I'm trying to make two puzzle pieces fit together.

The first piece is that Achieve -- a national organization known for the high quality of its consulting work -- has stamped Montgomery County's new math curriculum framework with its version of the Good Housekeeping Seal of Approval.

"The MCPS frameworks are rigorous and reasonable. If they are strengthened in several key areas, they can be on par with the best in the nation and the world," says Achieve.

The second piece of the puzzle is that elementary and middle school teachers all over the county are howling in pain over the new curriculum.

Can those two puzzle pieces fit together? Sure: Just turn one of them upside down, and you'll see how. The curriculum framework is a general statement of what should get taught, and when. With some important reservations and recommendations for improvement, Achieve says Montgomery County has the right idea. The curriculum plan begins in elementary school and progresses logically through the grades, ending with standards that, for the most part, are comparable to the best in the world.

But Achieve wasn't looking at the day-to-day curriculum and materials that the county has produced -- and from every report, they are not as thorough and helpful as the overall framework. In other words, the theory may be fine, but the practice doesn't match up.

Getting the theory right, by the way, is not a negligible accomplishment. A good theoretical framework is important, because it gives everyone a clear idea of what should get done and in what time frame. But the practice is where the teaching and learning all gets done, and the county has a long way to go before it does that right.

For one thing, the curriculum materials have been sent to teachers this year one at a time. They have been teaching units with no idea what the next one will look like. The next unit might arrive a day or two before the teacher is supposed to teach it, or even after it was supposed to have started.

Such uncertainty is a real hardship for teachers, and for their students.

In addition, the mostly inadequate math training of elementary and middle school teachers becomes absolutely critical when teachers are asked to do something new and different. Experienced and knowledgeable teachers can sometimes make up for a lack of training with their savvy and skills, but even they seem to be having trouble with the new curriculum.

In theory, next year should be better. With the curriculum in place, the county will presumably be able to iron out any bugs in the hurriedly written materials and provide the teachers more training.

But this way of doing things is too slapdash. The curriculum materials should have been developed, put in place and tested to make sure students learned more than with the old ones before the school system shoved them into all the schools. The medical motto "First, do no harm" should also be the motto of educators.

And the school system should have provided much more training and support than it has. Last month I mentioned that College Gardens Elementary School in Rockville has had success using math materials developed in Singapore, and has raised the achievement of students across the board in that school ["Without Better Elementary Teaching, High School Math Will Never Add Up," Jan. 2]. The fact is, however, that the materials alone are not responsible for that success, because other schools have seen little or no effect from using the materials. The key to success at College Gardens was the extra training and support the school gave its teachers, including having a designated math resource teacher who helped them use the materials, fashion lessons and talk through the issues that arise in any curriculum.

That's the kind of support teachers need. Schools and school systems ignore such needs at great peril to students.

But I want to get back to the Achieve report. Montgomery County hired Achieve to look at its math curriculum framework, its English and language arts curriculum framework, and the Algebra I, Geometry I and ninth-grade English exams.

Achieve is unfamiliar to most people but well known to those who follow education. It was founded in 1996 after one of those education summits that presidents and governors have been holding fairly regularly since the famous "Nation at Risk" report of 1983, which declared that the country's schools were experiencing a "rising tide of mediocrity."

A nonprofit organization, Achieve is a partnership among the nation's governors and corporate leaders to support educational standards that are very clear about what we want children to know and be able to do at every level of their education, and then to refashion schools so that all children meet those standards.

The organization's board of directors includes, for instance, Govs. Gray Davis (D-Calif.) and Bill Owens (R-Colo.) and chief executives of major corporations, such as Boeing Co. Chairman Philip M. Condit and Intel Corp. Director Craig R. Barrett. Achieve is supported mostly by corporate donations and fees it receives from its studies. Montgomery County, for example, paid it $195,000 to study the curriculum frameworks and high school exams. (For more information about Achieve and to see a copy of its report on Montgomery County, go to www.achieve.org.)

Because of work by Achieve and many other organizations, pressure for educational standards is building across the country, though it is uncertain whether they'll succeed, and the process of applying them is difficult and messy.

The main standard for schools traditionally has been time: All students received 12 or 13 years of schooling, six hours a day, 180 days a year, after which the schools spat them out with wildly divergent results -- some couldn't read, whereas others were doing original research in rocketry. The standards movement has been working to make sure that without putting a ceiling on learning, schools make sure all graduates can read and write well and know enough to prepare them for citizenship in a complex society.

The reason the transition is messy is that it challenges entrenched ideas that some children can't learn well because they are poor, or African American, or because they come to school speaking a language other than English, or for any number of other reasons. The standards movement holds that, given enough time and expert support, all children can learn well, and that it is up to the schools to provide them with that time and support.

In its report on Montgomery public schools' curriculum framework, Achieve was doing the nitty-gritty work of making sure that the county's standards are worthy of the name. In other words, if students learn everything they are supposed to in the math and English curriculum, will they have a good grounding in those fields and be prepared for postsecondary education and training?

For the most part, Achieve says yes. But, as I mentioned before, there are important caveats, and they are worth mentioning.

I'll talk about English another time. But in terms of math, Achieve warns against something I have written about before -- that Montgomery County seems determined to accelerate kids through a dizzying curriculum without making sure they have fully developed what mathematician Liping Ma calls a "profound understanding of fundamental mathematics." Here is what the Achieve report says: "Acceleration should not come at the expense of depth (having students develop conceptual understanding and learn to apply mathematics to new situations, even while they gain numerical and procedural fluency)."

Further, it says, "MCPS indicators tend to be written at a . . . specific level, making them appear to be more procedural -- as opposed to conceptual -- in nature. Although making the indicators clear, specific and measurable is very important, it also can create a 'checklist' mentality where teachers are concerned more with coverage than they are with cultivating deep understanding of mathematics."

Achieve is being tactful -- after all, it might want to do more work for Montgomery County in the future -- but its warning is quite clear that children need to understand fundamentally how numbers work and not just rocket through a curriculum.

In addition, Achieve finds that although the county schools' framework for geometry is excellent, the framework for algebra is not good enough -- that kids should be learning more algebraic concepts earlier and the Algebra I curriculum framework should be beefed up if kids are going to be prepared for college-level work.

Montgomery County deserves praise for initiating a serious look at its curriculum frameworks, and if it fixes the problems identified by Achieve, it seems to me that it will be time to call off the fight about what should be taught and start making sure that teachers get all the support they need to teach it.

That means teachers need high-quality curriculum materials that have proved to be effective; time to think about those new materials and discuss them with other teachers; and any additional training they lack to make them work.

Homeroom appears every week in Montgomery Extra. Send questions, opinions and issues that you would like to see discussed to Homeroom, The Washington Post, 51 Monroe St., Suite 500, Rockville, Md. 20850. The fax number is 301-279-5665. Or e-mail homeroom@washpost.com.To see previous columns, go towww.washingtonpost.com, click on the Education page and look for Homeroom under Education Columnists.