Sunday, 27 February 2011

Missed Opportunities

Thinking back over my recent blog on moving west, I realize I have missed an opportunity to make a point that tends to be missed when folks talk about statistics. I have just sat through a number of repeated meetings about education and school improvement in which people tried to compare complex distributions of students and subjects by comparing them with a single number. That single number was almost always an average of some kind, although the distinction between mean and median tends to be lost in the drive to have "a number" to use to compare distributions. It seems that people who would question anything you say will suddenly fall silent at statements like, "15 is more than 12, thus God exists." In the nation of the mathematically insecure, the person who quotes numbers has great power. Even if the numbers are purely "Potemkin" numbers created on the spur of the moment... "I have here in my hand a list of 205—a list of names that were made known to the Secretary of State as being members of the Communist Party and who nevertheless are still working and shaping policy in the State Department".
We are a people in love with, and easily duped by statements of quantity. Averages are our favorite pieces of mis-information. But the center, the average location of population, for the 300,000,000 plus US citizens is in an empty field with the nearest residence over 100 yards away (my estimate, not an exact measure). The center of mass of the US population is near a city with a population of about 65 people in an area with a population density of about ten persons per square mile.

It is in some ways very typical of the wide planes areas to the west, which with exceptions in a few big cities have population densities from 0-10 persons per square mile. Only a few miles to the north-east, south-east or north-west the population density rises to over 1000 per square mile in St. Louis, Memphis, and Kansas City.

About 1/3 of the total population of the United States lives in one of ten urban centers, and almost one in ten live in one of the two population centers of New York/Newark in the East and Los Angeles/Long Beach in the west. Both are a long way in both physical and metaphorical distance from the quiet little field south-west of Plato, Missouri.

The US population that is nestled mostly in areas of high density is centered in a grassy field beside a tiny stream near a small quiet village in a very sparsely populated rural area.

Centers are generally a weak measure to describe the great variability of a population in any measure. It's not that averages are bad... it's just that...well, averages ARE bad.

One Stop Service

As tax time approaches, more and more businesses in the US seem to be getting into tax preparation. I found this ad, with more, at Mary O'Keefe's Bed Buffaloes in Your Tax Code... and someday I will have the courage to ask Mary where that name comes from.[See Mary's response in the comments]

Saturday, 26 February 2011

Friday, 25 February 2011

Moving West, WCYDWT

Just a few miles south of I-44 near Ft Leonard Wood lies the little town of Plato Missouri, population 65. If you head south out of town on county road 5870 (Turley Road) for about 1/2 a mile you come to a sharp right followed in a few hundred feet by a sharp left. At that moment you are at the side of Roubideau Creek, and very near the population center of the United States. (I estimate it to be across the creek in a field on the other side.)

Every ten years a new population center of the country is determined by geographers at the Federal Census Bureau, using data from the latest recently completed national census.

The Center for Land Use Interpretation (your tax dollars at work) report:
The official Center of Population for the USA is determined mathematically as the place where an imaginary, flat, weightless and rigid map of the United States would balance perfectly if all the 310,570,000 or so residents counted in the 2010 census were of identical weight. The previous official population center of the USA, based on the results of the 2000 US Census, is 2.8 miles east of Edgar Springs, Missouri. It moved 35 miles west-southwest from a point near Steelville, Missouri (its position determined by the 1990 census). The new Population Center, near Plato, is 25 miles southwest of Edgar Springs, reflecting the general migration of population from the wintery urban northeast to the sunbelt of the Southwest. Currently the federal government is working with local representatives at Plato to determine a location for a monument to mark the new center in a place that is accessible to the public.

A nice map from Wikipedia shows the population center for every census since the first census in 1790.

It raises several interesting questions for a multi-curricular investigation.. for example, note the change from 1920-1930 or 1940-1950 as compared to 1930-1940. Now I read "The Grapes of Wrath", and all those Okies heading west should have made a big jump, ...shouldn't it? But it seems prosperity brought a greater incentive to move west than poverty.

And what accounts for the really small jump from 1910 to 1920? Yes there was a war on...but there was in the forties also.

If you are interested in expanding this analysis, you might explore another site operated by USA Today which has separate offerings for population centers of various descriptions (whites, Blacks, American Indians (NOT Oklahoma if that was your guess)etc..). You can also pop open a page with statistics about the various states and the nation as a whole, but it doesn't include the 2010 census results. The data includes some interesting (read scary) data on poverty numbers by age..

So, the next time you are cruising west on 44 across Missouri, set the old Tom Tom on Plato, Missouri and swing by. Cannon's Cafe on Hwy 32 in Plato should have Coffee on, and if not Weber's is just up the road. Can't vouch for the pie yet, but when I go through they will almost certainly give me a big slice for free for this unsolicited plug. Are you listening Plato?

I also found an animated link for US centers, but only up to 2000..  At the same site I found another animation for the MEDIAN center of population of the US.  Very different type of motion, and the current median is located well east of the average position... still back in Indiana (although the 2010 census may put it over the line into Illinois, or maybe down into Kentucky... I'm rooting for a location that will be, like me, near beautiful Possom Trot, Ky. 
Addendum:  After Arjen's post (below) I found the following on a Wikipedia page about Centers of population:


In Finland, the point of minimum aggregate travel is located in the municipality of Hauho.[4] It is moving slightly to the west and south every year because people are moving out of the periphery areas of northern and eastern Finland.


In Germany, the centroid of the population is located in Spangenberg, Hesse close to Kassel.[5]

Great Britain

A centre of population in Great Britain did not move much in the 20th century. In 1901, it was in Rodsley, Derbyshire and in 1911 in Longford. In 1971 it was at Newhall, South Derbyshire and in 2000, it was in Appleby Parva, Leicestershire.


The centroid of population of Japan is in Gifu Prefecture, almost directly north of Nagoya city, and has been moving east south east for the past few decades.[9] More recently, the only large regions in Japan with significant population growth have been in Greater Nagoya and Greater Tokyo.


The demographical center of Sweden (using the median center definition) is Hjortkvarn in Hallsberg Municipality, Örebro county. Between the 1989 and 2007 census the point moved a few kilometres to the south, due to a decreasing population in northern Sweden and immigration to the south.[10]

Thursday, 24 February 2011

Usury vrs Interest

I have been browsing through some articles and books about "commercial" arithmetic and its history. Along the way I realized I had not understood the shades of difference in the meaning of usury and interest.

In the middle ages commercial arithmetics seldom discussed compound interest. This was because under canon law it was illegal to charge for the use of money.
Geoffrey Poitras of Simon Fraser University describes the situation in his "From Commercial Arithmetic to Life Annuities:
The Early History of Financial Economics, 1478-1776."
While partnerships could be used to disguise the payment of simple interest, the explicit recognition of a ' profit on profit' payment could bring the sanctions of canon law upon those requiring the receipt of such a payment. For flagrant violation, these sanctions could include ex-communication and even banishment. If such payments were made, and there is some evidence that payment of compound interest was a regular business practice at the time of the Treviso, such payments were made in silence.

It is obvious from the resurrected documents of Chuquet's "Triparty" that compound interest was known, and used, but never openly until later.

It was this church proscription against payment for the use of money, usury, that created the term. Once more in the words of Poitras, "More precisely, under canon law interisse (from the Latin verb "to be lost")
was acceptable while usura (from the Latin noun "use") was not. Compensation could be charged for a loan (mutuum) only if it was a reimbursement for a loss or expense, no net gains were permitted."

Wednesday, 23 February 2011

Subtraction by Adding-On

I'm sure most middle school teachers have encountered mistakes like the one in the picture above due to students using an algorithm to replace understanding subtraction.
Many years ago I became alarmed when I saw it happening frequently in my Alg II classes on a test question that appeared in a testing program used at the school... it wasn't in my syllabus, but still... these are bright kids. I had always been aware of a few students who were dependent on their calculators for very trivial questions, but in following up on the particular subtraction issue, I became aware of how many of my students had almost no skills in what I call "mental math".

Now in my history I had been a regional director for the math council, and had talked with elementary teachers at conferences and in school visits about many areas of math outside my personal teaching experience, and on a few occasions when it was requested, I laid out a suggested topic map for units on teaching subtraction by the method commonly called adding-on using an idea I called number-roads. (not original, but I don't know from whom I claimed it)

In essence the idea was to develop facility with tens compliments and hundreds compliments as part of the pre-development and then use the "roads" to allow something similar to skip-counting, but with variable skipping, to get the differences between numbers. In my high-school classes I dispensed with "number roads" and simply encouraged the use of "linking numbers" to pave the way for counting on.

A student given 307 - 128 would be encouraged to set up the problem with the numbers spaced apart more than is typical, and then put a few easy linking numbers in between. These would depend on the student, and would change as they got better at mental math, but might include 130, 150, 200, 300. They could then make little gap-markers between each pair (2 for the gap between 128 and 130... twenty more for 130 to 150...etc) and simply add the gaps...

When I met resistance by students who felt they were fully functional with conventional algorithms, I would make them solve a couple of problems with time or distance involving bases other than ten. If they handled these, I would assume they understood and used subtraction well enough to do well with whatever approach they used.

I was amazed at how many of the students became fond of challenging themselves with doing the problems mentally after awhile. We would do two or three problems in the review at the beginning of each class and they often became quite skilled. When we had guests and they wanted to show off I would pick a problem which particularly favored counting on, something like 1004 - 397, but it really became unnecessary as they quickly progressed.
Their interest gave me a beautiful lever when we got to working with polynomials. I would show that (10x+5)^2 allowed you to mentally calculate the square of numbers ending in five... and 38x42 could quickly be viewed as (40-2)(40+2)... We would continue to the end of the year with a few moments of mental drill each day, and the kids who returned for pre-calc and calculus seemed to retain the skills for the most part.

I don't teach Alg II anymore, and still mention mental math once in awhile in pre-calc, but I've never gone back to the regular teaching of mental math skills.

I wonder if counting-on is taught in the elementary grades at all, or how subtraction is approached.
I did find a UK educational site where the term "counting on" was used to describe an approach very much like what I have taught...and they even have an interactive Excel spreadsheet to teach the method.

Tuesday, 22 February 2011

"Borrowing" in Subtraction, A Brief History

Subtract joins two easy to understand roots, the sub which commonly means under or below, and the tract from words like tractor and traction meaning to pull or carry away. Subtraction then, literally means to carry away the bottom part. The "-" symbol for subtraction was first used in Germany as markings on barrels to indicate those that were underfilled. Around the 1500's it began to be used as an operational symbol and it became common in English after it was used by Robert Recorde in The Whetstone of Witte in 1557.

The term subduction was often used in older English books up until about 1800. John Wallis uses the term in his "Treatise on Arithmetic", 1685 in describing subtraction... "Supposing a man to have advanced or moved forward, (from A to B,) 5 yards; and then to retreat (from B to C) 2 yards: If it be asked, how much he had advanced (upon the whole march) when at C? Or how many yards he is now forwarder than when he was at A? I find (by subducting 2 from 5,) that he is advanced 3 yards. Samuel Johnson's 1768 dictionary defines both terms, but includes "substraction" as part of the definition of subduction.

In the same dictionary, Johnson defines both subtract and substract, but for subtraction, the reader is referred to "see substraction" so I assume that was the more common term.


The method of subtraction commonly called Borrowing or decomposition seems to go back at least to the 1200s. In The Art of Nombrynge by John of Hollywood (Sacrobosco) subtraction was taught "entirley like the method of today, 'borrowing' and all." [E. R. Slight from "The Craft of Nombrynge, Mathematics Teacher, Oct. 1939].  Robert Recorde used the term in The Ground of Arts (1543) and Denniss and Smith (Robert Recorde, the life and Times of a Tudor Mathematician) describe the term as "a term that had already gained some currency".   The word, "borrow" may not have been commonly used until around 1600 as the earliest listing in the OED associated with subtraction is "[1594 BLUNDEVIL Exerc. I. (ed. 7) 91] Take 6 out of nothing, which will not bee, wherefore you must borrow 60." The borrowing of 60 suggests the exercise may have been about time. Here is a link with an image of a page from the Arithmetic of John Ayres in 1695 in which the word borrow is used in subtraction, although the method is more like what Ross and Pratt-Cotter (below) call the "equal additions" method.

Susan Ross and Mary Pratt-Cotter [Subtraction in the United States: An Historical Perspective, from The Mathematics Educator] show that prior to about 1940 in the US there were three common approaches to subtraction in arithmetic texts. The Borrow method, a method they call "equal additions" which also seems to date back to the 15th century (and is probably a logic based alternative to the borrowing approach), and a method of subtracting by adding on from the subtrahend, which is sometimes called the Austrian algorithm. (images of all three types can be found in the link above to Ross and Pratt-Cotter's document)
The research by Ross and Pratt-Cotter indicated that before 1937 there were few illustrations in American textbooks that show any physical "marking through or numbers being rewritten". Their work states that almost overnight, after a study by William Brownell, "most textbooks used the decomposition (borrow) method for describing borrowing in subtraction, and the use of the crutch described by Brownell became very popular. Today this method of subtraction is used in most textbooks that teach subtraction." The study also states,
Only one example was found, from a text published in 1857(Ray's Practical Arithmetic), where markings were used to keep track of the renaming process. This was done in only one problem in the text, with all other problems worked without any markings. Brownell was not aware, however, of any textbook employing this technique.
This statement, which I assume to be true, and the existence of a clear example of the "borrow" with markings in an 1898 copy of Gill's Oxford and Cambridge Practical Arithmetic , shown at right, make me suspect that the borrowing crutch first appeared in England and then made it's way to America. {if you have knowledge of any earlier appearance in textbooks in ANY country, please write and if you include a digital image, I will sacrifice a student in your honor)

I posted a request for information about texts or other sources of the use of the "crutch" and received the following from Ralph Raimi of the University of Rochester:
" I entered kindergarten in 1929, ten years before the Brownell article, and while I don't remember distinctly just which grade introduced me to the "borrowing" scheme for subtractions, it was surely in my schoolwork by 1933, probably 1932, in the Ferry School, Detroit, Michigan; and the crutch pictured in Ballew's example was standard procedure for us. We did (and every grocer did, too) the corresponding thing when adding a column of figures, as a grocer would do on his brown paper bag, listing and summing the item prices in a vertical column on the side of the bag before filling it with the items themselves. If the sum of the right-hand column (in cents) was, say 126, he would enter the 6 below the line, as part of the ultimate sum, and enter the 2 above the top of the tens column and the 1 above the top of the hundreds column, etc. In practical commercial sums and differences the place of the decimal point was implicit (dollars and cents) and disregarded in grocery stores until the end. In school we were careful to make vertical things *beautifully* vertical, and to preserve the decimal points throughout. Of course this made no difference, except to our understanding. Later, probably by the fifth grade, we were encouraged to *imagine* the crutch in a subtraction problem; writing it down was a sign of weakness, akin to moving the lips while reading. I'm sorry I have no documents from that era in my education, but I do know that my work was always supervised by my older brother, who was five years older than me, and he never showed any surprise at anything I did in arithmetic, so I imagine that by 1925 or 1926 he had also been learning to write subtractions (also in Ferry School) in the same way. Some years ago I was studying a facet of the history of the Detroit Central High School in the period 1898 - 1950 (it was debating clubs that interested me, not math lessons, but no matter), and I found that the Detroit Public Library had several archives of random materials of no particular importance that they had filed under the names of certain (dead)teachers, but cross-referenced so as to make it easy to know what they were. How those particular teachers, or their heirs, got these memorabilia into the library I can't imagine, for the files contained only old school newspapers, club meeting minutes, letters and so on, and the teachers themselves had not been notable; but I believe that if you go to any big city public library and ask for archives of local school teachers of a certain era (1900-1940, say), you might find a sheet of homework or a set of exams or answers, written out in that teacher's hand, or a student's, in some one of them. That might tell you more about the arithmetic style of the time than even the popular textbooks would. "
It may be that the use of the "crutch" markings were commonly taught, but not found in books because disagreement about whether they should be used. In The Teaching of Arithmetic by Paul Klapper (1934), he gives an example both with and without the markings, and calls the form without the markings the "recommended form --- no 'crutches' should be permitted." The very use of the word crutch seems to confirm Professor Raimi's assertion that the marks were viewed as a weakness to be avoided or overcome.

However in the article Klapper states that, "This method is the favorite of many teachers who hold that it is very simple because it can be demonstrated objectively with dimes and cents and that it can be habituated quickly. Others are opposed to it because it requires a second set of number facts -- the subtraction combinations." The evidence seems to suggest that the use of a the borrow markings were common in America well before the publication of Brownell, but it may not have been common in textbooks because, as stated by both Professor Raimi and the Klapper book, it was viewed as a weakness.

I have also found another early use of supplemental marking of a problem. This example, using the equal additions method, comes from a 1873 copy of Charles S Venable's A Practical Arithmetic. Here is a copy of the paragraph from page 25

And I have just found a copy of a 1773 Encyclopedia Brittanica that has "borrowing" used in the process of subtraction

Sunday, 20 February 2011

Hard Homework

From   Tales of Statisticians..

Neyman was invited to Berkeley in 1938. Berkeley, as he put it, was "tabula rasa" - no statistical study at all then existed. None of these deeply rooted London antagonisms: he was getting in at the foundations.
Neyman's mention of two unsolved problems, including the Gosset/Student problem, in a Berkeley lecture in 1940, would have been of little consequence had George B Dantzig not been late to that particular class. He mistook the unsolved problems on the blackboard for a homework assignment, and wound up solving both of them. Neyman took the initiative in getting Dantzig's papers published, and thus launched Dantzig's career. Dantzig's discovery is legendary, and Neyman's generous response deserves to be legendary also: not all teachers will tolerate a student who discovers something in their presence, and few of those will take the initiative in getting that student's work published.

Love these stories...

Thursday, 17 February 2011

The Sin of David

In the bible stories in Samuel and Chronicles, God sent a plague on the people of Israel because of David's "numbering the people". It is not clear what the word meant in this context, or why God was displeased, but thousands of years later religious people fought to defeat the idea of public numeration (census) based on the fear that it would bring God's wrath.
[from 2nd Samual 24]
"David was conscience-stricken after he had counted the fighting men, and he said to the LORD, "I have sinned greatly in what I have done. Now, O LORD, I beg you, take away the guilt of your servant. I have done a very foolish thing."
11 Before David got up the next morning, the word of the LORD had come to Gad the prophet, David's seer: 12 "Go and tell David, 'This is what the LORD says: I am giving you three options. Choose one of them for me to carry out against you.' "
13 So Gad went to David and said to him, "Shall there come upon you three [b] years of famine in your land? Or three months of fleeing from your enemies while they pursue you? Or three days of plague in your land? Now then, think it over and decide how I should answer the one who sent me."
14 David said to Gad, "I am in deep distress. Let us fall into the hands of the LORD, for his mercy is great; but do not let me fall into the hands of men."
15 So the LORD sent a plague on Israel from that morning until the end of the time designated, and seventy thousand of the people from Dan to Beersheba died.
Not sure why God couldn't have chastised David more directly instead of killing 70,000 other folks.... or why his son Solomon was not punished later for a similar counting is not clear, "Solomon took a census of all foreigners in the land of Israel, like the census his father had taken, and he counted 153,600."  2nd Chronicles 2:17.  He would have surely known of his father's experience...and he was, afterall, as wise as ....well... Solomon. 

Similarly God's retribution seems not to fall on a later enumeration reported in Ezra 2:1-34 where many of the people were not only counted, but named by name...."Now these are the people of the province who came up from the captivity of the exiles, whom Nebuchadnezzar king of Babylon had taken captive to Babylon (they returned to Jerusalem and Judah, each to their own town, 2 in company with Zerubbabel, Joshua, Nehemiah, Seraiah, Reelaiah, Mordecai, Bilshan, Mispar, Bigvai, Rehum and Baanah):
   The list of the men of the people of Israel: .....

So why the singular focus on David's sin became so common is unclear to me..but it lasted well into the 18th century. 

In a letter from John Winthrop to Sir Nathaniel Rich he reported that the people of the Colony had not been formally numbered, by reason of " David's example," but there were believed to be more than 4000 of them. 

In The "Triumph of Numbers" by I B Cohen he quotes Winthrop letter with the colorful reasonin that, "David's example sticke's somewhat with us."

From the same source I found, "In 1712, in a letter to the Lord of Trade, the Governer of New York blamed the imperfections of the census of 1712 on the fear of God's wrath and, in a report, claimed that an earlier count had been followed by excessive sickness in the colony." 

Wednesday, 16 February 2011

Euler's Gem....Descartes' Pearl?

One of the great lost papers of mathematics was the Progymnasta de solidorum elementis [Exercises in the elements of solids] of Descartes.  It is in this paper that he did, or did not, depending on who's argument most impresses you, first give the famous theorem by Euler on the relation between the faces, edges and vertices of polyhedra.

In 1649 Descartes went to Sweden to serve as the tutor of Princess Christina.  After his death he was buried in Sweden, but his possessions were sent back to France, but the box containing his manuscripts fell into the river.  Many of them were rescued, and this particular one was recopied by Leibniz.  Afterward, the original seems to have been lost, and Leibniz copy was undiscovered until 1860.  The manuscript is unquestionably the first known study of polyhedra.  It certainly had something close to Euler's famous V+F=E+2, at least to the modern eye.  It was just as certainly not known to Euler or any other mathematician of the period. 

So did Descartes discover Euler's Gem?

Here is what Ed Sandifer Writes about the paper:

So, what did Descartes do? He studied something closely related to Euler’s formula for the sum
of the plane angles of a polyhedron. In Descartes’ time, people had a concept of a solid angle called the deficiency. The deficiency of a solid angle is the amount by which the sum of the plane angles at the solid angle fall short of four right angles. In the case, for example, of a solid right angle, formed by three right angles, the deficiency will be one right angle. For a cube, which contains eight solid right angles, the total deficiency is eight right angles. Descartes’ main result is that this always happens:

Theorem: The sum of the deficiencies of the solid angles of a polyhedron is always eight right angles. It is an almost trivial step from this to Euler’s theorem, that the sum of the plane angles is four times the number of solid angles, less eight right angles, that is 4V – 8 right angles.
Descartes’ other interesting result is more subtly related, but still remotely equivalent to V – E +
F = 2. Descartes writes:  "Dato aggregato ex omnibus angulis planis et numero facierum,
numerum angulorum planorum invenire: Ducatur numerus facierum per 4, et productum addatur aggregato ex omnibus angulis planis, et totius media pars erit numeris angulorum planorum."
.. Given the sum of all the plane angles and the number of faces, to find the number of plane angles: The number of faces is multiplied by 4, and to the product is added the sum of all the plane angles, and the half part of this total will be the number of plane angles.

It is easy, but not obvious, to transform this rule into Euler’s V – E + F = 2,

In his "Proofs and Refutations", Imre Lakatos believes that the small step from Descartes to Euler was not so small a step in the period of Descartes.  Descartes failed to seize upon the concept of dimensionality that Euler grasped, a connection between the zero dimensional points, the one dimensional edges and the two dimensional faces.
Here is how Lakatos stated it, as quoted in Descartes Mathematical Thought by Chikara Sasaki

Tuesday, 15 February 2011

The Origin of "Read Euler..."

Some years ago in my high school class I quoted the famous "Read Euler" quote with the preamble, "As a great mathematician once said, ... and then the quote."..

A couple of days later a student mentioned laughingly that another student had thought the quote was created by me (???a great mathematician???).  The student paused for a minute and then responded, tentatively, "You didn't..(long pause)....did you?".

I was reminded of this because I have recently been learning more about Euler reading articles by Professor Ed Sandifer.. and especially enjoyed a footnote about the origin of the phrase on his article on Euler as a Teacher.

Let us start with the Great Quotation, dubiously attributed to Laplace by Guglielmo Libri about
Lisez Euler, lisez Euler, c'est notre maître à tous.
We traditionally translate this as
Read Euler, read Euler. He is the master of us all.
This gave Bill Dunham a title befitting his most excellent book, [Dunham 1998] but there are
other ways to translate it. Because maˆ itre =   master, teacher
and "notre … à tous" can mean "of us all" or "notre" can be assigned to modify "maître", leaving "à tous" to mean "all things", other valid translations include:
Read Euler, read Euler. He is our master in all things.
Read Euler, read Euler. He is the teacher of us all.
Read Euler, read Euler. He is our teacher in all things.

The footnote is about the quote at the beginning:
 Libri was a scoundrel, a forger, a book thief and an indifferent mathematician, [Rice 2003] but he did write a decent history of mathematics. In Libri's defense, note that he claims that he heard these words "de sa propre bouche", from Laplace's own mouth, not that Laplace actually wrote them down. [WikiQuote]

Monday, 14 February 2011

Math and Politics

Politics and math
I received a nice e-mail from Dan MacKinnon, a Canadian math/computer teacher (who writes a nice recreational math blog)  after my blog about Karl Marx and Mathematics.  
He wrote:
I enjoyed your short post on Karl Marx's mathematics.
I first heard about Marx's mathematical work when I was a student at Dalhousie University in Halifax. While I was there, I heard a story that that back in 1970 a prof there was pushed out by the admin because he was using Marx's stuff as the basis for a course he was teaching on Real Analysis. I wish I knew the whole story - what made it more interesting was that the prof was F.W. Lawvere (pretty famous Category Theorist) and he was pushed out during the October Crisis (a terrorist incident in Montreal, 1970), which was used as a pretext to get rid of a number of radicals and undesirables in a lot of Canadian institutions.   [MY INSERT- I have found online that  “Dalhousie University in 1969 set up a group of 15 Killam-supported researchers with Lawvere at the head; but in 1971 it terminated the group. Lawvere was controversial for his political opinions, for example, his opposition to the 1970 use of the War Measures Act, and for teaching the history of mathematics without permission. (?boy they could lock me up any day?) But in 1995 Dalhousie hosted the celebration of 50 years of category theory with Lawvere and Saunders Mac Lane present.”   Not sure how long it took to be “pushed out”.]
In connection with this this story, I was told that politics and mathematics go together surprisingly often. In the early days of Category Theory, this area of mathematics was perceived as "leftist" - even Saunders Mac Lane's famous book, "Categories for the Working Mathematician" used "working" with a slightly political nuance. I was also told that while category theorists were perceived as progressives, set-theorists were perceived as reactionaries. I have no idea whether or not these supposed political distinctions among mathematicians is true today, or if they were ever true.

Mathematicians getting in trouble because of their political/religious views is not a new idea... as I found in this old cut from the introduction to a geometry textbook.. In this case, one might suggest that bad politics lead to good math.  

And one of my favorite math stories is  from George Gamow's autobiography and is about the Nobel Laureate, Igor Tamm.
 "Here is a story told to me by one of my friends who was at that time

a young professor of physics in Odessa. His name was Igor Tamm (Nobel
Prize laureate in Physics, 1958). Once when he arrived in a neighboring
village, at that period when Odessa was occupied by the Reds, and was
negotiating with a villager as to how many chickens he could get for
half a dozen silver spoons, the village was captured by one of the
Makhno bands, who were roaming the country, harassing the Reds. Seeing
his city clothes (or what was left of them), the capturers [sic]
brought him to the Ataman, a bearded fellow in a tall black fur
hat with machine-gun cartridge ribbons crossed on his broad chest and
a couple of hand grenades hanging on the belt.
'You son-of-a-bitch, you Communist agitator, undermining our Mother
Ukraine! The punishment is death.'
'But no,' answered Tamm, 'I am a professor at the University of Odessa
and have come here only to get some food.'
'Rubbish!' retorted the leader. 'What kind of professor are you ?'
'I teach mathematics.'
'Mathematics?' said the Ataman. 'All right! Then give me an estimate of
the error one makes by cutting off Maclaurin's series at the nth term.
Do this, and you will go free. Fail, and you will be shot!'
Tamm could not believe his ears, since this problem belongs to a rather
special branch of higher mathematics. With a shaking hand, and under
the muzzle of the gun, he managed to work out the solution and handed
it to the Ataman.
'Correct!' said the Ataman. 'Now I see that you really are a professor.
Go home!'
Who was this man? No one will ever know. If he was not killed later, he
may well be lecturing now on higher mathematics in some Ukrainian
I tell this story every other year or so to my physics students when
they cannot be bothered to remember the form of the remainder in Taylor

I imagine that as long as you do math, or teach math in a public environment, we will be subject to political influences.  I’m not sure it is always bad..... but....

Pi, e, and Gamma....and Euler

Pre-statement for some of my students who have not met the very important constant gamma...

As you know, the harmonic series, h=1 + 1/2 + 1/3 + 1/4 + ..=  diverges to infinity, but very slowly. For a finite number of terms, the summation is very nearly the same as ln(n). The error in this approximation is considered an important constant in several areas of mathematics and is called the Euler-Mascheroni constant, or more often, gamma and has a value of about .577.
I came across a paper by Professor Ed Sandifer that he wrote back in 2007 in which he says, "Sam Kutler, now retired from St. John’s College in Annapolis, once pointed out that there are three great constants in mathematics, , e and , and that Euler had a role in all three of them. Euler did not discover e or , but he gave both of them their names. In contrast, Euler discovered, but did not name , the third and least known of these constants."

He goes on to point out that Euler and Mascheroni, who names are both often used for the constant, referred to it as A or C.

Jeff Miller's excellent web site on the first use of math symbols gives several of the sources which seem to be based on one mistake being passed on and quoted without verification:

According to William Dunham in Euler, the Master of Us All (1999), Mascheroni introduced the symbol γ for the Euler-Mascheroni constant. Dunham's source is "On the History of Euler's Constant," by J. W. L. Glaisher, which appeared in 1872 in The Messenger of Mathematics. In the paper, Glaisher does not specify where Mascheroni used the symbol, but seems to imply it is in Adnotationes ad Euleri Calculum Integralem, which Glaisher indicates in a footnote is a work he has not seen but which is referred to in volume 3 of Lacroix's Differential and Integral Calculus.
According to the CRC Concise Encyclopedia of Mathematics (2003), Euler used γ in 1781.

Professor Sandifer finally concludes that probably credit should go to an 1835 article by Carl Anton Bretschneider, the first physical source that seems to include the use, but it seems that Bretschnieder also thought Euler had used it first.

Here is page 260 from [Bretschneider, Carl Anton, Theoriae logarithmi integralis lineamenta nova, Journal für die reine und angewandte Mathematik, (Crelle’s journal) vol. 17, pp. 257-285, Berlin, 1837].

I do have to disagree mildly with one point in Professor Sandifers paper.   It was not Euler who named the ratio of the circumference to diameter of a circle as Pi.  I will stand by what I have written before that "The first known use of the symbol π for its present purposes was in 1706 by William Jones, an English mathematician, although it was the use of the symbol by Euler that brought it its permanency."  Jone's use can be found in “Synopsis Palmariorium Mathesios”. 

Since my students assume that all geometry dates back to the ancient Greeks and Egyptians, I should point out that a special pat on the back might go to J Christoph Sturm. Florian Cajori points out that "perhaps the earliest use of a single letter to represent the ratio of the length of a circle to its diameter" occurs in 1689 in Mathesis enucleata by J. Christoph Sturm, who used e for 3.14159....
si diameter alicuius circuli ponatur a, circumferentiam appellari posse ea (quaecumque enim inter eas fuerit ratio, illius nomen potest designari littera e)."


Sunday, 13 February 2011

Homework Confusion

John Cook over at the Endeavour wrote a quick blog about a homework conversation with his daughter about physics homework:

My daughter and I were going over science homework this evening. A ball falls for 10 seconds. What is its final velocity?
JC: So how fast is the ball going when it hits the ground?
RC: Zero. It stops before it bounces back up.
JC: Well, how fast is it going just before it hits the ground?
RC: They didn’t ask the almost final velocity. They asked for the final velocity.

It reminded me of a two students working on a related rates problem in class,

It was a typical -------A 6 foot tall man is walking away from a 15 foot tall lamppost at a ......find the rate at which his shadow is growing. 
After about ten minutes of the one who "got it" trying to explain, they called me over. 

So I asked,  "Have you drawn a picture?" 
She replies, "That's the problem, how can you draw a picture if  you don't know where the sun is at?"
Me, confused, "????"  THE SUN??? 

Her.." Yes, the shadow will depend on the angle of the sun won't it?

ME.....OHhhhhhhhhhh.... OK... It's night-time... now read the problem....

HER... read...mumble,,, read... OHhhhh...... why didn't they say that...

Friday, 11 February 2011

Cursed by XKCD

Why Can't I get it out of my mind...

Damn You XKCD....

Wait, why didn't I think of this sooner...


ahhhhhh, better.

Chances Are???

Lovely Math Stories...

 What are the odds..... In 1948, the mathematician Warren Weaver, writing in The Scientific Monthly, suggested that the chance of a fair coin landing on its edge when tossed was about 1/109
But...just came across this story in my notes from someone I don't know named Robert Matthews who apparently knows the professor in question:

On October 9, 1972, the mathematician Dr Jeffrey Hamilton from Warwick University wanted to show his students the effect of chance by tossing a coin. Taking a 2p coin out of his pocket, he tossed it, then watched as it hit the floor, spun round and came to rest on its edge.
Prof Hamilton tells me that dozens of students witnessed the amazing event, and after a stunned silence they all broke into wild applause. As well they might, for you don't need to be a distinguished Cambridge mathematician to postulate that none of them will see such an event again.
  As I tell my stats kids, unusual things happen all the time, they just aren't going to happen to YOU!.... usually

Thursday, 10 February 2011

Infinite Series and Proofs w/o Words

Came across a nice post at irrational cube about proofs- without words and infinite series....

 here is a teaser, and a link to the  post.

Sometimes I inadvertently come up with a math problem and can’t seem fall asleep until I solve it.  So it happened last night.
It started with my unit on limits.  I plan on giving the students a math problem in which they fill in half of a square, half of the remaining area, and so on and so forth until they see that they get arbitrarily close to filling in the whole thing.  Mathematically, this means that
\displaystyle \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + ... = 1
\displaystyle \sum_{n=1}^{\infty} \frac{1}{2^n} = 1
I made it into a proof without words for the visually inclined:
(note: the different color shading is to differentiate iterations)

and finishing up with .......
 \displaystyle \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3} + \frac{1}{x^4} + ... = \frac{1}{x-1}

Really nice... so check it out

Wednesday, 9 February 2011

The Origin of FOIL for Binomial Multiplication

Recently alerted by Dave Renfro to a discussion on the Math Forum about the first use of the acronym FOIL for the multiplication of binomials.

Robert Hansen wrote , "While reviewing Shute's Elementary Algebra book I came across a reference to FOIL during the discussion of products and factoring of polynomials."

Dave Renfro responded with, "I was also curious when I saw Robert Hansen's post earlier
today, so I did a quick google check  and found
FOIL used in William Betz's book "Algebra for today: second
course: Book 2" from 1931"

Jonathon Crabtree found an earlier use of the term in an earlier edition by the same author:
 " FOIL also appeared 'Algebra for Today' written by William Betz first published by Ginn and Company in 1929.
QUOTE "first terms, outer terms, inner terms, last terms. (The rule stated above may also be remembered by the word FOIL, suggested by the first letters of the words first, outer, inner, last.)"
As William Betz had been writing math books for more than a decade by this time, it's likely 1929 was not the first time FOIL was written about."

I have renewed my search for the origins of Foil, and a little more about William Betz, which I will add to this as I do not have enough for much of a bio page for him.
In 2015 Thony Christie, a very knowledgeable math historian, wrote that he had never hear of the term Foil, and had to look it up.  His historical experience, and the fact that he has lived and worked in both England and Germany for extended periods suggest that this was an American creation, not an import.
I also heard from Shemsi alhaddad, who grew up in Kuwait,  is an associate professor of math. He writes, "not in Arabic (that I know of). Just taught as the distributive property.."  in the 1980s. "It may be different now. I didn't hear of anything like FOIL or Sohcahtoa until I moved to the U.S"/
Pat Grew , a Canadian math teacher, was aware of its use; " My intro was in Ontario Curr in '84 as a teacher."

Here is an early bio in the Orient, the yearbook of East High School in Rochester, New York where he was head of the Math Dept, and eventually the Assistant Principal as well..

He is also listed in the Admin and Mathematics Dept in the faculty of the 1935 Orient.  There is also a Martha Betz listed in the Modern Language Faculty, and a student named Betz on the baseball team picture,(pg 48) but it now seems they were not relations.
The 1940 census says he was married to a Elsie Betz, and had two children living in the house.  since the oldest was only 12 in 1940.  William lived until at least 1959 when he is listed, still at the same address, in the phone book. 

If anyone has an earlier link to this use of FOIL, or about the life of William Betz,  please advise.

Tuesday, 8 February 2011

Doctor Dolittle and Algebra

Just came across this quote from Hugh Lofting's "The Voyages of Doctor Dolittle".  The Doctor asks his friend Bumpo how he got on at Oxford, and he replies..

“I liked it all except the algebra and the shoes. The algebra hurt my head and the shoes hurt my feet. I threw the shoes over the wall as soon as I got out of the College quadrilateral this morning; and the algebra I am happily forgetting very fast...."

Reminds me of conversations with my calculus class...

Report Card Time

Well, report card time again... And along with grades comes the difficult task of trying to find a quick short comment that summarizes 9 weeks of a student's life into a single sentence; Always hard to do. It is a time of year that brings out the humor in teachers, and I got the following from a colleague who, I assure you, meant them only in jest.

OK, so most of my kids did well, and I didn't have to use anything even close to the comments below, but I did chuckle at several of them, and one or two actually made me think of students I had known. The source said they were actual comments sent out by New York City teachers, but I hope that is just one more of those urban legends that is totally false. Whatever the source, I hope you can enjoy them for the tongue-in-cheek, sometimes too close to true, humor that prompted them.

These are actual comments made on students' report cards by teachers in
the New York City public school system.

1. Since my last report, your child has reached rock bottom and has
started to dig.
2. I would not allow this student to breed.
3. Your child has delusions of adequacy.
4. Your son is depriving a village somewhere of an idiot.
5. Your son sets low personal standards and then consistently fails to
achieve them.
6. The student has a "full six-pack" but lacks the plastic thing to hold
it all together .
7. This child has been working with glue too much.
8. When your daughter's IQ reaches 50, she should sell.
9. The gates are down, the lights are flashing, but the train isn't
10. If this student were any more stupid, he'd have to be watered twice
a week.
11. It's impossible to believe the sperm that created this child beat
out 1,000,000 others.
12. The wheel is turning but the hamster is definitely dead.

And One I got form a comment from "false friend"...
13)  Effort has improved: Eyes occasionally flutter open.

OK, there must be a thousand of these 0ut there, so send me your additions to the list....


Monday, 7 February 2011

Follow up to Sum of Squares Problem

A shiny "Attaboy" award to Gasstationwihtoutpumps for the correct answer to both Lewis Carroll Problems. 

Herer is his solution again. 

The first result is moderately interesting:
(a+b)^2 + (a-b)^2 = 2 (a^2 + b^2)

But the second one is then trivial
(a+b)^2 + (a-b)^2 + a^2 + b^2 = 3 (a^2 + b^2)

Do you have some extra constraints for the second problem to make it non-trivial?

For most folks of my generation, raised on lots and lots of factoring problems, the first problem was one of a pair that jumped off the pages of algebra books so often they were almost memorized.  The sum and difference of (x+y)2 and (x-y)2 produce the  results 2( x2+y2) and 4xy; both of which seemed to occur in textbook problems of my youth.  
Had Gas... not already told me he had always had problems with memory I would think he had grown up on the same factoring fare and had also memorized them.  Perhaps he is more like Dodgeson (Carroll) who apparently wrote about re-discovering this in his diary "towards the end of his life" as he too did not recall it.  As a puzzle maker, he probably thought the addition of a third multiple would disguise a common factoring and it was this second problem that appeared in his "Pillow Problems".

 Since my own memory also is very fragile, I was fortunate to find the information in Edward Wakeling's "Lewis Carroll's Games and Puzzles". 
Carroll also included a somewhat more difficult probability problem in his "Pillow Problems".  Since I can't make the first problem more difficult for Gas.... I will give the advanced version of this one. 

An urn holds B Black and W White balls.  One is selected at Random and placed in a bag without your knowledge.  Then a White ball is added to this bag.  A random draw from this bag produces a white ball.  What is the probabilty the original ball was White?

Sunday, 6 February 2011

Sums of Squares Problems

Now even my ex-students are trying to hang me up with "Pythagorean-Like" Problems.  Today I got these two in an email. Since the problems date back to Lewis Carroll (whom  my student's assume I must  have gone to school with since I am so very old).

 I knew the problems and the solutions so I will leave them for now as problems for the reader, and post a solution in a day or two; hopefully with a litlte of the history. 
They are suitable for HS level so everyone is invited. Feel free to submit your solutions.

For any two positive integers X and Y, twice the sum of their squares can be expressed as the sum of two squares.  For example 2(42 + 72) =    2(16+49) = 2(65) = 130..but 130 = (112+32)   

And the followup??? Prove that three times the sum of two squares can be expressed as the sum of four squares. 


Saturday, 5 February 2011

A Problem with a Plethora of Pleasing Results

Two squares, and it is almost Pythagorean.....

I had a student ask me this one. He thought I might want to add it to my almost Pythagorean list... I do, but could I prove it's true?   He asked:
If one square is inside another with corresponding sides parallel and the vertices of the inner square are connected in order to the sides of the outer square, then the sum of the squares of the distances of the connections from opposite vertices will be equal.  For the image below,  GD2 + EB2 = FC2 + HA2

We played around at the board for awhile and finally proved it with nothing fancier than the Pythagorean Theorem and simple algebra.  My method was to draw a horizontal segment (labled R for right shift) to mark how far the inner square is moved from the left edge of the outer square and then a vertical one, D, for down-shift. Then with lables of L for the large squares side and  S for the small squares side (not creative, but easy to remember) I could write the length of all four segments.  (DG)2 for instance would be D2 + (R+S2. We expanded out the squares of all the binomials and trinomials (and he corrected me when I wrote D for R or got a negative as the product of two negatives) and then crossed off mathching parts until everything was gone..  the two expansions were the same alphabet soup of algebraic variabless.  QED. 

One student witnessing our work commented that we had covered the board and never written a number.  I seized the moment to explain what power that was, and began to expound on the implications for possible generalizations... since L, S, R, and D could be any positive (actually two could have been negative) value, and had no relative order or size, S did not have to be smaller than L... and R did not have to be smaller than L... conclusion??? It must also be true that the squared distances are equal even if the one square is not inside the other.  It could be partly inside, or totally outside and either could be larger. 
AND.... pausing for breath... S could be very, very small..... approaching zero, so the degenerate case would  have the smaller square as a another corollary :
For a square and a point on the same plane, the square of the distances from the point to one pair of opposite vertices will have the same total as the square of the distances to the other pair. (Why have I never seen this theorem?... Is it in textbooks?)    And I propose that the condition that the point is in the same plane is unnecessary.  In fact that seems almost trival since it only adds twice the square of the perpendicular distance from the point to the plane of the square... if my mind is working correctly this morning.  If that is true, that also means that for any square based pyramid, the squares of the sums of the lengths of two opposite edges from the base to the vertex will also be equal.

Now that I was on the roll, I kept playing after the kids wandered away...

What about if the sides of the squares are not parallel.  Playing around with Geogebra, I have determined that that is not a necessary condition.  The original theorem is true for any two squares in the plane however oriented... that is, they do not have to have their sides parallel.  (proof needed, submit yours and save me some time, I'll amend this part)

what about rectangles.??..... If we inscribe a square in a rectangle, then the degenerate case (the inner square is a point) is still true, but with a square and a rectangle the vertex to vertex sums are not equal... in fact, I have discovered that they will differ by a constant value (no matter where the square is or how it is rotated ) that is equal to the product of twice the side of the square times the diffence in the sides of the rectangle.  This property, however, is not preserved when the square is rotated inside the rectangle.  I think that there would be a different constant difference for any orientation of the corresponding sides of the square and rectangle (untested wild conjecture).

While trying to search for the history of the problem, I found a similar relation about parallelograms In the MAA reprint of Jacques Hadamard's Geometry, a similar problem appears on page
I worked out this "constant distance" and found that the difference  is equal to twice the product of the length times the width times the cosine of the acute angle between them, 2*L*W Cos(theta) ("Holy Law of Cosines, Batman!").  I think this proof would be suitable for HS students.  Again, there seems there would be no need for this point to be in the plane of the parallelogram. 

If we replace the square by a rhombus, and sum the squares of the distances from a point to a pair of opposite vertices, I believe the two sums will differer by s^2.  I have not  had time to prove this  yet (offers accepted). 

While I'm in the mood to make conjectures, If the two squares lie in parallel planes then the sum of squares will almost cerainly still be equal. I imagine a physical construction of the two squares in the plane, with elastic chords connecting the vertices.  When I lift one square out of the plane and translate it along a line perpendicular to the original plane I am adding a distance of 2* h^2 to each pair of vertices sum.  Since the two squares could have been in any orientation originally, it seems that no special precaution about the orientation of the squares would be needed.  This would seem to have application to skew prisms, frustums of pyramids and skew frustums (I'm not sure we have such a definiton in geometry, but I mean a frustum in which one base is not similarly aligned with the other when projected on the same plane. 

 I suspect that if the two squares are not even in parallel planes, the sum of the squares of segments containing alternate vertices will be equal also.  No time to attack that yet.

A first postscript:  It seems that for any even sided regular polygon, the sums of squares of distances from a point to pairs of opposite vertices would be equal also since any two pairs of these opposite verties would form a rectangle.   Likewise, if two even sided regular polygons have opposite vertices connected in sequential order, the same property is preserved as for squares. 

Friday, 4 February 2011

A Geometrical Theorem

I came across the following beautiful result in Clifford A  Pickover's Wonderful Math Book.
It got me interested, and so I set out to see what I could find.

The article below appeared in The Quarterly Journal of Pure and Applied Mathematics: Volume 2 - Page 38  in 1858. 

The simply titled article, by an almost unremembered Cambridge mathematician, must rank as one of the most unusual and interesting in all of math. Although demonstrated with a circle, the problem states that if you construct a chord on any closed curve which is smooth and convex, and divide the chord into two lengths p and q, with a point X; as the chord is moved around the circle the point will generate a path that will always cut out the same area between the locus and the outside boundary of the curve.... No matter what shape the original curve was..and that area is the area of an ellipse with semi-major axes of length and q..... yet there is no ellipse involved in the problem.  ....Or is there.

The motion is reminiscent of a drawing instrument called a trammel.  See Archimedes Trammel at Wikipedia.

In this image the trammel has the scribe, or drawing point, on the exterior of the two guide points where the theorem above presumes the scribing point is interior to the two guides, but that locus will also form an ellipse when drawn on two straight lines for the guides. 

Many students may already have seen that ellipse, or at least a fourth of one in illustrations of the Falling Ladder Problem which often shows up in related rates problem in calculus texts. For this illustration a fixed point on a ladder is followed as the foot of the ladder slides away from the wall and the top of the ladder slides down the wall.  An interactive image for those  who have never seen it is at the Geometers Sketchpad site.

But this means that in some cases, the idea that the outside curve must be smooth (differentiable?) is not true.  If you moved the chord along a square, or rectangle with sides greater than the length of the chord, then it would curve out four quater-ellipses around each corner of the rectangle, with a total area of    
I'm trying to think of a illustrate this theorem with an irregular curve on Sketchpad or Geogebra, but  the task has overwhelmed me to date... if you do it, send me a copy and I will link it here
History stuff

So anyway, it made me wonder about  this mysterious guy, Hamnet Holditch.  It turns out he must have been incredibly bright.  He was Senior Wrangler at Cambridge, and also won the Smith Prize.  In England, this is a big deal, and folks make a fuss about it.. they did then, they do now.

He went on to be a fellow of Gonville and Caius College and hold a number of positions.  But as the passage below indicates, he "held" them rather loosly

From Mr. Hopkins' Men: Cambridge Reform and British Mathematics in the 19th Century.

As it turns out he was from the area of King's Lynn in Norfolk, not far from where I live in  the UK.  I found a little more about him online.

Henry Hillen's e-book, "History of the Borough of King's Lynn".was born at Lynn (1800); he was the son of George Holditch,
described as pilot, beaconer and harbour-master, to whom the Society
of Arts awarded a gold medal for the invention of life-saving beacons
(1833); modifications of whose designs are still erected upon the
treacherous sands of the Lynn Deeps. The son was for three years
instructed by the Rev. L Coulcher, at the Grammar School.
When eighteen years of age, he was admitted pensioner
of Caius College, Cambridge (6th February 1818); he gained
his B.A. (senior wrangler and Smith's prizeman) in 1822,
when, as a mark of esteem, he was presented with the
freedom of Lynn; his M.A. in 1825, and was senior fellow from
1823 to his death. Probably, in view of a fellowship, he was
baptised late in life at St. Michael's church, Cambridge (17th March
1823). He held various responsible college offices, for example,
Hebrew and Greek lecturer, bursar (1828), etc. He was remarkable
for his extreme shyness. Owing to some slight, perhaps more
imaginary than real, he absented himself for many years from Hall
and Chapel, and was thus known to a few only. Hence a junior
fellow, taking him for an interested strnngcr, ])olitely shewed /lim
round the college ! He delighted in angling, and spent his summers
in Scotland or Wales. He died in college the 12th of December 1867, and was
buried at North Wootton.