WASHINGTON — In a child's eyes, rebuilding New Orleans couldn't be simpler. Put the city on a hill.

Such an image, drawn in crayon by a New Orleans grade-schooler, inspired one of the most powerful designs in an exhibition of visionary postKatrina architecture now at the National Building Museum here. The show is called "Newer Orleans: A Shared Space."

The child, listed only as "Courtney S.," imagined people walking to safety up a hill they didn't have. After seeing the drawing, the Dutch architectural firm MVRDV devised a multistory school tucked inside a grassy, man-made mound. All the usual functions — classrooms, cafeteria, library — are stacked within the hill, which would double as a playground. Places that children would congregate are well above sea level. Windows and balconies poke out of the sides like spokes on a sputnik. It's a funky-looking structure, but entirely buildable.

The point of the exhibition is to propel visitors beyond catastrophe. As the wall text promises, this is "architecture that can give New Orleans hope."

Six proposals — shown through intricate models, renderings, blueprints, sketches and politically charged factoids — offer ways in which innovative design can revive a sense of community and build a stronger connection with nature.

In the re-imagined city, mangrove trees flourish in a vast, welcoming park; levee embankments are recast as picnic grounds; and civic pride finds expression in a public library so wildly creative that it would supplant the Superdome as the city's most notorious architectural icon. And children get a safe haven for a school.

The exhibition was organized in six weeks by a trans-Atlantic brain trust including Reed Kroloff, Tulane University's dean of architecture and a member of the New Orleans recovery team; Aaron Betsky, an American who heads the Netherlands Architecture Institute in Rotterdam; and Tim Griffin, editor of Artforum magazine.

"There is an absolute necessity for visionary thinking," says Betsky, former curator of architecture at the San Francisco Museum of Modern Art.

The exhibition is tightly focused. A select group of high-profile U.S. and Dutch firms was asked to address three specific problems: the design of an elementary school in a poor neighborhood; the restoration of City Park, an unkempt and nowcontaminated public green in central New Orleans; and the creation of an architectural symbol as powerful as the Washington Monument.

The American firms are Morphosis, Hargreaves Associates and Huff+Gooden Architects; chosen from the Netherlands were UN Studio and West 8 Urban Design and Landscape Architecture, along with MVRDV. All are highly regarded for unconventional approaches to structures, urban design and the re-invention of landscapes. There is no expectation that any of the designs will be realized, or even that any of the designers might become part of the actual rebuilding, although one can always hope.

The search for a symbol to represent the future produced an exploded ziggurat by UN Studio. The staggered form would house a multimedia library and civic offices while incorporating a vertical garden unlike any seen in the Garden District.

The central park flooded by Katrina becomes a major opportunity for psychic healing in the conceptual plan by Adriaan Geuze of West 8. The land would be desalinated and restored in phases, serving as a tree farm for the city and later becoming an environmentally sound delta of serpentine creeks crossed by 1,000 pedestrian bridges.

The path of the hurricane would be marked with a progression of tidal pools, which he calls the Katrina Trace Memorial.

Morphosis, the firm of 2005 Pritzker Architecture Prize winner Thom Mayne, tackled the real-world issue of who should live in the new New Orleans. Mayne adopts the highly controversial idea that the city should shrink to its most prosperous neighborhoods. Maps show a densely populated zone on high ground surrounding the French Quarter. Flooded areas extending from the Ninth Ward north and west along Lake Pontchartrain are marked for a "strategic return" to wetlands.

The blueprint is grounded in research that Mayne did for a client before Katrina struck. He determined that the city lacked the tax base to sustain, much less rebuild, infrastructure. "Radical subtraction" would alleviate the burden on city services, the plan suggests, while the central core could be revived as a cultural destination. The cost of establishing Category-5 levees has been put at $32 billion, at the least. Morphosis compares that figure with the projected $9 billion cost of buying out displaced residents.

The opposite position is taken by George Hargreaves, a master planner and past chairman of the Landscape Architecture Department at Harvard's Graduate School of Design. He notes that 70 percent of homes in the area that Morphosis consigns to wetlands were owner-occupied. That suggests to Hargreaves that if people could be made to feel safe, they would return. In his view, the opportunity to restore the city's "soul" outweighs the cost.

"Newer Orleans" opened in Rotterdam in January. The exhibition was brought to Washington through the efforts of Dutch Ambassador Boudewijn J. van Eenennaam, who traveled to New Orleans in November and organized a tour of Dutch flood protection systems for Louisiana officials in January.

Betsky hopes the exhibition, which will travel to New Orleans, New York, Chicago and Louisville, Ky., can generate discussion beyond the problems of New Orleans. Cities without defining elements such as Washington's Mall, Chicago's Millennium Park or Seattle's new public library could find strong arguments here.

As for schools, they serve as safe havens for children everywhere, but unfortunately, designs rarely live up to their potential to foster well-being.