Monday, October 24, 2016

Mo' Monic

If look at any list of elementary algebra topics, or any book's table of contents, etc., you'll probably find that all of the subjects can be called out by name except for one single exceptional case, which is always expressed in symbolic form. For example, from the College Board's Accu-Placer Program Manual, here's a list of Content Areas for the Elementary Algebra test:

Do you see it? Or, here are some of the section headers in the Pearson testbank which accompanies the Martin-Gay Prealgebra & Introductory Algebra text:

Or, here's a menu of topics and quizzes from the algebra site:

I could repeat this for many other cases, such as: the CUNY list of elementary algebra topics, tables of contents for most algebra books, etc., etc. It's weird and to my OCD brothers and sisters surely it's a bit distracting and frustrating.

There should be a name for this. The funny thing is that, to my current understanding, there's a perfectly serviceable name to make the distinction that we're reaching for here: "monic" means a polynomial with a lead coefficient of 1. So I've taken to, in my classes, referring to the initial or "basic" type (\(x^2 + bx + c\)) as a monic quadratic, and the more general or "advanced" type (\(ax^2 + bx + c\), \(a \ne 1\)) as a nonmonic quadratic. My students are accustomed to being expected to know proper names for everything, and so they pick this up as easily as anything else, and without complaint. Thereafter it's really much easier to reference the different structures by their proper names.

Now: I must admit that I picked this up from Wikipedia and I've never, ever, seen it used in any mathematics textbook at any level. Perhaps someone could tell me if this is new, or nonstandard, or inaccurate. But even if that weren't the right term to distinguish a polynomial with lead coefficient 1, there should still be a name for this structure. We really should create a name, if necessary, and I'd be prone to make up my own name for something like that.

But "monic" fits perfectly and is delightfully short and descriptive. We should all start using "monic" more widely, and I'd love to start seeing it in major algebra textbooks.

Monday, October 10, 2016

Natural Selection of Bad Science

Smaldino and McElreath write a paper which asserts that the problem of false-positive papers in science -- especially behavioral science -- is getting worse over time, and will continue to do so as long as we reward quantity of paper outputs:
To demonstrate the logical consequences of structural incentives, we then present a dynamic model of scientific communities in which competing laboratories investigate novel or previously published hypotheses using culturally transmitted research methods. As in the real world, successful labs produce more ‘progeny,’ such that their methods are more often copied and their students are more likely to start labs of their own. Selection for high output leads to poorer methods and increasingly high false discovery rates. We additionally show that replication slows but does not stop the process of methodological deterioration. Improving the quality of research requires change at the institutional level.

Quotes Campbell's Law: "The more any quantitative social indicator is used for social decision-making, the more subject it will be to corruption pressures and the more apt it will be to distort and corrupt the social processes it is intended to monitor."

Review at the Economist.

Monday, October 3, 2016

Euclid: The Game

A marvelous little game that treats Euclidean construction theorems as puzzles to solve in a web application:

Play it here.

Hat tip: JWS.

Monday, September 26, 2016

When Blind People Do Algebra

From NPR:
A functional MRI study of 17 people blind since birth found that areas of visual cortex became active when the participants were asked to solve algebra problems, a team from Johns Hopkins reports in the Proceedings of the National Academy of Sciences.

This is not the case with the same test run on sighted people. Which is interesting, because it serves as evidence that the brain is more flexible, and can be re-wired for a greater multitude of jobs, than previously believed.

Read more here. 

Monday, September 19, 2016

NY Times: Stop Grading to a Curve

An excellent article by Adam Grant, professor at the Wharton School of the University of Pennsylvania:
The more important argument against grade curves is that they create an atmosphere that’s toxic by pitting students against one another. At best, it creates a hypercompetitive culture, and at worst, it sends students the message that the world is a zero-sum game: Your success means my failure.

Read the full article here.

Monday, September 5, 2016

Epsilon-Delta, Absolute Values, Inequalities

Working through the famed "baby" Rudin, Principles of Mathematical Analysis. (Which was not the analysis book I used in grad school: we used William Ray's Real Analysis).

First thing that dawns on me is that I'm shaky on following the formal proofs of various limits. I decide that I need to brush up on my epsilon-delta proofs, and go back to do exercises from Stein/Barcellos Calculus and Analytic Geometry. I find that I get stuck outside of simple limits of linear functions (say, with a quadratic).

Second observation is that these proofs use extensive algebra involving absolute value inequalities, and my trouble is that, in turn, I am shaky with these to the point of being blind to a lot of very basic facts. Kind of frustrating that I teach basic algebra as a career but as soon as as absolute values and inequalities enter the picture I'm nigh-helpless.

Third observation is that this explains the copious section in college algebra and precalculus texts on absolute value inequalities (which are skipped in the curriculum at our community college). Previously my intuition had failed to see the use for these; but it's precisely the skills you need in analysis to write demonstrations involving the limit definition.

A few key properties on which I had to brush up (and would argue for preparatory exercises if I was teaching/scaffolding in the direction of analysis):
  • Subadditivity: \(|a+b| \leq |a| + |b|\). Equivalent to the triangle inequality. 
  • Partial Reverse Triangle Inequality: \(|a| - |b| \leq |a - b|\). A bastardized name of my design. It follows from the preceding because \(|a| = |a - b + b| \leq |a-b| + |b|\), and then a subtraction of \(|b|\) from both sides. Gets used at the start of almost all my limit proofs. 
  • Multiplicativeness: \(|ab| = |a||b|\). Which leads to manipulations such as: multiplying both sides of an equation by a positive number allows you to just multiply into an absolute value; squaring both sides can be done into the absolute value, etc. 

More at: Wikipedia.

Discussion of general limit exercises: StackExchange.

Friday, August 26, 2016

Crypto Receipts for Homework

An interesting idea to a problem I've also experienced (student claiming they submitted work for which the instructor has no record): Individual cryptographic receipts for assignment submissions.