# Ending the Math Wars in a Treaty of QAMA

As a country, we are often at war. If it’s not against Germany, England, terrorism, or Grenada, it’s the war on poverty (that’s gone so well), the war on cancer (ditto), and, of particular interest to me, the Math Wars, which have been raging for decades. On one side, the traditionalists insist on drilling and back to basics, “on behalf of sanity and quality in math education.” On the other side, the reformers insist on conceptual understanding using computers and calculators, to “promot[e] the rational reform of mathematics education.”

Both are half-right and half-crazy. As the reformers say, students need to understand what the mathematics means. Students whose word problem for “6 x 3 = 18” is of the form “There were 6 ducks, and 3 more showed up, so 6 times 3 is 18,” understand little. (See “Children Learning Multiplication, Part 1,” in the articles by Professor **Thomas C. O’Brien**.) As the traditionalists say, using computers for everything leads to needing a calculator to compute what 6.5 x 10 is.

However, there’s a tool to combine the merits of both sides: the Quick, Approximate, Mental Arithmetic (QAMA) calculator. (I don’t have any financial interest in the company, or in any calculator or computer company, alas.) Mine just arrived. 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.

The calculator is skillfully programmed for learning, for its error tolerance depends on the difficulty of the calculation. When I ask for the tangent of 80 degrees, my estimate can be up to 20 percent in error. At the other extreme, for “6.5 x 10” the estimate needs to be precisely 65. (On the other hand, its strictness is sometimes crazy. The logarithm of 1001 is 3.000434 … However, 3, my first guess, or even 3.0006, which makes an error of 0.006 percent, is not accurate enough!*****)

If the QAMA calculator is the only calculator a student uses—and who needs more, for **Wernher von Braun** designed moon rockets with only a slide rule—she cannot help but think. With each use, rather than becoming dumber to the point of needing a calculator to compute 6.5 x 10, she learns mathematics ever more deeply.

The 30 Years’ War ended in the Peace of Westphalia. Could the 30-years-long Math Wars end in a Treaty of QAMA?

***Update (7/16/12)**: After an informative discussion with the inventor, I understand why the calculator is so picky here. What I learned was that the calculator mostly allowws a much wider tolerance in logarithm calculations. For example, with log(4000), which is 3.602…, it accepts 3.5. But when the user asks for log(1001), and the calculator demands high accuracy, the user must learn a new idea: the series approximation for the natural logarithm and how to convert the approximation to a base-10 logarithm. Difficult, but educational!

## Bryan H

Thanks for the heads up. As a junior high math teacher who guides students to understand estimation this calculator could be great!

## Eric M. Jones.

I have just two words for you:

RPN

## Erik Dallas

I can only use my HP Revers Polish Notation calculator, as I can touch type and it is intuitive at this point. But RPN is not as fun "learning" as this thinking estimator requirement. QAMA would solve the, I need to borrow your calculator to multiply? What problem do you have? 2x2. No you don't get to borrow my calculator for that... Thinking is important, and for more complex problems and for not making mistakes it is essential that you have an estimate in mind so that you can self-check computation errors.

## DanSanto

Is your math word problem straight? You might want to check it again.

"Students whose word problem for “6 x 3 = 18? is of the form “There were 6 ducks, and 3 more showed up, so 6 times 3 is 18,” ...."

That word problem is not the same as the straight math problem. The word problem would evaluate to 6 + 3, not 6 x 3. Or maybe it was intended as "...and 3 more GROUPS showed up....", but then that would evaluate to 6 x 4.

Or was the incorrect math problem the point, and I'm just being dense?

## Kyle

The point was that students need to understand math concepts well enough to know that 3 more ducks showing up means 6 + 3 not 6 x 3. But I don't think that means you're dense.

## Jen

The math wars are far too entrenched for this to end them, sadly, but I love it. The 10 year old is going to be getting a present, though he'd prefer a beyblade...

## Tap Estes

Bret Victor has some really interesting thoughts about math instruction...

http://worrydream.com/KillMath/

## Ian M

Being able to conceptualize and estimate may help with simple math concepts such at what is 9.5^3? Close to 1000 but definitely less than 1000. More than 9^3 (729). Closer to 9^3 than 10^3. This could all be easily visualized or explained.

What about complex problems like what is 3.2^-pi?

How can a student conceptualize that?

## Jason

I personally fall in the middle on the Math Wars, but I think the argument that the conceptualists would make on your 3.2 ^ -pi question is that we shouldn't be asking kids to solve such a problem if there is no conceivable application. Why might pi be useful in an exponent?

I would say that there are some math concepts where (and some people, including myself, for whom) learning the calculation is easier than the concept. I currently encounter these types of problems in financial math where you can calculate integrals and derivatives following set rules or by applying known patterns. However, wrapping your brain around the actual application takes a lot of practice. I would prefer to teach kids some calculation techniques so they will have the tools ready when they later study higher applications like physics, economics, finance, etc.

## Seminymous Coward

e^(i*pi) + 1 = 0 is an example of pi in an exponent being both useful and deeply enlightening.

## oldtaku

That is seriously fantastic. Probably the single most important math skill you can have is the ability to estimate reasonably well. We had an entire class on it in engineering school, and it's one of those things that just keeps on giving. But not just for engineers, if consumers could do that it would be huge.

Years ago being able to do simple arithmetic accurately might have been the most important, but now we just let the calculator or computer do the precision work - but the estimate in your head should sanity check it.

## Bradman

Why is this not a $1.99 app? You know, 1 dollar times 99...

## Eric M. Jones.

Yes, but how long did the 100-years War last?

(Ans.: 116 years from 1337-1453)

## Barry Garelick

Saying that the traditionalists in the math wars are half crazy is a simplification. I am on the traditional side, and while I advocate for procedural fluency and mastery of facts, I also maintain that procedure and conceptual understanding work in tandem. I don't advocate memorization alone. Traditional math done poorly is not a representation of traditional math. I realize this is not what your article is about, but thought I'd just enlighten you a bit.

## Michaluk

This calculator is brilliant. As someone who teaches engineering majors, for who, arguably, good estimation skills are even more important than for people in my field (physics), I find the state of fundamental mathematics knowledge to be incredibly poor. Students will routinely give negative numbers for a problem that can clearly only be positive, give answers that are off by, literally, dozens of orders of magnitude, give nonsense units, etc.

## Jen St. Onge

I wouldn't think this should replace a calculator, but it's pretty cool. Generalization and estimation are always good to learn for practical purposes.

It would be neat if it could provide relationships to other calculations. For example, if I ask what 10+10 is, it would be cool to have a hint button that tells me 10+10 = 2x10.

## David Hemmer

I think you should have another conversation with the inventor about the log example. The one he gave assumes the student is pretty sophisticated in taking calculus. What if he's a high school sophomore learning about logarithms without any calculus? This is going to be awfully frustrating.

## Digital Workshop

Fantastic concept. And the mere cognitive dissonance of a recalcitrant calculator will get brains more active in itself.

I was going to say this would be better done in software because it could learn and adapt it's accuracy avoiding some of the issues debated above - but then that would lose much of that productive dissonance.

Still might see if we can try something similar with our educational resource examples. Hmm - Today will not be the day I expected.