Thursday, June 10, 2021

The Most Irrational Number

This is a fascinating discussion about the Golden Ratio by Jordan Ellenberg appearing in the online magazine Slate. 

Something totally new to me was the "barcode" representation of a number and how it can be used to determine some sense of the irrationality of an irrational number.


Wednesday, June 9, 2021

A Simple Way to Solve Any Quadratic Equation

I stumbled across this last December but had no time to investigate it until now. The technique is from Po-Shen Loh of the Department of Mathematical Sciences, Carnegie Mellon University. Relevant articles:

·        Math Genius Has Come Up with a Wildly Simple New Way to Solve Quadratic Equations (a bit of hyperbole, but what can you expect from the website

·        A Different Way to Solve Quadratic Equations

·        A Simple Proof of the Quadratic Formula

·        Quadratic Method: Detailed Explanation

·        Quadratic Method: Related Work

In Elementary and Intermediate Algebra classes we spend a huge amount of time on the factoring of polynomials, particularly second-degree polynomials with integer coefficients, as a way of solving certain quadratic equations; i.e., those with rational number solutions. Of course, in practical applications, the coefficients are not integers and the solutions are almost never rational numbers. So, we then talk about radicals and complex numbers, completing the square and eventually work our way up to the quadratic formula as the ultimate way to solve any quadratic equation. What Po-Shen has described is a single technique that can be taught at the beginning when the solutions are rational numbers, but will still work on quadratic equations with real number and even complex number solutions. In the above articles, he shows (including several videos) how to justify and teach the technique right after the student has learned how to multiply two binomials and has been introduced to the idea of a quadratic equation.

The basic idea in practice for solving any quadratic equation of the form Ax2 + Bx + C = 0 is:

1.     Take the quadratic equation and put it in the form x2 + Bx + C = 0.

2.      Let r = (-B/2 - u) and  s = (-B/2 + u) where u is any complex number. [note that r + s = – B]

3.      Now note that r x s = B^2/4 - u^2. Set B^2/4 - u^2 = C and solve for u. r and s are the solutions to the original quadratic equation.

Note that in addition to solving any quadratic equation, the technique also obviously leads to the factorization of the second-degree polynomial: P(x) = (xr)(xs).

Friday, May 21, 2021

Should we split standard 16-week semesters into two 8-week semesters?

Inside Higher Ed's "Confessions of a Community College Dean" blog posed this question recently. Matt Reed gives an excellent description of the plusses and possible challenges of such a move. He also comments that given the population community colleges tend to draw from, such a move could result in higher throughput and better student success.  I encourage all to read his blog post.

Apparently there has been little research on this, though several colleges are known to have moved to this format.

I took special note of this because in the spring 2021 semester I taught one section of Intermediate Algebra in the 16-week format and one section of the same course in an 8-week format. Same assignments, lecture notes, tests, same number of class meetings, etc. In other words, the two classes were taught identically, the only difference being the length of the "semester".

While two classes are obviously too small a sample to draw any conclusions, the difference in results was staggering:

The 16-week class

The 8-week class

I would love to hear any real world examples of schools that have gone to 8-week semesters and what the experience has been.

Monday, May 17, 2021

I'm Baaaaaack!

 It is hard to believe it has been 8 years since my last post.

A lot of things have happened, both personally and professionally. Some highlights and low lights ...


  • We moved into a great new house 6 years ago on small man-made lake. Much time and money has gone into decorating and remodeling both inside and outside.
  • During the last 5 years, we lost both of my parents, both of my in-laws, and our wonderful Husky, Mariah. We racked up the frequent flyer miles going coast to coast dealing with end-of-life issues for both sets of parents.
  • I am officially old, having turned 70 in 2017.


  • Five years ago, I accepted a fulltime teaching position at American River College in Sacramento, California.
  • I was recently granted tenure.

Of course both currently and in the recent past, we are all dealing with the coronavirus pandemic (more on this in a future post) and its effects on our personal and professional lives.

Here in California, we mathematics instructors are also dealing with the legislator's AB-705, which basically outlawed remedial education and it's fallout. The legislature decided they knew more about teaching mathematics and English then the professionals. When this happens, it almost never turns out well. 

That's it for today. I have grand plans for the future of this blog, but then I also had grand plans 8 years ago.

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.