Monday, April 22, 2013

Teaching/Learning Mathematics

There are two components to learning mathematics: learning skills and learning conceptual thinking.

Skills are learned through practice and drill. The fact that you understand the concept of parallel parking does not mean you can actually do it. You have to practice and practice and practice some more until you can do it. Once a skill is mastered, you never lose it; you may need some refresher practice after a long time of not using it, but that is all. Think bike riding. Skills are relatively easy to teach.

Conceptual thinking is a completely different thing.

The sequence of math classes from primary school through college form a continuum. At the lower levels, it is mostly about learning skills. At the advanced levels, it is mostly about conceptual thinking. In between, it is a mixture of both.

Monday, April 1, 2013

Resurrecting California’s Public Universities

So, I pick up the Sunday Review section of the New York Times and there on the editorial page is an editorial with the above title.
Quoting from the editorial:
"The same California State Legislature that cut the higher education budget to ribbons, while spending ever larger sums on prisons, now proposes to magically set things right by requiring public colleges and universities to offer more online courses. The problem is that online courses as generally configured are not broadly useful. They work well for highly skilled, highly motivated students but are potentially disastrous for large numbers of struggling students who lack basic competencies and require remedial education. These courses would be a questionable fit for first-time freshmen in the 23-campus California State University system, more than 60 percent of whom need remedial instruction in math, English or both.
"The story of how the state’s fabled higher education system got to this point is told in a troubling analysis by the Public Policy Institute of California, a nonpartisan think tank."
For a variety of reasons, our state legislature is reflecting society at large in trying for the quick fix to every problem that presents itself. This NYTimes editorial and others need to be understood as not antagonistic to online classes in general, but rather against using them in the wrong setting. To the extent MOOCs in mathematics have been successful, they are successful in the more advanced classes where students are already skilled in the basics and are organized, disciplined and motivated.

Online classes at the community college level can be a tremendous aid for the many students with family and job responsibilities who find it difficult to regularly attend on ground classes. However, they are not suited for students missing any of the above mentioned characteristics. The legislature, if it continues to pursue this short-sighted course will only hasten the long slow slide of California's higher education system from the envy of the world to just another state education system fallen on hard times.  In my opinion.

Sunday, March 31, 2013

Stephen Wolfram at SXSW

His complete talk as a streaming video as well as a slightly edited transcript is available on his blog.

It is impossible to listen to him talk and not come away excited and with many ideas to pursue.

The 100th Annivsary of Paul Erdos' Birth

March 29: There is an amusing article on about Paul Erdos on what would have been his 100th birthday. I actually met him very briefly while a first year graduate student at the University of Colorado. He was treated somewhat like a rock star by the math department; when he took walks around the campus he would be followed by adoring faculty and graduate students.

Tuesday, August 21, 2012

New GeoGebra Routines

There is a new version of the GeoGebra Arithmetic routines at There are several new routines and all routines have been updated to use the latest version of GeoGebra.

Sunday, October 16, 2011

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: 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 to see some way cool examples and then try it out for yourself IMNSHO.


Monday, August 1, 2011

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; however, I think you need to be a subscriber to read the whole article (unless the college has a site license).