Konstantin Kakaes's argument essentially boils down to three elements:

- Many math teachers who use technology do so ineffectively.
- Many students who are taught mathematics with calculators don't have a good grasp of basic arithmetic, or other "traditional" mathematics.
- A few teachers teach math successfully without calculators to some students, where "successfully" is defined as "according to traditional criteria."

Therefore, teaching math with calculators results in students who can't do math.

Although I would agree with #1, I would take it further: for as long as there have been math teachers, there have been many ineffective math teachers, with or without technology. As Kakaes himself acknowledges late in the article (when he claims that software won't be able to teach children "any time soon"), teaching (anything) is a complex process. Teaching math requires actually understanding math, and people who understand math have

*always*been in short supply, in and outside of the teaching profession. So a different, simpler explanation for the failure of students to learn math is that there aren't a lot of excellent (or even mediocre--see my previous post) teachers out there teaching math.
A bigger problem with Kakaes's argument is that, if it were true, we would expect to see declining math achievement in the U.S. In fact, the opposite is true. TIMSS and NAEP scores have been rising steadily for the last twenty or so years--the exact time period in which calculators became standard equipment in high school mathematics classrooms. To mention one statistic, the number of students in the U.S. passing the AP Calculus BC exam each year--half of which is no-calculator--is now more than five times the number who even

In fact, I would argue that calculators have made possible one of the great sea changes in mathematics education in the western world. In 1960, there was a dropout rate of 27%, and of the 73% of U.S. students who graduated from high school, very few took any math beyond geometry or trigonometry, which was still a course offered at many colleges. In 2009, there was a dropout rate of 8.1%, and of the 91.9% of U.S. students who graduated from high school, something like 50% (77% in 2004) had trigonometry or higher. Put differently: we are now in a world in which about half of U.S. students are expected to learn substantial amounts of advanced algebra and trigonometry before graduating from high school. Technology makes it possible, as great teachers like my friends John Benson and Natalie Jakucyn, to name two have shown, to

*took*the exam in any year in the 1980's.In fact, I would argue that calculators have made possible one of the great sea changes in mathematics education in the western world. In 1960, there was a dropout rate of 27%, and of the 73% of U.S. students who graduated from high school, very few took any math beyond geometry or trigonometry, which was still a course offered at many colleges. In 2009, there was a dropout rate of 8.1%, and of the 91.9% of U.S. students who graduated from high school, something like 50% (77% in 2004) had trigonometry or higher. Put differently: we are now in a world in which about half of U.S. students are expected to learn substantial amounts of advanced algebra and trigonometry before graduating from high school. Technology makes it possible, as great teachers like my friends John Benson and Natalie Jakucyn, to name two have shown, to

*increase*students' access to higher mathematics. With technology, it's possible for a student who doesn't know how to add fractions to learn what a derivative is, what it means, and what you can do with it--and how to let a computer do the computations that he needs to use the derivative in an actual application.
Finally, Kakaes never engages what is, to me, the central question technology poses to the mathematics teacher, namely, what of the traditional pencil-and-paper mathematics is worth teaching? Kakaes writes:

If you learn how to multiply 37 by 41 using a calculator, you only understand the black box. You’ll never learn how to build a better calculator that way.Besides the inaccurate alarmism of his example--even calculator-active elementary school curricula like

*Trailblazers*and

*Connected Math*expect that students will be able to multiply two-digit numbers by hand (and explain their computations, a higher cognitive skill than was demanded in my day)--he proves too much. If it were necessary to teach

*everyone*a skill to ensure the supply of programmers able to create machines in the future, we would presumably also teach the following:

- Computation of decimal approximations of square roots, using the "two digits at a time" method found in old textbooks, or continued fractions, or the Babylonian method.
- Approximations of transcendental functions using Taylor series.
- Approximations of trignonometric functions using matrix multiplication (faster and better for most angles, actually).
- Approximations of transcendental functions using tables and linear interpolation.

But while there's a pedagogical value to each of these (my advanced students think that continued fractions are pretty cool, as any number theorist will attest), we just don't teach them anymore. Why would we? It's inconceivable that anyone would need to know these values accurately

*without*a calculator, and while Kakaes is correct that many university math departments are stuffed full of old-fashioned mathematicians, even they use calculators (actually, Mathematica or Maple) to do these problems--and expect their students to do the same. My point is just that we all agree that there's a line to be drawn between what math students should be able to do by hand and what they can (and should) use a calculator for--we're only arguing about where that line is.
That line is porous at best. Zalman Usiskin has pointed out (in his NCTM Yearbook article on technology), even paper-and-pencil algorithms are technology, every bit as much as computer software. Old-fashioned long multiplication, as I've pictured it at right, is one:

This "killer app" version is fast and correct if you actually do it right--but many students find it hard to understand, hard to apply consistently, and--in practice--extremely inaccurate, because the most common errors (not shifting the second row over, for example) actually have huge effects on the results. Other algorithms (partial products, estimation with corrections, etc.) are not as fast, and (sometimes) only produce approximate answers. But students can actually understand and explain them, and apply them correctly. And if you really need a completely accurate answer quickly, in a grocery store or at a worksite, wouldn't you do what I do -- that is, pull out the calculator on your phone?

Kakaes does raise some valid points. Technology by itself isn't the sole indicator of high-quality math instruction: there's lots of low-quality math instruction with technology (just as there's lots of low-quality math instruction without technology). Promethean boards do not raise outcomes by themselves. And (as Sugatra Mitra says, in his argument for why technology can be transformative for the poorest children), for kids in affluent districts (which, by his standards, is much of the U.S.), the marginal impact of any given new technology might be quite low. But as Zalman argues, the question is never "should we teach students to use technology?", but "which technologies should we teach students to use?" That question--not this fake "can Johnny learn math with a calculator?" question--is where the discussion should start.

Kakaes does raise some valid points. Technology by itself isn't the sole indicator of high-quality math instruction: there's lots of low-quality math instruction with technology (just as there's lots of low-quality math instruction without technology). Promethean boards do not raise outcomes by themselves. And (as Sugatra Mitra says, in his argument for why technology can be transformative for the poorest children), for kids in affluent districts (which, by his standards, is much of the U.S.), the marginal impact of any given new technology might be quite low. But as Zalman argues, the question is never "should we teach students to use technology?", but "which technologies should we teach students to use?" That question--not this fake "can Johnny learn math with a calculator?" question--is where the discussion should start.

**Update:**A version of this article is now a posted response on Slate!