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.


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?


Remedial Math at CUNY (NYTimes, 2011)

Here's a clear-eyed and concise article from the New York Times back in 2011, "CUNY Adjusts Amid Tide of Remedial Students", regarding remedial math classes at CUNY (where I work), mostly focusing on LaGuardia Community College (a different school than my own). Similar information to stuff we know from elsewhere, but I didn't note it at the time, and I wanted to document it here. Some highlights:
  • Nationally, about 65% of incoming community college students need some form of remedial education (2:1 ratio of math to reading). 
  • At CUNY, about 75% of students need some remediation.
  • In NYS, fewer than 50% of graduating high school students are ready for college or careers.
  • In NYC, the proportion of prepared high school graduates is only 23%.
  • At LaGuardia, 40% of all math classes taught are remedial.
  • Cost of remediation at CUNY doubled in the last 10 years to $33 million.
  • About 25% of CUNY community college freshman graduate with a degree after 6 years. (Nationwide it's about 35%.)


San Jose State Suspends Udacity Experiment

News this weekend that San Jose State in California has suspended its experiment with Udacity offering low-level courses for pay and college credit and requirements:


Key point: "Initial findings suggest that students in Udacity courses performed poorly compared with students in traditional classes." Note that this is broadly in line with the prediction I made here several weeks ago, in the post titled Online Remedial Courses Considered Harmful, something that I considered to be a fairly easy and obvious call. I asked the question, "We'll see how quickly MOOCs such as UDacity, and those partnering, paying, and linking their reputation with them, re-learn this lesson", and I'd have to say that this turnaround was faster than I would have guessed at San Jose State. Perhaps they will agree with the earlier experiment at the Philadelphia school where it was concluded, "The failure rates were so high that it seemed almost unethical to offer the option" (see link to my earlier post above).

The last paragraph of today's news story reiterates my own views, which I've written about here on numerous occasions: "Educators elsewhere have said the purely online courses aren't a good fit for remedial students who may lack the self-discipline, motivation and even technical savvy to pass the classes. They say these students may benefit from more one-on-one attention from instructors."

A few other points: "Preliminary results from a spring pilot project found student pass rates of 20% to 44% in remedial math, college-level algebra and elementary statistics courses." Now, it would be much better if this success rate were broken down individually for each of these several classes. I might guess that the 20% success rate is specifically for the remedial math course? That does seem marginally lower than most remedial courses where the success rate seems to be around one-quarter or one-third.

Also, the article says, "In a somewhat more promising outcome, 83% of students completed the classes." This seems unsurprising, given that students are paying $150 out-of-pocket for the course. This completion (but mostly failing) rate is about in line with the remedial courses that I teach, where students are similarly paying, meeting an absolute requirement by the  college, and have no real academic penalty for failing (the course grade does not affect GPA, for example).

Perhaps charitably we might say that the $150 expense level is lower than standard college teaching costs, and perhaps someone might think it's a reasonable return on investment, even granted a lower success rate (although maybe not when accounting for student time spent). And we might also be suspicious of (a) whether this is the actual Udacity expense, or if they're operating at a loss to establish the market, and (b) the quality of the assessment at the end, when there's a clear incentive to make it easy to pass and the Udacity statistics final I've seen in the past was almost comically trivial.

Supposedly this suspension is for re-tooling and analysis of possible improvements. "The courses will be offered again next spring, [San Jose State Provost Ellen Junn] said." We shall see.


Proof of Approximating Radicals to the Closest Integer

The in-house textbook that my college uses for basic algebra classes does an interesting thing -- as part of the introduction to radicals, it goes through approximating a whole-number radical by comparing it to the nearest perfect squares. An example from the book:
Example 2: √3000 is closest to which integer?

Solution:... [after some preliminary estimates] Try between 50 and 60, (55)2 = 3025, still a bit too high. Try (54)2 = 2916, now a little too low. Thus √3000 is between 54 and 55, but closer to 55 since 3025 is closer to 3000 than is 2916.
So I think we all agree that in a case like this, the radical is clearly between the two integers indicated (since the radical function is monotonic). But the additional step of saying which of the two it's closer to is not done in all textbooks. Here's another example (not our school's textbook). Let's clearly state the claim being made here:
Claim: If x is closest to n2, then √x is closest to n. 
Above, "closest" means the minimum distance from x to any n ∈ ℕ. This claim gave me a squirrelly feeling for some time, and with good reason; it isn't true for arbitrary x ∈ ℝ.
Counter-example: Consider x = 12.4. It's closest to the perfect square 32 (distance 3.4 from 9, versus 3.6 from 16). But the square root is actually closest to the integer 4 (√12.4 ≈ 3.52).
Now, let's characterize the kinds of numbers for which the claim in question won't work. For some integer n, take the cutoff between it and its successor, n+1/2 (i.e., the average of n and n+1). Any x below this value is closer to n, while any x above it is closer to n+1. Under the squaring operation, this cutoff gets mapped to the square-of-the-average (n+1/2)2 = n2+n+1/4.

On the other hand, consider the cutoff  between the squares of the integers in question. Any x below their average is closer to n2, while any x above the average is closer to (n+1)2. This average-of-the-squares is ((n)2+(n+1)2)/2 = (n2+n2+2n+1)/2 = (2n2+2n+1)/2 = n2+n+1/2.

So you can see that there's a gap between these two cutoffs, and in fact it's exactly 1/4 in all cases, no matter what the value of n. If you pick x in the range n2+n+1/4 < x < n2+n+1/2, then x will be closer to its ceiling of n+1, but x itself will be closer to its floor-square of n2. Specifically, the problem cases for x are anything a bit more than the product of two consecutive integers (also called a pronic or oblong number), exceeding n(n+1) = n2+n by a value of between 1/4 and 1/2. Since n2+n is itself an integer (ℕ closed under add/multiply), we see that any x in violation of the claim must be strictly between two consecutive integers, and thus cannot itself be in ℕ.

In conclusion: While the claim in question is not true for all real numbers, it is a trick that does happen to work for all whole-numbered values of x. How important is that? Personally, I'm pretty uncomfortable with giving our students an unverified procedure which can leave them thinking that it works for any number under a radical, when in fact that's not the case at all.


Why Z-Scores Have Mean 0, Standard Deviation 1

This article is aimed at introductory statistics students.

Statistics, as I often say, is a "space age" branch of math --many of the key procedures like student's t-distribution weren't developed until the 20th century (and thus helped launch the revolution in science, technology, and medicine). While statistics are really critical to understanding modern society, it's somewhat unfortunate that they're built on a very high edifice of prior math work -- in the introductory stats class we're constantly "stealing" some ideas from calculus, trigonometry, measure theory, etc., without being explicit about it (the students having neither the time nor background to understand them).

One of the first areas where this pops up in my classes is the notion of z-scores: taking a data set and standardizing by means of z = (x − μ)/σ. The whole point of this, of course, is to convert the data set to a new one with mean zero and standard deviation (stdev) one -- but again, unfortunately, the majority of our students have neither the knowledge of linear transformations nor algebraic proofs to see why this is the case. Our textbook has a numerical example, but in the interest of time, my students just wind up taking this on faith (bolstered, I hope, by a single graphical check-in).

Well, for the first time in almost a decade of teaching this class at my current college, I had a student come into my office this week and express discomfort with the fact that he didn't fully understand why that was the case, and if we'd really properly established that fact. Of course, I'd say this is the very best question that a student could ask at this juncture, and really gets at the heart of confirmation and proof that should be central to any math class. (Interesting tidbit -- the student in question is a History major, not part of any STEM or medical/biology program required to take the class.)

So I hunted around online for a couple minutes for an explanation, but I couldn't find anything really pitched at the expected level of my students (requirements: a fully worked out numerical example, graphical illustration without having heard of shift/stretches before, algebraic proof without first knowing that summations distribute across terms, etc.) Instead, I took some time the next day and wrote up an article myself to send to the student, which you can see linked below. Hopefully this careful and detailed treatment helps in some other cases when the question pops up again:

(Edited: Jan-9, 2015).


Institutionalized Score Mangling

For some reason, there's been a bunch of stories of schools secretly boosting near-failing grades recently. A few that come to mind:

  1. Just this weekend -- Hempstead High School on Long Island (somewhat near me) has a scandal of regular boosting failing 63 and 64 scores to passing 65's in any class from grades 6-12. Apparently this has been done for some number of decades, and the Deputy Superintendent defends it as customary at their school and others (although it was done in secret and not any documented policy). Other schools nearby deny that they engage in the same practice.
  2. Early last month, an Indian student attending Cornell University accessed and mined the data from the Indian national high school exams from the last year, and found that the scores being reported were very clearly manipulated in some secret way, as there were irregular gaps in the achieved scores across all subject areas. In particular -- none of the scores 32, 33, or 34 were achieved by any student for any subject in the entire country, whereas 35 is the minimum to pass.
  3. Less publicized (but perhaps more dramatic) is the fact that New York State Regents Examinations are in some sense getting easier, as the high school system brags about increased graduation rates at the same time as their graduates needing remedial instruction in college reaches around 80%. Someone who really ought to know told me that the scores on the exams are effectively mangled by administrators in Albany, i.e., a 45% raw performance is reported as a passing scaled score of "70" and so forth.

All of this certainly seems really bad to me in a first-pass "smell test" of credibility. It just seems like any kind of secret score-mangling is a foul wind that carries with it lack of transparency, disbelief in results, corruption, etc.  Interestingly, a great many commentators at Slashdot (around the Indian story) said things like "this is done everywhere, if you don't understand it then you don't know anything about teaching", which is false in my experience. But apparently the motivation is frequently to avoid conflict and time spent around complaints over barely-failing scores. Some other institutional strategies I've seen or heard about to deal with this issue:
  • Those who miss passing by 5% get to immediately take a re-test. I haven't seen this, but I've heard it said of other universities.
  • Those who miss passing by 5% get a one-week refresher seminar, and can then re-test on the final. A somewhat more subtle version of the preceding which is used where I teach at CUNY for math remediation.
  • Keeping both scores and the passing criteria itself secret -- reporting only pass-or-fail results for the test. This was done in the past at my college, allegedly to forestall complaints over scores. It's pretty much my least favorite option, because it just made everyone involved confused and upset over the secret criteria and unknown scores.

Now, I'm always in favor of maximal transparency, honesty, and confidence in any kind of process like this. But in some cases I've found myself to be a lone voice for this principle. Is this kind of secret score-mangling an acceptable social massaging of high-stakes testing, or is it the harbinger of corruption and non-confidence in our institutions? Do we even have any choice in the matter anymore, as educators or citizens?


The War on Structure

Here is a partial list of subjects that have been seemingly expunged from primary and high school in the fairly short time since I was in those institutions:
  1. Phonics
  2. Multiplication tables
  3. Algorithms for addition, multiplication, long division, etc.
  4. Grammar
  5. Logic
What do all of these topics (and more) have in common? They are all structure-oriented. They require attention and focus to a fairly small number of rules about the proper order and organization of written symbols. The are all detail-oriented. To the extent that American education-school philosophy is to disparage such issues under a "holistic" approach, I reject that as terribly flawed and respond that, "you need both!".

It's particularly aggravating that several of the issues cut to the very heart of why our writing systems in numbers and words are originally designed as they were: to support simple adding, multiplying, and division procedures by hand; to easily convert spoken sounds to writing by way of phonics. Lose sight of that, and you lose the very essence of those systems of writing.

So doesn't this explain why remedial algebra is the single-most devastating course in the university curriculum, preventing about half of all community college students from ever graduating? After all, it's now the first time in their academic career that students are finally forced, inescapably, it attend to the detail and structure of things. And they've been set up for failure; when our students cannot actually parse the structure of a sentence (don't know what a verb is, what clause connects to what, how to diagram a sentence, etc.), then it's impossible for them to translate word/application problems to math (to say nothing of actually solving them). And it's very sad and heart-wrenching to watch.

What is particularly galling about all this is that this has occurred precisely at the same time as the world around has become more driven by machines, computers, technology, and an increasingly technocratic government structure. All of our young people carry computing devices at all times as a very intimate part of their lifestyle, but their understanding is at about the same level as a cargo-cult. This is why "STEM" academic careers are held out as some kind of bizarre alien life-form that the normals cannot hope or imagine crossing into. To the extent that we have removed the capacity to understand structure from our students, we are making them unavoidably victims of the highly technocratic society that controls their lives, without any hope of understanding it.

Obviously this painful disconnect between prior classes and the introductory algebra course cannot last -- and all signs are that, long term, the algebra class will likely be removed as a requirement even for a college degree (my prediction). And thus a mile-high iron wall will be put in place between the unlearned masses and the elite who are educated in the "real deal" of structure, mathematics, language, and computer skills.


SMBC Today


Thanks, SMBC!


Online Remedial Courses Considered Harmful

Online remedial courses are inherently absurd. Even though this is UDacity's (for example) great-white-hope moment, as it has begun offering elementary algebra courses for college acceptance in California (link), it practically beggars the mind that this will help the crushing wave of need that students in such courses evidence.

The reason why is that everything that online courses do well is precisely the opposite of what remedial students need. We know that online courses require a higher level of discipline, dedication, and self-starter initiative than in-person courses do. Online courses are inherently tougher to follow than live courses. You also need a certain technical proficiency just to interface with the platform (and occasionally troubleshoot problems). This is all well and good for high-functioning academic-types who fundamentally love to learn on their own.

But our remedial students have none of that. One of the first overwhelming problems is that they don't have self-discipline in schedule or study habits -- frequently helping with this is itself part of the remedial math course. And they don't like the subject; surveys routinely show an overwhelming and long-seated hatred for the discipline; often a large proportion of a remedial class doesn't show up for the very first class. Minor technical problems will routinely frustrate them and throw them off completely.

Frankly, what the remedial student needs is clear -- if we were serious about getting these students educated, then they would need more individual, one-on-one interaction to address their deep level of need (not less). They need a personal touch to get them over their often pathological resistance for the technical subject matter. They need personal tutors -- but the cultural structure is not one that is interested in paying for that.

Here is a quote, based on the reform experience at a community college near Philadelphia, as they focused effort on the classes with the highest failure rates throughout their college, and in many cases improved their statistics by as much as half (based on interventions such monitoring no-shows on the first day, requiring early evidence of class participation, academic probation communication procedures, etc.):
In some cases, Hayden said, the college's analysis has led officials to believe that some courses were being offered in inappropriate formats. For instance, several of the highest failure rates were in online developmental courses (around 60 percent) -- and various reforms didn't budge those numbers. So the college has ended online remedial education. "The failure rates were so high that it seemed almost unethical to offer the option," Hayden said. (link)

This quote inspired me to write this post tonight. "Unethical to offer the option" (for online remedial education) seems about right. We'll see how quickly MOOCs such as UDacity, and those partnering, paying, and linking their reputation with them, re-learn this lesson.


Facebook Removes Downloads of Your Posts

This post isn't exactly about math, but it is technical in nature, so I figured I'd get it out there. As part of my regular data-backup process, I routinely download my information archives from whatever online presences I can, such as Facebook (which I've been on since early 2010), Google Blogger (this blog you're reading right now), etc. Obviously on Facebook the thing that I'm most interested in is what I actually write, which are usually called "wall posts" (as opposed to photos or media, which I retain locally anyway). Once in a while I've found it very useful to pull up the downloaded posts file and search it for some particular bit of info, contact, or date. What I seem to have discovered is that sometime in the last few months, Facebook silently and completely removed our ability to download that "wall posts" information.

This first dawned on me the other day when I used the Facebook "Download Info" process (Gear icon > Account Settings > Download a copy of your Facebook data), and tried to search for a particular post. Well, the normal file was just entirely missing. You can see the difference below in the downloaded archive from March 2013 versus the download from June 2013. The file "wall.html" -- which actually contains all of my posts and is by far the largest data file in the old archive -- is missing from the new archive.

Now, initially I thought this was some kind of temporary glitch. (In the three years that I've been on Facebook, occasionally  the "wall.html" file has mistakenly contained just a few days worth of posts. Or for several months in 2012-2013 the download seemed to just fail completely any time I tried to use it.) But if I now go to the top-level "index.html" of the download, then I find that the reference-link to the wall posts has also been removed there ("Wall" used to be the second link but is now missing; before-and-after shots below):

So at this point I poked around a bit on Facebook's information pages, and landed on the page where they supposedly tell you what information is included in this and other resources ("Accessing Your Facebook Info", which I find at this link). This page currently describes three repositories of information: (1) "Downloaded Info", the archive which I've described above, (2) "Expanded Archive", another download which includes more transaction and login information, and (3) "Activity Log", which is an online-only manipulation of the Facebook timeline (not part of any download). What I see here is that "Your Posts" is now categorized under "Activity Log" (and note that this entry is also out of alphabetical order, possibly evidence of some change, and making it a bit harder to find in the list):

So what this means is that Your Posts, the things you've actually written on Facebook, are no longer included in the "Downloaded Info" which allegedly includes all of your info (and did from at least 2010-2013). The posts are not in the "Expanded Archive", either (I checked to be sure... it has dozens of files including Ad Clicks, Apps, Facial Recognition Data, Poke Data, Relationship Info, etc. ... but no wall posts). The "Activity Log" in which they are now categorized is online-only at Facebook, can't be downloaded, doesn't show all your posts at once, and can't be searched unless you know the date of the post that you're looking for in advance.  (I considered trying the Wayback Machine to find a date when this was altered on the help page, but of course Facebook bricks off any internet crawlers by way of its robots.txt file.)

In short: Facebook seems to have silently locked up everyone's personal posts in their system, with no way to get them out or search them, without any comment or notification of the switch that I can find anywhere online. The "Download Info" process screen itself remains unchanged, so potentially people could keep using it, not knowing that the largest and most fundamental type of data, their posts, has been stripped out of the archive.

Perhaps equally disturbing is how this hunt highlighted for me the fact that Facebook makes it totally impossible to search your own data (in any way other than tedious manual scrolling). I had flatly assumed that any digital entity would have this capability, if perhaps in a difficult-to-find location or UI. But Facebook apparently doesn't let you digitally search your own information in any way, and now they've removed the capacity to archive your information outside their system, too. Perhaps if attention is brought to this matter they might reverse course (as in some past cases) and restore the ability to truly "Download Info" from the largest and most fundamental aspect of your personal account.

Or can you now find any other way to download all the wall posts that you've written on Facebook?

Edit 6/13/13:  About a week now after I first posted this, and a fresh download does include the "wall" posts file. If this was a bug (see comments from Facebook associates, below), then we much appreciate this being resolved and hope the function sticks around in the future, too. Thanks!


Semester-End Teachable Moments

At the end of a semester, math students become extremely eager (or anxious) to know about the details of how their grade will be calculated, what their chances are of a particular grade, and what they need on the final exam in order to achieve a particular grade. In the past I would just fire back the answer to such questions in order to have time for other matters, but now I realize that this may be the most fruitful "teachable moment" of them all. Such questions are, after all, algebra questions, and so any of our remedial-or-above students really should be able to answer them for themselves -- and nowadays I require them to do just that.

It's particularly advantageous, because the customary reaction to the incessant "what good is this math for?" question is usually to introduce application/word problems into a course, but frequently to students this looks even more tangential, abstract, and unearthly than the original math it was meant to demonstrate (particularly for any students lacking background familiarity with the given application area, which is almost guaranteed to be most of them). But here we have a concrete example of intense interest that is being brought up by the students themselves -- and therefore it represents a matchless opportunity to hammer home exactly what the utility of basic arithmetic and algebra is, in a way that is hopefully intense and thus memorable.

Likewise, I used to presume this was trivial for higher-level students and so I'd scribble out the math quickly to not be mutually bored, but then I'd find that even my college algebra and statistics students were stunned by what I was doing. So that's the kind of basic thing that warrants time to make totally ironclad. Two cases that come up in my remedial classes:

(1) The university elementary algebra final exam has 25 questions, and at least a 60% score is required to pass the class (among other requirements; link). Common inquiry: "How many questions do we need right to pass the final?" So my answer is now to write on the board "60% of 25" and ask the students to translate that to math as a word problem, and then compute the decimal multiplication by hand. Some are very rusty, but parts of that process are obviously on the final itself; so, good review.

(2) In my classes, I usually compute the weighted total for the overall grade by taking 15% of a quiz average, 50% in-class test average, and 35% of the final exam. Common question near the end of the course: "What do I need on the final exam to get a B grade?" (or whatever). So my response now is to say, "Well you're asking me an algebra question, and you should be able to solve that yourself", writing on the board "W = 15%Q + 50%T + 65%F", and then assisting them in substituting the decimals, desired W, and known Q and T values. Then I tell them to apply the basic solving process (likely with a calculator), and once they know F, then they have the answer to their question.

In fact, I feel that this latter case is such a golden opportunity that I've modified my class procedures in at least two small ways to highlight it. (a) I used to have a policy where I might possibly replace one test score with the final exam if it was significantly higher; but so as to make the test average definitely known prior to the final, now I just drop one test score outright for everyone (which is nicely handled by our Blackboard grade center). (b) I actually spend a half-hour block on this very topic in the early part of the course, as a prime application of the basic algebraic solving process; I get some resistance due to the longer-seeming equation (compared to simpler drill exercises), but -- you'll get resistance anyway for any application problems, and when it truly comes up at the end of the semester suddenly it seems a lot more relevant.

So in summary -- I used to think that these questions were trivial and uninteresting, but it turns out they're very much not. Most of my students, either in an algebra class or thereafter, can't recognize such inquiries as a basic application of algebra that they should be able to solve. Instead of firing off the answers as an aside so as to cover more literal coursework, I now take the opportunity to leverage that intense interest into making it abundantly clear what kinds of important questions can be translated to math and solved by algebra.

Can you think of other inquiries that you commonly answer in grading or course procedures, that are really opportunities for basic math reviews in disguise?


Punished for True Math

Reading some Mathoverflow the other day, I ran into some truly blood-boiling recollections in a discussion of "Examples of common false beliefs in mathematics" (mostly at the research level, but the comments diverged), such as that "Many students believe that 1 plus the product of the first n primes is always a prime number". Recollections such as these (link):
When I was 11 y.o. I was screamed at by a teacher and thrown out of class for pointing this out when he claimed the false belief stated (it wasn't class material, but the teacher wanted to show he was smart). I found the counterexample later at home. I didn't let the matter drop either... I knew I was right and he was wrong, and really had a major fallout with that math teacher and the school; and flunked math that year. – Daniel Moskovich May 5 2010 at 1:19   

@Daniel: Sorry to hear that. When my daughter Meena was the same age (11), her teacher asserted that 0.999... was not equal to 1. Meena supplied one or two proofs that they were equal, but her teacher would not budge. Maybe this is another example of a common false belief. – Ravi Boppana May 5 2010 at 2:59   

@Daniel: I've heard a worse story. A college instructor claimed in Number Theory class that there are only finitely many primes. When confronted by a student, her reply was: "If you think there are infinitely many, write them all down". She was on tenure track, but need I add, didn't get tenure. – Victor Protsak May 5 2010 at 5:38   

This false belief leads to a proof of the Twin Prime conjecture: For every n, (p1p2⋯pn−1,p1p2⋯pn+1) are twin primes, right? – David Speyer May 6 2010 at 15:50   

Daniel, about the same age, I was asked to leave class for claiming that pi is not 22/7. The math teacher said that 3.14 is an approximation and while some people falsly believe that pi=3.14 but the true answer is 22/7. Years later an Israeli newspaper published a story about a person who can memorize the first 2000 digits of pi and the article contained the first 200 digits. A week later the newspaper published a correction: "Some of our readers pointed out that pi=22/7". Then the "corrected" (periodic) 200 digits were included. Memorizing digits of pi is a whole different matter if pi=22/7. – Gil Kalai May 11 2010 at 5:45
I guess I had the good luck to not ever have such a completely horrible math instructor, because I think any of these cases would have made me completely lose my mind. (As an aside, I will say that it's routine in my classes that I'll have to disabuse people of the idea that pi = 22/7... perhaps this is more common in Israel, as Prof. Kalai above is, and many of my students are from.) Have you ever seen someone punished, yelled at, or thrown out of class for actually expressing true basic math facts?


Parity of Zero

Did you know: As much as half the population doesn't know that zero is an even number? And that this can cause problems in cases like gas-rationing based on license plates' last digit (as happened here in NYC last fall after Hurricane Sandy).



Village Voice on CUNY

Here's a very nice cover story from NYC's alternative newspaper, the Village Voice, basically on the subject of my math teaching job at a CUNY community college (and more generally, community colleges across the country):


Some highlights:
  • Enrollment at CUNY community colleges is up 33% in the past 5 years
  • CUNY has seen a 40% drop in per-student funding from the state in the last 20 years.
  • 80% of NYC public school grads who enroll in CUNY need remedial-level instruction
  • Just 14% of public school grads pass the CUNY algebra placement exam
  • Only 20% of remedially-placed students have advanced to a for-credit class 2 years later
  • Only 1 in 4 remedially-placed earn any degree after 6 years.

Regarding NYC public high school statistics: "The numbers are 'better'—there are more graduates—and yet, in an endless loop of absurdity, these students get to college only to be told they haven't finished high school."

Regarding NYC's Harry Truman High School: "Truman currently boasts an A grade from the city. Yet only 10 percent of its graduates are able to enter CUNY without remediation."

Regarding a new pre-matriculation START program which takes small classes and gives detailed basic math instruction: "That process sounds an awful lot like what we used to think of as 'teaching.'"


Everyday LCMs

Here's an exercise that I'm planning to use in my remedial arithmetic class in the near future. The question is: For each of the following number ranges, state (i) the LCM (least common multiple), (ii) some everyday examples that use that LCM, and (iii) an explanation of why that number is convenient.

(a) {1, 2, 3}.
LCM is 6. Examples: Six-pack of soda, beer, donuts, etc. Convenient because you can divide them evenly whether you have one, two, or three people.

(b) {1, 2, 3, 4}.
LCM is 12 (a dozen). Examples: 12-pack of beer, dozen eggs, hours on a clock, etc. Convenient because you can divide them evenly among either one, two, three, or four people (or dishes or periods).

(c) {1, 2, 3, 4, 5}.
LCM is 60. (And see next exercise.)

(d) {1, 2, 3, 4, 5, 6}
LCM is also 60. Examples: 60 seconds in a minute, 60 minutes in an hour, 360 degrees in a circle (6×60), etc. Convenient because you can divide them evenly into one, two, three, four, five, or six periods, as desired. (See also: Babylonian numerals.)


Explaining Proportions

A common basic math exercise is to set up and solve a proportion (equivalence of two ratios, i.e., fractions), often in the context of some word problem. The funny thing I recently discovered (updating lecture notes for the spring term) is how there's usually a complete absence of explanation on why you're doing this, or justification for why it makes sense to do so. In fact, I flat-out couldn't find any explanation for the procedure in any of the resources that I have available to me at the moment. Here are some examples:
Writing proportions is a powerful tool for solving problems in almost every field, including business, chemistry, biology, health sciences, and engineering, as well as in daily life. Given a specified ratio (or rate) of two quantities, a proportion can be used to determine an unknown quantity. [Elayn Martin-Gay, Prealgebra & Introductory Algebra, 3rd Edition]

We can use proportions to solve applied problems by expressing a ratio in two ways, as shown below. For example, suppose that it takes 8 gal of gas to drive for 120 mi, and we want to determine how much will be required to drive for 550 mi. If we assume that the car uses gas at the same rate throughout the trip, the ratios are the same, and we can write a proportion. [Marvin Bittinger, Intermediate Algebra, 9th Edition]

Proportions are typically used when you want to solve for an unknown. Let's look back to our car example. In the last section we found we could drive 120 miles on 4 gallons of gas. We want to find out how many miles we could drive on 10 gallons of gas. This information is displayed in the table below. The value we want to determine is represented by an x in the table above. We can find this value by setting up a proportion. This is shown on the right. [Syracuse University Mathematics Tutorial, retrieved 2/13/13 -- the first of several Google searches I looked at]
In each case (and there were numerous others), that is the entirety of the explanation of why you'd want to set things up in a proportion. To my mind, each of them are extremely sketchy. And like my own lecture notes up until recently, they have a tendency to start off with a sample problem first; they say something like, "take this and set up a proportion like so", then go through the solving steps. But I've become highly sensitized to that fact that if I can't start out with a simple explanation as to why the mechanics of a certain procedure make sense (in this case, setting up the equation a certain way), then that's an indication that I don't fully understand what I need to answer questions on the subject, and need to rectify the situation.

Here's how I put it in my brief lecture notes nowadays -- Problems involving a constant rate can be set up as a proportion. For example: If 2 boxes of cereal cost $10, then how much do 6 boxes cost? One way of looking at it is this: The cost of one box is 10/2 = 5 dollars, so the cost of six boxes must be 5∙6 = 30 dollars. But another way of putting it is that, if we turn both of these into divisions, then the result is the same; i.e., 10/2 = 5 and 30/6 = 5. Therefore we could set up the original problem as a proportion, being careful to line up like units, e.g.: 10/2 = x/6 [dollars/boxes] → 60 = 2x [cross-multiply] → 30 = x [divide by 2]. And again we see that the total cost is $30. 

Observations: The "constant rate" here is specifically the price point of $5 per box of cereal, which is a reasonable and common-sense assumption we're making in the solution, that the price-per-box is the same for the two transactions (barring some kind of bulk discount, say). But note that the proportion method is not the only way we could solve this problem, and in fact it has some very notable disadvantages: (a) we don't ever see the actual "constant rate" itself in the calculation ($5 per box in the example above), and (b) in the intermediary step it produces a much larger number than anything that existed in the original problem (the 60).

So for me, I frankly find the proportions method pretty unintuitive, and in my own work I rarely turn to it as a first choice in solving strategy. Particularly if I have a calculator or computer available, then I find it easier to do the divide-first-and-multiply-second method, as given initially above (and this strikes my students as far more understandable, i.e., actually seeing the constant rate price-point). Or alternatively, you could divide the box numbers first (6/2 = 3), and then intuit that the dollar amounts would have to be increased by the same factor, i.e., a product of 10∙3 = 30 in the given example (again, dealing with smaller and mentally-manipulable numbers along the way).

That said, the proportions method does indeed have some specific advantages. Ones I can think of immediately are: (a) it encapsulates the entire problem into a neat, concise, and attractively symmetric piece of equation writing; and (b) if you're working by hand, and there's going to be decimals in the final answer, then the decimal work is minimized and only appears in the very last division step (as opposed to dealing with it twice, in my divide-and-then-multiply method). This latter feature is of course devalued the more that cheap computation devices become ubiquitous, and is similar in that regard to a lot of other methods which trade off a large intermediary value so as to delay working with cranky divisions, fractions, and decimals (for example: the "calculating formula" for standard deviation, etc.). Perhaps, then, the proportions method is already something of a legacy dinosaur in that regard; I know that for my own work, I find more utility in actually seeing the constant rate I'm dealing with itself identified in the middle of the workflow.

Can you think of any other advantages to the proportions method for these types of elementary problems?


Graphing Mistake

You would not believe how often I see this mistake in a basic algebra class: