Wednesday, May 25, 2016

Jose Bowen's Tales from Cyberspace

A few weeks ago I went to a CUNY pedagogy conference at Hostos Community College. It featured a keynote speech by Jose Antonio Bowen, author of the book Teaching Naked, which is nominally a manifesto for flipped classrooms, in which more "pure" interactions can occur between students and instructors during class time. Weirdly, however, he spends the majority of his time waxing prosaically about how incredible, saturated, future-shocky technology is today, and how we must work mostly to provide everything to students outside of class time using this technology.

Here's how he started his TED-Talky address that Friday: He contrasted the once-a-week pay phone call home that college students would make a few decades ago ("Do dimes even exist anymore?") with the habits of college students today, supposedly contacting their parents a half-dozen times daily. In fact, he claimed, his 21-year-old daughter will actually call him for permission to date a young man when she first meets/starts chatting with him online. She supposedly argues in favor of the given caller by using three websites (shown floridly by Bowen on the projector behind the stage):
  • She has started chatting with the man on Tinder.
  • She has looked up his dating-review score on Lulu.
  • She has examined his current STD test status on Healthvana.
Now, that's heady stuff, and of course the audience of faculty and administrators "ooh"'ed and "ahh"'ed and "oh, my stars!"'ed in appropriate pearl-clutching fashion. Review dates and look up STD status before a date online? Kids these days -- we're so out of touch, we must change everything in the academy!

But this presentation doesn't pass the smell test. First of all, we should be suspicious of an adult daughter supposedly interrupting her real-time chat to "get permission" from her father. That's just sort of ridiculous. Admittedly at least Tinder really is a thing and you can chat on it; that much is true. (Although Bowen presented this as the daughter and a friend communally chatting to two guys together, which is not a group event that can actually happen.) But worse:

The dating-review site Lulu doesn't actually exist anymore. In February of this year (3 months ago), the site was acquired by Badoo and the dating-reviews shut down. If you go to the link above you'll realize that the whole site is offline as of this writing. (Link.) And:

You can't access anyone else's STD result on Healthvana. Yes, Healthvana is a site that allows you to quickly access and view your own STD results without returning to a doctor's office to pick them up. But it's only for your own results, and it requires an account and password to view them after a test. Obviously there are all kinds of federal regulations about keeping medical records private, so it's not even conceivable that those could be made available to the general public on a website. One might theoretically imagine a culture in which one pulls up your own STD records on a phone and shows it to someone you're meeting -- but there's no evidence that actually occurs, and of course it's strictly impossible in Bowen's account, in which his daughter had not yet physically met with her supposed suitor. (Link.)

That "Reefer Madness"-like scare-mongering accounted for the first 30 minutes of Bowen's hour-long presentation, at which point I couldn't take anymore bullshit and I got up and left the auditorium. In summary: The half of Bowen's presentation that I saw was entirely fabricated and fictitious, frankly designed to frighten older faculty and staff for some reason that is opaque to me. Keep that in mind if you pick up his book or see an article or presentation by Mr. Bowen.

How did I get clued in to the real situation with these websites, after my BS-warning radar first went off? I asked some 20-year-old friends of mine, who immediately told me that Lulu was shut down months ago, and Healthvana was nothing they'd ever heard of. Crazy idea, I know, actually talking to people without instantly fetishizing new technology.

Friday, May 13, 2016

Schmidt on Primary Teachers

Dooren et. al. ("The Impact of Preservice Teachers' Content Knowledge on Their Evaluation of Students' Strategies for Solving Arithmetic and Algebra Word Problems", 2002) summarize findings by S. Schmidt:
Nearly all students who wanted to become remedial teachers for primary and secondary education and about half of the future primary school teachers were unable to apply algebraic strategies properly or were reluctant to use them. Consequently, they experienced serious difficulties when they were confronted with more complex mathematical problems. Many of these preservice teachers perceived algebra as a difficult and obscure system based on arbitrary rules (Schmidt, 1994, 1996; Schmidt & Bednarz, 1997).

Friday, May 6, 2016

Noam Chomsky: Enumeration Leads to Language

One of my favorite videos, including a bit where famed linguist Noam Chomsky theorizes that a mutation in the brain regarding “likely recursive enumerations, allowed all human language”.

I have a possibly dangerous inclination to mention this on the first day of an algebra class when we define different sets of numbers, which is possibly a time-sink and a distraction for students at that point. But still, this is the guts of the thing.

(Video should be starting at 33m25s.)

Monday, May 2, 2016

What Community College Students Understand

Reading this tonight -- Givvin, Karen B., James W. Stigler, and Belinda J. Thompson. "What community college developmental mathematics students understand about mathematics, Part II: The interviews." MathAMATYC Educator 2.3 (2011): 4-18. (Link.)

They make the argument that prior instructors' emphasis on procedure has overwhelmed students' natural, conceptual, sense-making ability. Now, I agree that terrible, knowledge-poor, even abusive math teaching in the K-6 time frame is endemic. But I'm a bit skeptical that students in this situation have a natural number sense waiting to be un-, or re-, covered.
  • In my experience, students in these courses commonly have no sense for numbers or magnitudes. Frequently they cannot even name numbers, decimals, places over a thousand, or know that multiplying by 10 appends a 0 to a whole number.
  • Givvin, et. al. assume that students "Like all young children, they had, no doubt, developed some measure of mathematical competence and intuition... ". I'm pretty skeptical of this claim (and see little evidence for it.)
  • One task is to check additions via subtractions, and quiz students on whether they know that either of the addends can be subtracted in the check. Admittedly, this is an unusual task: usually we take addition (or multiply or exponents) as the base operation, and later check the inverse via the more basic one -- for which the order definitely does matter. (Because addition is commutative but subtraction is not, etc.) So it's unsurprising that students' intuition is that the order matters in the check; as usually applied, it does. 
  • Another student multiplies two fractions together when asked to compare them (obviously nonsensical). But it's easy to diagnose this: many instructors teach equating the fractions and then cross-multiplying them, and seeing which side has the higher resulting product; in this case the student scrambled the cross-multiplying of equations with multiplying fractions. Which highlights two things: teaching mangled mathematical writing as in this process leads to problems later; and the whole idea of cross-multiplying is so striking that it "sucks the oxygen out of the room" for other visually-similar concepts (like multiplying fractions). 
  • The authors state that "These students lack an understanding of how important (and seemingly obvious) concepts relate (e.g., that 1/3 is the same as 1 divided by 3)." Not only is this not obvious, but I can repeat this about every day for a whole semester and still not have students remember it. Just this semester I had a student who literally couldn't repeat it when I just said it about five times.
  • The importance of "combining like terms", which is essentially the only concept under-girding the operation of addition (and subtraction and comparisons) -- in terms of like units, variables, radicals, common denominators, and decimal place values -- is highlighted here. I don't know how many times I express this, but I'm doubtful that any of my students have really ever understood what I'm saying. I wouldn't be surprised but some students could take a dozen years of classes and never understand this point. Which is dispiriting.
  • The authors have some lovely anecdotes of students making a small discovery or two within the context of the hour-to-two-hour interview. This they hold at as a hopeful sign that discovery-based learning might be an effective treatment. But I ask: How many of these students will remember their apparent discoveries outside the interview? I find it quite common for students to have "A-ha, that's so easy!" moments in class, and then have effectively no memory of it a day or two later. "I do fine when I'm with you, and then I can't do it on my own" is a fairly common refrain.
  • Discussing concepts is Element Two (of three) in the authors' list of prescriptions. "A teacher might, for instance, connect fractions and division, discussing that a fraction is a division in which you divide a unit into n number of pieces of equal size. Alternatively, the teacher might initiate a discussion of the equal sign, pointing out that it means 'is the same as' and not 'here comes the answer.'". Sure, I offer both of those specific explanations regularly, almost daily -- they're essential and without them you're not really discussing real math at all. But many of my students can't remember those foundational facts no matter how much I repeat or quiz them on it. 
  • From my perspective, it almost as though most of my developmental students aggressively refuse to remember the overarching, connecting definitions and concepts that I try to share with them, even when they're immediately put to use within the scope of each daily class session. 
  • I can't help but feel that the distinction between "procedures" versus "reasoning" is an artificial, untenable one. The authors admit, "Even efforts to capitalize on students’ intuitions
    (as with estimating) often quickly turn to rules and procedures (as in 'rounding to the nearest')". I think this argues, perhaps, for the following: All reasoning is ultimately procedural. The only question is knowing what definitions and qualities of a certain situation allow a given procedure to be applied (even so simple a one as comparing the denominators of 1/5 and 1/8, for example). Even counting is ultimately a learned procedure.

While I don't seem to have access to Part 1 of the same report, the initial draft report has a few other items I can't help but respond to:
  • "'Drill-and-skill' is still thought to dominate most instruction (Goldrick-Rab, 2007)." This is a now-common diatribe (my French-educated partner is aghast at the term). But let's compare to, say, the #1 top scientifically proven method for learning, according to a summary article by Dunlosky, et. al. ("What Works, What Doesn't", Scientific American Mind, Sep/Oct 2013). "Self-Testing... Unlike a test that evaluates knowledge, practice tests are done by students on their own, outside of class. Methods might include using flash cards (physical or digital) to test recall or answering the sample questions at the end of a textbook chapter. Although most students prefer to take as few tests as possible, hundreds of experiments show that self-testing improves learning and retention." Which is a somewhat elaborate way of saying: Practice and homework.
  • "The limitations in K-12 teaching methods have been well-documented in the research literature... An assumption we make in this report is that substantive improvements in mathematics learning will not occur unless we can succeed in transforming the way mathematics is taught." I would not so blithely accept that assumption. What it overlooks is the perennial decrepitude of mathematical understanding by K-6 elementary educators. My argument would be that it doesn't matter how many times you overhaul the curriculum or teaching methodology at that level; if the teachers themselves don't understand the concepts involved, there is no way that even the best curriculum or methods will be delivered or supported properly.
  • "Perhaps most disturbing is that the students in community college developmental mathematics courses did, for the most part, pass high school algebra. They were able, at one point, to remember enough to pass the tests they were given in high school." But were they, really? A few years ago when I was counseling a group of about a dozen of my community-college students, as they left the exit exam and thought that they had failed, I stumbled into asking exactly this question in passing: "But this is totally material that you took in junior high school, right?" To which one student replied, "But there it was just about buttering up the teacher so he liked you enough to pass you," and the other students present all nodded and seemed to agree with this. The evidence that students are being passed through the high school system on effectively fraudulent grounds seems, to me, nearly inescapable.

Near the end of the Part 2 article, the authors appear to express a bit more caution concerning their hypothesis; a cautionary question which I'd be prone to answer in the negative:
For some students we interviewed, basic concepts of number and numeric operations were severely lacking. Whether the concepts were once there and atrophied, or whether never sufficiently developed in the first place, we cannot be certain. What we do know is that these students’ lack of conceptual understanding has, by the time they entered developmental math classes, significantly impeded the effectiveness of their application of procedures. (p. 14)
We hope that future work will seek to address questions such as whether community college is too late to draw upon students' intuitive concepts about math. Do those concepts still exist? Is community college too late to change students' conceptions of what math is? (p. 16)