A parabola is the locus of all points equidistant from a given point, the focus, and a line, the directrix.

The September 2011 issue of The College Mathematics Journal (published by MAA) contains an article by Dan Joseph, Gregory Hartman and Caleb Gibson titled Generalized Parabolas (available online if a member/subscriber or through jstor: http://www.jstor.org/pss/10.4169/college.math.j.42.4.275 if you have access to jstor). In their article they investigate what happens if you change the directrix in the definition above to a general curve, for example, a parabola (see example 3 below). The authors took an analytical approach, using Mathematica to find the equation of each generalized parabola.

I recognized GeoGebra could be used for a purely geometrical investigation. Go to http://www.burke-consulting.com/GeneralizedParabolas.html to see some way cool examples and then try it out for yourself IMNSHO.

John

Is mathematics invented or discovered?

I often tell students when introducing new (to them) concepts that when mathematicians hit a brick wall in pursuing some investigation, they invent something to knock down the wall. Mathematics is purely an invention of the human mind.

Another way of looking at the same thing is that the tool to knock down the wall was there all along just waiting for someone with the need to discover it. All of mathematics exists as a fundamental part of the universe independent of us.

In thinking about this, I realized that I use the terms “invent” and “discover” interchangeably when talking about mathematics, and yet they are not the same.

Theoretical astrophysicist Mario Livio has an article in the August Scientific American titled “Why Math Works” where he examines both sides of this philosophical argument … and comes down squarely in the middle.

The article is at www.sciam.com; however, I think you need to be a subscriber to read the whole article (unless the college has a site license).

Science News Article About the Status of P not equal to NP

Julie Rehmeyer has an article in Science News about the status of of P vs NP. The explanation is excellent and well worth several reads. Particularly interesting is how the collaborative possibilities of the Internet have come into play.

HP Labs Mathematician Claims P not equal to NP

Vinay Deolalikar, who is with Hewlett-Packard Labs, has sent to peers copies of a proof he did stating that P is not equal to NP, one of the Millennium Prize Problems. Discover Magazine has a good lay description of the problem. For those interested, the 98 page article containing the proof is online.

The Online Encyclopedia of Integer Sequences

Julie Rehmeyer's August Math Trek column in Science News talks about the Online Encyclopedia of Integer Sequences and its creator Neil Sloane. The OEIS currently has almost 200,000 integer number sequences in an online searchable database. This is a passion for Sloane begun in the mid-1960s while in graduate school.

From the article, "the OEIS ... provides a sequence’s full 'life story.' Along with listing the numbers that form the beginning of a sequence (sometimes hundreds of thousands of them), it gives all the different known ways to generate the sequence, lists references to the sequence in the scientific literature, links to any sites with information about it, cross-references related sequences, provides a graph of the sequence, and even offers a way to listen to the sequence."

It is very easy to lose yourself for hours in the OEIS so be careful ;-).

Stephen Wolfram on TED

If you haven't already seen it, spend an enjoyable 20 minutes listening and watching Stephen Wolfram's TED talk from February of this year. Note that it will probably take a lot longer than 20 minutes since you will find yourself frequently pausing to catch the Wolfram|Alpha search requests and Mathematica code. Well worth multiple viewings.

New Math Standards?

One of the little dramas being played out in California and probably elsewhere has to do with whether the state should adopt the K-12 Common Core standards for mathematics and English developed by a consortium of 48 states, including representatives from California. The Sacramento Bee editorialized that the state should adopt the US math standards, agreeing with 19 of the 21 members of the California Academic Content Standards Commission, the Governor's Office and most of the academic community. The editorial referred to California's "mile-wide, inch-deep" math curriculum as a problem the Common Core addresses. In an Op-Ed piece by the two dissenting members in which they claim the new standards "would gut the state's successful program", they also refer several times to "high-performing foreign countries" teaching to something like California's current "mile-wide" curriculum. Several days later in a point-counterpoint debate, the same arguments were raised.

In a letter to the editor published on July 28, I pointed out the real problem and the real difference between the US and "high-performing foreign countries" is the length of both the school day and school year:

Fix class time, not standards

Re: "State should adopt U.S. math standards" (Editorials, July 24) and "Proposed math standards unteachable" (Viewpoints, July 24): Both the editorial and op-ed column miss the point.

California's current curriculum is indeed "mile wide, inch deep." The reason it is an "inch deep" is because the school day and school year are too short and because students are not required to take mathematics through 12th grade.

"High-performing foreign countries" teach the same breadth of material, but they can teach it better because the school day and school year are longer and students have more years of mathematics instruction.

The state's current system is not a "successful" program, it is just broken differently.

No amount of changed standards or national commission reports will catch us up with the many countries ahead of us in math education; only the realization that, like learning to play well some musical instrument, learning mathematics requires many hours and years of study and practice.

- John Burke, Sacramento