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Archive for October, 2009

Factoring

We were discussing factoring in my Intermediate Algebra class the other day and happened upon the polynomial x4-81.

So, we factored the original expression into (x²+9)(x²-9) and then to (x²+9)(x+3)(x-3).

One student asked why we split up the x²-9, but not the x²+9, so we talked about how the x²+9 was prime whereas the x²-9 was factorable as the difference of two squares, just as we had factored the original x4-81 as a difference of two squares.

The student then went on to say, “So that means that x4+81 would be prime, just like the x²+9 was prime….”

I hesitated because I remembered seeing a textbook exercise somewhere (and I still can’t remember where) which factored a sum of x4 and a constant of some kind.

So, I mentioned this, said that we shouldn’t overgeneralize and promised to look into it.

After class, I went to Wolfram Alpha and typed in “factor x^4+81” and got back (among other things) the factorization of

(x²-3(√2)x+9)(x²+3(√2)x+9)

or, if we look at this in general terms,

(x²-bx+c)(x²+bx+c) where 2c=b²

Next, I worked to set this up so that b would be a whole number, in particular b=2 and c=2.  This leads to an expression of

x4+4

factorable as

(x²+2x+2)(x²-2x+2)

I then remembered the factorization I had seen in the textbook was as follows:

take x4+4 and add and subtract 4x²

x4+4x²+4-4x²

then factor the first three terms

(x4+4x²+4)-4x²

(x²+2)²-4x²

then factor as a difference of squares

(x²+2+2x)(x²+2-2x)

or

(x²+2x+2)(x²-2x+2)

Generalizing this a little bit, I came up with a form of x4+4n4 factorable as (x²+2xn+2n²)(x²-2xn+2n²) by adding and subtracting 4x²n².

This generaliztion comes from the relationship between “b” and “c.”  If 2c=b², then for both “b” and “c” to be whole numbers, b² must be even, and therefore must be an even perfect square (4, 16, 36, 64, ….).

If b² is even, then “b” itself must also be even…so

b=2n

b²=4n²

2c=b²=4n²

2c=4n²

c=2n²

And, because the final term in each expression (x4+81 or x4+4) comes from squaring “c”

c²=4n4

So, in general

x4+4n4=x4+4x²n²-4x²n²+4n4

or

x4+4x²n²+4n4-4x²n²

(x²+2n²)²-4x²n²

(x²+2n²+2xn)(x²+2n²-2xn)

or

(x²+2xn+2n²)(x²-2xn+2n²)

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Area Under a Parabola

I was giving a test in my Beginning Algebra classes this past week, and while I was proctoring, I began to page through John Coburn’s Algebra and Trigonometry textbook.  I came across an interesting picture indicating a formula to find the area under a parabola.

parabola area

So, the formula indicates that to find the area under a parabola when it is cut by a horizontal line, we simply multiply two-thirds by the product of the length of the line segment between the points of intersection and the distance from the horizontal line to the vertex.

My first instinct was to see if I could derive this formula myself.

So, I started with a general parabola and, for the sake of simplicity, I put the left intersection point at the origin and made the horizontal line the x-axis.  These choices don’t affect the generality of the derivation.

general parabola

The first thing I did was to integrate the generalized parabola

∫(a(x-h)²+k)dx      from 0 to 2h

This results in an answer of (2/3)ah³+2hk.

If I use the formula (2/3)(2h)(k) I get (4/3)hk.  I was confused and couldn’t see right away why they were different so I integrated the parabola as ax²+bx+c instead to see what would happen.

Using the assumptions, if we integrate

∫(ax²+bx+c)dx      from 0 to 2(-b/2a)=-b/a

We get (b³/6a²) -(bc/a).

If we use the formula using this method, we get

(2/3)(-b/a)(b²/4a-b²/2a+c)=(b³/6a²) -(2bc/3a).

Again, different.

So, I looked at the assumption that the parabola goes through the origin.

If this is true in the first case (where y=a(x-h)²+k), then

0=ah²+k

-k=ah²

Plugging back into the result I had from the first go-round

A=(2/3)ah³+2hk

A=(2/3)(-k)(h)+2hk

A=(4/3)(hk)

Which is what I got using the formula. OK!

Next, the ax²+bx+c method.  For a parabola with the point of intersection at the origin, c=0.

So,

A=(b³/6a²) -(bc/a)

A=(b³/6a²) -(b*0/a)

A=(b³/6a²) -0

A=(b³/6a²)

Using the formula, we got

A=(b³/6a²) -(2bc/3a), but with c=0

A=(b³/6a²) -(2b*0/3a)

A=(b³/6a²) -0

A=(b³/6a²)

So, again they reconcile nicely.  I was glad I realized this quickly otherwise it would have stuck with me for the rest of the day!

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Math Education

Yesterday’s New York Times had a story regarding the results of the 2009 Mathematics portion of the National Assessment of Educational Progress.  The results indicated that, at the fourth and eighth grade levels, American math skills have plateaued, with about 35-40% of students showing proficiency.

The Times also posted commentary from five individuals expressing their opinions about math education.

  • Bruce Fuller, professor of education and public policy
  • Lance T. Izumi, Pacific Research Institute
  • Holly Tsakiris Horrigan, parent of public school children
  • Richard Bisk, math professor
  • Barry Garelick, U.S. Coalition for World Class Math

The first one of these commentaries to catch my attention was from Holly Tsakiris Horrigan, a parent in Massachusetts.  She pinpoints what I think is an important problem in math education – the distinctly un-mathematical nature of many math textbooks.

If we want to improve mathematics education, we should banish nonsensical curricula like Trail Blazers, Everyday Math and Investigations and make sure that our teachers are properly educated and proficient in math content. These curricula substitute writing, drawing and calculator usage for solid math content, leaving children unprepared for more advanced math topics. The precious few children who go on to succeed in higher math after being subjected to one of these programs are those that had the benefit of substantial reteaching outside of the classroom.

Professor Richard Bisk, chair of the Mathematics Department at Worcester State College also points out the role that bad textbooks have played in the sorry state of America’s mathematical pedagogy.

To give students a firm foundation in math, we must start in the elementary grades by providing three things: a substantial improvement in elementary teachers’ knowledge of mathematics; a more focused curriculum that emphasizes core concepts and skills; and more challenging textbooks that teach for mastery and not just exposure….

Finally, many math textbooks in this country use the “spiral” approach, in which topics are not taught for mastery. Instead they are repeated year after year. I strongly agree with the report of the National Mathematics Advisory Panel, which said, “in elementary school textbooks in the United States, easier arithmetic problems are presented far more frequently than harder problems. The opposite is the case in countries with higher mathematics achievement, such as Singapore.”

It can be maddening to try to teach a “spiral” curriculum if you love the interlocking nature of mathematical skills and concepts.  I was given a spiral curriculum textbook to use when I taught high school and couldn’t make sense of it because it jumped all over the place without ever coming to any kind of resolution for any of the topics.

Now, it is possible to design a proper spiral curriculum in which ideas are revisited at higher levels in later courses, but only after a certain amount of mastery has been achieved for the topic.

For instance, in Elementary or Intermediate Algebra, students may study the graph of the parabola simply as the graph of a quadratic function.  They can see the solution(s) of the related quadratic equation by finding the zeros of the function.  Application problems involving projectiles can also be explored.

These ideas can then be built on in a Pre-Calculus course with a deeper analysis of the parabola as a conic section defined by its focus and directrix.  Completing the square and determining the focus and directrix of a parabola are more advanced skills that can be covered.  Further applications involving the 3 dimensional paraboloid of rotation for headlights, microphones and satellite dishes (as well as solar powered electical generators) can also be explored.

The same can be done with other topics.  But many of these textbooks change topics from day-to-day.  In order to give students the foundations they need to build future knowledge on, each unit should ideally contain three or four ideas that are related to each other.  Students can then spend several days on each topic, building a unit of study 10-14 days long (or whatever time period works for a particular situation).

Barry Garelick, co-founder of a parental advocacy group also stressed the importance of a cohesive curriculum:

The education establishment needs to understand that even process is based on skills and exercises, and a logical sequence of topics whose mastery builds upon itself. We need solid math curricula and textbooks that are based on the premise that procedural fluency leads to conceptual understanding.

The reader comments at the Times website for this story also discussed to this issue:

I have tutored elementary and Jr high school student for the past 10 years. I am appalled at the lack of math skills. When I tutored a program 4 days a week , one of the things I discovered was the lack of repetition and the lack of cohesiveness of the math lessons. One day, the kids did areas of a figure, the next, percentages, the next , something else. If they did not get the process in class, Oh well. I had 6th graders who could not do know the basic times tables and could not multiply 2 di[git] numbers.

Another commenter pointed out the differences between the American and Japanese curricula:

My son completed American (Main Line Philadelphia) and Japanese first grade curricula side-by-side. In the first grade, the topical coverage was similar but the Japanese curriculum presented the skills in a more logical order; later topics clearly built on previous topics. In the American curriculum, the topics were in a more scattered sequence. It was like building a brick wall with a brick here and a brick there with the expectation that a complete wall would result.

Though he also points out that the Japanese system isn’t perfect:

To be fair, the Japanese curriculum suffers from a belief that there is only one right answer.

In many cases there is only one right answer, but there are many different ways to get there!

Here’s another commenter discussing an important idea in math education:

You can’t teach something you don’t understand yourself. And “understand” means knowing not only how, but why.
Either all elementary school teachers need to learn the “why” of K-6 mathematics, or we need mathematics specialists in all elementary schools.

Li-Ping Ma published an excellent book in 1999 called Knowing and Teaching Elementary Mathematics that explores the mathematical knowledge of American and Chinese elementary school teachers.

For one thing, the Chinese teachers were generally math specialists.  As a result, they often had a more profound understanding of the topics.  This allowed them to understand WHY algorithms work the way they do and use this knowledge to effectively answer their students’ questions.

Ma points out that many of the Chinese teachers told her that they expect the students to “know how, but also know why.”

I first became aware of the problems in math education by doing an internet search on “Jaime Escalante” back in 1997 (I think I used Infoseek as Google didn’t exist yet!).  What I found was the story of Escalante’s ostracism from the Garfield High School math department he had made famous in “Stand and Deliver.” The story was posted at a website called “Mathematically Correct.”

I spent a lot of time reading the content at that site in the late 1990s.  A good summary of the state of math education at that time (which apparently hasn’t changed appreciably) can be found in the two-part article published in the Notices of the American Mathematical Society (Part I ; Part II) in 1997.

Professor H. Wu of UCal Berkeley also has many good articles about teaching mathematics and training math teachers as well.  A favorite of mine is Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education.

There is one commenter at the NY Times site with whom I disagree.  He is a science teacher who is concerned with his student’s math skills.  He feels that mathematics should only cover “real life” applications:

Math courses do not give kids the skills they need to do REAL math…the kind of math that can be applied beyond theoretical math situations. The curriculum is so disconnected from reality as to be useless….

We need to rethink math. It has to go from being a bunch of theoretical ideas to actual practical skills. I’m not talking about the old “Farmer John has a fence 300 yards long…”

The Farmer and the Fencing is actually a particularly rich application problem that involves many different areas of mathematics.  I have used this problem in classes ranging from 7th grade to Beginning Algebra to Differential Calculus, and, if a rectangular pen is split along the diagonal, the solution requires multi-variable Lagrangian methods.  Creating the pen of largest area regardless of shape leads to the Isoperimetric Inequality.

If mathematics is to be centered around problems, then they should be problems chosen by each student individually and the problems should either be sufficiently rich in their own right so that the connections to many other ideas become clear or they should be embedded as a piece of a larger research project.

Physicists and other researchers in the hard sciences often spend time studying the mathematics they need to know in order to understand particular areas of their research.  There’s no reason why students couldn’t approach mathematics in this way as well.

But, in a situation in which there are many different students whose different needs must be addressed in the same classroom, I believe that broad, generally applicable problems best serve the needs of such a diverse group.

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I read a review of the graphic novel Logicomix in the New York Times recently.

Logicomix is a graphic novel that dramatizes the work and lives of some of the most important mathematical logicians of the late 19th and early 20th century.

A review in the Financial Times explains:

First among them is Bertrand Russell, the English philosopher whose life story this is – at least as far as 1939. Also present are his fellow pioneers in the philosophy of mathematics: Alfred North Whitehead, with whom Russell sought, in the years before the first world war, to provide a logically rigorous, good-for-all-time foundation for mathematics; Ludwig Wittgenstein, the austere Austrian who argued that Russell’s project was misconceived; Kurt Gödel, Wittgenstein’s compatriot, who proved that it was; and assorted other pin-ups of higher mathematics – Cantor, Poincaré, Hilbert.

So, essentially, what was going on is that the British logician/mathematician Bertrand Russell, together with his collaborator Alfred North Whitehead, were trying to prove that mathematics was “self-contained.”  That is to say that they were trying to show that there is no “fudge factor” in mathematics.

It is understandable that someone might be sucked into believing that this is possible.  Math can be very reassuring in the apparently solid nature of its logical structure.  It sometimes appears as if there are no holes in the chains of reasoning upon which all of mathematics is built.

But, in reality, mathematics is and always has been a construct that has significant holes in its foundations.  Not being able to divide by zero may be the least of math’s logical dilemmas.

This is not to say that the lack of logical perfection makes mathematics useless.  Far from it, math has been and remains a very useful reasoning tool for humans.  In fact, one of the most amazing things to me about math is that Kurt Gödel used mathematical logic itself to demonstrate the holes that exist in the logical structure of mathematics.

Gödel, Escher and Bach

I was first introduced to these events in the history of mathematical logic through the book Gödel, Escher and Bach, which was written in 1979 by Douglas Hofstadter.

In the introductory chapter, Hofstadter explains the basic ideas of the work of Bertrand Russell and the Austrian mathematician/logician Kurt Gödel.  Russell and Alfred North Whitehead were working very hard in the early 20th century to establish a rock solid logical foundation for mathematics that would dispel all of the paradoxes and conundrums that both ancient and contemporary mathematicians had struggled with.  Russell and Whitehead’s work was eventually published as the three-volume Principia Mathematica.

The simplest example of the type of paradox that Russell and Whitehead were concerned with is the one that Hofstadter gives in Gödel, Escher and Bach:

That paradox is the so-called Epimenides paradox, or liar paradoxEpimenides was a Cretan who made one immortal statement: “All Cretans are liars.”  A sharper version of the statement is simply “I am lying”; or, “This statement is false”.  It is that last version which I will usually mean when I speak of the Epimenides paradox.  It is a statement which rudely violates the usually assumed dichotomy of statements into true and false, because if you tentatively think that it is true, then it immediately backfires on you and makes you think it is false.  But once you’ve decided it is false, a similar backfiring returns you to the idea that it must be true.  Try it!

This paradox is particularly vexing for mathematicians because mathematics relies extensively on the “Law of the Excluded Middle,” which assumes that a statement is either true or false.

In Gödel, Escher and Bach, Hofstadter explains that many mathematicians weren’t sure that Russell and Whitehead would be able to accomplish their task, so the mathematician David Hilbert set up a challenge to mathematicians to determine conclusively whether Russell and Whitehead were right or wrong.

In 1931, the Austrian mathematician Kurt Gödel demonstrated that Russell and Whitehead’s work was incorrect – or, if it was correct, it was incomplete.  This is the essence of Gödel’s Incompletness Theorem.  It says that no formal mathematical system can be both complete (“all-encompassing”) and logically consistent.

I was always curious as to why Russell felt that mathematics could be fully explained through logic, when it seems pretty clear that the world we live in is far more complex than the rules of logic seem to account for.

Logicomix explores Russell’s private life, both as a child and as an adult.  Russell’s family situation as a child was somewhat unsettled and he came from a family that had a history of madness.  It may be that Russell’s love of mathematics and craving for logical certainty was an understandable attempt to ground himself in a family situation that was somewhat unstable.

The scientific and mathematical results of this period of history are important milestones in our culture.  It was in the 1920’s that Werner Heisenberg, Niels Bohr and Albert Einstein were working through the implications of quantum physics and it was in 1935 that Schrödinger first conceived of his cat that was both alive and dead.

These ideas and conclusions put 20th century European and American science and philosophy on shaky ground, and mathematicians and physicists continue today to work through the implications of these ideas.

It is unclear to me whether or not it is possible to resolve the various paradoxes and seemingly nonsensical conclusions that arise out of this body of work.

I think that examining the scientists and mathematicians themselves as historical figures (as Logicomix does) can be helpful in creating a structure for inquiry into an area of knowledge that, almost by definition, lacks structure.

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This isn’t really mathematical, but there has been a flurry of articles recently about the financial crisis – both its lingering effects and the day to day issues it raised last September in the Federal Government and the large, money center banks and invesment banks.

Andrew Ross Sorkin has an article in the new Vanity Fair that is an excerpt from his book Too Big to Fail, which comes out later this month.  In it, he details the events of the third week of September 2008.

On September 7th, Fannie Mae and Freddie Mac were taken over by the government.  Bear Stearns had been taken over by JPMorgan Chase in May 2008 (with a $30 billion government guarantee serving as a spoonful of sugar to make the deal go down easier) and Lehman Brothers went into bankruptcy on Monday September 15th.

Sorkin’s story opens on Wednesday September 17, 2008.

The panic was already palpable in John Mack’s office at Morgan Stanley’s Times Square headquarters. Sitting on his sofa with his lieutenants, Walid Chammah, 54, and James Gorman, 50, drinking coffee from paper cups, Mack was railing: the major news on Wednesday morning, he thought, should have been the strength of Morgan Stanley’s earnings report, which he had released the afternoon before, a day early, to stem any fears of the firm’s following in Lehman’s footsteps….

Apart from the general nervousness about investment banks, he was facing a more serious problem than anyone on the outside realized: at the beginning of the week, Morgan Stanley had had $178 billion in the tank—money available to fund operations and lend to its hedge-fund clients. But in the past 24 hours, more than $20 billion of it had been withdrawn by anxious hedge-fund clients, in some cases closing their prime-brokerage accounts entirely.

“The money’s walking out of the door,” Chammah told Mack….

‘What’s wrong?” Mack asked in alarm as Colm Kelleher walked into his office later in the day, his face ashen. “John, we’re going to be out of money on Friday,” Kelleher said with his staccato British inflection. He had been nervously watching the firm’s tank—its liquid assets—shrink, the way an airline pilot might stare at the fuel gauge while circling an airport, waiting for landing clearance.

“That can’t be,” Mack said anxiously. “Do me a favor: go back to the financing desk—go through it again.”

Kelleher returned to Mack’s office 30 minutes later, less shaken, but only slightly. After finding some additional money trapped in the system between trades that hadn’t yet settled, he revised his prognosis: “Maybe we’ll make it through early next week.”

The article details the machinations of the major economic players in last September’s meltdown – Former Treasury Secretary Henry Paulson and then-Head of the New York Federal Reserve Bank and current Treasury Secretary Timothy Geithner, as well Morgan Stanley’s John Mack, Goldman Sach’s Lloyd Blankfein, JPMorgan Chase’s Jamie Dimon, Citibank’s Vikram Pandit, Bank of America’s Ken Lewis and many others.

Another article, this one a blog post from Chris Whalen at The Big Picture economics blog goes over some of the problems that came out of the wave of takeovers late last year as Bank of America took over Countrywide Mortgage and Merrill Lynch, Wells Fargo took over Wachovia Bank, and Bear Stearns was taken over by JPMorgan Chase.  Whalen is a professional bank analyst (who also worked as a republican congressional staffer in the 1990s) and his description is pretty heavy on banking jargon and details.

Oil Trading Denominated in Dollars

Also today, there is an article in the British newspaper The Independent written by Robert Fisk about a plan to sell oil using a “basket” of currencies instead of the U.S. dollar.

Ever since Richard Nixon removed the gold standard in the early 1970s, the U.S. dollar has remained a reserve currency for the world mainly due its status as the currency in which oil trading is denominated.  This is set to change.

In the most profound financial change in recent Middle East history, Gulf Arabs are planning – along with China, Russia, Japan and France – to end dollar dealings for oil, moving instead to a basket of currencies including the Japanese yen and Chinese yuan, the euro, gold and a new, unified currency planned for nations in the Gulf Co-operation Council, including Saudi Arabia, Abu Dhabi, Kuwait and Qatar.

Secret meetings have already been held by finance ministers and central bank governors in Russia, China, Japan and Brazil to work on the scheme, which will mean that oil will no longer be priced in dollars.

The plans, confirmed to The Independent by both Gulf Arab and Chinese banking sources in Hong Kong, may help to explain the sudden rise in gold prices, but it also augurs an extraordinary transition from dollar markets within nine years….

Ever since the Bretton Woods agreements – the accords after the Second World War which bequeathed the architecture for the modern international financial system – America’s trading partners have been left to cope with the impact of Washington’s control and, in more recent years, the hegemony of the dollar as the dominant global reserve currency.

This could devalue the dollar and make imports (like oil itself) more expensive.

In addition to the article, The Independent also included an editorial about the proposed change:

Last autumn’s global financial crisis set off an economic earthquake. And we are still feeling the tremors. The latest sign of the ground shifting beneath our feet is our report today of plans by Gulf states, China, Russia, France and Japan to end their practice of conducting oil deals in US dollars, switching instead to a diverse basket of currencies.

It is not hard to see the motivation for oil exporters to move away from the dollar. The value of the US currency has fallen sharply since last year’s meltdown. And fears are growing, in the light of a spiralling US government deficit, that a further depreciation is likely. They do not want to sell their wares in return for a currency with an uncertain future.

Corruption

Last on the list is a story about the New York State Pension Fund and a guilty plea in a case of corruption reportedly involving the New York State comptroller Alan Hevesi

ALBANY — Raymond B. Harding, one of the last of New York’s political bosses, admitted on Tuesday that he had accepted more than $800,000 in exchange for doing favors for Alan G. Hevesi, the former state comptroller; among the favors was a scheme to secure an Assembly seat for Mr. Hevesi’s son….

The case has focused on the state’s $116.5 billion pension fund, and how people close to Mr. Hevesi exploited their relationship with the former comptroller to enrich themselves. The comptroller serves as the fund’s sole trustee, a relatively unusual arrangement that gives him ultimate authority over what firms are allowed lucrative contracts to manage the fund’s money…

Mr. Cuomo’s office also said on Tuesday that Saul Meyer of Aldus Equity, a Dallas firm that consulted with the pension fund, had pleaded guilty to a similar charge that had been sealed since Friday. Mr. Meyer admitted to violating his fiduciary duty to pensioners in both New York and New Mexico and taking part in schemes allegedly orchestrated by Mr. Morris. He also faces four years in jail, but is cooperating with the investigation.

“These guilty pleas vividly depict the depth and breadth of corruption involving the New York State pension fund,” Mr. Cuomo said. “In one case, we see New York’s state pension fund looted to reward a political boss with hundreds of thousands of dollars in improper payments.”

“In the other, we see a pension fund adviser — the outside ‘gatekeeper’ who is supposed to safeguard the integrity of the pension fund process — recommending deals based on pressure from pension officials and politically connected people.”

This type of corruption is at the root of the problems in our financial system.  The recently released inspector general’s report on the investigation (or lack of investigation) into Bernard Madoff’s massive ponzi scheme details the failure of the Securities and Exchange Commission to look into repeated reports of fraud.

Cleaning up corrupt practices in both business and government must precede any meaningful resurgence of the U.S. economy.

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