Monday, August 25, 2014


Here in New York, it's back-to-school starting next week. Of course, if you're in the teaching profession (or really just know about it), you've probably been doing academic work and preparing for the fall semester throughout the summer and year-round. I'm in the very fortunate position that I'll have a new permanent position at CUNY this fall, so here's how I've spent my August:

I've developed a new website for practicing basic numerical skills that are prerequisite for algebra and other classes like statistics, calculus, and computer programming: I've written a few times in the past about the need for certain skills to be automatic, skills that have been taught but not mastered by most students who arrive in a remedial community-college algebra course, and therefore causes continual disruption and frustration when we're trying to deal with actual algebra concepts (links one, two). Like, for the algebra course itself: times tables, negative numbers, and order-of-operations. Or for a basic statistics course: operations on decimals like rounding, comparing, and converting to percent.

So what you get at the new site for each of these skills is a brief, 5-question quiz for each of these skills. Here's how I designed them:
  • Timed, so that students get a very clear portrayal of what the expectation is for mastery of each of these skills (15, 30, or 60 seconds per quiz). For example: sequential adding and counting on fingers for multiplications will not suffice.
  • Multiple-choice, so the site is usable on a variety of devices, including touch-screen mobile devices. For example: you can stand on a bus and drill yourself on a smartphone by just tapping with your thumb.
  • Randomized, so once you take a quiz you can click "Try Again" and get a new one, and drill yourself multiple times in just a few minutes each day.
  • Javascript, so the quiz runs entirely on your own device once you download the page the first time. The site doesn't require any login, accounts, server submissions, recording of attempts, or any transmitted or collected information whatsoever after you initially view the page.

This is something that I've wanted for a few years now, that no existing website really implemented and consolidated the way I wanted as a reference for students. Thanks to the new full-time track I feel I'm in the position to warrant developing it myself and leveraging it for numerous classes of my own in the future.

Feel free to check it out, and offer any comments and observations. If you feel it might help your incoming students this fall semester, give them the link and maybe it will lift a whole lot of our boats all at the same time. Do you think that will be of assistance?

Monday, August 11, 2014

When Are Parentheses Required for Substitution?

In my remedial algebra classes, on introducing the substitution of numerical values for variables, I've always said that it's safest to perform this substitution inside parentheses, especially for negative numbers. Of course, we all intuit times when that's not strictly necessary. So in this lecture I usually get one of the brighter students asking, "Exactly when is it necessary?". I've found this to be a surprisingly difficult question to answer. After a rather embarrassingly long consideration, here's what I've come up with.

Parentheses are basically required in the following two situations:
  1. Separating juxtaposed signs and numbers, and
  2. Collecting expressions with one operation under a higher-order operation that is not also a grouping symbol.

For situation #1, I'm assuming that we're not ever inserting new operational symbols like × or in cases of juxtaposed multiplication -- just the substituted expression and possibly parentheses. Parentheses are probably only needed for factors after the first one (i.e., after the coefficient).

For situation #2, we're mostly talking about multiplication and exponents, with some lower-order operation in the expression being substituted. Contrast with fraction bars (for division) and radicals, which have grouping built into the symbol, and thus no general requirement for new parentheses.

Here are a few examples of each. For the following, let x = 1, y = –2, z = ab, and w = a + b. Examples of separating juxtaposed signs and numbers:

Examples of collecting expressions with one operation under a higher-order operation:

We can explain the first example immediately above in that the negative sign acts the same as multiplying by –1, and therefore must be collected under the exponentiation operation. However, this does get slightly complicated by the use of the minus sign for both unary negation (i.e., multiplying by –1), and binary subtraction, which have different placements in the order of operations. For example, the following may be taken as a slightly ambiguous case:

Here, in substituting any numerical values at all for x and y, parentheses will definitely be necessary. However, this particular instance doesn't have juxtaposed numerals -- the real reason may be taken to be that without the parentheses, this would read as subtraction (lower order than the initial juxtaposed multiplication).

A few notes on specific cases of substitution:
  • If substituting one variable for another, then parentheses are never needed (the order of operations is clearly identical before and after).
  • If substituting a whole number, then only the situation of juxtaposed numbers after the coefficient can apply. Obviously a whole number has no written sign, and includes no operations to interfere with higher-order interactions.
  • If substituting a negative number, then any of the situations are possible. It does have an attached sign, may need separation from an advance factor (as above), and operates similarly to a multiplication (and thus needing collection under an exponent). 

Note that Wikipedia articles do show use of juxtaposed signs, e.g., 7 + –5 = 2, and discusses possibly superscripting the unary negation in elementary contexts and the computer language APL (link one, two), something that I've also seen on some calculators, in which cases parentheses would not be necessary. However, that's not something I've ever seen in textbooks (either college-level or otherwise), so I take that as nonstandard and not qualifying as well-written algebra. 

What do you think? Have I missed any important cases or examples?