The Quick, Approximate, Mental Arithmetic (QAMA) calculator could end the math wars, writes Sanjoy Mahajan on Freakonomics.

I typed “25 x 37? and pressed “=”. A short underline cursor flashed away on the bottom left of the screen, without offering an answer. Instead, it demanded an estimate. Like a skilled tutor, it answered my question with its own.

When I entered 100, it asked again. For how could two numbers, each around 30, multiply only to 100? When I tried 400 and even 800, I still got no answer. Only when I tried 900 did the calculator answer my original question and tell me the exact answer (925). By experimenting, I found that, in order to get the exact answer, the estimate must be at least as close as 814–an error of 12 percent.

Useful? Good enough to enable The Treaty of QAMA?

I don’t think Sanjoy understands the math wars. I think this calculator is an interesting idea, and I suspect that it’s use will help students get better at recognizing the size of the numbers they are calculating before relying overly on the calculator BUT that’s not the thing that math teachers are arguing about in the math wars.

The math wars are essentially about whether mathematics is a set of skills to be learned, or a creative discipline to be practised. It’s about a definition of what it means to do mathematics, and not so much about the tools being used.

No, this will not end the math wars.

I do *not* understand the education establishment’s obsession with estimation as a skill.

One does not become practiced at math because one is good at estimating. One gets good at estimating because one is practiced at math.

Skill at estimating is a symptom of proficiency, not a cause. But everywhere I turn, teachers and schools are trying to teach “number sense” and estimation skills (including some truly god-awful methods like FEE).

It’s just like in reading — people notice that good readers by and large are able to “make predictions” as they are reading, and now all of the sudden everyone has to stop and make predictions before they move on with the rest of the SSR, as if it were some sort of separate, specialized skill.

I can only wince and wait for the next thing that educators will want learners to pointlessly ape.

Two words: “Cargo cult”

Estimation is useful, but only works when the person has a reasonably accurate idea of how correct the estimate will be. In the math for dummies series, several authors show the pitfalls which can occur when using rounding (esp. for 2-3 digit numbers in calculations).

Sounds like this calculator idea is as brain damaged as the person who thought it up (at least to me)

>I do *not* understand the education establishment’s obsession with

>estimation as a skill.

Well, I can think of two reasons: first, it’s a valuable skill to have in the business world. The person who can best see the shape of a deal of any kind will have an advantage negotiating it. A person who can have an elevator conversation about numbers of some sort is going to have an advantage over those who cannot. Second, being able to estimate provides a quick and easy way to check your work for errors. Most engineers I know have a pretty good ability to spot a number that’s just wrong, somehow. I think that they are subconsciously estimating the approximate value of the number and noticing when it doesn’t fit.

This calculator won’t solve the biggest problem of calculator use: not recognizing when the user has made an input error. I use estimation to help me recognize a “garbage in, garbage out” mistake. If I accidentally entered 250 x 37 and got an answer of 9,250, I would immediately realize that couldn’t possibly be the answer I’m looking for because 25 x 40 is only 1,000. The way I’ve developed this “sanity check” estimation skill is through lots of practice doing pencil and paper calculations.

Calculators and spreadsheets are wonderful timesavers but their use should only come AFTER the student has mastered longhand calculations.

The calculator battle is only relevant amongs mid to high ability kids. Low ability kids will continue to press buttons until they get an answer. They won’t use the calculator as designed.

Michael, I don’t ever teach kids how to estimate. I teach them TO estimate by using their inherent math knowledge. Low to mid- ability kids will never remember formulas, so it’s very helpful for them to realize that they can think their way through to an approximation. They’re never going to use it in the real world, of course, but that’s true of the entire curriculum. Math in school exists to teach kids how to solve problems.