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tough financial choices. In Kyrene, for example, even as technology spending has grown, the rest of the district’s budget has shrunk, leading to bigger classes and fewer periods of music, art and physical education.

The district leaders’ position is that technology has inspired students and helped them grow, but that there is no good way to quantify those achievements — putting them in a tough spot with voters deciding whether to bankroll this approach again. “My gut is telling me we’ve had growth,” said David K. Schauer, the superintendent here. “But we have to have some measure that is valid, and we don’t have that.”

Since then, the ambitions of those who champion educational technology have grown — from merely equipping schools with computers and instructional software, to putting technology at the center of the classroom and building the teaching around it.

. The district’s pitch was based not on the idea that test scores would rise, but that technology represented the future.

For instance, in the Maine math study, it is hard to separate the effect of the laptops from the effect of the teacher training.

“Rather than being a cure-all or silver bullet, one-to-one laptop programs may simply amplify what’s already occurring — for better or worse,” wrote Bryan Goodwin, spokesman for Mid-continent Research for Education and Learning, a nonpartisan group that did the study, in an essay. Good teachers, he said, can make good use of computers, while bad teachers won’t, and they and their students could wind up becoming distracted by the technology.

Larry Cuban, an education professor emeritus at Stanford University, said the research did not justify big investments by districts. “There is insufficient evidence to spend that kind of money. Period, period, period,” he said. “There is no body of evidence that shows a trend line.”

“In places where we’ve had a large implementing of technology and scores are flat, I see that as great,” she said. “Test scores are the same, but look at all the other things students are doing: learning to use the Internet to research, learning to organize their work, learning to use professional writing tools, learning to collaborate with others.”

It was something Ms. Furman doubted would have happened if the students had been using computers. “There is a connection between the physical hand on the paper and the words on the page,” she said. “It’s intimate.” But, she said, computers play an important role in helping students get their ideas down more easily, edit their work so they can see instant improvement, and share it with the class. She uses a document camera to display a student’s paper at the front of the room for others to dissect. Ms. Furman said the creative and editing tools, by inspiring students to make quick improvements to their writing, pay dividends in the form of higher-quality work. Last year, 14 of her students were chosen as finalists in a statewide essay contest that asked them how literature had affected their lives. “I was running down the hall, weeping, saying, ‘Get these students together. We need to tell them they’ve won!’ ”

For him, the best educational uses of computers are those that have no good digital equivalent. As examples, he suggests using digital sensors in a science class to help students observe chemical or physical changes, or using multimedia tools to reach disabled children.

engagement is a “fluffy term” that can slide past critical analysis. And Professor Cuban at Stanford argues that keeping children engaged requires an environment of constant novelty,

“There is very little valid and reliable research that shows the engagement causes or leads to higher academic achievement,” he said.

computers can distract and not instruct.

t Xavier is just shooting every target in sight. Over and over. Periodically, the game gives him a message: “Try again.” He tries again. “Even if he doesn’t get it right, it’s getting him to think quicker,” says the teacher, Ms. Asta. She leans down next to him: “Six plus one is seven. Click here.” She helps him shoot the right target. “See, you shot him.”

building a blog to write about Shakespeare’

classmates used a video camera to film a skit about Woodrow Wilson’s 14-point speech during World War I

Professor Cuban at Stanford said research showed that student performance did not improve significantly until classes fell under roughly 15 students, and did not get much worse unless they rose above 30. At the same time, he says bigger classes can frustrate teachers, making it hard to attract and retain talented ones.

he resisted getting the interactive whiteboards sold as Smart Boards until, one day in 2008, he saw a teacher trying to mimic the product with a jury-rigged projector setup. “It was an ‘Aha!’ moment,” he said, leading him to buy Smart Boards, made by a company called Smart Technologies.

. Sales of computer software to schools for classroom use were $1.89 billion in 2010. Spending on hardware is more difficult to measure, researchers say, but some put the figure at five times that amount.

“Do we really need technology to learn?”