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.

Remedial Recommendations

So granted that the last blog post here was thinking about all the reasons why remedial college math classes in algebra are so tough (for students and teachers), I'm pleased to say that 3 weeks into this almost-all-algebra-remediation semester, things are definitely going the best for me in my decade-long teaching career. Here are some things that I'd say have had a clear, beneficial impact on my current semester:

1. Shorter class times. In the prior 8 years at CUNY, I have always had 2-hour long algebra classes, meeting twice per week (partly because I've mostly been part-time, teaching at night). For the first time, my classes are 1 hour long, meeting four times per week. This clearly works better for the endurance and attention available to the students. We're in, focused on one narrow topic, and finished before everyone gets too tired & cranky. This has been a pleasant and great surprise to me; definitely the biggest-impact of the semester. (Not that it would work for night students or part-time teachers, where the travel burden would be inefficient.)
2. Starter exercise pack. I expect students to have a copy of the textbook and be practicing exercises from it regularly, but very few do so (as noted last time). One problem is that students don't immediately have the textbook in the first week, as they're saving up, looking for a used copy, or having an old edition shipped online (as I explicitly encourage). This gap then sets the habit of them skipping my "practice" advice. What I did this semester is to copy a packet of "starter exercises" from the book, covering the first few weeks, with answers, so I can hand it out the very first day and explicitly point to what they can practice that very night. I've found this to be quite helpful in setting the precedent for regular practice; I've had more students than usual come to class with questions about problems, and this sets up a virtuous cycle of other students seeing it as expected behavior.
3. Tailored, trickier problems. In the past my routine was to lecture, then turn to the book and practice problems from the text with students. Partly due to the relatively small number of problems in our in-house text, about a year ago I went through the course and wrote custom exercises for every in-class topic. Generally I wrote these to be tougher than standard starting problems, and every single problem from the first integrates common stumbling blocks (negative numbers, one and zero coefficients, etc.). Among the advantages here are that (a) we're not totally boring the students who have seen the material before, (b) we're always dealing with problems similar to test items, and (c) we're spending time "triaging" all the trouble spots. These exercises are working very, very well for me. Textbooks usually start problems sets with very rudimentary "common sense" examples to get started, but granted the limited class time we have available, I would highly recommend skipping those low-level problems and immediately start working with at least mid-level exercises for every topic.
4. Ending with flex-time. There's probably a better name for this, but what I mean is: I end every class with a few exercises (one word problem or two pure algebra) and say, "This is the last thing we'll do today; show me the answers and you're free to go" (this being maybe 20-30 minutes before the end of the period). Then I circulate and check answers, give corrections or hints, etc. The better students push themselves to finish quickly and happily leave (thereby avoiding bored-irritated-distracted people in the room); the mid-level students get more time for feedback and cleaning up trouble areas (and also with less embarrassment or defensiveness from a roomful of people listening in); and the very weakest student gets some personal one-on-one time with me. I have to remember to give any homework or next-class directions prior to this point, of course. This was a great, semi-accidental find on my part. (And the flex-time mechanism works even better with 1-hour classes, since it happens twice as often as it would for my night classes.)
5. Surrendering on mobile devices. My remedial students commonly come in with smartphones running and earbuds in both ears throughout the entire class. Considering that my higher-level students practically never do this, in the past I felt it was my responsibility to model proper collegiate discipline and be very hardcore about having people shut off their devices at all times. Frankly, the resistance to this could be so fierce that it blew up into security issues on me a few times. So as stupid as it seems, this semester I've been letting people sit in class using phones and with earbuds in without immediately confronting them (unless they were directly interacting with me at the time). It seems to take some of the pressure off, and in some cases for students who are legitimately already on top of the information, it may reduce the boredom-irritation factor. On the one hand, it's dumb as all hell, but on the other hand I don't really have the tools to fix that problem on top of everything else.
6. Entering with a sense of joy. Not really new, but I try to remember to come into class with an upbeat attitude and thinking about how great it is to share the topic of the day with whomever's willing to listen. Obviously from the name of this blog you can tell that's not actually my most natural personality. But if I can, I try to shake as much crankiness off before stepping into the room. As the simply amazing film Monsieur Lazhar put it, "A classroom is no place for despair". That does seem to make things run more productively and with less general combativeness than some times in the past.

Do you have any tactics and strategies that work particularly well in the context of remedial college classes?

Reasons Remedial is Rough

Today is the start of my fall semester at CUNY, and my schedule is almost entirely teaching remedial algebra courses. (You know, the toughest course in the curriculum, that generally less than half students anywhere pass.) So as I think about introducing myself to my students this week, and trying to earn their trust that what I'm asking them to do is truly necessary and worthwhile, one question that sometimes pops up is, "Why do so many students fail at remedial algebra?"

The answer is that there's lots of reasons, and usually more than one for any given student. The philosopher Michel Foucault would call this state being "overdetermined" -- there's no single root cause we can ferret out that would fix everything. Without consulting hard data sources, here's a list of the top reasons that I see from my personal experience:

1. Lack of math skills from high school. Many students simply don't have the requisite skills from high school, or really junior high school (algebra), or in many cases even elementary school (times tables, long division, estimations, converting decimals to percent, etc.). This deep level of deficit is like sand in the engine when trying to learn new math.
2. Lack of language skills from high school. What's dawned on me in the last year or so, in the context of applied word problems, is that many students may actually be worse at English than they are at the basic math. Grammar isn't taught anymore, so students can't parse a sentence in detail, can't identify the noun or verb in a sentence, and so forth. This cripples learning the structure of any new language, algebra included.
3. Lack of logic skills from high school. No one teaches basic logic, so students can't automatically parse If/Then, And, Or, Not statements, which form critical parts of our mathematical presentations and procedures.
4. Lack of study skills or discipline. Almost none of my students do any of the expected homework from our textbook. (On the one hand, I don't collect or award points for homework, so you might say this is unsurprising; but my judgement is that the amount of practice students need greatly exceeds the amount of time I have to mark or assess it.)
5. Lack of time to study. Certainly most of our community college students are holding jobs, or caring for children, or supporting parents or other family members. The financial aid system actually requires a full-time course load for benefits; combine that with a full-time job -- really, the equivalent of two 40-hour jobs at once -- and you get a very, very challenging situation. (Side note: In our lowest-level arithmetic classes, I find that work hours are positively correlated with success, but not so in algebra or other classes.)
6. Untreated learning disabilities. This would include things like dyslexia, dyscalculia, ADD, etc. All I can do is speculate as to what proportion of remedial students would exhibit such problems if we instituted comprehensive screening. But I suspect that it's quite high. When students are routinely mixing or dropping written symbols, then disaster will result. Unlike other languages, concise math syntax has no redundancies to enable the "you know what I mean" safety net.
7. Emotional problems or contempt for the class. I put this last, because it's probably the least common item in my list -- but common enough that it shows up in one or two students in any remedial classroom; and a single such student can irrevocably damage the learning environment for the whole class. Some students who actually know some algebra start the course thinking that it's beneath them, and become regularly combative over anything I ask them to do, sabotaging their own learning and that of others. It's pretty self-destructive, and the pass rate for these kinds of "know-it-all" students seems to be about 50/50.

If you've taught similar courses, does that line up with your experiences? Have I left anything obvious out of the list?