Elementary School Mathematics Education Mysteries


Since these two problems are really the exact same problem, in two different forms, why not just use “x” to teach it, from the beginning, in elementary school, instead of using the little box? The two symbols have the exact same meaning!

To the possible answer, “We use an ‘x’ for multiplication, instead, so doing this would be confusing,” I have a response: why? Using “x” for multiplication is a bad idea, because then students have to unlearn it later. In algebra, it’s better to write (7)(5) = 35, instead of 7×5 = 35, for obvious reasons — we use “x” as a variable, instead, almost constantly. This wouldn’t be as much trouble for students taking algebra if they had never been taught, in the first place, that “x” means “multiply.” It’s already a letter of the alphabet and a variable, plus it marks spots. It doesn’t need to also mean “multiply.”

Why are we doing things in a way that causes more confusion than is necessary? Should we, as teachers, not try to minimize confusion? We certainly shouldn’t create it, without a good reason for doing so, and these current practices do create it.

These things may not be mysteries to others, but they certainly are to me.

[Note: for those who do not already know, I am a teacher of mathematics. However, I do not have any experience teaching anything at the elementary level. For this particular post, that’s certainly relevant information.]

My New Middle Initial and Name: A Mathematical Welcome-Back Gift from My Alma Mater


I just had a middle initial assigned to me, and then later, with help, figured out what that initial stood for. With apologies for the length of this rambling story, here’s an explanation for how such crazy things happened.

I graduated from high school in 1985, and then graduated college, for the first time, with a B.A. (in history, of all things), in 1992. My alma mater is the University of Arkansas at Little Rock, or UALR, whose website at http://www.ualr.edu is the source for the logo at the center of the image above.

Later, I transferred to another university, became certified to teach several subjects other than history, got my first master’s degree from there (also in history) in 1996, and then quit seeking degrees, but still added certification areas and collected salary-boosting graduate hours, until 2005. In 2005, the last time I took a college class (also at UALR), I suddenly realized, in horror, that I’d been going to college, off and on, for twenty years. That, I immediately decided, was enough, and so I stopped — and stayed stopped, for the past ten years.

Now it’s 2015, and I’ve changed my mind about attending college — again. I’ve been admitted to a new graduate program, back at UALR, to seek a second master’s degree — one in a major (gifted and talented education) more appropriate for my career, teaching (primarily) mathematics, and the “hard” sciences, for the past twenty years. After a ten-year break from taking classes, I’ll be enrolled again in August.

As part of the process to get ready for this, UALR assigned an e-mail address to me, which they do, automatically, using an algorithm which uses a person’s first and middle initial, as well as the person’s legal last name. With me, this posed a problem, because I don’t have a middle name.

UALR has a solution for this: they assigned a middle initial to me, as part of my new e-mail address: “X.” Since I was not consulted about this, I didn’t have a clue what the “X” even stands for, and mentioned this fact on Facebook, where several of my friends suggested various new middle names I could use.

With thanks, also, to my friend John, who suggested it, I’m going with “Variable” for my new middle name — the name which is represented by the “X” in my new, full name.

I’ve even made this new middle initial part of my name, as displayed on Facebook. If that, plus the e-mail address I now have at UALR, plus this blog-post, don’t make this official, well, what possibly could?

Proposed Mechanisms for New and Different Types of Novae

Theoretical New Type of Nova

The picture above shows a proposed model for the production of a sudden increase in the brightness of a star — or rather, what is apparently a single star, optically, but would actually be a suddenly-produced binary stellar system.

The yellow object is a star, the system’s primary, and it has high mass (at least a few solar masses), when its mass is compared to those of the brown dwarfs in the two highly elliptical orbits shown in blue. These brown dwarfs aren’t quite stars, lacking enough mass to fuse hydrogen-1, which requires 75 to 80 Jupiter masses, but one of them (the larger one) is close to that limit. The smaller brown dwarf has perhaps half the mass of the larger brown dwarf. Their high orbital eccentricities give them very long orbital periods, on the order or 100,000 years. In a very small fraction of orbits, both brown dwarfs will be near perihelion (closest point to the primary) at the same time, and, during those rare periods, the two brown dwarfs become much closer to each other than they are to the primary.

When the two brown dwarfs become close enough to each other, matter from the smaller one could be drawn, by gravity, into the larger brown dwarf, increasing its mass, at the expense of its smaller sibling. At some point, in such a system, the larger brown dwarf’s mass could then reach the threshold to begin fusing hydrogen-1, and “turn on” as a true star — a red dwarf. From Earth, this red dwarf would not be distinguishable from the system’s most massive star, shown in yellow, until much later, when the two moved further apart. There would, however, be a sudden increase in luminosity from the system as a whole. Unlike other types of novae, this increase in luminosity would not fade away quickly, for red dwarfs have very long lifespans. This would enable them, upon discovery, to be distinguished from other single-brightening stellar events. Confirmation could then come from resolution of the new red dwarf component, as it recedes from the primary, making detection easier.

For a variation on this mechanism, the primary star could be somewhat more massive, and the two large brown dwarfs could be replaced by two large red dwarf stars. The larger red dwarf could draw matter from the smaller one, until the larger red dwarf became large enough to cross a higher mass threshold, and brighten substantially, with its color suddenly changing to orange or yellow.

A problem for this model:  no such events are known to have happened. If they do happen, a likely explanation for their rarity is the likelihood that such orbits would be unstable, in a large fraction of similar cases, preventing the stellar-brightening event from having time to happen — in all but a few cases, none of which humans have (yet) both seen, and understood. If one of these things goes off nearby, though, we will learn about it quickly, for it will make itself known.

For another possible mechanism, there is another option:  remove the primary altogether, and let the two objects of near-threshold mass orbit their common center of mass directly. They could then create a new star, or brighter star, by the mechanism described, one which might even produce a detectable accretion disk. A actual merger of the two brown dwarfs, or red dwarf stars, would be a variation of this idea, and would presumably be more likely if the two objects had masses very close to each other, so that neither would have an advantage in the gravitational tug-of-war.