America's Choice (AC):
Since the first TIMSS results came out, has there been any improvement in mathematics achievement in the United States?

William Schmidt (WS):
The simple answer is no. It's very difficult to make the comparisons across '95 to '99 to 2003 because the number and the nature of the countries participating changed over those years. So a few reports might give the impression that the U.S.'s relative position has improved. But if you closely examine the data, there really is no change. The performance of the U.S. relatively, as well as absolutely, is pretty much the same as in '95.

AC: As you look across these studies and also at the PISA study, are there particular areas within mathematics where American students are weak or areas where they are strong?

WS: Consistently, one of our weakest areas has been geometry and the related area of measurement, since so much of measurement is geared to geometry. I don't believe there is any indication of any particularly stellar performance in any area of mathematics. In some we do a little bit better than in others, but I don't see any great strength.

AC: What do the international findings suggest about the way the American mathematics curriculum is organized? And how does this organization contribute to America's relatively poor showing on the assessments?

WS: When we initially did the work, we used the phrase "mile wide, inch deep" to characterize the American curriculum. But, analyzing the results further, we've learned that that is more the symptom than the root cause. The issue seems to be coherence: the degree to which topics are put together in the curriculum and sequenced across the years in an order that reflects the inherent structure of mathematics.

And we find this to be evident in the top achieving countries. Topics that are prerequisites for others occur earlier in the grades and the more complex ones that build on previous concepts only appear in the latter part of middle school. By contrast, in the United States, there still is a great deal of incoherence in a lot of state standards in mathematics. That's really a big issue, because it means that we don't have the rigor that many other countries do. That is, in the middle grades in top-performing countries it's about algebra and geometry, not about arithmetic. We don't have the focus that is the opposite of mile wide, inch deep.

AC: How do other countries support their curricula with materials?

WS: The instructional materials in most of these countries are driven by commercial enterprise, just as they are in the United States. The difference is, when you have a clearly articulated set of national standards, as most of these countries do, textbooks are written to those standards, because nobody would buy them if they weren't. Not having a clearly articulated set of national standards, we produce these 800-page textbooks with just about every conceivable topic at every grade level. They're not coherent and only the absolutely most thoroughly knowledgeable teachers are able to work their way through these books in a rational fashion — skipping parts they know they shouldn't do, and reorganizing some of the chapters.

AC: What can teachers do if they are not as well skilled, given this incoherent curriculum? How can they present material in ways to ensure that students learn it?

WS: I'm not sure that the answer lies directly with the teachers. Probably the responsibility lies with the districts.

District administrators need to focus on a well-articulated, coherent set of standards, then recognize that curricular materials are not synonymous with curricular standards. Using the materials and pretending they line up with the standards is folly. So administrators must think about their materials, then think about those standards, map the one on to the other, and say, "Okay, does this particular sequence in these materials make sense? If not, then change the sequencing. Are there holes in these materials? If yes, then how do we supplement or augment the textbooks?"

Some districts are doing just that, and they have the better opportunity for improving student learning, because they're trying to use their resources in a coherent fashion and then augment them when they aren't adequate.

AC: What other supports could districts or private providers offer to teachers?

WS: This comes back to professional development. We don't have adequate data to speak to it very clearly, but in general, the professional development in other countries is built around the particulars of the curriculum that's in that country. In a country with a national curriculum, professional development can be specifically geared to what it is that's supposed to be covered in a specific grade.

It's essential to help teachers understand what they need to know in order to adequately teach the topics they're supposed to be teaching.

But they really need to understand some of the topics that they're not teaching at their grade level in order to more adequately teach the topics at their grade level. For example, 5th grade teachers really need fairly sophisticated knowledge of the rational number system to effectively present fractions. Yet they say they're deficient in such knowledge. This is a vital area in which to work with the teachers.