (Mullis, Dossey, Foertsch, Jones, and Gentile, 1991).

In December 1994, Newsweek reported that "12.4 percent of new math Ph.D.s - the highest ever measured - had no jobs after graduation; the rate was 4 percent in 1981." Apparently we are doing a good job of producing Ph.D. mathematicians, and have a more than adequate supply. By this standard we are doing well. On the other hand, only 14 percent of US 13-year-olds are proficient in mathematics, and only one percent are "advanced" according to NAEP levels (Pashley and Phillips, 1993). Few are satisfied with this level of performance.

Whether we are first in the world, or tenth, or last; whether we are better or worse than we were twenty, thirty, or forty years ago, our expectations are not being met, and we must change school mathematics.

**How Do Students Come to Understand Mathematics?**

Facts about the nature of instruction that results in understanding are similarly elusive. The growth of mathematical knowledge can be viewed as a process of "constructing internal representations of information and, in turn, connecting the representations to form organized networks." (Hiebert and Carpenter 1992): "... [U]nderstanding can be viewed as a process of making connections ... either between knowledge already internally represented or between existing networks and new information."

Different views of how these connections come to be made lead to very different views of what mathematics teachers should do. Some believe teachers should reveal these connections directly to the students. Others believe teachers should pose problems that create opportunities for students to build the connections themselves.

The genius of the NCTM Standards is that they were written as political documents which large numbers of diverse individuals and organizations could agree to support. But the devil is in the details. At the next level of specificity--the level at which we make specific changes in curriculum, instruction, and assessment--consensus evaporates and our differences appear. My hope is that Project EXTEND will foster dialogs that challenge each of us to test our "facts" and opinions against the facts and informed opinions of others.

**References:**

Hiebert, J. & Carpenter, T. P. (1992). Learning and teaching with
understanding. In D.A. Grouws (Ed.), *Handbook of research on mathematics
teaching and learning* (pp. 65-97). New York: Macmillan.

Mullis, I. V. S., Dossey, J. A., Foertsch, M.A., Jones, L.R., & Gentile,
C.A. (1991). *Trends in Academic Progress.* Washington, DC: U.S.
Department of Education.

Pashley, P. J., & Phillips, G.W. (1993). *Toward world-class
standards: A research study linking international and national
assessments.* Princeton, NJ: Educational Testing Service.

Stigler, J. W., Lee, S. , & Stevenson, H. W. (1990). *Mathematical
knowledge of Japanese, Chinese, and American elementary school children.*
Reston, VA: National Council of Teachers of Mathematics.

*Donald L. Chambers is a researcher at the University of Wisconsin,
Madison. Previously he was Supervisor for Mathematics for the State of
Wisconsin. *