Yes, And...

This winter session I'll be teaching College Algebra, which I rarely do (once a year or less). Students are definitely sharper than in remedial algebra classes, which is a delight, but they're also more honed into "playing the game" of grades for their own sake. That is to say: I get more incessant "will this be on the test?" cries than I do in other classes.

One thing I'm doing new this semester is to give open-response tests (not multiple-choice), so that I have the option on grading issues of correct writing format and the like. Or really anything else that comes up as an issue. (As a counter-balance, I'll be giving tests with fewer but more complex questions.)

But in conjunction with that, I'm mentally prepping to to try to answer those inquiries with a "Yes, but more importantly..." response. Like: Q: "Will our writing be graded on the test?" A: "Yes, but more importantly, that's how you communicate math to other people, and it's what you should be prepared to read in a math book on your own." Or Q: "Will graphs be on the test?" A: "Yes, but more importantly, it's the faster way to estimate or double-check any answer and avoid mistakes." So it gets the somewhat irritating question out of the way in the first word, and more importantly, it explains why that's really of secondary importance at best. Kind of like in improvisational comedy where you're supposed to respond to any creativity on your partner's part with "Yes, and..." ("and" being logically equivalent to "but", of course).

Do you have any clever ways of dealing with cries of "Will this be on the test?

Automatic Drills

I think we all know that certain skills need "automaticity", that is, such thorough learning and practice that they become automatic, unconscious, instantaneous. For example: Recognizing the letters of the alphabet, reading standard vocabulary words, times tables, negative numbers, etc. If you don't have those basic things working unconsciously, then you inevitably get distracted and make mistakes trying to attend to larger, more full-featured problems.

But I've been thinking lately that the expectation and need for these most fundamental skills is often not communicated to our students; in the era that frowns on structure and drills for automatic knowledge, many of our students have never seen such a requirement assessed directly anywhere. Of course, I'm thinking of the times-tables drills that people my age did in the 2nd or 3rd grade, and nowadays may possibly be done in the 8th grade or high school by the more exceptional and dedicated teachers (so I hear).

Might it be the case that in any class, there's at least one specific skill that is expected to become automatic, even if many of us overlook communicating and drilling on that? For example, it's occurred to me that we might expect the following regular speed drills to take place:

1. In early grammar school -- Times tables.
2. In late grammar school/college remedial arithmetic -- Negative number operations.
3. In junior high school/college remedial algebra -- Matching a slope-intercept equation to the graph of its line.
4. In statistics -- Estimating the area under part of a normal curve, or interpreting confidence intervals and P-values. (?)

I don't know, that last one perhaps I'm reaching too much for a uniform rule throughout all my classes. But I am starting to consider a timed test for those automatic prerequisites on the first day of my classes, and repeated timed tests on the "new" automatic skill in each class.

What do you think? Have you used timed drills to communicate the expectation of automatic skills? And for anything other than times tables?

Branching Decisions in Algebra

Yesterday (as I write this) was a hard day for some students in my several remedial algebra classes. The lesson wasn't a long one (I was done lecturing about 40 minutes into the hour on the two topics), but about 1/3 to 1/2 of the class seemed to run into a brick wall in trying the final exercises on their own. The subject was basic factoring of polynomials, and after two days on the subject I had this combined procedure written on the board:

Factor completely process:
(1) Factor GCF if possible.
(2) Try DOS for binomial, or SQ for trinomial.

All of those terms had been defined previously and quizzed verbally many times on prior days (GCF = greatest common factor, DOS = difference of two squares, SQ = simple quadratic, i.e., x^2+bx+c). Now obviously, anyone who had missed the prior day or been significantly late (so as to miss one or more of the 3 core procedures) would be at a disadvantage.

But it appeared to me that the major roadblock was reading and implementing that direction in part (2): that is, following a logic branching procedure, making a decision on what to do next. For some of the more struggling students, I could stand by their desks and say something like: "Now you have two terms. That's a binomial. What should you try now?", and they either couldn't tell me or pick the wrong procedure. (And then one student could only squint and squint at the board and clearly couldn't read what was written there, I guess ever for the semester or any other class they've taken.)

And generally isn't that true for the hardest parts of the basic algebra class? Things like solving general equations (knowing what inverse to apply next for the problem, including exponents or radicals), identifying special products in a multiply exercise (FOIL, DOS, or square of a binomial), or even just following the written directions on any given problem (simplifying vs. solving vs. factoring vs. graphing, etc.). Maybe the weakest students are quasi-okay following directions for a few steps with straight-line flow of control (what do you call that?), but are unable to deal with any conditional branches or decisions along the way. (And of course this would be similar to the other great brick-wall of basic academics, namely computer programming.)

I was talking to colleague recently and said, "I really wish someone had taught our students basic logic at some point". And his response was, "Oh, logic is a very deep subject that is very difficult, I'm not sure how endless truth tables would help". (Oops, I didn't realize that he was a logician by research area.) But I responded: "All I'm looking for is that students can parse an And-Or-Not statement or an If-Else. Like if I write 'If the base is negative, then any odd power results in a negative', many students will make all odd powers negative at the end, by simply ignoring the first part of that statement." And he said, "Oh yes, I've been having the same problem in my classes lately..."

Is this a key part of our problem for students attempting to enter college for the first time, at the level of either algebra or computer programming? That they simply can't make branching decisions when required? (Personally, one change I'll make the future is to write my process as "If binomial try DOS..." so the decision is explicitly before the action, but I know from even a statistics course that I teach that many students still can't follow such a direction.) Is this intrinsic to the student, or is it evidence of high school academics that demanded mindlessness when following directions?

Keep Change Change

Here's another one of these stupid memory devices that I guess some pre-algebra instructors use to get their students to hobble through their class, but then put them on the wrong path later on. It's a reminder specifically for how to subtract a negative number: +9-(-4) = +9+(+4) = 13, or -3-(-6) = -3+(+6) = 3, stuff like that. The "keep change change" mnemonic supposedly gets them to cancel the two juxtaposed negatives (and not the one in the first term).

But like PEMDAS, this sets up a terrible habit, and masks the real meaning to the writing. The actual story is that a negative functions like multiplication, and flows left-to-right the same as we read in English. Yes, students in algebra are routinely stumbling over negatives in general and the subtraction most of all. But when I try to clarify it, usually some student now goes "oh, it's keep-change-change". Then I ask them to simplify an expression with three or more terms in it, like +9-(-4)-(+3), and at that point they have no idea what to do. They don't see that juxtaposed negatives are cancelling out, just like a multiply. The mnemonic that get them through pre-algebra with only two terms at a time was a waste, and has set them up for failure later on.

I've only heard this brought up by students in the last 4 years or so (not before that). Initially I suspected that the mnemonic was specific to where I teach, because the initials happen to be the same as our school. But when I do an online search it does show up in a small number of hits elsewhere -- well: actually just once at algebra-class.com and then once as an answer to a Yahoo question (possibly  those two items might be written by someone that went to our school?).

So my question: Have you ever heard of this "keep change change" nonsense anywhere else? Did you ever hear it before, say 2008?

Are Parentheses Multiplication?

Are parentheses multiplication? My remedial algebra students will pretty universally answer "yes" to this question; I guess they must be taught that explicitly in other courses. I'm pretty damned sure that the answer is "no", and I try to pound it out of them on the first day of the class.

Even professional researchers exploring common mistakes in algebra education are prone to saying "yes" to this question, for example:

Misconceptions: Bracket Usage -- Beginning algebra students tend to be unaware that brackets can be used to symbolize the grouping of two terms (in an additive situation) and as a multiplicative operator [Welder, "Using Common Student Misconceptions in Algebra to Improve Algebra Preparation", slide 7; references Linchevski, 1995; link]

But are parentheses a multiplicative operator? It seems clear that the answer is "no". Now clearly all of the following are multiplications of a and b: ab, (a)b, a(b), (a)(b), etc. But notice that the parentheses make no difference at all in this piece of writing. These are multiplications because of the usage of juxtaposition; any two symbols next to each other, barring some other operator, are connected by multiplication. Obviously, if there were some other written operator like + - / ^,  between the a and b it would be something different; but granted that multiplying is probably the most common operation, we read the absence of a written operator to indicate multiplication.

The chief problem with telling students that parentheses indicate multiplying is that they then routinely get the order-of-operations incorrect. Assuming a standard ordering ([1] inside parentheses, [2] exponents & radicals, [3] multiply & divide, [4] add & subtract), students want to perform multiplying with any factors in parentheses in the first step, before exponents. One of the first and frequently repeated side-questions I ask in my class is, in an exercise like "Evaluate 2+3(5)^2" -- "Yes or no, is there any work to do inside parentheses?" On the first day of algebra, almost the entire class will answer "yes" to this (and want to do a multiply), at which point I explain that the answer is actually "no". If there is no simplifying inside the parentheses, then the first piece of actual work will be to apply the exponent operation. And that's all that parentheses mean. (There is of course a multiplication here -- not because of the parentheses, but because of the juxtaposed 3, and it must take place after the exponent operator.) A majority of the class will pick up on this afterward, but not all -- some proportion of a class will continue to say "yes" and be confused by this particular question throughout the semester. (As another example, some students are prone to evaluate something like "(5)-2 = -10" for this and other reasons.)

Whether a factor is juxtaposed next to something in parentheses or not is irrelevant to the multiplication; parentheses are a separate and distinct issue. What say you? Have you ever said that the parentheses symbols actually mean multiplying?

Quaternion Anniversary

170 years ago today, Sir William Rowan Hamilton had the flash of insight on how to extend two-dimensional complex numbers to cover 3- and 4-dimensional space, in the form of quaternions -- in particular the rather sticky problem of how to make their multiplication work reasonably. This occurred while he was walking with his wife along a canal, to an academy meeting in Dublin, Ireland. And to insure that he didn't forget the insight, he famously took a knife and wrote the formula into the stone of Brougham Bridge as he walked underneath it.

This summer I had the good fortune to visit Dublin, and my partner and I took an afternoon to make the hike and find the plaque commemorating Hamilton's discovery. (It's about a 3-hour round-trip walk outside the city, beside the utterly enchanting Royal Canal. Quicker if you have a car, of course, but we do everything on foot.) Eureka!

You Are Now Entering a Region With a Logarithmic Scale

Armagh Observatory Astropark, Northern Ireland, UK.