A calculator yields answers but not necessarily understanding. Using one takes little or no proficiency in math, which may discourage students from acquiring that skill. So in a world where calculators are ubiquitous, how do you encourage mathematical understanding?
Ilan Samson, who works as a full-time inventor, believes he has the answer. His QAMA (Quick Approximate Mental Arithmetic) calculator can do arithmetic, powers and logs, trigonometry, and nested calculations. But it won’t do anything unless the user first shows some understanding of the process and its results. Designed, naturally enough, for students, QAMA has been adopted so far by some 60 schools in the United States and Europe, and by the IEEE Electron Devices Society’s Engineers Demonstrating Science program, which the society refers to as an Engineer-Teacher Connection (EDS-ETC). The goal of the society’s program is to engage young students in electrical engineering by demonstrating and giving hands-on experience of basic concepts.
The calculator’s name, QAMA, also means “how much” in Samson’s native Hebrew. Its job is to force students to think about “how much”—or at least in what ballpark—each calculated result should be. Thus, the idea is not only to get results but also to understand them.
“Math isn’t only about calculations,” Samson says. “Without quantitative comprehension, students are just manipulating numbers that don’t mean anything. If you cover the result a student has just gotten from a calculator and ask, ‘What kind of result did you get? Small? Big?’ They may not know because they didn’t absorb what the calculation represents.”
And when users get an erroneous result, usually from conceptual or keying errors, they’re likely not to notice there’s a problem—sometimes even arguing that the numbers on the display must be right, no matter how ridiculous.
HOW IT WORKS
With QAMA, students are forcibly reminded not to accept results for which they have no explanation; unless they show at least the glimmering of an explanation, they’ll get no results at all. Enter a calculation, and QAMA won’t show the result until an estimate is entered. If the estimate is reasonable, the display will then show the actual answer, plus the estimate for comparison. Enter an unreasonable estimate, and it will be erased from the display while the calculator awaits a better one. If the student can’t produce a decent estimate in five tries, QAMA will show the result, but only after a tiresome 40-second delay.
What’s a “reasonable” estimate? That, says Samson, was the main detail to be worked out during the 15 years of QAMA’s development: How accurate does the student’s estimate have to be for the calculator to accept it and show the result? “The tolerance for the estimate must always appear reasonable to the student, requiring high accuracy for simple calculations but with greater tolerance as the task becomes more difficult,” he says.
For simple calculations, like 6 x 7, only perfect accuracy will do. For calculating integer powers, such as for 232, the estimate must be fairly close. For calculating non-integer exponents, such as 232.1, the tolerance is “more generous,” says Samson, “because the QAMA calculator knows that estimating non-integer powers is much harder.”
For √753 x tan70, estimating, say, 25 for the square root and a bit over two for the tangent (gleaned from imagining the corresponding triangle) shows a decent understanding of each operation and of their respective magnitudes, so QAMA will accept an estimate of 60 and deliver the actual result, 75.39.
But tolerances get tighter when you switch QAMA’s mode from “Estimate” to “Each.” In this mode, where students estimate and then see each component of the calculation separately, the product of the components should be easier to figure because the multiplication in this example takes place after its components (27.44 and 2.747, respectively) are known.
The estimate requirements can be turned off when fast calculation rather than learning is required. But students can’t get off so easily. Randomly flashing LEDs on the device can alert teachers when QAMA is used this way in class.
As QAMA’s user’s manual puts it, “The objective is not a uniform level of accuracy, but a uniform depth of understanding. This, in some cases, can only be proven by a fairly accurate estimate, while in other cases a rougher estimate could provide the evidence.”
“If a student comes back with the right result from QAMA,” Samson says, “the teacher can rest assured that the student had an idea of what he was doing.”
And it works, according to David Weber, who chairs the mathematics department at Preuss School UCSD, a high school in San Diego. He found that students using QAMA “improved tremendously” on their ability to estimate mathematical calculations even for topics where it wasn’t used.
“Some said it really gave them a sense of ‘reasonableness’ when looking at their answers,” he continues. “One student even said, ‘I feel like I finally have a brain that works!’ It does help kids with longer calculations, and makes them responsible for understanding what ‘tan56’ or ‘log2.4’ actually means.”
“Of all my inventions,” concludes Samson, “this is my best.”
To watch how the QAMA calculator works, visit "Educational Calculator's Instructional Video."