I've much enjoyed the discussion about the multiplication posts, and I agree that calculating in one's head is a good skill to have. Even if there's a calculator handy, people are often reluctant to use it (perhaps overestimating the effort it would take, or not wanting to look ignorant or skeptical), and if one can do even approximate calculations in one's head, one can often spot errors that one would have otherwise missed. Also, my sense is that learning arithmetic is good mental training for other thought processes, but I may be mistaken on that.

Nonetheless, my point was different: **The important thing is not knowing how to multiply, but what, when, and why to multiply.** Once you understand what calculation to do to check something, you can do it easily enough on a calculator. But the calculator won't explain to you how to structure the calculation, or whether a calculation ought to be done.

That, incidentally, is one reason that I'm so frustrated when people say "I never minded math" (usually referring to arithmetic or at most algebra) "in school, but I just never liked the word problems." 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.

Related Posts (on one page):

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.

Having a more mathematical mindset, I feel, comes in handy in situations that require risk or efficiency assessment, even if you are not doing any actual calculation. i can't quite articulate it, but it definitely feels like arithmetic is a helpful part of the big picture.

Ahh, but not all is lost -- some of us more mathematically challenged do have the abilities to leabr formal Logic and apply THAT to the real world. Maybe the next best thing, but a workable alternative nevertheless.

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? And, while you're at it, how about the missing emotion-reading brain cells as well?

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".

agre=agree

leabr=learn

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)

I found that with spreadsheets I didn't have to know how to structure problems before calculating. I just construct a formula and test the answer and if it's wrong, I keep changing the formula untill I get it right. Since you can easily estimate and change things with spreadsheets and perform "double entry bookkeeping" on your formulas, it generally works out.

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!

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.

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

From dictionary.com

Arithmetic: The mathematics of integers, rational numbers, real numbers, or complex numbers under addition, subtraction, multiplication, and division.

Mathematics: a.The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols; b. A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement.

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.

World population: 6 billion (I should use 7)

US population: 300 million

US GDP: 10 billion (a bit low)

Federal budget: 3 billion (a bit high; pre-2006, I used 2 billion)

Days in a year: 400

Workdays in a year: 200 (sometimes I'll use 250)

Does anyone else have any of these?

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

Sorry.

2 + 2 =

whereas on RPN they are

2 enter 2 enter +

...which seems pointless until you try to calculate something more complicated, at which point it becomes immediately indispensible.

What boggles my mind is that the Windows "calculator" emulates this ridiculous interface. For a really great desktop calculator, download R from http://www.r-project.org/ . It's a very complicated and technical tool at its deeper levels, but for doing everyday calculations it's also super duper. Wish I had a version for my Treo!

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.

One work year in the United States: 2,000 hours (insert joke here about the country of your choice)

One year: 9,000 hours (a bit high) or half a million minutes, or 30 million seconds

One day: 1,500 minutes or 90,000 seconds

Tony:

I agree entirely about RPN. I think that I would agree entirely even if two of my grandparents hadn't come to the United States from Poland.

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)

""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)"

Deal. After I get my bar admission and win my first big case (and get that $10,000), I'll take the cure and being a prodigious mathematics career (with my newly created brain cells). Sorta opposite the career trajectory of EV.

Proofand 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.

People are very willing to tell you that they are not very good or even terrible with math.

Yet, those same people would be horrified if they found out a friend or

colleague couldn't read.

Why should the stigma be any less - or absent completely - for innumeracy than for illiteracy?

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.

Actually a gallon of water weighs about 8.3 pounds (@STP) but the above is close enough for government work.

(E.g., if you're multiplying by three, put down your third finger.)

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.

I always recommend that school kids learn simple algebra along with grammar, for they are both grammars and have similiarities. The other element missing in education and practice is the concept of unit arithmetic, which simplifies the formulation.

Example: Miles/hour * hour = miles The hour unit cancels because I know my simple algebra, so the correct forumla is pointed to me. What are the errors we always see in the media? Wrong units, billion, instead of trillion; hour instead of minute, etc.

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.

This will probably sound obvious and dumb to the more mathematically-inclined, but I just learned the other day how multiplication in binary "looks" exactly the same as multiplication in decimal. E.g., 10 X 11 = 110 in decimal, and 10 (2) X 11 (3) = 110 (6) in binary. Neat!11 x 11 = 121 in decimal; 11 x 11 = 1001 in binary. Needing to "carry the 1" makes the difference. That example is just a coincidence, like 2 + 2 equalling 2 x 2.

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 288.12 ^ 3 = 1728; 1 ^ 3 = 1

10 ^ 3 = 1000; 9 ^ 3 = 729

The irony!

BTW, I may be somewhat mathematically inclined but that doesn't stop me from making lots of mathematical mistakes of my own....

mostof 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.

Butif 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.

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.

Back when I took calculus in high school, Sr. Rita Cecile made us calculate integrals by counting rectangles. We had to take a piece of graph paper, plot the function by calculating a reasonable sampling of points, then count the square, and estimate the triangles at the function line. She only made us do this twice, but it was enough to give us two extraordinarily valuable pieces of knowledge. The first is that an integral really is the area underneath a curve. The second is that we really really really

wantedthe Fundamental Theorem to be true...cathy :-)

muchgreater than 123 x 456 + 789, or even 123 x 456 + 789!notrepeated addition; that's just how you calculate it. Multiplication is function composition. 3 times 2 means you have 2 three times. This seems completely trivial, until you learn to extend multiplication to non-numerical objects. Every physicist is capable of multiplying physical units by numbers and other units, and this definition remains unchanged when you consider multiplication in groups.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?

0 + 0 = 0

1 + 0 = 0 + 1 = 1

1 + 1 = 10

0 * 0 = 0 * 1 = 1 * 0 = 0

1 * 1 = 1

You do multiple digit addition/multiplication the same regardless of the base. Just remember to carry when the sum reaches the base (in this case, 2). Binary multiplication takes less memorization, but a heckuva lot more writing.

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?"For most everyday purposes when you don't want to physically punch the problem into a calculator, you can get a good estimate of how big an answer is by throwing out all the least significant digits. If you have to multiply 28,472 by 13,287, figure it's pretty close to 3*1 x 10^8 (actual answer:378307464) The biggest advantage to mental math is in getting a "number sense" of about how big the answer _should_ be so that you can pick out the obvious mistakes. You only need a calculator if you need to know the answer with a great amount of accuracy.

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

Too much math, not enough cents...

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

knowit. Similarly you mustknow" (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) schoolThat'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.

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~~________hourMy 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.

Because "stigma" constitutes a vio of the Americans with Disabilities Act. "innumeracy=mathematics disorder, DSM IV. "iiteracy" (peope who can't read)=dysexia.

SeaDrive, you want to talk to these people. I think there are some ancient societies (in Africa, maybe) thatarethought to have used base-12 counting systems. Certainly it survives in weights and measures, and in special words likedozen. 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

righthand to point to one of the 12 finger segments on thelefthand.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.I hope you don't ever try to measure a volume in square inches... there are an infinite number of square inches in a cubic foot, right?

I understood that you meant cubic inches in a cubic foot, but in a thread about math errors, why propagate more?

I mention this in part because I had a junior-high chemistry teacher who would routinely answer mismatched units questions with either "infinity" or "zero"... we learned to take that as a cue that we had a fundamental error in unit expression that we needed to fix before proceeding further. (I actually had someone at my table ask him how many cubic feet were in an acre, and he answered them with "zero"... perfectly straight-faced.)

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

All of life is word problems!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. They just weren't any good at stories and narratives.

These people certainly did not lack for any ability to solve "real world problems" -- but they didn't use an intermediate representation in

wordsduring the process of turning the real-world problem into mathematics. Some of them used wordless mental images, no doubt, but quite a number "saw" physical processes directly in terms of the math required to quantify them. Indeed, I had one professor who had profound insight into many-body quantum mechanics -- one of the best men in the world in this field -- but who was so famously inarticulate his students used to joke that he had never learned about nouns (he used pronouns and verbs, largely). Unless he could say it in equations, this man found it difficult to explain even the simplest of things. Needless to say, his social skills were not the best, but in solving certain "real world problems" he had no equal.I'm sure Mr. Volokh can rescue his argument by re-defining "word problem" to mean "any problem not originally expressed in math." But this is not what most people mean by "word problem." And I suspect

some(although by no means all) of the people Mr. Volokh finds frustrating, who are good at math but not at "word problems," are the brilliant engineers and scientists of tomorrow.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.

With very few exceptions, all the base systems from natural cultures are from five different bases: 2,5,10,20,60. There is almost universal agreement that these derive from human digits. The primary exceptions are systems of four or eight. One theory is that they used their four fingers, but not their thumbs; Another is that they used the spaces between their fingers. The latter has anthropological support in at least some cultures.

The book that I recommended above - John D Barrow's Pi In the Sky - has a fascinating chapter on this, and gives some interesting examples of other bases:

" ... there exist whole counting cultures which are based upon the human anatomy. One of the most elaborate examples was that practiced by the Torres Strait Islanders as late as the nineteenth century. Starting on the right side of their body, they touched first five fingers for the numbers 1 to 5, then the wrist(6),elbow(7), shoulder(8), chest(9), left shoulder(10), left elbow(11), and so on, down to the left fingers which took them to 17. Then they continued with the left little toe(18), through to the left big toe which takes them to 22, left ankle(23), left knee(24), left hip(25), right hip(26),and then back down the right side in the same way, finishing with the right little toe(33). One imagines that maths lessons might have resembled a form of St Vitus's dance."

There are also several variations of finger-counting that involve bending the fingers, some in elaborate ways (think of sign language).

Why 60 is often a natural base is something that there is no consensus on. Hypotheses involve astronomy (An early belief was that a year had 360 days, which divided by 12 equals 60.) Another has to do with geometry, although this is more improbable. The commingling of bases which is also not very probable. (The theory I read was on the mixing of base 6 and base 10, but it's entirely possible that Splunge might have read a different theory than me). Then the fact that 60 has a lot of divisors, although this is rather putting the cart before the horse. The truth is that it's all guesses as to 60 - which I may market as a slogan for a t-shirt. "IT'S ALL GUESSES AS TO 60"

Mary, I'll throw in the copyright to my t-shirt for another five grand, and sure you can keep the rights and profits - My generosity is legend.

A final thought: Imagine what feminists would say if male cultures had developed bases of 11?

"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

"Imagine what feminists would say if male cultures had developed bases of 11"

don't have to - I know per a friend of my wife's: "oh, that's just another male behavior that's [alliterative profanity] driven"

it's surprising that the origins of 60 as a base are so obscure given its prevalence in cultures (sec/hr, min/hr and min/deg, mph (:>)). but in any event, as a codger I'd recommend that your t-shirt should read:

IT'S ALL GUESSES TO 60