Sunday, 26 September 2010
Since I'm not sure how this graphic will work, here is a link to the original source, or just click on the image.
Friday, 24 September 2010
My friend and colleague, Dru Martin in Germany sent me a link that points out that the average cost of tuition at four-year colleges in the US was very closly keeping pace with the total government spending budget. In fact, the article states that the correlation coefficient between the two for the period from 1997 to 2009 is 99.4%.
Argue all you want about correlation and causation... 99.4% is more than a "nod and suggestive wink." Here is the full article for you perusal.
Wednesday, 22 September 2010
The student is then expected to see that the value of r is always in the interval between -1 and 1 inclusive. The fact that this is not even remotely obvious to the new student is indicated by the frequency with which this question is asked by AP Statistics teachers on the AP electronic discussion group. Because my stats course is still a mathematics course, I would like for my students to develop some understanding of the reason, rather than just accepting it is true because “the teacher said so.” Toward that end, and because I believe vectors are undernourished in the current curriculum, I tend to explain the reasoning using the dot product of vectors. Because of the number of equations involved, I have stored the paper (in docx format) here.
One reader also requested the link to the blog on Standard Deviation as Distance..
Tuesday, 21 September 2010
Just came across a link on the Mathforum's History page (not sure why it was there) but you can download a free chapter of "Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin" by Lawrence Weinstein & John A. Adam.
An American-British study has reported that the true number of plant species on the earth is far short of the 1,000,000 usually reported, and may be much closer to 400,000. They credit the overcount to "centuries of scientists naming 'new' plants that had already been discovered". That's a 60% drop if anyone is doing the math. Here is a brief summary from Newser.
The drastic reduction reminds me of the story of the French efforts to measure the earth, and in the process redraw the map of France more accurately, leading to someone's quote that, "Louis XV lost more land to his cartographers than his successors have to the Germans."
"We should add a final word. Leonhard Euler was a mathematician of the very first rank, yet he is almost universally unknown among the general public, most of whom presumably cannot even correctly pronounce his name. The same people who have never heard of Euler would have no trouble identifying Pierre-August Renoir as an artist or Johannes Brahms as a musician or Sir Walter Scott as an author. Euler's contrasting anonymity is both an injustice and a shame.
But what makes it all the worse is that Euler's counterpart among painters is not Renoir but Rembrandt; his counterpart among musicians is not Brahms but Bach; and his counterpart among writers is certainly not Walter Scott but William Shakespeare. That a mathematician with such peers-the Shakespeare of mathematics- commands so little public recognition is a sad, sad commentary.
So, readers are urged to toss this book aside and begin forming fan clubs, making banners, and otherwise spreading the word about one of the most insightful, most influential, and most ingenious mathematicians of them all: Leonhard Euler of Switzerland."
Perhaps it is even a greater shame that many elementary and middle school teachers are equally unaware of the "Master of us all".
Dweck, C., Caution: Praise Can Be Dangerous. In American Educator, Spring 1999.
Carol Dweck has written a book, Mindset (2006), which says about the same things this article does, at much more length. The article describes her thesis and her research much more concisely and (in my opinion) effectively. Her claim (proved by her research) is that praising a student’s intelligence makes them wary of harder tasks and of looking dumb, but praising their effort encourages them to tackle harder tasks and enjoy it. She also looks at people who think intelligence is fixed and compares them to people who think effort can change one’s intelligence. People with the second mindset are able to develop their potential much more effectively than those with a ‘fixed intelligence’ mindset.
You can read a little at the link at amazon. I found the first page really hooked me in.
Thanks to Sue for correcting me (in the comments), her blog has links to an article about the book, not the actual book..
Sunday, 19 September 2010
Way to go JD
Saturday, 18 September 2010
And the author offers that, "Given the level of math phobia present in American schoolchildren and the sorry state of financing for education, it's worth asking whether or not the trend lines of declining quality in education and increasing quality of educational games have already crossed for a significant portion of American students." And then admits he "made up" the graph to accompany the article. But wait, it gets better. Someone read his post and sent an "improved made up" graph (more wiggles means more realistic?). Am I expecting too much from MIT?
Read an interesting article with some informative graphs this morning. The article is "Evolution of National Nobel Prize Shares in the 20th Century" by J¨urgen Schmidhuber. It has a collection of graphs like the one above (showing all science prizes by country of citizenship) showing the percentage of Nobel Prizes won in different areas by the country of birth, and also by the country of citizenship at the time of the award.
They really worked at making it accurate, by allocating partial winners credit. I didn't know that many times the prize is awarded to three or more people, but not equally. One may get half and the other two a fourth, for example.
Here is the abstract: "We analyze the evolution of cumulative national shares of Nobel Prizes since 1901, properly taking into account that most prizes were divided among several laureates. We rank by citizenship at the moment of the award, and by country of birth. Surprisingly, graphs of this type have not been published before, even though they powerfully illustrate the century’s migration patterns (brain drains and gains) in the sciences and other fields."
Thursday, 16 September 2010
I found the link exploring a blog called Junk Charts by Kaiser Fung (see his book below). Here is what it was mostly about.
"The Wall Street Journal reported that the Ritz-Carlton brand of hotels has been hit worse in the slump than other brands in the Marriott family," and then this chart:
Now how much does that difference jump out at you looking at the graphs? For me you have to study it pretty closely to see it. But then he constructed a line graph of the same data...
You be the judge... (or as I often say to my students in imitation of a UK tv commercial..."compare the meerkat"0
"A British Man Placed a $10 Weather Wager That Could Have Netted Him $27 Million, but the Complex Bet Was Invalid"
"As snow enveloped much of the northern U.K. over Christmas, the 52-year-old graphic designer found the weather outside delightful -- and profitable. Mr. Bryant wagered that snow would fall in various regions of the country, and he placed what is known as accumulator bets, which can pile up winnings like the wind-driven snow. Mr. Bryant believed his bets had netted him about $27.5 million in winnings."
(In the UK, there are regulated betting shops in almost every small village where you can place bets on literally anything)
Accumulator bets work like this, The bettor places simultaneous bets. If the first comes through, the initial stake plus the winnings are placed on the second bet. If that comes through, the entire pile is placed on the third bet, and so on. If all parts of the wager are won, the winnings accumulate quickly. If even one part of the bet loses, though, there are no winnings.
"Mr. Bryant recalls going into his local shop and producing one of his two winning accumulator betting slips. He recalls that the manager "started doing the math, and turned a bit white." Then he placed a phone call to headquarters and reported that the ticket was invalid."
So much for agency law in England... but it is a nice story for stats teachers to use in class...and ask why the bookie would be willing to accept a bet on four football games with an accumulator bet, but not on snowfall in four postcodes in England.
People who have read and enjoyed accounts of Dr. John Snow's creative statistical sleuthing that led to the end of a cholera outbreak in London (and to the development of modern epidemiology) may enjoy this news story. (Read below if you are not familiar with Dr. Snow and his "Ghost Map")
It seems that an outbreak of legionaires disease in the small Spanish town of Alcoi was stumping the investigators looking for its source. For those who do not know:
"This microbe lives in fresh water nearly everywhere, and it becomes a problem only when inhaled as a fine spray or aerosol. (Legionella is harmless if you drink it.) Outbreaks are usually traced back to man-made supplies of warm water, such as water cooling systems, fountains, hot tubs, even showers." they investigated all the usual suspects, and came up empty.
At that point, perhaps inspired by the success of Dr. Snow, "In the meantime, an analysis of the patients' locations revealed an unusual geographic shift. The first two cases occurred in the northern part of the city; however, the following eight appeared elsewhere, in the Santa Rosa quarter. So investigators expanded their search to include moving devices that used water, such as street-sweeping machines."
At this point, road repaving in the Santa Rosa neighborhood was under way. One of the machines in use was a milling machine that ground up old asphalt. It carried a 528-gallon (2,000-liter) water tank to supply atomizers, which sprayed a mist intended to settle dust. The water in this particular machine did not come from the municipal water supply, but directly from a spring, unchlorinated and untreated.
"The suspicious machine, it turned out, had been at work in the northern part of the city around July 15 and in Santa Rosa starting July 31. On Aug. 21, investigators found the machine and took it out of service, although not in time to prevent the final case of Legionnaire's disease, which appeared two days later. "
You can read the report in the CDC's Journal of Emerging Infectious Disease fpr September, 2010... or at this Live Science summary.
For people who haven't read about Dr. Snow's work in breaking the hold of Cholera, I highly recommend the book below:
Tuesday, 14 September 2010
These were judged to be the funniest ten jokes told at the Edinburgh Festival Fringe...allow for British Humor (sometimes a little black, sometimes a little dry) and a slight cultural difference
1) Tim Vine - "I've just been on a once-in-a-lifetime holiday. I'll tell you what, never again."
2) David Gibson - "I'm currently dating a couple of anorexics. Two birds, one stone."
3) Emo Philips - "I picked up a hitchhiker. You've got to when you hit them." (OK, this one had MY vote)
4) Jack Whitehall - "I bought one of those anti-bullying wristbands when they first came out. I say 'bought', I actually stole it off a short, fat ginger kid."
5) Gary Delaney - "As a kid I was made to walk the plank. We couldn't afford a dog."
6) John Bishop - "Being an England supporter is like being the over-optimistic parents of the fat kid on sports day."
7) Bo Burnham - "What do you call a kid with no arms and an eyepatch? Names."
8) Gary Delaney - "Dave drowned. So at the funeral we got him a wreath in the shape of a lifebelt. Well, it's what he would have wanted."
9) Robert White - "For Vanessa Feltz, life is like a box of chocolates: empty."
10) Gareth Richards - "Wooden spoons are great. You can either use them to prepare food, or, if you can't be bothered with that, just write a number on one and walk into a pub…"
"As many as half of all the children identified as having special educational needs are wrongly diagnosed and simply need better teaching or pastoral care instead, a report published today finds." [emphasis added]
The report from the Guarian is here.
"Ofsted found that about half the schools and nursery providers visited used low attainment and relatively slow progress as their principal indicators of SEN. In nearly a fifth of these cases very little further assessment took place."
Response from the educational union was that the report was "insulting" (duh!) and wrong. Surely, there will be more to come.
Monday, 13 September 2010
The Mathematics and Multimedia Carnival number three is up at Math4allages... check it out...
Sunday, 12 September 2010
"ScienceDaily (Sep. 12, 2010) — As children in Western cultures grow, they learn to place numbers on a mental number line, with smaller numbers to the left and spaced further apart than the larger numbers on the right. Then the number line changes to become more linear, with small and large numbers the same distance apart. Children whose number line has made this change are better at remembering numbers, according to a new study published in Psychological Science, a journal of the Association for Psychological Science."
See the whole story here.Child's 'mental number line' affects memory for numbers
I found a bunch of nice images of fruits and vegetables at Inside insides
For example, if someone ask you what type of symmetry a watermelon has.. wait..before you answer, look at this.
Had some time to scan through some new (to me) blogs this weekend and one of the things I found was this graphic-visual-calculator by Nathaniel Johnston. It seems like an interesting tool for use by students at home because it is not ad-ridden or powered by Java so it loads quickly and works on almost any platform. It will calculate normal, Student's-t, chi-square and F distributions and do p-values for one or two tails, as well as areas around the mean on one or both sides. Check it out here.
My quick survey of his blog indicates he has lots of stuff on calculus as well, and computing..and he seems to be big on computer games and works them into problems and teaching. See his blog here.
Saturday, 11 September 2010
Almost two years ago I wrote about an experience that happened when I let my kids watch an old science fiction movie in class just before Christmas... The blog, and a followup requested by a teacher who admitted he wasn't really sure why the common "divide and multiply method worked... Here they are as a package...
The day before Christmas break one of my seminar students brought in the old (1951) video of "The The Day the Earth Stood Still". I worked at my desk as they watched, and about thirty minutes in they called my attention to ask if the math on the blackboard was "real". The Alien in the movie, Klatu(Michael Rennie), in the company of a young boy who lived in the house where he was renting a room, had entered the home of a professor who was supposedly knowledgeable about Astro Physics. I did not recognize any physics I knew from the brief shot of what looked like differential equations of no particular relation, but that could be my limited physics more than the actual images.I returned to work, but in a few minutes in another scene, Klatu is helping Bobby with his homework and the only line you hear is "All you have to remember is first find the common denominator, and then divide." My head pops up... what were they doing? "Common denominators" leads to thoughts of fractions, but almost no one teaches finding common denominators as a prelude to dividing fractions (which is sort of a shame because it makes division of fractions work like multiplication...the way kids think it should.) It works in fact, if you do not find the common denominator first, but sometimes the answer is as confusing as the problem.
When you multiply fractions, as every fifth grader learns, you just multiply top times top and bottom times bottom... 2/3 x 5/7 = 10/21. The fact that division works the same way is often missed, or misunderstood because it so often leads to nothing simpler... 2/3 divided by 5/7 is indeed (2 divided by 5) over (3 divided by 7) but that seems not to give the classic simple fraction we seek. For some fractions, it will work out fine... if 4/27 is divided by 2/3, the answer is (four divided by two ) over (27 divided by 3) = 2/9 and that is the answer you get by the method you memorized (but never understood, most likely) in the fifth grade.
But what if we follow the advice of the alien Klatu. If we convert 2/3 and 5/7 to fractions with a common denominator, we get 14/21 and 15/21, and if we divide top by top and bottom by bottom we get 14/15) over 1, which is just 14/15... job done...
I can imagine including some visuals and suggestive images to help it make sense... It is after all, just a reversal of the multiplication process. If we say "3 dogs times 5 = 15 dogs" then by division we should have the equivalent expressions that "15 dogs divided by 5 = 3 dogs." and just as naturally "15 dogs divided by 3dogs = 5" . Students who have learned (I've been in England too long, I just had to edit "learnt") that "eighths" and "fifths" are just units like "dogs" and "kittens" should then understand that 5 eighths divided by three eighths is just as clearly 5/3.
A few days after I wrote that blog I got a response that asked, more or less, "OK, why does the common algorithm work?"
This was my response
I want to make one comment about division of fractions that seems harder to visulaize than for general division, and then I hope to explain in simple terms just why "invert and multiply" works.
For every multiplication problem, there are two associated division problems; A x B = C begets C/A=B and C/B=A. Elementary teachers call these a "family of facts for C" (or did in the recent past.. educational language changes too fast for firm statments by a non-elementary teacher). So if we add units to one or both factors, appropriate units must be appended to the product. So how does this effect operations with fractions? Well if we have length, as in ANON's comment, then the division problem he states, "If we think of it as a piece of wood with length 2/3, then I believe the question is how many 5/7's are there in the piece of wood" he is dividing length by length to get a pure scaler counting how many pieces (or fractions of a piece) will fit into another. In the case he gives, the answer would be only 14/15 of a piece... becuase the 2/3 unit length is not quite enough to provide a 5/7 unit length piece...
The multiplication associated with this operation is then 14/15 of 5/7 units = 2/3 units... What about the other division in this family of facts... 2/3 units divided by 14/15 (a scaler here, not a length)will give 5/7 units length. What is this sitution describing? This seems the one most difficult for teachers and students alike. We all know what it means to divide a length into (by?) two pieces, but what sense does it make to divide it into 1/2 a piece.
We might try to make this clear to students by taking some common length (12 inches?) and see what happens if we divide it into (by) 8 pieces, then four, then two, then one, (each division is by half the previousl number)and look at the pattern of lengths. 12/8=3/2; 12/4 = 3; 12/2 = 6; 12/1= 12... I am confident most students could identify the next numbers in the sequence, 12/ (1/2) = 24, and 12/(1/4) = 48.
At this point, using whole numbers as divisors, the pattern for "invert and multiply" seems obvious, but this is far from a why for all fraction problems.
Let's look at one more case where we sneak in a related idea at the elementary level. Given a problem like 3.5 divided by .04, the student is taught to "move the decimal places enough to make the divisor (.04) a whole number. What we do is another problem (350 divided by 4) that has the same answer (87.5)as the original. Another why does that work that is not often explained.
What do the two operations have in common.... multiplication by one. In each case we have a division (fraction) operation and we simply mulitiply the fraction by a carefully chosen version of one that will make it easier to do. If we view 3.5/.04 as a fraction, then every fifth grader knows that multipliying it by one will not change its value. This is the core of what we do to find equivalent fractions... to get 3/5 = 6/10 we multiply by one, but expressed as 2/2... The decimal division problem uses the same approach... we multiply 3.5/.04 by 100/100 to get another name for the same fraction, 350/4.
Now to explain "invert and multiply" we just use the same idea... dividing fractions is simply fractions which have fractions instead of integers in the numerator and denominator. We want to multiply by one in a way that the division problem will be easier. But the easiest number to divide by is one,... so why not pick a number that changes the denominator of the fraction over a fraction to be a one... that is, multiply by its reciprocal. So for 2/3 divided by 5/7 we can write
(Pardon the duplication, I'm trying a new latex editor to replace some images... )
And I hope that makes it clear....
And the “Shameless Exploitation of Children for Shock Value In the Furtherance of Misinformation” Award Goes To….
…The Physicians Committee for Responsible Medicine. This group, which anyone can join for $20 dollars, has been denounced by the American Medical Association for “inappropriate and unethical tactics” and for giving medical advice that was “irresponsible and potentially dangerous to the health and welfare of Americans.”
So when the group launched a YouTube advertisement in July with young children talking directly to the camera about getting cancer from hot dogs – “I thought I would live forever,” said one child, “I was dumfounded when the doctor told me… I have late-stage colon cancer.” – you might have expected a firestorm of controversy to ensue. Wrong.
Though CNN noted that the ad was the work of “an animal rights group that wants us all to be vegans,” it also told viewers that there was “research” to support the link between hot dogs and cancer. “The problem experts believe it is in nitrites in the processed meats,” said CNN’s Elizabeth Cohen. “That's apparently what's causing the cancer link.” (Nitrites are present in salt and used in meat to prevent spoilage and protect against botulism. They are especially important for meats cooked at relatively low heat, such as hot dogs),
What the CNN report did not relay is that there are no substantiated links between nitrites and cancer, according to the American Medical Association. Or that we get 90 percent of our intake of nitrites from vegetables.
An example is in an article I just came across which suggests that all the drug counseling we do for teens may do more harm than good..Here is a quote
Increasingly, substance-abuse experts are finding that teen drug treatment may indeed be doing more harm than good. Many programs throw casual dabblers together with hard-core addicts and foster continuous group interaction. It tends to strengthen dysfunctional behavior by concentrating it, researchers say. "Just putting kids in group therapy actually promotes greater drug use," says Dr. Nora Volkow, director of the National Institute on Drug Abuse (NIDA).
The article is here, for those who want to see the other side of the story.
For teachers of statistics, this is one more case of correlation caused not by causation, but a common cause.
I found the article at a George Mason web site, which points out that the media headlines missed the point.
"Media coverage reveals a classic confusion between causation and correlation, as they implied that Viagra results in greater risk for older men. But if anything, the study suggested just the opposite: Men who are interested in Viagra have riskier sex lives.."
A while back I got a copy of Ian Stewarts "Cows in the Maze" to review, and did.
Today I was glancing through the links for one of the chapters, "Cone with a Twist", which describes a shape called a sphericon. If you make two equal cones with a 90o angles at the vertices and glue the two bases together then a cross-section cut through the two vertices forms a square, you can slice the shape with a plane through the two vertices, then rotate one 90o and reconnect them and the shape is called a sphericon.
It is sort of a Mobius Strip in 3d since it has one continuous surface...and it rolls with a wobbly gate that makes it fascinating.. See below.
If you want to make one of your own (or many) a nice net for the shape is found here.
The fun really starts when there is more than one.... This site helps you to see how they can roll around each other.
A prelude to set the mood: Name the three most common last names in the USA.
Playing around at the "Book of Odds" web site today and came to a paradoxical epiphany. Common last names are not that common.
What led to the conclusion was today's "odds of the day": The odds a baby will be born outside of a hospital was 1 in 115.6 (based on USA figures in 2004); which seemed pretty rare. Looking down my student list of just over 100 names there might be one who had been born outside of a hospital. (one wonders how many were planned, and how many were born in the taxi on the way to the hospital)
But glancing through the list of things that were approximately equally likely, one of them was having a last name of Smith (about 1 in 116.5). I think before this I would have thought, in general, that it is unusual for a baby to be born outside of a hospital, and not unusual for a baby to be born with the name Smith... turns out, the former is more probable than the latter.
I think this is an easy fallacy, and one my intro-stats kids make often. For example, they all know that in 100 flips of a coin, 50 heads is the most common outcome.. but they tend to let themselves think "most common" means it happens frequently, when in fact it doesn't. The probability of exactly fifty heads turns out to be a little under 8%.
By the way, the three most common surnames in the US according to Wikipedia were Smith, Johnson, and Williams (Smith is only number two in Canada).
On a more scary thought, the odds a person in a hospital will be diagnosed with an infection acquired while in the hospital is more than five times as likely, about 1 in 22.
So what is the probability that a baby born this week is born outside of a hospital and named Smith?
The Baby at top was neither of those.
Friday, 10 September 2010
It's been a pretty good week for the math blogging ego. My hit counter has been (relative to previous months) spinning like the national debt counter; and then today I got a note from some folks who maintain a list of "the 25 best Math blogs for college students" saying they had my blog on their list.
After a minute of supreme ego.. "Well, of course you did!", and a minute of critical doubt.."How could anyone keep up with all the math blogs out there to know which were best, .... and by what criteria would you judge and....yada, yada, yada?"...
Finally I clicked on the link to the list, and grew more quiet as I read down the list...
Yikes!! Fields medal winner Timothy Gowers is on the list, and he only made number three!!!!
Peter Woit, who wrote "Not Even Wrong" about String Theory and stuff is on the list.
Dave Richeson, who is chair of the department of mathematics and computer science at Dickinson College, and wrote "Euler’s Gem: The Polyhedron Formula and the Birth of Topology" (and is, in my mind, one of the best math expositors around) is on the list.
A group of eight bright kids from Berkley have a blog on the list....
A blog by the entire math dept at Nazereth College is on the list...
There's a blog on computational geometry that is even more perplexing than the one on string theory.....
And there is a high school teacher from a US military base in England..... oh wait, that's me.....
As the folks on Sesame Street used to sing, "One of these things is not like the others...."
So I spent a little while feeling very, very unworthy,.... but now.... now I think I'll go get a big cup of coffee and sit back in my comfy big chair and feel very, very smug for a little while... after all, it will probably take them at least a month to figure out what a mistake they've made..... meanwhile, just call me "Number 18 with a bullet"
Wednesday, 8 September 2010
David Bee sent a note to the AP Stats EDG with a link to an article in the NY Times by Charles Blow with the interesting graph above. Students should be able to find at least four variables described in the one graph.
Dave suggests that the US is an outlier... do you agree?
Monday, 6 September 2010
The symbol was, as you know if you read the blog, created by Robert Recorde in his "Whetstone of Witte", and published in London just before Recorde's death in 1558.
Recorde died in the King's Bench Prison at Southwark in London, most likely for being unable to pay his debts. Then, as now, teaching was not an economically rewarding position. Along the way, he had been a very successful writer of mathematics textbooks. His first arithmetic, "The Ground of Artes," was published around 1540-43. It became a very popular arithmetic and was reprinted over a dozen times. After his death, other mathematicians would use it as the base of their own books, making minor adaptations and keeping Recorde's well known name. One such edition by Edward Hatton bears the date 1699, almost 150 years after Recorde's death. Algebra, it seems, was a less popular topic. The "Whetstone of Witte", in which he introduced the "gemowye" lines of equality, never made it to a second edition.
If you remember that most publications in mathematics were in Latin, then the task of creating an English language Geometry or Algebra required the introduction of English terms for some of the technical terms needed. Recorde's "Pathway to Knowledge" was two decades ahead of Billingsly's translation of Euclid. In it he introduced many Saxon-English words for the Latin terms. A "poynt or prycke" was used for the point. One-hundred years later Newton would write that he used "pricked letters" to indicate a fluxion. "Sharp" and "blunt" corners were the translations from the Latin acute, and obtuse, to the detriment of many geometry students who would better remember and understand the Saxon terms. Recorde allowed that his "gemowe" or parallel lines (he used both terms) need not be straight. Those crooked copies of each other, like ss, he called "tortuous parallels".
Tangent lines were called "touche lynes" in an Anglo-Saxon translation. Vertical angles were referred to as "matche corners", and rectangles were "losenges or diamondes". If the parallelogram was oblique, it was called "diamonde-like".
None of these became popular terminology, most likely because the language of scholars continued to be Latin for over a century after his death. Although the terms "long square" for a rectangle and "diamonde" for rhombus appeared in the Billingsly translation of Euclid, little other evidence of Recorde's language inovations seem to appear.
Recorde was not totally adverse to using existing or foreign terms. He referred to "cooslike" numbers for variables, which had been introduced by Pacioli (1494) from the Latin "cosa" for thing, which had become very popular in Germany. The German algebraists were sometimes called cossists, and the study of algebra as "die cost."
Also, we should keep in mind that it was Recorde who first used the word sine in print in English.
Sunday, 5 September 2010
By the time they finish geometry most high school students have heard that it is impossible to trisect an angle with the traditional tools of a straightedge and compass (which should be referred to in the plural, but Americans seldom do). What they seldom hear, or see, is a way to trisect a general angle with a marked straightedge, or what the Greeks called a neusis (from the Greek word for "incline toward", apparently because of the way it was used). More than that, you can find a simple way to show the double angle formula for the cosine of an angle.
In the figure above, the angle to be trisected is EAB with the "3x" in there. To create the construction, use the marked segment (call this one unit) of the straightedge as a radius, and construct a circle with the radius of one with the vertex at the center of the circle.
Now put one edge of the straightedge on the point at E and slide the edge along this point until the marked segment touches the circle (F), and the endpoint is on the diameter of the circle(D). The angle ADF is 1/3 of the angle EAB, and has a measure of x.
To see that this is true, draw the radius AF... and the two perpendicular lines shown dotted. Note that both triangles ADF and FEA are isosceles. If we begin by letting FDA =t then EFA is an exterior angle for that triangle, and so it, and its congruent angle AEF both have measure of 2t. Now if FA is extended, the exterior angle F'AE would have a measure of 4t, and this exterior angle would consist of the vertical reflection of FAE with a measure of x, and the remaining original angle of 3x... so 4x =4t or t=h.
The drawing can now be used to derive the cos(2x). Notice that DG and AG are both Cos(x). Since DH and DA are the adjacent leg and hypotenuse of angle D with measure x, we know that Cos(x)*DA = DH. Now DA=1+cos(2x), and AC = cos(x)+ cos(x), so we get 1+cos(2x) = cos(x) *cos(x)+ cos(x).
This simplifies to 1 + cos(2x) = cos2(x) + cos2(x) or...
1 + cos (2x) = 2 cos2(x)....and with one more step..
cos(2x) = 2 cos2(x) -1.
You can actually extend this with a little algebra to calculate cos(3x).
The first part is a little quiet as there is a question from the audience (if you have cool video skills you could probably kill this part).
For folks who like projects that take them out of the classroom, an hour collecting data on the putting green might be cool. Pick a few distances like 3, 6, 9, 12, and 15 ft (that may be a real low percentage for the general public) and have each kid put once (or twice if you need more data) and then make a curve showing what percentage of the students hit each distance. Now do your analysis and curve fitting.
I wrote this a couple of years ago, and was reminded of it by someone who read my recent blogs (here, and here) about the "Crisis in Education". I thought it might be worth a second read, sooo...here 'tis:
I’ve been thinking a lot about how kids grow up today, and how it impacts on their education, which should not be surprising since teaching is what I do. I’m sure there has never been a generation that didn’t ask that same old question, “What’s the matter with kids today?” Several articles and a couple of TED broadcasts seemed to resonate with what I’ve been thinking lately.
Robert Siegler, a psychology professor at Carnegie Mellon University in Pittsburgh, has done some research that suggests that children’s early concept of a number line is more logarithmic than linear, and that’s bad for math achievement. As they grow older, they usually develop a linear conceptualization of the number line but it happens in bits, and how long it takes to happens seems to be closely related to how well they do later in mathematics. The ones who seem to develop later spend more time with video entertainment while the ones who develop earlier, and therefore have a better chance of being successful in math, are the ones who play card games and board games. There are exceptions to every rule, but most mathematicians tend to have a game aspect to their mathematical learning.
Sir Ken Robinson, is an internationally recognized leader in the development of creativity and innovation. In one of his TED talks, he suggests that creativity is as important in literacy, and we should treat it so. I think he is right, but I think the problem, or at least a part of the problem, is the lifestyle of young children today. Don’t take my word for it, watch the TED video below by Gever Tulley creator of the "Tinkering Schools".
Tulley thinks that our children are over protected, and suggest five dangerous things we should give our children (he admits he has none of his own). I’m going to give my two cents worth about some of these, so if you want to, just skip down and run the video first…
Tulley suggests that you let your children play with fire. He makes some suggestions about what they might pickup, but relapses back to a faith based mantra… You don’t know what they are going to learn, but they are going to learn.
His second dangerous recommendation for children is a pocket knife … OF course they will cut themselves... and while you are wondering if you were a terrible parent to give them something so dangerous, they are proudly peeling back the bandages to show their cut to their friends on top of the garage as they take turns jumping off the roof and onto the limb of the tree a few feet away...(you won't find out until they fall, and incredibly, many of us never do).
I still remember the wisdom of my father-in-law telling me the story of his experience as a child at a carnival near Perry, Michigan… after paying a dime at the tent that said “Life Saving Advice Inside”... they entered a string of other curious visitors. The line passed by an old guy with a pocket knife and a stick that was quickly turning into wood chips...as he repeated... "Whittle from you, never cut you”| and through the next tent flap which led... outside... I’m not sure how many times he told me that story in the forty years I’ve known him, but it was more than once for each of those years. Ten cents when you are a child for a story you could still be telling at in your 90's... a bargain...
The third dangerous thing he recommends is a spear. I’m not sure I see the benefit of a spear, but our brains do seem to be wired for throwing things, and when you let one part of the brain go soft, other parts seem to follow. So throw a spear or a baseball, girls and boys NEED to play baseball, even if they don’t compete in organized sports, a game of catch can build both body and mind....Throwing and catching are incredible analytic teaching methods visual acuity, 3 d analysis, and attention and concentration skills.
Number four was to give them your broken appliances to take apart. The land fill can wait a couple of days. Seeing what things look like and trying to figure out what they do is BIG.. and the curiosity is BIGGER..I suspect it develops a belief that they can KNOW difficult things.. and an awareness that it may not come all at once. I worry that kids today don’t seem to persevere at difficult tasks. I suspect that a kid who takes apart a toaster or spin dryer (with parental supervision even) will look at all the “black boxes” he encounters differently. They will always be wondering what might be “in there” and how it could work…. Did I mention that curiosity is GOOD!
And finally, about ten or twelve, let them drive the car. Find a good safe place out in a vacant lot, sit them in your lap if you have to, and let them control the steering wheel and any other parts they can reach while you roar across the lot at 15 mph. Being in charge of this huge piece of moving steel can give them a sense of control… Kids need to feel they have some control over their world… If you really believe that children are the future, you better help them feel they can be in control.
Here are a couple of extras that I think are important for life and math skills… Teach them to play games... NOT video games,, they'll do that on their own.. Teach them to play card games, board games... Monopoly, Parcheesi, Snakes and Ladders are GREAT childhood games, even for kids UNDER 8 years old.. Teach them to juggle, to ride a unicycle, to do card tricks, to play a musical instrument, to speak a foreign language. All those things take discipline and focus, both of which are essential for success in education and life. When my youngest son was learning to play soccer in elementary school, we would go out in the front yard in the evening and take turns seeing who could juggle the ball with our feet for more touches. It only took a month or so and he had gone past the best I would ever do. When a parent watches their kid go past them in anything, it’s a great feeling.. that’s the natural order of the universe, each generation gets a little better and smarter.. I worry if we are doing that lately.
And perhaps to add a new touch to the post, I'll add a few words from an Alan Jackson song...
A young boy, 2 hands on the wheel
I can't replace the way it, made me feel
And I would press that clutch and I'd keep it right
He'd say a lil' slower son you're doin' just fine
Just a dirt road with trash on each side
But I was Mario Andretti
When daddy let me drive
Or.... you can hear the whole thing compliments of the folks at CMT
Saturday, 4 September 2010
In a talk to the summer meeting at the University of Michigan, Professor K. P. Williams of Indiana University pointed out that, although the "National Committee entitled Reorganization of Mathematics in Secondary Education" in 1923 stated that "one year of mathematics should be required of all pupils in the high school".... Then he adds, "This thought was out of harmony with the ideas that were in control of secondary mathematics and could hardly be expected to win favor."
"Certain general aims for mathematical study were set forth and it was argued that they would not be advanced by having mathematics a universally required study at a time when the high schools have come to enroll so large a percentage of youths of high school age."
Addendum: A few days after I wrote this, I came across a journal article that said that in 1336 the University of Paris introduced a rule that no student should graduate without attending lectures in mathematics.
Another comment only a few months before in a review of a translation of Euclid's Elements comments, "So it is today, in all highly civilized countries, that geometry is taught, usually to girls as well as boys, but the textbooks lack the rigidity of proof ....., being softened to fit the feeble mentality of high-school pupils today."
I am reminded of a comic from a long while back who would ask at the end of each show, "Is there anyone I haven't offended yet?"
His concern was that high schools and universities were no longer requiring mathematics. "It appears that in seventeen states mathematics is no longer a required subject in high schools." He stated that in a survey of the state superintendents of instruction, they blamed poor teaching, poor textbooks, and "an unconvincing and ineffective formulation of the objectives of elementary mathematics." I wonder what the same type of survey would say today.
Later, he asks a question that could come, and frequently does, from many concerned educators today, "Does secondary mathematics... represent such a vitally essential educational element that it should be made mandatory in the curriculum of all normal pupils, even though many of them may never have occasion to use it for immediate, vocational use." How would you answer that question, and how would you justify your answer to someone who disagreed?
He goes on to give his answer to the question with three aspects of mathematics education that are most important, symbolic thinking, postulational thinking, and relational thinking. No "solving quadratic equations" or any other laundry list of "skills".
I recently wrote about the high number of college students who take remedial, not for credit, math courses in college. Many of them graduate with grade point averages of B or higher. This morning I searched around to see what I might find quickly about the present state of remedial courses... here are a few notes.
"An Arizona Community Foundation study released in January showed that half of Maricopa County's 2006 high school graduates who entered Arizona universities or colleges had to take a remedial math class and a quarter had to take a remedial English course." [By MIKE McCLELLAN, The Arizona Republic (Phoenix)]
In many areas of Arizona it was much worse. Higher education in the United States: an encyclopedia, Volume 1, By James J. F. Forest and Kevin Kinse reported that in Maricopa County Community College, "in 1997 up to 90% of entering freshmen required remedial mathematics courses".[pg 521] On the same page I read that the California State University System "which only admits students from the top one third of their high school class", regularly had over 40% of freshmen taking remediation courses in mathematics. The book also reported that in 1999 CUNY, after thirty years of a pro-remediation approach (following student protests in the 60s), began to deny entry to "underprepaired students". I am not sure what the current status of that policy is ten years later.
The two-semester remedial math program at LaGuardia Community College, City U. of New York, ... about 60% of entering students are required to take one or both of these courses.
From the Columbus (Ohio) Dispatch, "Statewide, 39 percent of about 52,000 first-year public college students took at least one remedial course in 2008, according to a state report released this year. Among all Ohio high-school graduates that year, more than two-thirds had taken a college-preparatory class."
Community colleges are struggling to solve the remedial math problem: How to get students who can’t find the lowest common denominator — or don’t know there is such a thing — to learn basic math skills and move on. For many, math is an insurmountable barrier, writes Elizabeth Redden on The Hechinger Report (and in Chronicle of Education).
"Only 31 percent of students placed into remedial math ever move beyond it, according to the Community College Research Center at Columbia University’s Teachers College.
Less than 25 percent of community-college students who take remedial or developmental courses earn a degree within eight years compared to 40 percent of non-remedial students
In her "Math Mama" blog, Sue VanHattum recently wrote about the start of her college course, "No one (3 sections, about 140 students total) passed the mastery test I gave on pre-algebra topics. (FIDO = fractions, integers, distributive property, and order of operations)."
Perhaps even more surprising was her discussion of having discipline problems with her college students. She has had more discipline issues in the first week of college classes than I have in a year of my advanced high school classes.
I admit I don't know where all these problems will take us. Many schools and states have started distance education programs where students don't even have to show up in person to take the final exam. Huge numbers of students are being home schooled, but I suspect they are getting a better than average education because of the selective sample who choose to raise their children this way. One more thing to look up...
Friday, 3 September 2010
The theorem is also sometimes called Soddy’s theorem because in 1936 Sir Fredrick Soddy rediscovered the theorem again, and then proceeded to write a really interesting poem about the relationship called “The Kiss Precise”.
For pairs of lips to kiss maybe
Involves no trigonometry.
'Tis not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.
Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the center.
Though their intrigue left Euclid dumb
There's now no need for rule of thumb.
Since zero bend's a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.
Soddy may also be known to students of science for receiving the Nobel Prize for Chemistry in 1921 for the discovery of the decay sequences of radioactive isotopes. According to Oliver Sacks' wonderful book, Uncle Tungsten, Soddy also created the term "isotope" and was the first to use the term "chain reaction".
In a strange "chain reaction" of ideas, Soddy played a part in the US developing an atomic bomb. Soddy's book, The Interpretation of Radium, inspired H G Wells to write The World Set Free in 1914, and he dedicated the novel to Soddy's book. Twenty years later, Wells' book set Leo Szilard to thinking about the possibility of Chain reactions, and how they might be used to create a bomb, leading to his getting a British patent on the idea in 1936. A few years later Szilard encouraged his friend, Albert Einstein, to write a letter to President Roosevelt about the potential for an atomic bomb.
The prize-winning science-fiction writer, Frederik Pohl, talks about Szilard's epiphany in Chasing Science (pg 25),
".. we know the exact spot where Leo Szilard got the idea that led to the atomic bomb. There isn't even a plaque to mark it, but it happened in 1938, while he was waiting for a traffic light to change on London's Southampton Row. Szilard had been remembering H. G. Well's old science-fiction novel about atomic power, The World Set Free and had been reading about the nuclear-fission experiment of Otto Hahn and Lise Meitner, and the lightbulb went on over his head."
Thursday, 2 September 2010
LONDON, England (CNN) -- God did not create the universe, world-famous physicist Stephen Hawking argues in a new book that aims to banish a divine creator from physics.
Hawking says in his book "The Grand Design" that, given the existence of gravity, "the universe can and will create itself from nothing," according to an excerpt published Thursday in The Times of London.
"Spontaneous creation is the reason why there is something rather than nothing, why the universe exists, why we exist," he writes in the excerpt.
Now, raise your hand and lift one finger for each person who probably will change their mind because of the book....and do what you wish with the left over fingers.
I have written frequently about relationships that, for one reason or another, reminded me of the Pythagorean Theorem. (see here, and here, and here for example).
After trying to write up all the ones I could think of, out of the blue I got a nice note from professor Robin Whitty who maintains the excellent "Theorem of the Day" web site. (If you are a high school teacher or student, it's a must see.) He had prepared a page on Des Cartes' Circle Theorem, and had included a link to my page on the subject (I told you it was a good site). It was one more "almost Pythagorean" theorem. It's easiest to right if we consider the bend of a circle as the reciprocal of the radius, b= 1/r. With that notation adjustment, then for four circles with bends b1, b2,b3,b4, Des Cartes' Circle theorem says that one-half the square of the sum of the bends is equal to the sum of the squares of the bends.
I came across a report on a South Eastern Section meeting of the MAA in an old American Mathematical Monthly (Feb 1934) and noticed a session by Professor C. G. Phipps of the Univ of Florida entitled, "Subfreshman Mathematics." The abstract for the talk said,
"The University of Florida is trying out a new plan to handle the student who is very poorly prepared in mathematics. At the end of the second week of school a simple test on algebra and arithmetic is given to all freshman classes. Those who are unable to score a certian percentage are put into a no-credit course where they are taught the fundamentals they should have learned earlier."
Dave Renfro, who sent me the article wondered if this might be the first no-credit remedial course in college math.
"Anyone, Anyone, Bueller?"
Also, I read a few years back that over 25% of college freshmen were taking such a course (math for no-credit) across the nation. Does anyone have a good source for a valid number in recent years?