Some of Katie Bungo's fifth-grade students question why they should puzzle over basic math facts the old-fashioned way when punching a few calculator keys will instantly yield an answer.
So the teacher at Horace Mann Elementary School in the Indiana Area School District is ready with a comparison: Calculator use is like her daily drive to school, a faster approach than walking, though it's no guarantee of success.
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"Driving doesn't necessarily mean I'm going to end up at my school if I don't know what I'm doing," she tells her class.
In the same way, calculators can't replace the basic understanding of math needed to know if an answer displayed on a screen makes sense. That's why she lets her students use calculators to speed up their math, but only after they show her they understand the concept behind the work.
"Calculators are not magic," she said.
More than 40 years after the invention of these hand-held devices that have changed the way math is taught, educators still grapple with their proper role.
Few advocate barring them from class until high school or college, said Henry Kepner, president of the National Council of Teachers of Mathematics. These days, debate centers on how best to use the now-ubiquitous devices so they encourage big picture thinking, without letting them become a crutch.
Pushing buttons runs counter to a die-hard view by some parents that math is best learned the way they were taught -- by drill and memorization alone. Some still see calculators as inevitably de-emphasizing those traditions.
But some teachers such as Ms. Bungo say keeping calculators away from students would bog them down by requiring too many routine calculations by hand. And it would put them at a disadvantage on tests, such as the Pennsylvania System of School Assessment, which allow calculator use for some problems.
"We don't use clay tablets anymore. We don't use tree bark to write on. Technology keeps advancing, and we need to acknowledge that," said John D. Baker, a professor of mathematics education at Indiana University of Pennsylvania.
Educators say geometry and other areas of math can be introduced to students sooner because of calculators. For instance, being able to display the movement of triangles or other shapes on a graphing calculator allows motion geometry to be more easily tackled as early as middle school, rather than high school, Dr. Kepner said.
Students can focus more on spotting trends and less on worrying about mistakes in single calculations.
"What does a quadratic function or equation look like? I can talk about what happens if I change some of the symbols," said Dr. Kepner, who taught middle and high school math for 12 years and is a professor of math education at the University of Wisconsin-Milwaukee. "I can talk about what happens to the graph."
Imagine the tedium students would face if they performed compound interest calculations without a calculator.
"If they did it by hand, we'd come to a stop for several minutes while everybody did the work," he said.
Tom Reardon, a high school teacher in suburban Youngstown, Ohio, remembers showing up in 1993 for a seminar on graphing calculators, convinced he had little use for the device. But he became awestruck watching it display graphically the distance that a batted baseball would fly, once values were inputted for the ball's velocity, its angle of ascent and wind speed.
That's when it hit him, said Mr. Reardon, who recently retired from his high school job and now leads professional development classes on calculators, some underwritten by calculator maker Texas Instruments. "I can't have my students not see this. This is fun," he said.
Yet for all the classroom math exploration that calculators open up, do the devices help or hinder one's learning of basic addition, multiplication, division and subtraction facts?
On the one hand, research "doesn't really support" the view held by detractors that calculators make students less likely to memorize those basic facts, said Robert Siegler, a Carnegie Mellon University professor and member of the National Mathematics Advisory Panel, which suggests ways to enhance American students' math skills.
Nor does the research that the panel considered bolster proponents of calculator use.
"The research suggests that it neither helps nor hurts learning of basic arithmetic," he said. "There is some evidence that it might do a little bit of good for learning functions and problem-solving skills, but basically research stopped (around) 1986."
That was largely before the emergence of graphing calculators, now commonly used in classes such as algebra and calculus to plot graphs and tackle multiple equations simultaneously on a larger screen than a basic calculator has.
Some panel members said calculators are so common in classrooms that making research comparisons have become more difficult.
Nevertheless, the math panel last year recommended that the nation pursue additional research on the effect calculators have on students' computation, problem-solving and conceptual understanding.
Calculators have come a long way since 1967, when Texas Instruments developed a prototype. It weighed almost three pounds, a behemoth next to some basic calculators today weighing less than an ounce.
Still, the early device with just 18 keys and four functions (addition, subtraction, multiplication and division) was revolutionary. At the time, calculations were done on tabletop models that weighed more than 50 pounds, required electricity and cranked out results on spools of paper tape.
The debate about ease versus understanding that has ensued ever since has spread to the Internet.
This spring, those who teach college math were abuzz over implications of WolframAlpha, a Web site, wolframalpha.com, and computational knowledge engine that does calculations at warp speed. Its search-engine-style question bar lets anyone ask for, and receive, instant answers to complex problems.
For instance, type in the query "What is the square root of pi?" and in seconds, a decimal approximation with 58 digits appears, above a set of equations.
Some say what's important at any grade level is structuring lessons and exams so students think about the underlying meaning, rather than giving them credit for reciting numbers spit out by a display screen.
"It doesn't make any sense to ask children (on an exam) to multiply 123 times 456 if they have a calculator," said Dr. Siegler. "What's the point? You're not testing their knowledge of multiplication. All you're testing is whether they know how to use the calculator."
Being able to estimate answers becomes more important as a safeguard against hitting a wrong button. Educators say calculators should not be allowed to become a substitute for understanding.
That can happen if a calculator "is used to guess answers, rather than solving for an answer using the relevant concepts and processes," said Tamara Lakins, an associate professor and past chair of the math department at Allegheny College.
Dr. Siegler said a better use for calculators is on problems that can't be solved merely by the calculation itself, ones in which the real challenge is knowing what steps on the calculator to take.
Mr. Reardon said calculators can even sneak some math in on someone who's not excited by the subject but enjoys gadgets.
"They get their thumbs going," he said. "They'll learn some math almost in spite of themselves."
Bill Schackner can be reached at email@example.com or 412-263-1977.