CTK Insights

The Spanish Civil War (1936-1939) attracted volunteers from about 55 countries who knew the dangers they were facing in that bloody conflict. Nevertheless, they came in substantial numbers to join the ranks of the Popular Front. The following passages have been plucked from an article A Prologue to WWII in the Jerusalem Post by Ervin Birnbaum:

DESPITE THE conspicuous presence of Jews in International Brigades, Jewish participation in the fighting has generally not been acknowledged.

In-depth research, especially in the last 10 years, has proven that the extent of Jewish presence in that crucial war was truly impressive. Though Jews were only 10% of the Polish population, 45% of the Polish volunteers - 2,250 out of 5,000 - were Jewish. Jews, 4% of the US population, formed 38% of its volunteers. In France, 0.5% of the population and 15% of the volunteers were Jews. Britain, with a Jewish population of 0.5%, had 11% to 22% Jewish volunteers. Palestine had a Jewish contingent of 500, 498 Jews and two Arabs.

The visit of President Bush on the occasion of the 60^th anniversary of establishment of Israel received a wide news coverage. There was a persistent speculation that the president will bring along some parting gifts, perhaps as a substitute for releasing Jonathan Pollard from incarceration that lasted well beyond any reasonable and moral norms.

One possibility discussed was a new radar system that would secure Israel from the Iranian nuclear threat:

The Bush administration appears set to offer Israel a powerful radar system that could greatly boost Israeli defenses against enemy ballistic missiles while tying it directly into a growing US missile shield.

President George W. Bush is expected to discuss the matter during a visit to Israel on Wednesday to mark the 60th anniversary of the Jewish state amid mounting US concerns about perceived threats from Iran, sources said.

This is “probably the No. 2 issue” on Bush’s agenda for the visit, second only to the Middle East peace process, said Rep. Mark Kirk, an Illinois Republican who has spearheaded calls in Congress for tighter US missile-defense ties with Israel.

What would be the benefit? Read on:

The system Bush may offer is known as a forward-based X-band radar and has been described by US officials as capable of tracking an object the size of a baseball from about 2,900 miles away.

It would let Israel’s Arrow missile defenses engage a Shahab-3 ballistic missile about halfway through what would be its 11-minute flight to Israel from Iran, or six times sooner than Israel’s “Green Pine” Radar is currently capable of doing, Kirk said on Friday.

How to divide evenly 5 apples between 6 boys if you are only permitted to cut an apple into not more than 4 pieces?

 The answer is remarkably simple and the approach serves an excellent motivation for the process of adding two fractions.

 First divide 3 apples into halves giving each boy a half, 1/2. Then divide the remaining 2 apples into threee pieces each giving each boy one third , 1/3, of an apple. At the end of the day, each boy will have 1/2 + 1/3 of an apple meaning at least that they all have the same amount. But if that is the case, each of the boys is bound to hold 5/6 of an apple. The conclusion just pops out:

1/2 + 1/3 = 5/6.

 Surprisingly, this result can be discussed even before the addition of fractions has been introduced. To motivate further, students may be asked to suggest similar problems or exchange similar problems with each other.

 This is a natural extension of a demonstration of equivalence of fractions. How to divide evenly 4 apples between 8 boys. Observe, that whatever the process is chosen, if correct, it would give each boy 4/8 of an apple. On the other hand, cutting each apple into two halves gives eight equal parts, 1/2 an apple each. So each boy receives 1/2 of an apple. It follows that 4/8 = 1/2.

 Combining the two ideas leads to the rule

1/2 + 1/3 = 3/6 + 2/6 = 5/6.

Quite a natural development.

It was quite a while, yes.

Since the last posting I happened to obtain the new book Sacred Geometry by Tony Rothman and Fukagawa Hidetoshi. The book is exceptional in the breadth and depth of its coverage of Japanese mathematics starting with its origins in China and evolving into wasan, especially during the period of seclusion. I have reviewed the book elsewhere.

Curiously, the authors never met; we learn about how they manage to write the book from Rothman’s Preface. There is also a Foreword by Freeman Dyson, which is of interest in its own right. The previous Fukagawa’s book was co-authored by Dan Pedoe, who happened to be Dyson’s teacher who, in turn, is a colleague of Rothman. This is how Rothman has been introduced into the sangaku in the first place. So Rothman never met Fukagawa, but Dyson did and he tells us the story of the meeting right in the Foreword:

Fukagawa Hidetoshi has been a high-school teacher in Aichi, Japan, for most of his life. During school holidays he has spent his time visiting temples all over Japan, photographing sangaku as works of art and understanding their meaning as mathematical problems. He knows more about sangaku than anyone else in the world. Unfortunately, in the hierarchical academic system of Japan, a high-school teacher has a low rank and is not highly respected. He was not able to interest high-ranking professors in his proposal to publish a book about his findings; without support from the academic establishment, his work remained unpublished and unknown. After many years he finally found a publisher outside Japan, with the help of Daniel Pedoe.

… In 1993 I was invited to Japan to give lectures at Japanese universities, and I finally had a chance to meet Fukagawa in person. Dan Pedoe made the arrangements for our meeting. My academic hosts expressed surprise that I should wish to speak with a “lowly” high-school teacher, and tried to cut my visit with him short. They allowed me only a few hours to spend with him, visiting a temple where some outstanding sangaku are preserved and an abacus museum where we could see other artifacts of indigenous Japanese mathematics. I would happily have stayed longer, but my hosts were inflexible. Since then I have stayed in touch with Fukagawa as he continued to make new discoveries and deepen his understanding of the historical context out of which the sangaku emerged.

Interesting, isn’t it? So the teacher in Japan is a fellow of a “low rank and is not highly respected.” Hmm, how does this jibe with a standard model that blames low teacher’s standing and their low salaries for the US underperformance in international math education studies. Have not Japanese 4^th graders out performed their US counterparts in the TIMSS 2003 and TIMSS 1999? Have not the 8^th graders?

It is really quite easy to get confused.

Now and again business leaders volunteer an advice on math education. The latest I came across was freely shared by Lee Iacocca in his new book Where Have All the Leaders Gone?;

Lee may not know much about education but he sure knows a lot about catching attention:

Am I the only guy in this country who’s fed up with what’s happening? Where the hell is our outrage? We should be screaming bloody murder. We’ve got a gang of clueless bozos steering our ship of state right over a cliff …

The book is not exactly about math education. From the excerpts, Lee is worried about the absence of leadership in industry, the bloody war in Iraq and other causes as well. As an expert PR person, he does not shun misrepresenting the facts be that intentionally or for the lack of knowledge. Concerning the No Child Left Behind program he blames the president:

He (President Bush) also ran on the No Child Left Behind program, which he proclaimed as his proudest achievement while governor of Texas. Only after Bush managed to push the program through Congress as a federal mandate did we learn that the Texas record was not exactly sterling. An inquiry into the Texas No Child Left Behind program revealed widespread test-rigging and numbers-fudging by educators and administrators.

I am no fan of the NCLB initiative which, in my view, was bound to be a failure from the very beginning. And I am not about to absolve President Bush from the responsibility. However, in all honesty, the whole congress embraced the program as everyone who cares to remember may recollect. The December 18, 2001 resolution passed Senate voting with flying colors: 87 YEAs against 10 NAYs. Pushed by the pangs of the leadership responsibilities, with the lackluster performance of the US students in international studies at the back of their minds, and prompted by the business leaders who worried about the future competitive edge of the country, the US Congress showed an overwhelming support for the program. This is an indisputable fact, but apparently not so for Lee Iacocca.

The former Ford’s and Chrysler’s CEO has more to offer than his indignation with the President’s program:

Teachers today have a brand-new problem to worry about-getting shot in the classroom. As I write this, I’m looking at three school shootings just in the last week-even though most schools have metal detectors. I think maybe a little tough love is in order-and a lot of people are going to scream, but hear me out. Why don’t we say that every kid has a right to go to school in this country-until the first time he shows up with a gun, a switchblade, or a little white bag of coke. Then we write him off. Send him packing. Think of it as a form of educational triage.

Here’s the way I see it. There are some kids who will make it no matter what you do or don’t do. Then, there’s the large majority who need a lot of help to make it. And finally, there are some who just can’t be helped, and who suck up all of the resources and attention like a black hole.

I am somewhat ambivalent about this approach. There is no point in speculating whether Lee Iacocca lost any sleep when, as a CEO, he had to fire thousands of workers. These were entitled to some benefits and, in any event, were grown-ups that could be assumed to know how to take care of themselves and their families. But what do you do with the kids in the streets? Thrown out of school, they may no longer be in a position to bring a knife to a class. But who may expect them to part with the blade in the street?

To be fair, some of the advice one gets from the book very acceptable to me. For example,

A word to parents: The biggest favor you can do for your kids is to have plenty of books around the house. Read to them, read around them, be a family that reads. (And if you’re not such a good reader yourself, it’s never too late to learn.)

Here, I join my voice to Lee Iacocca’s, although I can’t recommend his book to have around. Perhaps, too, the US system of education should be modeled after some successful business organization. I do not have a definite opinion on this account. But two things I am sure about. First, the US schooling was never so bad as to hamper the societal or industrical progress in the country. This nonwithstanding the high pitched concerns of the math educators and business leaders. And second, as I strongly believe, the only way to improve the system of education is through a systemic change on all levels from kindergarten to college by emphasizing the development of interest and motivation as opposed to the skills and the necessity of tomorrow’s job market. Wasting years to retain a few basic facts is meaningless regardless of the educational philosophy under which those facts are acquired. A good start for the required change would be to admit a simple truth which is that very little math knowledge is required for a successful and fulfilling life. As Lee Iacocca has put it, There are some kids who will make it no matter what you do or don’t do. And the fact is, I believe, no more is actually necessary. (See my Manifesto.)

Raymond Smullyan starts his What Is the Name of His Book with a story of how he was fooled by his older brother:

(One) morning, my brother Emile (ten years my senior) came into my bedroom and said: “Well, Raymond, today is April’s Fool Day, and I will foll you as you have never been fooled before!” I waited all day long for him to fool me, but he didn’t. Late that night, my mother turned to Emile and said, “Emile, will you please fool the child!” Emile then turned to me, and the following dialogue ensued:
Emile: So, you expected me to fool you, didn’t you?
Raymond: Yes.
Emile: But I didn’t, did I?
Raymond: No.
Emile: But you expected me to, didn’t you?
Raymond: Yes.
Emile: So I fooled you, didn’t I?

A good example of a benefit drawn from older brothers and a nice introduction into the art of logic and logic argument.

Reference

1. R. Smullyan, What is the Name of This Book?, Simon&Schuster, NY, 1978.

The final report of the math advisory panel is out. At 120 pages long, it may take time to fully analyze the document. Do not know if I am going to do that. My interest in the document has faded upon reading a few pages of the Executive summary. Here is a couple of examples.

The fellows make a glib misuse of statistics in the way of which the members of the panel - mathematicians and math educators - could not help but be aware.

… Algebra is a demonstrable gateway to later achievement. Students need it for any form of higher mathematics later in high school; moreover, research shows that completion of Algebra II correlates significantly with success in college and earnings from employment. In fact, students who complete Algebra II are more than twice as likely to graduate from college compared to students with less mathematical preparation.

The members of the esteemed panel are certainly aware that correlation does not necessarily entails causation. Brighter, more persevering and better motivated students take algebra. The same group of students goes on to graduate from college. Taking algebra in high school may be helpful in college (but hardly in a Liberal Arts one) but there is in no way to arrive at such a conclusion in the manner the advisory panel avers.

The panel’s report emphasizes the importance of Algebra:

Although our students encounter difficulties with many aspects of mathematics, many observers of educational policy see Algebra as a central concern.

Indeed! How may it be reasonable to form an educational policy based on the opinion of many observers? The state of California is on the record of having tried the policy of making algebra a prerequisite for high school graduation. Why not to base the report on the results of that experiment?

The panel did not point a finger to any particular failure of the educational system:

This Panel, diverse in experience, expertise, and philosophy, agrees broadly that the delivery system in mathematics education—the system that translates mathematical knowledge into value and ability for the next generation—is broken and must be fixed. This is not a conclusion about any single element of the system. It is about how the many parts do not now work together to achieve a result worthy of this country’s values and ambitions.

The report offers recommendations which, when implemented, will allow the many parts of the system combine into a workable whole. I could not detect in the report any attempt to outline the many parts mentioned above or explicitly say what is wrong with each. As a matter of fact the panel did not recommend to change the system. I somehow doubt that a bad system could be fixed solely by recombining the parts.

References

1. Foundations for Success: The Final Report of The National Mathematics Advisory Panel

A particular case of the Isoperimetric Theorem says that among all rectangles of a given perimeter, the square has the largest area. (This is of course equivalent to the claim that among all rectangles of a given area the square has the least perimeter.) This particular case of the general theorem is so simple that it was considered suitable for the NJ 4^th graders preparing to take the NJ ASK4 test in mathematics. The problem has been posed in the following form:

Mr. Lynch has 80 feet of fencing that he will use to make a garden. His garden can be either a square or rectangular in shape. He wants to choose the shape that will give him a larger planting area.What shape garden will have a larger area?

How is a fourth grader supposed to sove this problem? Have they ever proved a general statement. My son who not accidently is in the fourth grade found the area of the square - 400, and the area of the 30×10 rectangle, which is 300, and correctly concluded that square is the answer to the problem. He was greatly disappointed when told that there are other rectangles he had not checked. We have eventually worked out three solution,s starting with a purely algebraic one and ending with an intuitive geometric solution. Judging from my son’s reactions the latter was quite intuitively clear to the boy.

Solution 1

If p is the fixed perimeter of the shapes at hand, a - the side of the square - equals p/4: a = p/4. For a rectangle of the same perimeter, the two adjacent sides, say u and v, add up to half of it:

If, say u exceeds the side of the square a by some length x, then the other side, v, is shorter than a by the same amount:

The area of the rectangle with sides u and v equals uv, or, in our case,

Which actually proves the statement of the theorem. Indeed, a² is the area of the square of perimeter p = 4a, whereas a² - x² is the area of a rectangle of the same perimeter for which one of the sides is longer, the other shorter (by the same amount x) than the side of the square. Since x² ≥ 0, a² - x² ≤ a², with equality only when x = 0, i.e. when a rectangle becomes a square.

The problem of Mr. Lynch is solved with p = 80.

Solution 2

Let’s draw a square and a rectangle of the same perimeter:

and put the two together.

The area of the square is the sum of the red and yellow areas, that of the rectangle is the sum of the red and blue areas. The red being common to both, it is clear that the area of the square is greater than the area of the rectangle if and only if the red area exceeds the blue area. But this is quite simple. The area of a rectangle is the product of the lengths of two adjacent sides. If a is the side of the square, then the yellow area equals x×a whereas the area of the blue region is x×(a - x). And, since, a exceeds a - x for x > 0, we are again done.

Solution 3

Observe the the blue and yellow rectangles and the dotted square in the upper right corner. What can you say about their areas. Quite obviously,

which clearly implies

Reference

1. ASK 4 Success: Work-A-Text. Mathematics, Instructivision, Inc., 2006

The National Mathematics Advisory Panel, appointed by President Bush in 2006 has arrived at some conclusions.

First of all, the panel that included representatives from both sides of the great math education divide has declined to take sides, saying the group agreed only on the content that students must master, not the best way to teach it. (This is wonderful and reminds one of the remarkable turn in the NCTM’s policy which, in response to an urgent need to clarify the ambuguities in their Principle and Standards for School Mathematics, has issued a document titled Curriculum Focal Points.)

Second, the content the panel has agreed on appears to focus on rather trivial needs of commonly required skills:

The draft report urges educators to focus on “critical” topics, as is common in higher-performing countries. The panel’s draft report says students should be proficient with the addition and subtraction of whole numbers by the end of third grade and with multiplication and division by the end of fifth. In terms of geometry, children by the end of sixth grade should be able to solve problems involving perimeter, area and volume.

Students should begin working with fractions in fourth grade and, by the end of seventh, be able to solve problems involving percent, ratio and rate. “Difficulty with fractions [including decimals and percents] is pervasive and is a major obstacle to further progress in mathematics, including algebra,” the draft report says.

Naturally, the need for another reform stems from the deep seated concern for the future of the United States’ position in the world order. “Without substantial and sustained changes to the educational system, the United States will relinquish its leadership in the twenty-first century,” reads a draft of the final report, due to be released next week by the Department of Education.

Francis Fennell, president of the NCTM and a panel member, said the group’s specific recommendations could help parents determine whether their kids are on the right track.

I wonder whether the kids were on the wrong treck, to start with. While this is common to talk disparagingly of math education in the US, some facts belie the trend. R. Rothstein mentions that

Each year since 1969, Gallup has asked Americans to “grade” their public schools. In the most recent survey (written in 1990s - AB), only 23% percent of parents of public school children gave the nation’s schools a grade of A or B, 46% gave a grade of C and 20% gave grades of D and F. But when the same public school parents were asked to grade the schools their own children attended, they had a different view: nearly three times as many, 64%, gave grades A and B, another 23% gave a grade of C, and only 11% gave a grade of D or F.

Curiously, the Panel is said to have reviewed 16,000 research publications and public-policy reports and heard testimony from 110 individuals. At the end of the day, however, the group said it could find no “high-quality” research backing either traditional or reform math instruction. So it looks like the Panel that consisted of 19 eminent mathematicians and educators has settled to what appears to be a common sense approach. The idea, once again, is to teach numeracy rather than mathematics (for a distinction see, for example, Mathematics and Democracy.)

For a reflection on the meager results and the future effects of the Panel’s deliberations I recommend a recent column by Keith Devlin that highlights a 2002 article by a mathematician and practicing math teacher Paul Lockhart, A Mathematician’s Lament.

Reference

1. R. Rothstein, The Way We Were?, Century Foundation Press (September 1998)
2. L. A. Steen, Mathematics and Democracy: The Case for Quantitative Literacy, Nced (December 2001)

Fermat’s point was the first remarkable point in a triangle discovered after a hiatus of a thousand years. One way to obtain the point in ΔABC is to construct equilateral triangles A’BC, AB’C, ABC’ on the sides of ΔABC and pointing away from it and join the segments AA’, BB’, CC’. The three segments are equal, concurrent at Fermat’s point, and forming successive 60° angles with each other.

All equilateral triangles are similar as are all squares. The configuration of squares formed on the sides of a right triangle came down to us via Euclid’s I.47, the first of his proofs of the Pythagorean proposition. This became known as the Bride’s Chair. More than a thousand years after Euclid, M. Vecten found that the configuration of squares on the sides of an arbitrary triangle, too, possesses a good deal of engaging properties.

In particular, the are three points (the diagram shows only one) where four straight lines meet forming successive angles of 45°. The curious thing about this is the fact the observation that has been made a few days ago - in the year 2008 - by Douglas Rogers, is a very direct, nay, immediate consequence of the properties that have been known for more than 100 years. This is certainly a remarkable feature of Vecten’s configuration having four concurrent lines spaced evenly; the fact holds for any triangle. Of the four lines, two are equal and in the remaining pair one is twice as long as the other.

It also remain true if the squares are drawn inwards the triangle.