Monday, November 4, 2013

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?


6 comments:

  1. Hello there, I teach remedial math at the high school level. My students crave the mindless!

    I recently realized that my students try to memorize every problem they ever see. In this way they try to match every test question to their mental database of problems. When they can't remember the solution they give up, regardless of the difficulty level, because they failed to successfully retrieve the answer.

    I attribute this to their inability to comprehend the word 'Context'. I'd be interested to know if your students now 'Context'. It is a difficulty concept to teach. I am trying to get them to recognize that mathematical techniques or strategies or tools are always the same ... it is the context in which you use them which changes.

    They don't get it. They continue to treat every question as being novel. If they understand context then I believe that they will stop this memorization game of theirs.

    Here's hoping.

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    1. Hi, Rando -- You've definitely hit close to the very essence of the problem there. I know some people define math as "identifying patterns". My attitude is that the whole point of these basic courses may be to try and introduce people to the power of abstract thinking -- which is to say, by recognizing basic patterns, how much less there is to memorize. (And those patterns may be expressed concisely with the language of variables, etc.)

      Funny story: I have a one-sheet summary of the entire remedial algebra course that I give out the first day, and remind students about from time to time (definitions on one side, procedures on the other). Nonetheless, I frequently get a student complaining that "there's so much to memorize!"... to which I think "but what other course can fit entirely on one sheet of paper!?".

      The other thing I've thought of recently is how there's an inherent complication that in math we're always inherently applying the information, not just regurgitating it. We don't usually ask a test question like, "What is the power rule?" (as would happen in most other classes), we ask, "What is (x^5)^3 simplified?". Sometimes I wish that we had time for basic tests of the actual rule/pattern statements themselves, so as to focus students on the fact that those patterns are really the whole point of what they're supposed to learn. Do students actually commit my one-sheet summary to memory? Almost never, because technically I don't ever ask a test question on those rules themselves. (To say nothing of word-problems, which are really a third layer of applying knowledge in a math class.)

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  2. Hey Delta - You've mentioned something just now with your thoughts regarding how we test our students that was a major topic of discussion at a recent workshop. We discussed the six levels of cognition from "Bloom's taxonomy", the first three being Recall, Understanding, and Application.

    The presenter spoke at length about a systemic problem wherein we teach to the Understanding level, expect Recall, and test Application. He was arguing for exactly the same thing that you are describing; if we teach at the Recall then we should test at the Recall level ... and so on. To do otherwise, he argued, would be unfair to the student and problematic for the teacher. It would be problematic for the teacher because students give us Recall answers on an Application question and we give them part marks. Really, it should be a Recall question that gives marks for Recall answers and a separate question for Application which we only grade for Application. The students should be informed which questions is which.

    Word problems probably subscribe to one of the remaining three levels, but I am still a mechanical user of all of these concepts and can't speak to them any more than to share them.

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    1. Yeah, I basically agree with exactly that. Very funny that you should be discussing the same thing. The next major problem is then evolving those standard math test expectations, when there's some much inertia around institutional and standardized testing.

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  3. See, I would absolutely think they need to be taught logic, preferably at the junior high school level. Contrary to what your colleague said, I find that the drilling in basic logical concepts to be very valuable to underclassmen. And in my experience, they tend to like the truth tables.

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    1. Yes, I do agree with that, and that matches my experience as well. Well put.

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