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%.)

http://www.nytimes.com/2011/03/04/nyregion/04remedial.html

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:

http://www.latimes.com/news/local/la-me-0719-san-jose-online-20130719,0,4160941.story

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:

WhyZHasMean0Stdev1.pdf

(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?