Prof Volokh, you are right on point regarding the importance of knowing how, when, and why to use arithmetic in real world situations.

You write something that I find intriguing:

"My sense is that learning arithmetic is good mental training for other thought processes, but I may be mistaken on that."

I wonder if there are any neuroscientific studies that might substantiate that hypothesis and identify specific mental tasks that are aided through mastery of mental arithmetic.

The more you reveal this "arithmetic" power to the public, particularly its application to the real world, the greater the risk becomes that it could fall into the hands of the terrorists, to deadly effect.

Word problems are the answer to what has got to be the most annoying question a teacher ever hears: "But when are we going to have to use this stuff in real life?" As such, I could never really fathom why people didn't like them. I felt like they made the problem both more concrete and more interesting.

123 x 456 + 789 = 56877!

"Understanding how to translate the real-world condition into numbers is the important skill, not the calculation."
This reminded me of an interesting fact: Negative numbers were invented by accountants, not mathematicians. (This is mentioned in Jacques Barzun's 'From Dawn To Decadence, 1500 To the Present').
Since many of the readers of these posts appear to have a strong interest in math, let me throw out that the best general book on mathematics that I ever read is John D Barrow's 'Pi In The Sky'. This is not so much a book for the pure math lovers, but a wonderful and even literary exploration into all the philosophical problems surrounding math.
Back on topic, maybe the reason so many of us went off on a rant about doing math in our heads is that we JUST DID NOT WANT TO ACCEPT that people who graduate high school can't figure out how to calculate how many minutes are in a year and whatnot. "Where ignorance is bliss, 'tis folly to be wise".

"Now, for the next scientific breakthough -- can anyone figure out how to inject the missing genetic mathematical material in to the brains of non-number oriented autistics?"
Mary, What luck! I just discovered this in my lab and will mail you the patent for only $10,000 !!! (Limited time offer)(Patent comes with proof of purchase)

Estimating calculations has saved me from making errors like the one you uncovered in this post. It's a fairly easy skill to master and cultivate.

Solving problems without paper and pencil is a gift. It's been a marvel to me that my father who was educated only to possibly third or fourth grade in a tiny village in Europe, instinctively understood mathematics and could come up with correct answers directly out of his head while I had to slog through formulas to come up with the solution. He didn't have the words to explain the process -- the best he could do was to say the answer was there in his head.

You're right that even grade school number learning is referred to as mathematics. Arithmetic has disappeared.. Third graders learning the multiplication tables are told they are doing math, or even worse, their maths!

You should go back and read some of the horrid word problems in schools these days (or at least, 10 years ago).

The real world I can handle (quite well, if I do say so myself). Math course "word problems" are a psychosis all of their own.

My mother has been teaching piano for almost 50 years. She is of the school of thought that certain physical exercises improve your math and logic abilities. I have no idea whether this actually works.

People often refuse to use even the simplest addition in their everyday life. I am always amazed when my girlfriend goes shopping and does not keep even a rough estimate in her head of how much her purchase should cost. Doing this has saved me money countless times due to double scanning or mislabeling. It just seems like common sense to me.

And by the way at what point does the "math" taught in schools become something that should properly be called math.

I'm one of those who can do reasonably intricate arithmetic in my head; this may well be, at least in part, a consequence of having obtained nearly all my schooling before calculators priced less than half a typical head-of-family wage were available. I agree 100%, however, that the important thing is not knowing how to multiply, but what, when, and why to multiply (or, of course, add, subtract, etc.)

The important thing is not knowing how to multiply, but what, when, and why to multiply.

Amen.

It's almost a cliche to state that students often have to re-learn mathematics to succeed at more advanced mathematical applications. I'd gone through this process myself. Much of this re-education concerns the acquisition of skills involving precisely the understanding E.V. implies. We, in the U.S., don't seem to provide an effective bridge between basic and 'advanced' maths.

I suspect that this is partly due to an exaggerated presumption of the innateness of mathematical ability - that such ability is a discrete heritable trait similar to the acuity of visual recall or proprioception. You'll notice this presumption by remarks on the 'irony' of prominent mathematicians and scientists relating that they were frustrated by rote mathematics but excel at abstract mathematical logic. Similarly the often hyper-pedantic mode of math instruction presented in college level sequences seems justifiable only if you assume that these courses are intended to speak to 'natural' mathematicians.

And so, by assuming that all good mathematicians 'just get it', we neglect to provide a thorough education in the foundational concepts upon which advanced maths are built - thus the bias confirms the result.

Alkali: I think you have a typo, and mean "trillion" for the US budget and GDP.

Alkali:

I have a few:

One employee-year costs about $150,000, including benefits, a share of management, equipment, etc.

Area of the United States: 3 million square miles without Alaska, 3.5 with.

As Frank Drackmann notes, one mile on foot burns 100 calories. One mile on a bicycle burns about 35 calories. One (non-light) beer (or one non-diet soft drink) contains about 150 calories (a little low, but it's helpful to remember that 2 beers=3 miles on foot or 9 to 10 on a bicycle.

Not strictly mathematical, but helpful (came in useful recently in correcting the alumni magazine of a major private university in Silicon Valley): one acre = one football field (the NFL kind, not a soccer or rugby pitch) without the end zones.

One other thing, that doesn't quite match the others: if your total doesn't match by a factor of nine (or ninety, or 900), it's likely that you transposed two numbers. If, for example, the difference is 18, you transposed two numbers with values apart by two, e.g., a 3 and a 5. If it's 45, you transposed two numbers with values apart by 5, e.g., a 7 and a 2.

The important thing is not knowing how to multiply, but what, when, and why to multiply

I think it is also important to know with whom to multiply.

Sorry.

I had a math teacher in high school who insisted on teaching us how to solve real life problems using algebra and arithmetic.

I still use his technique for multiplying 2 digit numbers in my head. (hint: same technique as solving (ax+b)(cx+d)).

I am always scribbling down simple little algebra problems and solving for X to figure out the formula I need. Maybe it's doublechecking the correct sales tax rate, or maybe it is refiguring out the conversion of celsius to farenheit. Recently I took a Paramedic class, and it was converting from Lbs to Kilogram to calculate dose rates for various medications.

Life is full of the power of numbers. I have loved your commentary on it.

alkali:

Rule of thumb Farenheit = 2*Celsius+30.

Correct: Farenheit = (9/5)*Celsius+32.

Rule of thumb is easier to calculate and is reasonably accurate from about -10 C to 30 C. (error is less than 4 degrees F)

ooops, typo: being=begin

Was there anybody else who saw the movie Proof and had to confirm (in his/her head, while watching the movie) that 1729 really is the sum of two different numbers cubed? (10^3 + 9^3 and 12^3 +1^3)

Cubing 12 in one's head isn't too difficult if there's a break in the dialogue--the one sticky part being the addition of 1440 and 288.

Cubing 12 in one's head isn't too difficult if there's a break in the dialogue--the one sticky part being the addition of 1440 and 288.

It's also one of those numbers that you remember if you attended Catholic schools earlier than, say, 1965. It's the volume in square inches of a cubic foot.

Silicon Valley Jim, I had never heard the factor of nine trick. That's neat. It reminds me of how delighted I was as a child learning that you can multiply by nine on your fingers (and that all multiples of nine add up to nine). For anyone who has never encountered this one:

• Hold up all ten fingers.
  
• Counting from left to right, put down the finger of the number you want to multiply by nine.
  (E.g., if you're multiplying by three, put down your third finger.)
  
• The resulting two groups of fingers represent your tens (on the left of the finger that is down) and ones (on the right) places.

Thus, 3 x 9 = 2 fingers on the left (3rd is down) 7 fingers on the right = 27

To this day I look at my hands if my brain isn't coming up with the multiple of nine fast enough for me.

To go along with the shortcut for multiplying nines, there is a trick for multiplying 11 as well. But it only works with two-digit multipliers. First, you "split" the multiplier. That is, just take the number from the tens column of the multiplier and put it in the hundreds column of the answer, and leave the number from the ones column in the multiplier in the ones column in the answer. Then, add up the number from the tens column of the multiplier and ones column of the multiplier, and put that in the tens digit in your answer (don't forget to carry over to the hundreds if necessary). And there's your answer.

For example, 11 x 36. Split the 3 and the 6 to get 3_6. Add 3 + 6 and put it in the tens column to get 396. 11 x 36 = 396. Or 11 x 87. Split them up to get 8_7. Add 8+7 to get 15 and add that to the tens column. Carrying the one makes your new number 957. 11 x 87 = 957. Piece of cake.

Our ancestors did most of their math in their heads, perforce, and this helps illustrate why some of our weights and measures seem strangely "illogical" to the modern world with its heavy emphasis on decimals and calculators.

The obvious example is the hour of 60 minutes, which can be exactly divided into halves, thirds, quarters, fifths, sixths and tenths. By contrast the "logical" decimal choice of 100 minutes per hour can be divided into halves, quarters, fifths and tenths. Not quite as useful if you have to do division in your head.

Another interesting example is Daniel Fahrenheit's thermometer scale, which modern folks often consider less "logical" than Anders Celsius' scale in which the difference in temperature between freezing and boiling water is set to 100 C. Fahrenheit, in his older scale, originally set the difference between freezing water and human body temperature equal to 64 F.

Why 64? Imagine making a thermometer: you fill the tube with your working liquid (mercury, or in Fahrenheit's day alcohol). You put it one of your references (say freezing water) and mark the level. Then you put it in another (in your mouth, for Fahrenheit's scale) and mark the level again. Now you need to divide the space between into equal divisions. If you have to make 100 divisions, you need a precision steel ruler. But if you have to make 64 divisions, you just need a compass and straightedge, because you just need to divide the distance exactly in half 5 times (2^6 = 64).

Furthermore, Fahrenheit took advantage of an interesting coincidence to set the freezing point of water at 32 F, and assign a third scale point, the temperature of a stable mixture of ice, salt and water, to 0 F. Since 2^5 = 32, this distance, too, can be marked off very easily with compass and straightedge.

Hence Fahrenheit's scale lent itself far easier to the manufacture of accurate thermometers at a time when precision instruments were very expensive.

There are often similar reasons partly underlying other "nondecimal" subdivisions of our traditional measurement units. Basically, it turns out powers of 10 are not a very convenient way to subdivide things, inasmuch as they have at best a merely typical number of factors smaller than themselves. For example, among two digit numbers 10 has a slightly lower than average number of factors smaller than itself (2 versus 2.8). More convenient numbers are 12 (4 factors), 16 (3 factors), 24 (6 factors), 36 (7 factors), and of course the champ 60 (10 factors).

Hardly a surprise, therefore, that we have 12 inches per foot, 16 ounces per pound, 24 carats in pure gold, 36 inches in a yard, and vide supra 60 minutes in an hour.

Oops, I should add that Fahrenheit later redefined the upper end of his scale to set 212 F equal to the boiling point of water at 1 atm, and that shifted human body temperature (he used axillary) from 96 F, the original, to about 97 F. Also, I should mentioned that you put the new thermometer under your arm, not in your mouth, for the second reference, since oral temperature is higher, about 98-99 F.

As for why he didn't use the boiling point of water right away, he did not orginally know why the boiling point varied as much as it does. Only later did folks realize the importance of atmospheric pressure, i.e. altitude and weather. Once this was understood, and it was possible to measure barometric pressure, one could defined the boiling point at 1.000 atm (760 mm Hg) and it would be reproducible.

If you Google "number of seconds in one year" it will tell you.

Yes, very clever. A factorial joke.

Telford Work-
Thanks for the correction; that's what I get for believing something I read on the internet! So how does one (no pun intended) multiply binary on paper? It does work sometimes to just do it like decimal (101 (5) x 10 (2)= 1010 (10) binary &101 x 10 = 1010 decimal); is there an intuitive way to do it right?

Knowing what to multiply when -- and what quantities NOT to multiply and when NOT to multiply them -- is also important in framing a statistical model. Failing to think about this in common-sense terms can result in a very implausible model, and possibly serious artifacts when the model is fitted to real-world data. I argue here that this flaw applies to Groseclose and Milyo's much-discussed investigation of media bias.

As an example this link from Yahoo Finance says the World's Biggest Photovoltaic Plant in Nevada "will develop a total of 36 MWs of PV projects in Nevada, enough energy to power 36,000 homes". With division (a little off topic) we see that they're really counting on energy conservation since that gives each home 1000 Watts to use anyway they want! That's about enough to run a hair dryer (on low). Most new houses have 100 amp service which provides ~11000 Watts so these guys are off by at least an order of magnitude as we used to say in m(ath) school

In the new blog age (blogage?) maybe they'll need pay more attention to multiplication in j-school and f(inance)-school.

Tony and Silicon Valley Jim (re RPN):

I agree entirely about the superiority of RPN, and my conviction came about in an odd way. Many years ago, my company bought a number of Wang desk calculators, and by pure dumb luck, the very first one purchased wound up on my desk. Therefore, I was forced to learn its "backwards" computation logic, that I found out much later was called RPN. I seemed to have a lot less trouble picking it up than most of my other colleagues; perhaps RPN happens to be hard-wired into my brain. Also, the Wang calculator was the first and only electronic device I've ever seen that used Nixie tubes for its display. It was no doubt pretty inefficient, but my gosh, they looked cool!

Was there anybody else who saw the movie Proof and had to confirm (in his/her head, while watching the movie) that 1729 really is the sum of two different numbers cubed? (10^3 + 9^3 and 12^3 +1^3)

I didn't see Proof, but I know that not only can it be expressed as the sum of distinct cubes two different ways, but it is the lowest such number that can be so expressed.


My father came close to physically striking me when he learned that in high school I was still doing subtraction the New Math way ("borrow 1 from the tens place and change it to 10 in the ones place...") -- fine for theory, but in practice so much easier to increment the subtrahend.

He also was well familiar with the 9 rule for accounting errors that Silicon Valley Jim mentions, which he promptly explained works because "9 is a magic number". (He claims he got through GI basic training with his sanity intact by calculating and memorizing the powers of various numbers, up to 9 raised to the 9th power.)

Word problems in my older son's 3rd-grade class are of the form "Carmen says that 32 is divisible by 5 because 3 plus 2 equals 5. Why is she wrong?" and he wants to answer "Because she's retarded!" The kid is going to go places.

Unlike my daughter, I'm afraid. In the 5th grade she's still trying to "remember" and "figure out" her times table. I repeat to her what a co-worker's aero prof said (about the equations of flight): "Do your 'remember' your name? No, you know it. Similarly you must know" (the fundamental equations of your field of study, or your times table.) I fear she gets this from her mother, who once described an exercise ball as being "about 2 feet high and 3 feet across". (And yet I understood what she meant: it's about as high as object she'd describe as 2 feet high, and about as wide an an object she'd describe as 3 feet wide, and to hell with the relationship between all diameters of a sphere. Just as at exactly the same height, I'm as tall as men who claim to be 5'8", while she's as tall as women who claim to be 5'6".)

tfellow writes:
With division (a little off topic) we see that they're really counting on energy conservation since that gives each home 1000 Watts to use anyway they want! That's about enough to run a hair dryer (on low). Most new houses have 100 amp service which provides ~11000 Watts so these guys are off by at least an order of magnitude as we used to say in m(ath) school

That's peak usage (100 amps) versus average usage. Both are important. I was concerned about an increase in my electric bills (turns out it was the new dehumidifier, not the new meter, oh well) so I got all my old electric bills off the supplier's web interface, and divided kilowatt-hours by 24 times the number of days in the billing cycle, to express my usage as average kilowatts drawn, and sure enough my typical around-the-clock average draw is about 1 kilowatt. (Or for the less numerate, 1 kilowatt-hour per hour.) Up to 1.3 kw when the dehumidifier is running. (Efficiency is kw-h per liter of water condensed, and it's rated IIRC 0.8, and it's rated 50 pints per day, and I'm emptying the 2.5 gallon bucket about twice per day -- I think that all worked out.)

Of course you can't use that daylight-only plant alone to power all those homes, but you can factor out that many homes from the expected fuel consumption of your existing generation plant.

One of the most important everyday math tips I learned is a trick to help me keep track of the units. For example, somebody says he can throw a ball 100 feet per second, and you want to convert that to miles per hour. You may already know that there are 5280 feet in a mile and 3600 seconds in a minute. But do you multiply by 5280/3600? 3600/5280? 5280x3600? 1/5280x3600?

Here's how I do it (forgive my lousy HTML):

100_feet___60_seconds___60_minutes______1_mile___68.2_miles

--------_x_----------_x_----------_x_---------_=_----------

__second____1_minute_____1_hour______5280_feet________hour

My high school math teacher Mr. Metzger (see, after all these decades I still remember the names of the teachers who were important in my life) taught us this trick. Treat the units as if they were part of the number, multiplying and dividing and canceling them as appropriate.

SeaDrive, you want to talk to these people. I think there are some ancient societies (in Africa, maybe) that are thought to have used base-12 counting systems. Certainly it survives in weights and measures, and in special words like dozen. The Sumerians and Babylonians used base-60, of course, which is why we have 60 minutes in the hour, vide supra, and 360 degrees in a circle. I've read an unconvincing argument this had something to do with a compromise between ancient base-12 and base-5 counting systems.

I believe most experts think a people get their base from how they count on their fingers (and toes, maybe). You can get base-5, base-10, base-12 and base-60 counting systems all from how you count on your fingers (base-20, like the Mayans, you presumably get from counting on your fingers and toes). Base-5 and base-10 are obvious. Base-12 you can get by using each of the three segments (outer, middle, inner) in each of the four fingers on a hand. Base-60 you can get by using each of the five digits of the right hand to point to one of the 12 finger segments on the left hand.

Barry:

Touche. You're absolutely right. Homer nods, and all that.

Jeez, Splunge -- "liberal artsy nonsense"! You sure know how to wound a math-computer science major, one who was pretty good for his age at math long before he got into law and other mushy stuff. Life is word problems in the sense that the sentences immediately following that one elaborate:

All of life is word problems! A problem in the real world never comes to you as "figure out 123 x 456 + 789." Understanding how to translate the real-world condition into numbers is the important skill, not the calculation. And while most school-level word problems are usually only a very first step in that direction, if people can't master that first step, their knowledge of raw arithmetic or algebra won't help them actually deal with the world.

Some people may be so good at translating real-world conditions into numbers that it happens seamlessly. But few are, and those who are, I suspect, mostly took some time to learn it.

If you don't want to view "real-world problems" as essentially "word problems," in the sense of problems that beginning students originally think about using words, and then translate into numbers, that's fine. But it seems to me that equating the two is pretty sound, and hardly nonsense, liberal artsy or not.

Splunge,

"This is liberal artsy nonsense. I knew plenty of people in college (MIT) and graduate school (Berkeley) who were poor at what most people mean by "word problems," but they were nevertheless gifted engineers and scientists."

Can't agree with you on the engineering (no idea about pure math or pure science). If I'm talking to the RF engineer (I'm a Mech. E) about a new invention, we need to start from concepts, words and verbal descriptions to work toward the correct formulation of the relevant equations.

My latest patent came about after a long brainstorming session which produced nothing but a clear, one sentence description of what was required to solve a particular problem. Only then could we start to generate ideas (verbal and 'sketches on napkins'). The math came later when it was time to decide 'how big does A need to be to get us B'.

RGT