On Piflars

In coordination with the week's theme of grammar -- seen on StackExchange: English Language & Usage:

Apparently in Slovenian, there is the single word "piflar", which derogatorily means "a student who only memorizes instead of truly learning". What would be the best comparable word for this in English?


Gruesome Grammar

A week or so back we observed the rough consensus that basic arithmetic operations are essentially some kind of prepositions. Coincidentally, tonight I'm reviewing the current edition of "CK-12 Algebra - Basic" (Kramer, Gloag, Gloag; May 30, 2015) -- and the very first thing in the book is to get this exactly wrong. Here are the first two paragraphs in the book (Sec 1.1):
When someone is having trouble with algebra, they may say, “I don’t speak math!” While this may seem weird to you, it is a true statement. Math, like English, French, Spanish, or Arabic, is a secondary language that you must learn in order to be successful. There are verbs and nouns in math, just like in any other language. In order to understand math, you must practice the language.

A verb is a “doing” word, such as running, jumps, or drives. In mathematics, verbs are also “doing” words. A math verb is called an operation. Operations can be something you have used before, such as addition, multiplication, subtraction, or division. They can also be much more complex like an exponent or square root.

That's the kind of thing that consistently aggravates me about open-source textbooks. While I love and agree with establishing algebraic notation as a kind of language -- as possibly the single most important overriding concept of the course -- to get the situation exactly wrong right off the bat (as well as using descriptors like PEMDAS, god help us) keep these as fundamentally unusable in my courses. This, of course, then leaves no grammatical position at all for relations (like equals) when they finally appear later in the text (Sec. 1.4). So close, and yet so far.


Link: Smart People Happier with Fewer Friends

Research by people at the London School of Economics and Singapore Management University that smarter people are happier with fewer friends, and fewer social interaction outings. Downside: The researchers are evolutionary psychologists and seek to explain the finding in those terms. Also: Uses the term "paleo-happiness".


Poker Memory

Maybe 15 years ago, I went to the Foxwood poker tables (vs. 9 other people), got pocket Queens, and had an Ace come up on the flop. I maxed out the bet and lost close to $100. So the other morning I woke up and the thought in my head was, "I really should have computed the probability that someone else had an Ace". Which was 1 - 44P18/47P18 = 1 - 0.225 = 0.775 = 77.5%. Sometimes my brain works glacially slow like that.


Reading Radicals

In my development algebra classes, I push radicals further forward, closer to the start of the semester than most other instructors or textbooks. I want them to be discussed jointly with exponents, so we can really highlight the inverse relation with exponents, and that knowledge of the rules of one is effectively equivalent knowledge of the other. Also: Based on the statistics I keep, success on the exponents/radicals test is the single best predictor of success on the comprehensive, university-wide final exam.

There are, of course, many errors made by students learning to read and write radicals for effectively the first time. Here's an exceedingly common category, to write something like (\(x > 0\)):
$$\sqrt{16} = 4 = 2$$
$$\sqrt{x^8} = x^4 = x^2 = x$$

Any of these expressions may or may not have a radical written over them (including, e.g., \(\sqrt{4} = \sqrt{2}\)). That is: Students see something "magical" about radicals, and sometimes keep square-rooting any expression in sight, until they can no longer do so. This is common enough that I have few interventions in my mental toolbox ready for when this occurs in any class:
  1. Go to the board and, jointly with the whole class, start asking some true-or-false questions. "T/F: \(\sqrt{4} = \sqrt{2}\) ← False. \(\sqrt{4} = 2\) ← True." Briefly discuss the difference, and the location of \(\sqrt{2}\) on the number line. Emphasize: Every written symbol in math makes a difference (any difference in the writing, and it has a different meaning).

  2. Prompt for the following on the board. "Simplify: \(3 + 5 = 8\)." Now ask: "Where did the plus sign go? Why are you not writing it in the simplified expression? Because: You did the operation, and therefore the operational symbol goes away. The same will happen with radicals: If you can actually compute a radical, then the symbol goes away at that time."

That's old hat, and those are techniques I've been using for a few years now. The one new thing I noticed last night (as I write this) is that there is actually something unique about the notation for radicals: Of the six basic arithmetic operations (add, subtract, multiply, divide, exponents, radicals), radicals are the only binary operation where one of the two parameters may not be written. That is, for the specific case of square roots, there is a "default" setting where the index of 2 doesn't get written -- and there's no analogous case of any other basic operator being written without a pair of numbers to go with it.

I wonder if this contributes to the apparent "magical" qualities of radicals (specifically: students pay more attention to the visible numbers, whereas I am constantly haranguing students to look more closely at the operators in the writing)? Hypothetically, if we always wrote the index of "2" visibly for square roots (as for all other binary operators), would this be more transparent to students that the operator only gets applied once (at which point radical and index simplify out of the writing)? And perhaps this would clear up a related problem: students occasionally writing a reduction as a new index, instead of a factor (e.g., \(\sqrt{18} = \sqrt{9 \times 2} = \sqrt[3]{2}\))?

That would be a pretty feasible experiment to run in parallel classes, although it would involve using nonstandard notation to make it happen (i.e., having students explicate the index of "2" for square roots all the time). Should we consider that experiment?


What Part of Speech is "Times"?

What part of speech are the operational words "plus", "minus", and "times"? This is a surprisingly tricky issue; apparently major dictionaries actually differ in their categorization. The most common classification is as some form of preposition -- the Oxford Dictionary says that they are marginal prepositions; "a preposition that shares one or more characteristics with other word classes [i.e., verbs or adjectives]".

Here's an interesting thread on Stack Exchange: English Language & Usage on the issue -- including commentary by famed quantum-computing expert and word guru Peter Shor:


Veterinary Homeopathy

A funny, but scary and real, web page of a homeopathic-practicing veterinarian who seems weirdly cognizant that it has no real effect:
How much to give: Each time you treat your pet, give approximately 10-20 of the tiny (#10) pellets in the amber glass vial, or 3-7 of the larger (#20) pellets in the blue plastic tube. You don't need to count them out. In fact, the number of pellets given per treatment makes no difference whatsoever. It is the frequency of treatment and the potency of the remedy that is important. Giving more pellets per treatment does not in any way affect the body's response. The pellets need not be swallowed, and it doesn't matter if a few of them are spit out. Just get a few pellets somewhere in the mouth, then hold the mouth shut for 3 seconds.