Monday, December 5, 2016

Observed Belief That 1/2 = 1.2

Last week in both of my two college algebra sections, there came a moment when we had to graph an intercept of x = 1/2. I asked, "One-half is between what two whole numbers?" Response: "Between 1 and 2." I asked the class in general for confirmation: "Is that right? One-half is between 1 and 2, yes?" And the entirety of the class -- in both sections, separated by one hour -- nodded and agreed that it was. (Exception: One student who was previously educated in Russia.)

Now, this may seem wildly inexplicable, and it took me a number of years to decipher this. But here's the situation: Our students our so unaccustomed to fractions that the can only interpret the notation as decimals, that is: they believe that 1/2 = 1.2 (which is, of course, really between 1 and 2). Here's more evidence from from the Patricia Kenschaft article, "Racial Equity Requires Teaching Elementary
School Teachers More Mathematics"  (Notice of the AMS, February 2005):
My first time in a fifth grade in one of New Jersey’s most affluent districts (white, of course), I asked where one-third was on the number line. After a moment of quiet, the teacher called out, “Near three, isn’t it?” The children, however, soon figured out the correct answer; they came from homes where such things were discussed.

Likewise, the only way this makes sense is if the teacher interprets 1/3 = 3.1 -- both visually turning the fraction into a decimal, and getting the division upside-down. We might at first think the error is the common one that 1/3 = 3, but that wouldn't explain why the teacher thought it was only "near" three.

The next time an apparently inexplicable interpretation of a fraction comes up, consider asking a few more questions to make the perceived value more precise ("Is 1/2 between 1 and 2? Which is it closer to: 1 or 2 or equally distant?" Etc.). See if the problem isn't that it was visually interpreted as decimal point notation.


Friday, November 4, 2016

The Math Menu

A quick thought, spring-boarding off Monday's post: A constant debate in math education is whether students should be directly-taught mathematical results, or spend time (like a mathematician) exploring problems, looking for patterns, and coming up with their own "theorems" (in Mubeen's phrasing "own the problem space").

Here is a hypothetical equivalent debate: What is supposed to happen in a restaurant -- Does food get cooked, or does food get eaten?

Obviously both. But the majority of people who visit the establishment are clientele who do not come to the restaurant in order to learn how to cook; they come for an end-product which is used in a different fashion (for consumption and nourishment). If someone expresses interest in becoming a chef themselves then of course we should encourage and cultivate that. But if some group of chefs become so self-involved that they demand everyone participate in cooking for a "real" restaurant experience, then surely we'd all agree that they'd gone off the deep end and needed restraints.

So too with mathematicians.


Monday, October 31, 2016

Scary Stories

A pair of scary math-education anecdotes by Junaid Mubeen, for your consideration:
  • How Old is the Shepherd? When 8th-graders are asked a short question with absolutely no information about age whatsoever, 3-in-4 will report some numerical result anyway. Repeated in numerous experiments. Watch a video.

  • I Can't Believe It's Not Unproven. Mubeen's 12-year-old nephew comes home with a math problem that can't be solved; he is shown a proof of that fact, and agrees to all the steps and the conclusion. Nephew spends the rest of the evening trying to find an answer anyway.

I don't really agree with Mubeen's rather broad conclusions at the end of the first article. But we can all agree this is a terrifying outcome!


Monday, October 24, 2016

Mo' Monic

If you look at any list of elementary algebra topics, or any book's table of contents, etc., then you'll probably find that all of the subjects are referenced 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 MathGuide.com 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 know they must learn proper names for everything, and so they pick this up as easily as anything else, and without complaint. Thereafter it's much easier to communally 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.