A comment to the follow-up to my "A Little Multiplication Could Have Gone a Long Way" post says:

What? You mean not everyone memorizes useless conversions like that there are 1440 minutes (or 86400 seconds) per day? What is this country coming to?

I'd have let this slip, but given that the whole thread was about multiplication -- and that I'm a math geek -- I just couldn't resist. First, knowing how many minutes there are in a day, it turns out, is not useless: Among other things, it would help journalists and press release authors avoid errors like the one I was blogging about.

But second, here's a secret -- *you don't have to memorize the conversions*. Even if you don't remember the conversion, you can still figure out how many minutes there are in a day, whenever you need to (for instance, if you want to check whether the item you're about to publish is accurate). How, you might ask? What occult science will give me this magical power? Why . . . *multiplication*!

In fact, you don't even know how to *do* multiplication, since there are, I'm told, electronic devices that can do it for you. All you need to know is that such an operation exists, and that it can be deployed to solve immensely difficult problems like the "how many 5-minute increments in a day" one? (To be fair, it also helps knowing about multiplication's partner in crime, *division*.)

As it happens, I do remember a rough estimate of the number of seconds in a year, partly because one runs into these "every X seconds/minutes Y happens" -- 30 million, or (for a better approximation) 10 million pi for math geeks. I don't remember the number of minutes or seconds in a day. But I am so learned that the numbers are nonetheless available to me *whenever I please*. And *you too* can have this fearsome power . . . .

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I wrote to him and noted that such a figure would indeed make for a major headline - and that he probably meant to express this in the standard format of events per 100,000 population.

"Oops! Good catch." he replied.

After all, it stands to reason that you are less skeptical towards positions or issues you agree with. Which would mean more instances where you don't even stop to think at all.

lookslike, for instance, Avogadro's number, but in reality has nothing to do with it. True story, happened two days ago, while cooking. 6.02 was a guess at the ratio intended by the instructions (Thai shrimp paste, don't make it without ventilation). Lucky for me, my partner tends to enjoy^H^H^H^H^Htolerate non sequitur math and physics.It is said that math geeks tend to have a rich inner life, in the pejorative sense. I won't speak for others, but it is true of me.

1 day / 5 minutes =

For instance, I heard a story the other day that the Republican caucus, on voting for a new whip to replace someone removed because of claims of ethical impropriety, had to re-do the first round of voting when they ended up with more votes than there were members of the caucus (true story.) I submit that this sort of thing wouldn't happen if they were better at math... :->

And that is what is most amazing about Mr. Volokh, a prodigy at age 15, an amazing accomplishment he rarely talks about. A feat a math-impaired autistic really admires. My mother was a math geek, but not by age 15, although she topped her classes in then-called atomic physics. All of it completely beyond the comprehension of myself, sad to say.

"In fact, you don't even know how to do multiplication, since there are, I'm told, electronic devices that can do it for you. All you need to know is that such an operation exists, and that it can be deployed to solve immensely difficult problems like the "how many 5-minute increments in a day" one?"

Even the prodigy's who really have the know how will never convince those standardized test designers, this time I am referring to No Child Left Behind (but by analogy to others), who have implemented a policy to prohibit those electronic devices and the idea a peson can be very competent by simply utilizing one while knowing about the issue.

But then, who ever agrees with anything I have to say.

I heard a story the other day that the Republican caucus, on voting for a new whip to replace someone removed because of claims of ethical impropriety, had to re-do the first round of voting when they ended up with more votes than there were members of the caucus (true story.) I submit that this sort of thing wouldn't happen if they were better at math...How to put this delicately... do you think that's a failure of math skills, or, perhaps, um, ethics? "One person, one vote" isn't something that requires precalc, afterall. I suppose procedure could be at fault, but they have been doing this for a rather long time, and if process is at fault, that's more damning than cheating.

I don't intend to put words in Ross's mouth, but I think the smiley that you omitted in your quotation of his post implied that he too believed it was actually more a problem of ethics than of calculation.

My guess is that Prof. Volokh knows about the Asimov story, and consciously intended this post as an homage to it (and him).

Day-Petrano:

I think students should understand how to do arithmetic calculations before being allowed to use calculators. Otherwise, their sense of numbers will be so poor that they will not even be able to multiply by 1 in their heads (true story).

I don't intend to put words in Ross's mouth, but I think the smiley that you omitted in your quotation of his post implied that he too believed it was actually more a problem of ethics than of calculation.Erm, color me embarrassed. I completely missed the smiley, not just in copy-paste, but in reading. Back to school for reading comprehension for me, and Sorry, Ross.

They didn't do the math wrong. They wondered why they had an extra vote and were considering a redo when someone remembered that the Puerto Rican Delegate (Luis Fortuno) who is a Republican could vote in the caucus though not on the floor. See here:

"We were pleased to learn we're not corrupt, just stupid," one congressman said after the caucusing.

And they should be pleased, since OTOH more and more politicans ARE corrupt, while OTOH, stupid Americans, a growing constituency, need representation too...

Fishbane: Glad to see you saw the Smiley in the end...

" 'Aub, How much is nine times seven?'"

" Aub hesitated a moment. His pale eyes glimmered with a feeble anxiety. 'Sixty-three,' he said."

"Congressman Brant lifted his eye-brows. 'Is that right?'"

"Check it for yourself, Congressman."

After explaining that he simply memorized the simple, single-digit multiplications, he goes on to show how he can multiply any length of numbers through a simple process, which leads to consternation ("'Computing without a computer', said the president, 'is a contradiction in terms'"). And of course he explains that once upon a time, computers had to be programmed by people, and these rules are the basis of how computers themselves operate.

On the one hand, Volokh might be read to making a different point since he is arguing that journalists can use a calculator, but what I think is the real point is that figuring how many minutes are in a day and similar operations is actually not very difficult. Any half-educated journalist should be able to write on paper 60 x 24, or even get a very close approximation in his head (24 x 6 and add a zero). And given the frequency with which these statistics are wildly off-base, any halfway decent reporter will run through a several second calculation in his mind to see what the actual figures per year are. The fact that journalists repeatedly print these off-the-wall ridiculous statistics does not speak well for them, when it only takes a few seconds to run the numbers even if one runs them approximately in their head. (i.e., if one can't figure out 24 x 6, just take 25x6= 150, take 1500 x 365, and see if that seems at least about right. Clearly, many journalists don't even do a rough calculation).

What bothers me is not that Americans are innumerate. What bothers me is that numbers are cast into different units for purely political reasons, when their actual relevance to an individual could be either very high or very low. And as you point out, the actual number is not nearly so important as the sense of panic it generates, so nobody bothers to check the figures or think about them in any depth. It's just empty blather, like so much in American politics.

It's more of an issue for the fundamentally innumerate since the wide availability of calculators (in cell phones, pens, etc.) has convinced supposedly educated people that they really don't need to have any

for how numbers work. Last week, I yet again found myself explaining to yet another attorney that if you're doing a "pro tanto community property share" calculation on a piece of real estate in Southern California, and one of the numbers that goes into the multi-step calculation is anfeel, the fact that your calculator yields "$XXX,X66.66" doesn't mean that you really can calculate it that precisely...appraiser's estimate of the value ten years agoBTW, Fishbane, if you didn't remember your chem constant trivia, your enjoyment of dumb puns like Trader Joe's "Avocado's Number GuacaMOLE" would be lessened... (they did start out, IIRC, just west from Caltech....)

rfgs

nothow to actually do the calculations or even to understand them.I once taught an analytical chemistry lab. Two students turned in their report (% K2CO3 out of a mixture of K2CO3 and Na2O3). They reported over 100% and asked what was wrong with the computer program they used! When they asked me I checked their data (at a glace I could tell that they messed up the experiment and that that is what the data actually said).

It turns out that they didn't even know

whythey were doing what they were doing. I told them to figure it out on the whiteboard. They protested that they couldn't do so because they didn't have the computer with them! I forced them to do it by hand, even going to far as to stop them from using the calculator. When they protested that they couldn't do basic arithmetic with "big numbers" I just made them do some basic rounding (a concept that was completely alien to them).I finally gave up and forced them to watch me do it by hand and explain

whyI was doing what I was doing.Scary, that these were students who were going to graduate in the spring from college with degrees in chemistry.

Sigh...

formerpart intended to be sarcastic, not the latter. For what little it's worth at this point, Idon'tconsider those conversions useless; and yes, wrote fully aware that knowing a few of these conversions would help reduce such errors (albeit not prevent them -- never underestimate the hazards of an empty coffee pot).Perhaps my knowing c in furlongs per fortnight is useless... but I know that one (about 1.8 trillion) without need of further reference, just as well as 86400, 1440, 5280, 2.54, 2.205, 6.022E23, and 0.6931, and 0.47712... and feel those others should be easily recognized by anyone. (Yes, I am a math geek, too.)

Mathematics is fundamental to the nature of the world, even in politics. (For any not already familiar, look up "Arrow's Impossibility Theorem".) But if people don't even bother to become comfortable and familiar with something as basic as conversions, what happens when anything harder needs to be decided? I swear, modern technological civilization is starting to look to me more and more like a statistical fluke by the day.

I think students should understand how to do arithmetic calculations before being allowed to use calculators. Otherwise, their sense of numbers will be so poor that they will not even be able to multiply by 1 in their heads (true story)."

I agree with you -- if they are capable. But between a rock and a hard spot of either (A) the person just does not have that capability, and can never do it unassisted by electronics, and (B) the person is able to do it with assistance of electronics, wouldn't it be preferable in that circumstance to opt for B?

I was unable to manually do statistics at all, but got an A+ both semesters via computers. It was a required course. Either it would have prevented my degree, or I had to utilize electronics.

If I had the capability, I would not have hesitated to go your preferred course. I cannot do word problems in mathematics, either. No capability to think that way. Some autisics are very numbers oriented, I am not and no amount of effort to achieve this is possible. Both of my parents were mathematics and physics geniuses. What can one do?

I went to law school.

The first day of training, I was concerned about how my employer wanted me to explain what the APR on a tax refund anticipation loan meant (since I suspected an explanation of "if we were charging you interest over time for an entire year, at this pace..." would just confuse and upset the customers) and I was assured, with shocking self-confidence on the part of my trainer, that there was no chance of any customer ever asking. Most of them are surprised to see that we include a list of each charge included in the loan amount when I tell them. And nearly all of them decline any offer for me to read them that list when they can't find the check stub, trusting that I'll do the subtraction for them -- even though they'd been yelling about some small sum of money they thought was missing from their check, a minute earlier.

On the other hand, it's quite nice to let the power of subtraction calm upset customers; if only I didn't have to walk them through calculations like $3565 minus $2500, I'd still have faith in humanity.

As a math teacher, I find that point the most important, and the most often missed. To relate the metric lengths to the American ones, all I need to remember is that the 440 yard "dash" I ran in high school (I could never "dash" the whole distance, but the good ones can) is now the 400 meters. Everything else proceeds from that fact. To relate the metric weights to the American ones, I remember 2.2 lbs per kilogram. As far as the strange American area units, I grew up knowing that there are 640 acres in a "section" (a square mile.) I still use that one to convert some "square feet" measurement into fractions of an acre.

An easy skill for students to learn is to cancel units, so they don't even have to puzzle out whether to multiply or divide.

Trigonometry is another great example. I always tell my students, "Where else can you take a 3-credit class where the only things you need to memorize are one formula you already know (the Pythagorean Theorem) and six simple definitions?" Everything else we just build from scratch. (OK, that's a bit of an exaggeration. It also helps to know that C=2*pi*r and A = pi*r*r, but that's about it.)

With the BAC measurements, the one that kills me (and would kill the drinker) is that students frequently report that their buddy "blew a 1.2" (instead of a .12); even if we assume the "percent" we still have a problem.

ismultiplication.SLS 1L: While Dworkin "explicitly disavowed" that formulation of her view, her "Intercourse" is full of statements that can reasonably be parsed as such, e.g., "Intercourse is the pure, sterile, formal expression of men's contempt for women. . ."

rfgs

A human being has a body that is inviolate; and when it is violated, it is abused.

. At the same time, the penetration is taken to be a use, not an abuse; a normal use; it is appropriate to enter her, to push into ("violate") the boundaries of her body.A woman has a body that is penetrated in intercourse: permeable, its corporeal solidness a lie. The discourse of male truth--literature, science, philosophy, pornography--calls that penetration violation. This it does with some consistency and some confidence. Violation is a synonym for intercourseShe is human, of course, but by a standard that does not include physical privacy. She is, in fact, human by a standard that precludes physical privacy, since to keep a man out altogether and for a lifetime is deviant in the extreme, a psychopathology, a repudiation of the way in which she is expected to manifest her humanity.

There is a deep recognition in culture and in experience that intercourse is both the normal use of a woman, her human potentiality affirmed by it, and a violative abuse, her privacy irredeemably compromised, her selfhood changed in a way that is irrevocable, unrecoverable. And it is recognized that the use and abuse are not distinct phenomena but somehow a synthesized reality: both are true at the same time as if they were one harmonious truth instead of mutually exclusive contradictions.Intercourse in reality is a use and an abuse simultaneously, experienced and described as such, the act parlayed into the illuminated heights of religious duty and the dark recesses of morbid and dirty brutality. She, a human being, is supposed to have a privacy that is absolute; except that she, a woman, has a hole between her legs that men can, must, do enter. This hole, her hole, is synonymous with entryThis is admittedly, pretty obscure stuff; I've read it repeatedly and am still not entirely clear on what Dworkin really MEANT. The reading of it as "intercourse= rape", while it may be wrong, isn't particularly farfetched...

Frank Drackmann: "Why can't you divide by 0? And don't say cause its against the rules, its ok to multiply by 0, so why is division verboten?"

Some good (and a couple of OK but not-as-good) explanations are at: http://mathforum.org/dr.math/faq/faq.divideby0.html

I would suggest that division isn't really forbidden, just that since the answer is "undefined", it makes that operation kind of pointless - "illegal" is just good shorthand to keep folks from doing it and thinking it means something.

As for numbers and "calculating machines" in general, the late SF author Hal Clement (aka Harry Stubbs) was a chemistry teacher who, when this subject came up, liked to recount an anecdote where one of his students multiplied instead of divided by Avogadro's number (a big number that defines the number of particles in a mole). The student was calculator-dependent, and could not figure out by just looking at the number that he had calculated enough of a substance to more than fill a galaxy rather than a test tube.

Since the husband has been teaching physics to pre-meds for the better part of a quarter century, he has many terrifying stories of student innumeracy. The most memorable one was for a homework problem to calculate how long it takes the space shuttle to orbit the earth. The student wrote down something times ten-to-the-minus-thirty-six seconds. She was a pre-pharmacy major -- now there's a comforting thought: a pharmacist who has no intuition about a dose being so tiny as to be ineffective, or so large as to kill every living thing on the planet. They had a measurement lab at the beginning of physics I where the students took a meter stick and measured the doorway three times and calculated the mean and standard deviation. They would write down their data -- 1.0115m, 1.0114m, 1,011.6m, add them up, divide by three to get 337.8743m, calculate a standard deviation, turn in the lab with nary a thought about the absurdity of a 300 meter wide doorway.

cathy :-)

My favorite personal story is when I took a real estate broker's test. One of the questions was how many square feet in an acre. Some, of course, memorized that number. But it is one of those numbers that I still don't see much need for, so I did the opposite, squared 5,280 (sq. feet per section) and divided by 640 (acres per section) for 43,560 sq. feet per acre. And I didn't even have to do it by hand, since they let us use our trusty HP-12Cs to compute who owed whom what on the closing statements.

Let me also concur about trig. I took a trig test a couple years after graduation from college, and couldn't remember any of the trig equations. But it was fairly easy to derive them on the fly, given that I had a math degree by then. I got a 100% on the test, while completing before the time limit. At least a math degree is useful for something.

As for Eugene's post, hilarious. Although I'd argue that multiplication isn't math, it's arithmetic. The difference being that it's not math until there aren't numbers involved anymore.

Bruce Hayden : "squared 5,280 (sq. feet per section) and divided by 640 (acres per section)"

Now there's some strong evidence you probably come from flyover country. "Section" is a term that will get you blank stares from 99+% of the population within a few hundred miles of either coast...

(A Minnesota native in New England who would have computed ft/acre exactly the same way…)

And "mole" is correct.

The new boss asked, "What's stoichiometric?"

_____

My opinion, based working with a fair sampling of each, is that lawyers as a group are much better at math than MBA's. In particular, many marketing types have very little number sense.

But, for something completely different, I'd like someone to tell me that the ice cream makers' estimate that Americans eat 1.5 billion gallons of ice cream a year, "enough to fill the Grand Canyon," is wrong. It's only a math problem in the way that Archimedes' challenge was--and I don't think we can fill the whole canyon and measure the water on the way out.

Umm, no not really.

I know the number of feet in a mile (easy).

I know the number of square feet in a "builder's acre" (40,000), and that a "real" acre is a bit more, but not the exact number.

I know that a square mile (a section) is 640 acres.

Kindly explain how I can get to sq ft/acre with an easier calculation.

The Grand Canyon is around 52 billion cubic yards in volume according to an estimate of the U.S. Geological Survey I found. That is 10,502,649,350,649 liquid gallons (10.5 trillion gallons) according to an online conversion tool.

At 1.5 billion gallons a year, it would take only a bit less than 7,002 years to fill the Grand Canyon, according to my calculation. They are not even close.

S/B "too"

No problem. Some coworkers were recently discussing random dishonesty like this in many advertisements -- annoying, but not quite worth a complaint -- and it piqued my curiosity.