Maybe he was brushing his teeth with Ben-Gay

Galluccio claimed he had used a toothpaste that mistakenly triggered the chemical breath test machine installed in his home. Galluccio failed the test the first time he was required to take it. Nestor scornfully sentenced Galluccio to one year’s imprisonment.

Perhaps I am too harsh on our State Reps.  But please allow me the smallest amount of schadenfraude for seeing such an idiot finally be put behind bars.  Maybe the next question we can ask ourselves is how many of our State Reps end up with a criminal record upon exiting Public Service?

The reason why the public is so angry with the TSA

In studying the media’s coverage, officials have come to conclude that a slow news week, combined with the president’s being overseas and Congress being out of session, created the perfect storm of bad coverage.

Yeah, that’s the ticket.  It’s not the fact that this stuff catches only the stupid terrorists or that it does not work against cavity bombs.  And when someone tries to set off a cavity bomb, to what sort of collective punishment will we be subjected?

An unintuitive limit

Problem: Let \(f(n)\) = the number of zeros in the decimal representation of \(n\).  For example, \(f(1009)=2\).  For \(a>0\), define



\(L=\displaystyle\lim_{N\to\infty}\frac{\log S(N)}{\log N}\).

Solution: The key is to recognize that the sum \(S(N)\) is best evaluated according to how many digits are in \(N\).  As an example, suppose \(N=9\): none of the single-digit numbers between 1 and 9 have zeros, so the sum \(S(N)\) is equal to 9.  Within the two-digit numbers, note that there are 9 numbers with 1 zero [e.g., 10, 20,…,90] and the rest [90-9=81] with no zeros; in this case \(S(99)=9 a+81+S(9)=9 a+9^2+S(9)\).  For three-digit numbers, there are 3 possibilities: the number can have 2, 1, or no zeros.  There are 9 such numbers with two zeros.  Numbers with 1 zero include 101 and 110; note that there are 2 different numbers resulting from two non-zero digits and one zero digit.  Given that there are \(9^2\) combinations of digits, it is clear that there are \(2\times 9^2\) three-digit numbers having exactly one zero digit.  Finally, there are \(900-162-9=729=9^3\) three-digit numbers with no zeros.  Therefore,

\(S(999)=S(10^3-1)=9 a^2+2\times 9^2 a+9^3+S(10^2-1)\).

The pattern here is clear to the [mathematically inclined] observer:

\(S(10^k-1)=9 a^{k-1}+\binom{k-1}{1}\times 9^2 a^{k-2}+\ldots+\binom{k-1}{k-2}\times 9^{k-1} a+9^{k}+S(10^{k-1}-1)\)


\(S(10^k-1)=9 (a+9)^{k-1}+S(10^{k-1}-1)=9 \displaystyle\sum_{j=0}^{k-1} (a+9)^{j}\).

Summing the series, we arrive at the following:

\(S(10^k-1)=9 \frac{(a+9)^{k}-1}{a+8}\).

In evaluating \(L\), observe that the limit \(N\to\infty\) is equivalent to \(k\to\infty\). In this limit, \(\log N \approx k\). We therefore get the final result:

\(L=\log (a+9)\)

Bilious Rubin

I don’t normally participate in making fun of anybody’s name: under most circumstances, people do not have control over their names and bravely go through life with them.  That said, I must make one example for Jennifer Rubin of Commentary, a persistently hypocritical blog writer on the Contentions site who is little more than – no, nothing more than – a shill for the GOP.  Of course, there’s loads of those, what makes her special?  As I will explain below, it is the sort of bile in her writing, the incredibly obvious double standards she has for people, that drives me nuts.  And someone who spews bile like this and has a last name of Rubin…I cannot resist.

Anyway, Jennifer Rubin  has been at the top of the shrieking heap against the Park51 Islamic Community Center, 2 1/2 blocks from NYC.  I’m not going to post any of her rants about it as anyone reading this post has heard it all before: the acknowledgment of the “right” to build, but the necessity to be “sensitive” to the 9/11 families [some of whom are Muslim, and others of whom support the project].  Plus, ugly questions about the funding source for the center and the personal leanings of Imam Rauf, the spiritual leader of the Cordoba Initiative, the group behind the project.  All of it is bullshit, and further commentary on it by me is a waste of your time.

No, I want to lay before you the sort of character this political marionette is or has become.  I don’t know her personally of course, so I can only judge her from her writings.  And believe me, judging from her writings, she is a unique character: an anti-Semite who accuses other Jews of being anti-Semites.  I do recognize how bizarre this charge is, and how easily it can be turned back on me.  But seriously, you must see what I mean to believe it.

I posted before about an article Rubin wrote this past January called “Why do Jews hate Palin?”  To summarize, Rubin looooves Palin because Palin loooooves Israel, despite the fact that Palin’s stated position on Israel aligns with fundamentalist Christians and Jews and is a sure path, according to every Mideast expert, to the destruction of Israel as a democracy and a haven for Jews worldwide.  Jews hate Palin in droves not because of Palin’s propensity to lie, or be divisive, or to be a dumbass.  No, Jews hate Palin because they hate who she is.  Palin is sexy, Jews like frumpy.  Palin is blue collar and has worked jobs Jews won’t touch.  Palin’s child is in the military, Jews never do that.  Palin has 6 (or 5) children, Jews never have that many kids [oog, unless their frum, then they’re OK].  Jews read, but they have been misinformed about Palin, who loved to read as a child, and they simply don’t appreciate “instinctual” leadership anyway.  O, and Palin has a Downs baby, while Jews love to abort.

If you think my interpretation of her article is extreme, it is not: people who are paid to notice these things for a living say the same damn thing.

To its credit, Commentary publishes readers’ letters, and has the author respond.  The letters chosen, of course, are chosen from the milder lot [the practice of any magazine], but the charge remains and Rubin responds:

Other readers…found in my article echoes of anti-Semitic tropes or fodder for anti-Semites, especially with respect to characterizations of “elitism” or “intellectualism.” But there is, I would suggest, nothing remotely anti-Semitic about the observation, supported by Tom W. Smith’s 2005 Jewish Distinctiveness in America: A Statistical Portrait and ahost of other data, that Jews are more educated than the population at large. Nor is there anything controversial in observing that the proportion of Jews in intellectual professions is higher than the proportion of Jews in the population. Simply observing an antipathy felt by a disproportionately well-educated, highly credentialed group for a politician with a different persona and background is not a value judgment on either half of the equation. Rather, it is a candid recognition that Palin and most American Jews simply don’t share a common perspective or life experience.

This is, of course, garbage.  Yes, the document she cites goes into the themes she mentions, but there is a huge difference.  Smith’s document, at 154 pages, is a carefully written sociological study [and, at the risk of attaining my wife’s ire, I must point out it lacks, despite heaps of table of results backing the conclusions, no decent summary of the statistical validity behind those conclusions] whose validity is does no use here to question.  David Harris, the head of the American Jewish Committee who published the survey, says the following:

Why should we care about all of this—other than for “bragging rights” or a parlor game? We care because the numbers reveal an underlying strength of the American Jewish community: Despite our declining share of the overall American population, a high intermarriage rate, and a growing geographical dispersion, Jews have been able to retain a distinctive profile which bespeaks a unique core Jewish identity. Furthermore, Jews have embraced certain broad values, such as belief in the importance of education and in expressive individualism, that seem to have resonance for other Americans as well.  Why should we care about all of this—other than for “bragging rights” or a parlor game? We care because the numbers reveal anunderlying strength of the American Jewish community: Despite ourdeclining share of the overall American population, a high intermarriagerate, and a growing geographical dispersion, Jews have been able to retain a distinctive profile which bespeaks a unique core Jewish identity. Furthermore, Jews have embraced certain broad values,such as belief in the importance of education and in expressive individualism,that seem to have resonance for other Americans as well. [Emphasis mine.]
That is, Jewish identity is something of which the American Jewish community should be proud.  Further, it is meaningful that American Jewish values have become American values as well.
The importance of this statement cannot be overemphasized with respect to Rubin, whose article bemoans these values because, let’s face it, it makes her an outlier with respect to her hero-goddess Palin.  She turns these values on their head as negative and anti-American.  That her article was published in Commentary and not the Occidental Quarterly is surreal.
Now, for my cri de coeur.  Peter Beinart, who has  stood up and questioned the role of American Jewish leadership in its stewardship of young Jewish Americans’ values, wrote with regard to the Park51 project:

And oh yes, my fellow Jews, who are so thrilled to be locked arm in arm with the heirs of Pat Robertson and Father Coughlin against the Islamic threat. Evidently, it’s never crossed your mind that the religious hatred you have helped unleash could turn once again against us. Of course not, we’re insiders in this society now: Our synagogues grace the toniest of suburbs; our rabbis speak flawless English; we Jews are now effortlessly white. Barely anyone even remembers that folks in Lower Manhattan once considered us alien and dangerous, too.

To which Rubin, of all people, writes:

As for Beinart’s second paragraph, it is an unfortunate example of the bile that can be splattered on Jews by Jews, with nary an eyebrow raised by elite opinion makers. Had Pat Buchanan, to whom Beinart lately bears an uncanny resemblance, accused Jews of walking with Father Coughlin, or had Al Sharpton (before becoming part of polite liberal company) referred to Jews as “effortlessly white,” I imagine all sorts of elites would be throwing a fit. But now it is par for the course.

I imagine she writes this with a straight face, for she clearly feels she has acquitted herself of the same accusation laid onto her by her intellectual superiors.  [I also love her use of the word “bile”.]  But all Beinart is pointing out is that the language used by many Jews is exactly analogous to that used by their former tormentors in years past.  And Beinart is not the only one to make this observation as well.

So, on the one hand, Rubin uses an anti-Semitic canard to someone who is attacking anti-Semitic language in discourse regarding American Muslims.  On the other hand, she is happy to take findings from an AJC report about “Jews” [ill-defined, but I assume they know of whom they speak] and turn them on their head to expose the other-worldliness of those values because they result in a severe dislike of Sarah Palin.

Anyone not lobotomized should only read Jennifer “Bilirubin” Rubin’s yellow journalism with derision.

Hoisting the world by a string

Problem: I have a rope, it fits around the equator exactly once. I add 10 cm to the rope, attach the ends, and pull up. How high off the ground can I pull the rope?

Solution: The length \(L\) of the rope before adding the 10 cm is \(L=2 \pi R\), where \(R\) is the radius of the earth and is about \(6.4 \times 10^6\) meters.

After adding the 10 cm, you give the rope a stretch in the center.  When the rope is stretched to full tautness, the result will be most of the rope hugged against the equator, with 2 straight pieces, each tangent to the equator.  Thus, we have the right triangle as pictured above.  We are interested in solving for the height \(h\).

We proceed by considering the rope after the addition of the 10 cm which will now be denoted as \(\Delta L\).  When pulled up, the rope hugs the earth outside the points of tangency.  Denote the points of tangency on the earth as corresponding to an angle \(\theta\) from the vertical, and the length of one of the two sections of rope not attached to the earth as \(y\).  The following relation holds:

\(L+\Delta L=2 (\pi – \theta) R+ 2 y\),


\(y^2=(R+h)^2-R^2=2 R h+h^2\).

Use the fact that \(L=2 \pi R\) and \(\theta=\tan^{-1} \frac{y}{R}\), and define \(w=\frac{\Delta L}{2 R}\) and \(z=\frac{y}{R}\).  The above equation is then rewritten as


where we must solve for \(z\).  Of course, this is a transcendental equation that cannot be solved exactly, but it is clear that, since \(w\) is very small, then \(z\) must also be small, and we can get places by expanding the transcendental function in a series:

\(z-\tan^{-1}z=\frac{1}{3} z^3-\frac{1}{5} z^5+\frac{1}{7} z^7-\ldots\)

The best way to use this series is to consider the first term, with all higher-order terms being some error:

\(z-\tan^{-1}z=\frac{1}{3} z^3+O(z^5)=\frac{1}{3} z^3 \left [ 1+O(z^2) \right ]\).

Then, to lowest order, we obtain the following:

\(z=\left ( 3 w \right )^{\frac{1}{3}} \left [ 1+O(w^{\frac{2}{3}}) \right ]\).

Now, to lowest order,

\(z=\sqrt{2 \frac{h}{R}} \left [ 1+O \left ( \frac{h}{R} \right ) \right ]\),

so that we now have an approximate solution and an error estimate:

\(h=\left ( \frac{9}{32} R \Delta L^2 \right )^{\frac{1}{3}}+O\left [ \left ( \frac{\Delta L^4}{R} \right )^{\frac{1}{3}} \right ]\).

Note that the solution involves the radius of the earth, which is a very large number compared with the rope extension of 10 cm.  The result will then be surprisingly large; the first-order term is, using the values given above, about 26.2 meters.  The error term, on the other hand, is on the order of 0.001 meters, or 0.1 cm, and can safely be called negligible.  However, for larger extensions, we can simply expand the series solution further until the error estimate is within acceptable bounds.

The ADL is a joke, and Peter Beinart is right

A long, long time ago, in a galaxy far, far away…[August 19, 2007, to be exact]

I wrote my first Facebook note, taking aim at the Anti-Defamation League for firing a local leader named Andrew Tarsy.  Why?  For carrying out the ADL’s mission in Watertown, MA, of “secur[ing] justice and fair treatment for all.”  In this case, the Armenian community, which had declined to take part in the ADL’s programs because Abe Foxman, the head of the ADL, continued to insist that the massacre of 1.5 million Armenians at the hands of Turks in 1915 did not constitute a genocide.  Given that the ADL is expert at such things, you would imagine it would stick up for the historically harassed Armenians, whose quest for justice has been roundly trivialized.  But, no such luck.  In fact, the ADL roundly fought hard against any efforts on part of the Armenians to have the historical record corrected.

Why?  Because Turkey, in 2007, was a friend to Israel.  And friends of Israel can never do wrong.  Foxman “consulted Elie Wiesel” and invoked Morganthau the Senior in performing the bee-like dance that required him to help our Turkish friends avoid the Genocide label.  [Of course, never mind that Raphael Lemkin, the originator of the term “genocide”, was motivated by the…, um, genocide of the Armenians to invent the word “genocide”.  Lemkin, by the way, was a Jew whose life would be horribly affected by a genocide against his own people.  The whole brutal story can be read in A Problem from Hell by Samantha Power.]]

Which brings me to current events, and we can now guess where this is heading.  Fast forward to June 2010, and here we have Turkey, who has been acting in a rather hostile way toward its old friend Israel.  The Gaza flotilla, for example, was Turkish in origin; the IHH which ran the operation is Turkish; and Turkish PM Recip Erdogan, rather than wondering how Turkish citizens sailed off for a hostile encounter with an ally, is making serious threats.

So, who knows how this will play out.  But what interests me is the integrity of our Jewish-American institutions, like the ADL.  And, gee, what do you think would be the position of the ADL on the Armenian genocide today?  Hmmmm…:

Today, far from being an asset for Turkey, the American Jewish community appears to becoming a potent foe of Turkish interests in Washington.

On Tuesday for example, the Anti-Defamation League issued a press release calling on the State Department to designate the IHH, the Turkish charity that helped organize the free-Gaza flotilla as a foreign terrorist organization. In Turkey, the IHH has been praised as a group of peace activists and humanitarians.

“In terms of the Jewish community and Israel, neither one of us wants to throw it away and hope it is not over,” Mr. Foxman said. “But every day there is another provocation. Every day the Turkish government goes out of its way to be insulting to Israel and another link is broken.”

Morris Amitay, a former executive director of AIPAC who has also represented Turkey, was more blunt.

“If someone asked me now if I would try to protect Turkey in Congress, my response would be, ‘You’ve got to be kidding,'” he said.

So, watch out for the Armenians now getting Congress to finally recognize their genocide by the Turks in 1915.  Because, for the ADL, truth and justice only means being on the right side of Israel.  And Peter Beinart is dead right: our American Jewish institutions have failed us.

Danny Ayalon can go fuck himself

Danny Ayalon is the Deputy Foreign Minister of Israel.  Both he and his boss, a racist from the Yisrael Beitanu [Israel Our Home] party named Avigdor Lieberman, represent an unfortunate trend of Israel being seen as less and less likable to its allies.  But it is Ayalon who is in my line of fire this week, for several reasons.

Ayalon made the news last month because of an incident which a skilled diplomat could have handled discretely and in a dignified manner.  Turkey, with whom relations with Israel have been going down the tubes, had one of its stations broadcast a program with anti-Semitic themes.  A quiet complaint and message of understanding the balance between freedom of expression and placing Jews in danger would have been appropriate.  But Ayalon, representing an increasingly tone-deaf Israel, made a mess of things:

Footage of Mr Ayalon urging journalists to make clear the ambassador was seated on a low sofa, while the Israeli officials were in much higher chairs, has been widely broadcast by the Israeli media.

He is also heard pointing out in Hebrew that “there is only one flag” and “we are not smiling”.

In an interview with Israel’s Army Radio on Tuesday, Mr Ayalon was unapologetic.

“In terms of the diplomatic tactics available, this was the minimum that was warranted given the repeated provocation by political and other players in Turkey,” he said, according to Reuters.

One Israeli newspaper marked the height difference on the photo, and captioned it “the height of humiliation”.

That, however, is not even my main problem with him.  Massachusetts Congressman Bill Delahunt is part of a contingent of Democratic lawmakers in Israel, traveling as part of a mission sponsored by liberal lobby group J-Street.  I will say that I have been warming to J-Street, although I do not like the arrogance of its identification as “pro-peace, pro-Israel”.  [So is the Zionist Organization of America…it just has vastly different definitions of these terms than does J-Street.]  Standard protocol is that such a delegation, if desired, may seek out an audience with Ministry officials.  However, Alayon, because he feels that J-Street is not in fact “pro-Israeli“, has locked the delegation out from any such meetings:

“We were puzzled that the Deputy Foreign Minister has apparently attempted to block our meetings with senior officials in the Prime Minister’s office and Foreign Ministry – questioning either our own support of Israel or that we would even consider traveling to the region with groups that the Deputy Foreign Minister has so inaccurately described as ‘anti-Israel,'” Delahunt continued.

“In our opinion this is an inappropriate way to treat elected representatives of Israel’s closest ally who are visiting the country – and who through the years have been staunch supporters of the U.S.-Israeli special relationship.”

Ayalon has clearly associated himself with the rabid right.  The combination of this tone-deaf outlook and utter incompetence as a diplomat is making for an explosive combination.  How dare he insult a delegation from the United States because their politics do not dovetail with his!  It is men like Ayalon that will be the downfall of Israel.  For that, he can go fuck himself.

Meanwhile, Michael Oren, in contrast to his dumbass administration colleague, is wisely making peace with J-Street.  Good for him.

An improper double integral

Problem: Evaluate the following double integral.

\(I=\displaystyle\int\limits_{-\infty}^{\infty}dx\int\limits_{-\infty}^{\infty}dy\; e^{-x^2-y^2-(x-y)^2}\)

Solution: This problem was taken from the collection “Berkeley Problems in Mathematics“, Problem 2.3.3.  Two solutions are given, neither of which are close to [stylistically] mine, which I give below.

First, change to polar coordinates; that is, \(x=r \cos \theta\), \(y=r \sin\theta\).   Using the Jacobian \(dx\,dy=r\,dr\,d\theta\), we get

\(I=\displaystyle\int\limits_{0}^{2\,\pi}d\theta\int\limits_{0}^{\infty}dr\,r\;e^{-2\,(1-\sin \theta\, \cos \theta)\,r^{2}}\)


\(I=\displaystyle\frac{1}{4}\,\int\limits_{0}^{2\,\pi}\frac{d\theta}{1-\sin \theta\,\cos \theta}=\frac{1}{4}\,\int\limits_{0}^{2\,\pi}\frac{d\theta}{1-\frac{1}{2}\sin \theta}\).

Now, there are two ways I can think of to go about evaluating this latter integral.  First, we can Taylor expand and hope for the best; it turns out that the resulting series converges to something well-known, but you have to be an expert at recognizing such things.  [I so hate solutions that require a deus ex machina like that.]  The other way is to convert to complex variables and use the Residue Theorem. [I’m afraid, however, if you are not familiar with the Residue Theorem, then we are back to a deus ex machina. But one has to draw the line somewhere, I guess…]

So, consider the following integral:

\(J(a)=\displaystyle\int\limits_{0}^{2\,\pi}\frac{d\theta}{1-a\,\sin \theta}\;:|a|<1\).

Observe that

\(\displaystyle \sin \theta=\frac{1}{2 i}\left (e^{i \theta}-e^{-i \theta} \right )\).

The trick is to recognize that we are integrating over the unit circle \(C\).  if we let \(z=e^{i \theta}\), and transform to an integral over \(z\), then the result is the following:

\(J(a)=-\displaystyle\frac{2}{a}\,\displaystyle\oint\limits_C \frac{dz}{z^2-i \frac{2}{a}\,z-1}\).

Recall that a residue of a function \(f\) at \(z=z_0\) is equal to

\(\mbox{Res}(f;z_0)=\displaystyle\lim_{z \to z_0} (z-z_0)\,f(z)\),

with the Residue Theorem stating that, for a function \(f\) having simple poles \(\displaystyle\{z_n\}_{n=1}^{N}\) within the simple closed curve \(C\), then

\(\displaystyle\oint\limits_C dz\,f(z)=i\,2\,\pi\,\sum\limits_{n=1}^{N} \mbox{Res}(f;z_n)\).

To compute \(J\) using the Residue Theorem, we must compute the roots of the quadratic in the denominator of the integrand.  These roots are at

\(z=z_{\pm}=\displaystyle\frac{i}{a}\left (1 \pm \sqrt{1-a^2} \right )\).

Note that \(\displaystyle |z_{+}|>1\), so that we need only consider the root \(z_{-}\).  Hence,

\(J(a)=-\displaystyle\frac{2}{a}\,i\,2\,\pi\,\frac{1}{z_{-}-z_{+}}=\frac{2 \pi}{\sqrt{1-a^2}}\).

Finally, the result is

\(I=\displaystyle\frac{1}{4}\,J \left (\frac{1}{2} \right )=\frac{\pi}{\sqrt{3}}\).

Is it cheating to use a symbolic math computer to do your homework?

Fascinating demonstration given by Conrad Wolfram of Wolfram Research at TEDx, concerning the question of whether or not one cheats by using Wolfram Alpha to do your integrals for you.

The short answer to the question is that there is cheating going on, but not in the way someone who asks this question would think. The gist is that, as Wolfram claims, about 80% of math education consists of hand computations: computing integrals, derivatives, limits, roots, matrix inverses, etc. But not only is this all incredibly boring, but it also ill-prepares students for the real mathematical challenges out there. Really, the challenge is to teach students how to translate real-world problems in business, engineering, etc., into a mathematical language. Once the pure computation problem is set up, then a machine like Wolfram Alpha can turn the crack and generate data. The remaining challenge is to figure out how to interpret the data, and such an interpretation does not lend itself to a black/white solution.

Another point that Wolfram makes is that calculus should be taught a lot earlier than it is now. When, he does not say, but he makes the case that there are concept in calculus, namely the limit, that a “3 or 4 year-old” could grasp. He points to a terrific visual example of using inscribed polygons to approximate \(\pi\).  The greater point is that math education in the US needs a radical reshaping, and that computers are crucial in this reshaping.  The cheating done, in the meantime, is not by the students, but to the students, because they are being told that the computational tools they will use int he real world to solve problems are viewed as verboten in school.

In my opinion, Wolfram has a number of terrific points and his demonstration is valuable and should be viewed by anyone with an interest in math education.  But ultimately, Wolfram’s proposals would create a generation of students with too much trust in the computer, and by extension the people who program the computer.  One must remember that Wolfram is in the business of providing computational engines, and the stock of his company rises if the people behind his company are seen as the gatekeepers to a mysterious technology.  It is not unlike the trend of making automobile engines more computerized and less able to be worked on by average people.  By saying that the messiness of computation is boring and turns off students, we increase the reliance of math professionals on the computer and leave out the crucial skill of checking the computer for errors.

I have had quite a bit of experience with this issue in my work at IBM.  In semiconductor lithography, one of the main challenges is to simulate the physical processes involved in imaging circuit patterns on a wafer.  The calculations involved in this simulation are extremely complex and very heavy-duty.  We did rely on software packages to do a lot of this, but most of the time, the models we built with these software packages led us astray.  Was it a problem with the data, or was the computer lying to us, or, even more subtly, was the computer telling us the truth but we were making false assumptions about that truth?  The problems in trying to answer these questions were severe: taking data on the few running machines we had was expensive and getting time was difficult.  The software vendors were always too busy to answer our difficult questions about the integrity of their computational models.  The only practical way to deal with this was for IBM to have someone who could devise simple tests that would reverse-engineer the engine’s algorithm and assess from where mistakes were coming.

That someone was invariably myself, as I had all the necessary background, both from my schooling and my work experience.  I knew how to look under the hood.  More importantly, I knew how to derive the equations that went under the hood.  And many of these equations weren’t simple expressions that could be typed into Wolfram Alpha.  Rather, such equations required careful geometrical reasoning and pattern matching that was difficult, if not impossible, with which to trust such a tool as Wolfram Alpha.  In fact, I found it best to be completely distrustful of the computer as I was building my test cases.  These test cases would be designed so as to be hand computable, yet nontrivial.  Once these test cases were designed and computed, then the diagnosing of problems could commence.

Furthermore, without someone to understand how to plumb the depths of how computations are done, we would not get users who can diagnose incorrect results at the chip level.  That’s right, recall Intel’s Pentium FDIV error.  Finding this error took a forensic approach to computation – an approach that none of us would have in Wolfram’s world, as none of us would deign to even think about so lowly an operation as division.  And, irony of all ironies, Wolfram’s flagship product, Mathematica, has not been without its own problems over the years – not just standard-issue software bugs, but incorrect algorithms.

As to the point Wolfram makes that calculus can be taught a lot earlier – making allusions to 3 or 4 year-olds.  I’m not so sure.  Yes, the basic calculus concept of the limit is easy to grasp, but beyond the most superficial level it is essentially a deus ex machina.  Further, applying those limits to sequences and series involves the culmination of everything a typical calculus student has learned.  Sloppy analytical techniques leads to an inability to solve problems, even if the calculus concepts are well understood.  I have a terrific example of this from my days as an undergraduate tutor in the Math Dept at UMass.  I used to sit in the calculus drop-in centers for students taking the business calc [Math 127/128 for those of you who know of which I speak].  Now, I admit, this was not the calculus that one with serious mathematical curiosity took, but still.  Anyway, at some point in time, the students were required to perform double integrations of polynomials over 2 variables, and come up with a number as an answer.  The drop-in center got real busy with folks who were simply perplexed.  A typical conversation would go like this:

  • Me: So, tell me, what’s troubling you?
  • Student: I can’t do these integrals!
  • Me: Well, why don’t you do this one in front of me, and let’s see what’s wrong.
  • Student: OK.  So first I do the integral over y…is that right?
  • Me: Yes.
  • Student: Now I do it over x.  is that right?
  • Me: Looks good.
  • Student: Now I plug in the limits and…it gives me a different answer than what the answer key tells me.
  • Me: that’s because you added wrong.  1/2 – 1/3 = 1/6, not what you wrote.
  • Student: huh?  I don’t understand?
  • Me: Do you know how I got 1/6?
  • Student: No.

So, what we learn here is that the student understood the mechanics of integration, but couldn’t add fractions.  How is such a student supposed to comprehend a result from Wolfram Alpha?

So, I disagree that the mechanics of computation are best left to the experts.  I do think that there is a place for learning the mechanics of a root solve, or an integration – in fact, many, many such operations – as a part of math education.  I do agree that computers should play a greater role in math education, and perhaps elements of calculus could be taught earlier.  But hand computation is essential if we are going to educate a class of people ready to question authority.