WASHINGTON - In a darkened Algebra II classroom, all eyes were on an illuminated graphing calculator projected three feet high on the white board as students studied a series of graphs and talked about absolute value functions.

The weightless image of a TI-84 Plus Silver Edition graphing calculator is a far cry from early typewriter-size calculators that weighed 55 pounds and plugged into an outlet. This year marks the 40th anniversary of the moment that revolutionized not only the calculator but also the way students learn math. It was 40 years ago that three Texas Instruments scientists shrank that monstrosity and created the hand-held calculator.

In math classrooms, calculators mechanized finger counting, pencil-and-paper calculations and slide rules. Now students move from calculators that can add, subtract, multiply and divide to graphing calculators in later grades. Future students will use programmable devices that show algebraic formulas, graphs and word problems on the same screen.

But as the technology continues to advance, a question remains: Are the devices helping or hurting students? Educators are deadlocked over whether calculators are helping create a more numerate society capable of claiming the next technological breakthrough or making students technology-dependent and mathematically insecure.

The United States lags in international math exams. Top performers, including Singapore and China, put more emphasis on mental math and memorization and introduce calculators to the curriculum later than the United States does, said Tom Loveless, director of the Brown Center on Education Policy at the Brookings Institution, who has researched calculators and student achievement.

Given a shortcut, Americans embraced it. Educators pride themselves on circumventing repetitive, drill-based learning and instilling creativity and curiosity in students.

"We can jump past the grunt work and get to more sophisticated levels of analysis," said James Rubillo, executive director of the National Council of Teachers of Mathematics.

But the concern is that students rely on calculators too much.

Elizabeth Korte, a math teacher at Stone Bridge High School in suburban Loudoun County, Va., said calculators can be a huge help to understanding math trends and principles. But only "in their place," she said.

"You should memorize your basic math facts. Period. The end," she said. "If you are comfortable with mental math, it's easier to do higher math."