Tall, Dark, and Mysterious

8/8/2005

Out of the mouths of babes

File under: Those Who Can't, Queen of Sciences. Posted by Moebius Stripper at 7:04 pm.

The student I’m tutoring, explaining why he didn’t do the last five questions of the homework, each of which defined a new term, and then required students to investigate how it applied to a few given functions:

These questions seem like they should have been in the lesson, not the homework. It seems like they’re trying to teach me something new.

God forbid.

The result of this unfortunate run-in with new material in the homework, of all places: we took a half hour break from going over new mathematical content, to spend on developing the skill of reading mathematics. Teaching this is easier than it sounds, when you’re starting from zero and dealing with straightforward material: today’s lesson consisted of me asking him to read the definition, and then following up with such prompts as “ok, so now what is the question asking?”, “What should we do?”, and “How can we do that?”, all of which he answered correctly. Not bad for someone who reaches for his calculator every time he is required to add one single-digit number to another.

“You did all of that by yourself,” I pointed out when we were finished. “Why couldn’t you do it last night?”

“You were here this time,” he replied.

“Yes, and I prompted you,” I agreed, “but the questions I asked you weren’t the slightest bit leading. I just told you to read the question, and then I asked you what the question was asking, and how to go about answering it. Basically - I asked you the stuff that the question itself is asking you to do. And you did that all - correctly.”

He thought for a minute. “I guess,” he said. He then paused, and looked up as my statement registered: “So I’m supposed to be doing that whenever I see a question I don’t know how to do?”

30 Comments

  1. You, and all your fellow math bloggers, never cease to render me speechless with these transcribed conversations. I so, so wish you were making them up.

    - sheepish — 8/8/2005 @ 9:31 pm

  2. heh!
    You have a gift for writing in words the specific teaching issues I deal with each day. I have never had a conversation like this with my students, but I do know that this is constantly what I am trying to get my students to appreciate.

    (High school policy was to NEVER give students a homework question they were unable to already complete by following the algorithm in class, guh)

    - Ronald — 8/8/2005 @ 11:56 pm

  3. Heck, I remember being flabbergasted in a GRADUATE linear algebra class, where we were given an open-ended homework question from the textbook. (e.g., if Ax = 0, (with certain dimensions of x, A, or other characteristics), what can you say about A?) One guy complained that he couldn’t tell when he was finished answering the question.

    The only good thing about that situation was the guy remonstrating was a masters student, not a doctoral student. Still, I can just imagine how fun he would be to have as an advisee, especially once he started work on a thesis.

    - meep — 8/9/2005 @ 2:18 am

  4. Oh boy, if I had a nickel for every time this exact same scenario has come up in office hours… the student can do excellent mathematical thinking as long as you are sitting there, even if you’re just getting him to ask the right questions and not leading him. But if you are not sitting there, the wheels fall off. I’ve often thought about going to the local FedEx/Kinko’s and having a life-size cardboard figure of myself made for each of my students, to prop up next to them while they are doing homework, so they’ll feel like I’m there in the room with them. Then maybe eventually they could wean themselves off the surrogate.

    - Robert — 8/9/2005 @ 4:27 am

  5. LOL, Robert! Some days I think my cardboard surrogate would be just as effective if I propped it up in front of the lecture and left, because no matter how often I tell them that Technique A only applies in Situation B, never in Situation C, they will inevitably try to apply Technique A to Situations C, D, and E, but when faced with Situation B they will say they have no idea what to do.

    - Wacky Hermit — 8/9/2005 @ 6:58 am

  6. Meep - I was taught that the correct attitude for a sceintist is that you are never finished answering the question. I can see how that would send a lot of undergrads (and Master’s students) into a tailspin. I guess that applies in mathematics, too. The only question is how valuable are your successive answers, and how long to you want to beat the deceased equine?

    - John — 8/9/2005 @ 7:32 am

  7. As someone about to start a Ph.D. program in UCLA, HOW CAN YOU TELL when you are finished answering the question?

    Those questions asking ‘what can you say’… one could EASILY get 10+ pages on it (depending on how technical you get)

    - mmailliw — 8/9/2005 @ 12:43 pm

  8. Ah, the cardboard cutout. I’ve found that when I explicitly say “Don’t do X”, the number of students doing X increases. I have taken to writing down the incorrect method X on the blackboard, and encircling it with a line through it, like in the no-smoking sign. My students might hear me selectively (”don’t do blah” gets processed as “do blah”, GOD ONLY KNOWS WHY), but hopefully they will not see their notes selectively.

    True story - my first term teaching precalc, a LARGE majority of my students were under the impression that (x+y)^2=x^2+y^2. After I came across this error around forty times on the first quiz, I took ten minutes out of class explaining why it was wrong, explaining the distributive property, and going over how to expand that quantity. Next quiz, almost as many students made the exact same mistake again. So I spent another ten minutes addressing it, and warned the class - “You’ve been told this twice so far. If I see this error again, you will get a zero on the question.”

    Four students (only! yay) made the mistake again. They got zeroes. One kid had the gall to tell me that he thought that zero was really harsh “for such a small mistake”. Oh, I had fun with that one.

    But about “So I’m supposed to be doing that whenever I see a question I don’t know how to do” - here’s what floors me: I mean, I KNOW that math requires skills that are unique to the subject. I know that being good at English and history and art doesn’t necessarily mean that one will be good at math. However, one might have a fighting chance at becoming proficient in mathematics if one availed oneself of certain skills, such as BASIC LITERACY, that are acquired elsewhere. I mean, shee.

    - Moebius Stripper — 8/9/2005 @ 2:57 pm

  9. I’ve found that one should NEVER EVER right down something which is incorrect on the board. NEVER. NOT ONCE. NOT EVEN IF YOU IMMEDIATELY CROSS IT OUT.

    Given that roughly 10-20% of what you say and do in class actually makes it into the students’ brains, the probability that writing (x+y)^2=x^2+y^2 and then saying that it’s wrong will be interpretted as (x+y)^2=x^2+y^2 is close to 1.

    As for punitive grading, I once told students that if they misspelled “commutative” on an open-book open-notes quiz they would receive NEGATIVE points on the quiz. Much to my surprise, one of the students did just that. And was immediately granted his negative points.

    - vito prosciutto — 8/9/2005 @ 3:37 pm

  10. My theory is that students believe that everything is a linear operator from R to R. The square of (x + y) = square(x) + square(y), just as cos(x + y) = cos(x) + cos(y), and e^(x + y) = e^x + e^y. Teaching them to integrate and differentiate just makes it worse.

    I recommend restricting yourself to functions that are only linear operators in the future.

    - Xn — 8/9/2005 @ 5:21 pm

  11. Yeah, I learned that lesson about writing wrong stuff on the board the hard way. (Damn reverse psychology! DAMN IT TO HELL! Ah, I mean…good reverse psychology! Good, good!) Which makes it really really difficult to address common errors while teaching - students seem to lose the ability to process the “not” operator when they’re in a math class. (As well as the “if/then” construct, and the “read the goddamned question before answering it” principle.)

    Xn - yes, everything is linear. However, I’d not previously considered your approach to dealing with this misconception. It seems as good a solution as any.

    - Moebius Stripper — 8/9/2005 @ 8:37 pm

  12. As someone about to start a Ph.D. program in UCLA, HOW CAN YOU TELL when you are finished answering the question?

    As someone who just finished a Ph.D. program at UCD, I found out I had finished answering the question when the third and final member of my dissertation committee said he was ready to place his signature on the cover page.

    Sorry to give such a utilitarian answer, but grad school is a weird beast with its own rules, varying from school to school. Some of the best practical advice I received came from the same hold-out committee member who recommended early in the process that I write a table of contents for my dissertation and write a detailed outline before I ever began writing it. Working on that clarified in my mind where I needed to begin and where I hoped (hoped!) to end up. The resulting document filled out in a close approximation to my intial plans, the big question being what would go in the conclusion blanks as my research results were collected and analyzed. Still, a happy ending. The preliminary outline gave me the guidance and focus I needed since no member of my committee wanted to lead me around by the nose the way MS’s student wanted to be led.

    - TonyB — 8/9/2005 @ 10:46 pm

  13. MS, have you also had problems with students integrating (1 - x^2)^(1/2)? My students always replaced that by (1 - x).

    - Eric Jablow — 8/10/2005 @ 6:15 am

  14. On the issue of students not reading the directions, I remember a quiz from 5th (?) grade. The first line of the instructions was something like, “Read all the instruction before beginning the test.”

    After that were a series of variously difficult and embarrassing tasks (I recall one as “Stand up and shout your name as loudly as you can.”) Not surprisingly, there were lots of people standing up and shouting their names.

    The last instruction was, “Disregard all other instructions, put your name at the top of the page, and hand in the test.”

    I don’t know how effective the exercise was at increasing the reading of directions, but I know that I would have been mortified had I not followed the directions. Perhaps public humiliation is part of the answer.

    - Doug Sundseth — 8/10/2005 @ 12:19 pm

  15. Doug,
    I had that same quiz (”don’t do steps 11-14″ or some such) in Home Economics class in 7th grade to teach us to read the whole recipe before starting to cook.

    - Paul — 8/11/2005 @ 10:02 am

  16. At my previous college, we did an unscientific study of the relationship between our students’ SAT scores and their performances in our math classes, and we found that the SAT verbal score had a much higher correlation with student success in math than the SAT math portion did. I think there really is some kind of intimate connection with reading skill and college math performance and the comments about basic literacy here seem to scratch the surface of something really deep.

    My students have serious problems with basic processing of written and verbal information — not just in a math problem, but in things like the syllabus and directions given to them for course assignments. Sometimes it’s that they let the sound or words into their bodies but it doesn’t register with them; other times they are simply choosing not to listen or read in the first place. They seem conditioned to want to get through the class with maximum speed and minimum “distraction” from something that isn’t getting the right answer. I blame standardized testing. (But then I blame that for everything.)

    - Robert — 8/11/2005 @ 1:27 pm

  17. Doug - I remember doing that exact quiz.
    I watched my friends humiliate themselves… it was fascinating.

    - Ronald — 8/11/2005 @ 5:27 pm

  18. I tutored math while knocking out prereqs at community college, and had much the same conversation with frustrated students. Interestingly, I saw this much more in remedial classes with unskilled workers. Eventually, I got the point that I would have them put away everything but one really basic problem, take a new piece of paper, and have them take notes on our conversation about solving th e problem. We wouldn’t actually solve the problem, we meta-solved the solution to the problem.

    Those people thought I was an idiot and an ass until I saw them the next week. Then they smiled at me and asked many fewer basic questions.

    (What I find most fascinating about the group who didn’t have these sorts of questions. They were people who spend all day making unique things, whether they be bakers or construction workers. In working this way, one eventually adopts one’s own problem-solving methods. These people would have questions, but they would be really weird, off-base things. So weird, in fact, that I would have to look at their work to see how they got there.

    Invariably, it came down to the fact that they bummed down to a set of primitives operations and transformations and worked up from there. When you take one small wrong step and follow it out, you wind up waaay out there, but through a seemingly-valid sequence. I’m exceedingly careful in my proofs now, and try to write them a few days before they’re due. That way I can come back and read them with a fresh mind to make sure I didn’t detour through left field myself.)

    - Josh Myer — 8/11/2005 @ 8:44 pm

  19. I got the same “read all instructions… disregard all other instructions” sheet in… sixth grade. I think it was biology class. But I was (and still sometimes am) one of those people who starts working on something before reading the directions. So I and one other person were standing up, reciting our name out loud, and so on. I thought we were working ahead of everyone else… until I got to the end of the worksheet. So that day, I learned that I can make a fool of myself when I don’t read directions.

    - Matt — 8/11/2005 @ 8:55 pm

  20. I remember getting that worksheet, several times in fact. But I knew something was up when you read like the first question and it was something like clap your hands or something.

    I, at the time, was kind of shy. So I kept reading while everyone else was clapping and what not. I skipped to the last question, because it’s a habit of mine to sometimes work from the back to the front (I do this with books sometimes, reading the last chapter before I get half way through). Well, what do you know, that last question asks me to do nothing. So I sit in my chair and laugh at what the other questions are. I waited patiently for everyone to say they were finished and had a good time in the process.

    I’ve told the math profs in our department to distribute this “test” on the first day, but they insist on having a seminar on who they are and what study skills you’ll need for the class.

    The second time I got that set of instructions though, I just read the top, read the last question (to make sure it was the same worksheet) and sat back and had fun. They learn quickly (hopefully).

    I’ve thought about making a big deal of the first test, two parts, etc. etc. The first part you can’t use a calculator on and that paper would be the first part. Then they’d have to switch papers with you to give you the second part. Make the first few problems math problems and the last one is “do nothing (none of the above problems), turn in this sheet”, etc. It could be a review exercise. But that might take some effort to pull of or get approval through the department, just depends.

    - Vanes63 — 8/12/2005 @ 11:17 am

  21. Am I the only one who’s going to ‘fess up to clapping, saying my name out loud, and so on, when I was giving the directions-following test?

    Honestly, though, my students’ problem isn’t that they start working on a problem before they finish reading it. It’s that they never finish reading it, and they never start working on it. As far as I can tell, they scan for keywords that match the examples they saw in class, and if they don’t find any, they wait for me to explain how to do that specific flavour of question.

    But this is all old news.

    Robert - very very interesting, about the verbal versus math SAT scores. But I think that if one were to correlate English skills with mathematical success in my classes, you’d find a negative correlation - but that would be confounded by the fact that many of my strongest students were the East Asian exchange students, many of whom spoke very poor English. (Often I’d have a student call me over to their desk in the middle of a test and ask a question such as “What’s a rectangle?” And then I’d draw the appropriate quadrilateral, and they’d thank me, and then write up a beautiful, correct solution.) However, perhaps as a result of their poor English, these students were more inclined to read over the entire question carefully rather than scanning for keywords - something that the weaker anglophones do regularly.

    - Moebius Stripper — 8/12/2005 @ 2:58 pm

  22. just a crazy idea that popped in my head this morning while reading the paper : the same techniques and skills that allow you to read the paper in a reasonable amount of time (instead of, say, all day) are the ones that are misapplied to dealing with math questions.

    So, continuing with this crazy idea, it might be that the students are inundated with too much stuff to deal with, and this is the sign of the breaking down of their ability to cope.

    It’s just a strange, off the cuff idea, so please don’t tear me to too many pieces.

    - Sam — 8/13/2005 @ 10:35 am

  23. Maybe, Sam, but I’m skeptical, because a) I’m not convinced that a majority of my students read newspapers, or anything else much, and b) if you were given a full day with nothing in your possession besides a newspaper, would you still skim it? Even if you were going to be tested for comprehension at the end of the day? One of my most frustrating experiences last year was having half of my students leave a test forty-five minutes early or more, and leave several questions completely blank.

    - Moebius Stripper — 8/13/2005 @ 2:28 pm

  24. Sam has a point here. I think the current generation of high school and college students is used to getting their information in disconnected bits, all streamed into their sensory receptors without any pretense of coherence, rather than expecting structure and meaning in a text and seeking that out when they are presented with something like a mathematical story problem. However I’d say that this represents not so much a breakdown in their ability to cope with too much information, but a basic void in their ideas about what information really is.

    In other words, I think this is partly a cognitive problem but also partly a cultural problem.

    - Robert — 8/14/2005 @ 12:37 pm

  25. I don’t understand point b) of your reply to Sam’s point.

    I think that his idea was that our culture selects for skills in rapid skimming and speed-reading to the point where such behaviour becomes habitual and is employed even in situations where more detailed reading is necessary.

    That would provide a more charitable explanation of the problem than that the students are just lazy, but my instinct is not to be quite that charitable.

    - saforrest — 8/14/2005 @ 12:41 pm

  26. I’m becoming convinced, Robert and saforres. Especially this part - However I’d say that this represents not so much a breakdown in their ability to cope with too much information, but a basic void in their ideas about what information really is. My observations that probably a majority of my students see math as a collection of disconnected bits that don’t in any way fit into a whole, has been well-documented. And regarding Sam’s speculation, I think this is part of something bigger - namely, that we live in a culture that values efficiency over everything else. We buy cheap things instead of good things. We choose magazines that we can read quickly over books that we can learn from. And something that I just don’t know how to deal with in my teaching is the extent to which my students FREEZE when need to read math. Half the time, they barely need for me to do much of anything beyond read the question to them.

    - Moebius Stripper — 8/14/2005 @ 3:23 pm

  27. Yeah, there is an aspect of people learning info as crystallized bits, carefully separated from every other bit of info in the universe.

    I totally blame all of this when natural philosophy got broken up into physics, math, chemistry, etc.

    And I doubly blame computer science for leaving math departments. And the removal of geometry from calculus.

    GRAH! I’m feeling curmudgeonly.

    As per usual, Dickens had something to say to this (as he had something to say about Enron… and people think every “new” scam is particular to this point in time, as opposed to something that bubbles up from human nature…): In =Hard Times=, at the very beginning, a “model” classroom is presented, where the lesson is the definition of a horse. There is a charity student in the class, who was the daughter of circus performers, and she knew horses very well from a practical point of view. But she was unable to answer the headmaster’s question, and a “know-it-all” gives a dictionary definition, a pure crystal of “fact” that is completely useless to actually =doing= anything with horses.

    It also reminds me of reading the first “encyclopedias” from the Middle Ages, and their math entries in particular. Did you know that a quadrilateral is a figure of four sides, all of which are of equal length, and all sides intersecting at right angles? I can just imagine how well they did with Euclid’s Elements. The concept of proof had regressed quite a bit at that point, and it took quite a few centuries of work before the Europeans got back to knowing standard logic, as a culture.

    - meep — 8/15/2005 @ 7:43 am

  28. The clueless, and those who teach them

    Tall, Dark, and Mysterious has far, far more patience than I. Her description of the hurdles required to get EI benefits in British Columbia is painful just to read, never mind actually deal with. And if the folks on the…

    - Number 2 Pencil — 8/16/2005 @ 5:40 pm

  29. Having seen the exercise sheet discussed - and several varieties on it - over and over again in boy scouts, with gaming associations and numerous times in school, one thing I distinctly remember is one with a slightly flawed main instruction. The main instruction read something like “Perform each instruction in turn”, with the first instruction being “Read all other instructions before proceeding”. The result was that clapping, saying your name et.c. was the (or at least one) legal way to proceed.

    When I pointed it out to my parents (who were giving the sheet out at the moment), they completely failed to understand my point.

    - Michi — 8/22/2005 @ 3:14 am

  30. teaching carnival #1

    Teaching Carnival is devoted to gathering select blog entries related to teaching issues in higher education. Below you will find the first installment. (Be sure to have a look at the common sense words of advice for readers.) Caleb McDaniel…

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