Thursday, April 05, 2012

Unclear on the Concept

I was subbing in a 6th grade recently and the warm-up question for the math lesson (directly from the teacher's manual) was something like this: A certain state has chosen to use the following format for their license plates: a single letter, followed by 5 digits. How many different license plates could they make that start with the letter A? The students were supposed to write their answers on their individual white boards and then show them to me to verify their answers.

The first answer I got was 28. I was completely baffled. Then came the other answers 15, 59, 25, etc., etc. NONE of the answers was even above 100, let alone near the correct answer. When I told the class (of 29 students) that their answers were all way too low, they started guessing above 100. But their guesses were completely (to me) random.

I decided to give them a hint: if they could use only 1 number after the A, you would have license plates A0, A1, A2, A3, ..., A9 - for 10 license plates. With two digits, you would have A00, A01, A02, ... A99 - giving you 100 license plates. They still didn't get it.

Pedagogically, I was so baffled by their lack of understanding, that I missed a golden opportunity to ask them what their reasoning was. I wish I had asked. "My bad", as they say. But now I am left wondering how they could possibly have thought that 28, 15, 59, or 25 could be anywhere near reasonable. This was a charter school, where the kids had to be delivered to the building in cars by their parents every day and picked up at the close of the day. How could they possibly think that 28 license plates starting with A, 28 starting with B, etc., would be enough? There are nearly that many cars at that one school in one day.

I was left thinking that the thinking habits of those kids were pretty bad.

Then came the lesson. It was on the number of degrees in specific turns. Though this wasn't in the teacher's manual, I had them stand up and demonstrate turning clockwise and counterclockwise. That isn't nearly as intuitive was it was in the days of all analog clocks. Our digital kids nowadays don't seem to have quite as much familiarity with the rotation of the hands on clocks. So I had them practice. First we established that a full turn, either CW or CCW was 360 degrees. Then that a quarter turn was 90 degrees and a half turn 180.

As I expected, a lot of the students mixed up CW and CCW, until we had practiced quite a bit. What surprised me a little was that several students refused to participate at all. This is a fairly strict and structured charter school and the non-compliance was unexpected. I didn't make a big deal of it, however, since it was an unplanned part of the lesson. I was more interested in the fact that this was actually a fairly difficult exercise for 6th graders.

That was just about the whole lesson - that and a few word problems. I was not terribly impressed with the curriculum, but this is a curriculum that I am not terribly fond of, anyway, so I am not going to name it. I was more interested in the seeming lack of comprehension of the students. They could do the rote problems, but the applications seemed to baffle them.

I would love to blame it on the curriculum, but I am not so sure that that is the problem. I have seen similar things with other curricula. What seems to me to be more evident is that kids are not particularly interested in making sense of things. They are willing to learn the arithmetic procedures - essentially just memorizing "how to do the problems", but they have very little (no?) interest in understanding why things work as they do. How have we gotten such disinterested kids? Was it always this way?

I remember hundreds of years ago, when I was a child, that I wasn't particularly interested in math. I could do the problems reasonably well, but the mathematics behind the arithmetic wasn't compelling to me. I don't remember if it was taught. I just remember that I was good at math, but, to me, that meant that I was good at arithmetic.

Then came my own 6th grade. The math teacher taught us about number bases - the reason behind carrying and borrowing when you get to the number of the base. It was, for me, a whole new ball game. Math became much more interesting. But that kind of enlightenment was sporadic. I remember asking several times in calculus classes how calculus was used, but I usually got either completely useless answers (It is used in everything!) or vague answers (In physics it is used to derive the laws of motion).

So now, I am wondering: it is counterproductive to try to explain mathematical reasoning to young children? Perhaps they just need to learn to do arithmetic very well. Then, in middle school or high school, with a bit more mature brains, they should take a class called number theory - and learn the reasoning behind the algorithms.

It is scary, though, to think that kids have so little practical understanding of math that they can't see the unreasonableness of the answer 28 for the number of license plates starting with A and having 5 numbers following the A.


  1. Laura, first, I'm delighted to find your blog - as you posted on Linked In (which turns out to have an 'interest for me' rate about 0.5%, but maybe that's enough).

    I am in my 34th year of teaching, the first 20 elementary, and the last 14 in middle school, primarily 6th grade, and except for last year, always including math. I teach in a public school in Ann Arbor, Michigan with very bright and capable students and an excellent math curriculum.

    And yet, I can totally relate to your comments.

    My working hypothesis is that we are being pressed to teach too much, too fast, too soon, and too abstractly.

    By 6th grade, many students think they're hot stuff, but what they do well is memorize. Others feel hopeless because they don't. Neither sees math as necessarily making sense - the very thing that I find math is designed to do - help students see that yes, the world can make sense.

    But we're still spending too much time teaching students to do what a $5 calculator can do, rather than providing ...

    1) concrete experiences (I'm delighted that you got kids up and turning to show 360º in a circle; though perhaps you were fighting an archaic battle with CW and CCW :)

    2) Skill and practice to build Number Sense - from Place Value understanding (yes, working in other bases is revelatory for students) to to Estimating Skills, and many many more.

    3) Conceptual Understanding, rather than tricks. I don't allow my students to move decimal points, say "two negatives make a positive," or any other shibboleth.

    4) Questioning and Reflecting (yes, you did miss an opportunity, though I doubt your students that day would have been able to stumble through an answer - they have learned that guessing serves a purpose, if not one we aim for). Oh, students hate having to explain why what they're doing works, but they are so much the stronger for it.

    From your description, I would guess that the students you worked with had little of these. Hence the mindless responses, and lack of participation.

  2. I understand that the nomenclature of CW and CCW may be archaic, but I disagree that the concept is. It makes a difference which way you turn. Perhaps the nomenclature should be replaced with the "right hand rule", which would also be applicable to physics.

    The thing that confuses me about your idea of too much, too fast, too soon, and too abstractly is that this particular curriculum is specifically designed to teach very little each day and review a lot. But it still produced kids who were completely incompetent at the license plate problem. Since I am a sub, I can actually perform an experiment. If I sub in other 6th grade math classes and if I have a chance, I am going to try the license plate problem with other classes. It will be interesting to see if other classes respond differently.

    My favorite two negatives make a positive analogy is the swimming pool problem: filling the pool in a given amount of time or emptying the pool. Time is either future (+) or past (-); filling is +, emptying is -.

    I love using manipulatives and getting the kids up to move made sense to me, but, as a sub, using manipulatives is very tricky. They usually end up being used inappropriately. Interestingly, I don't think regular classroom teachers use them much either. They are usually stowed away in inaccessible or dusty places.

    If I had had the time, I think I would have eventually thought to ask the kids about their reasoning in the license plate problem. But, as a sub, I feel tremendous personally imposed pressure to follow the lesson plans and teach the lessons as they are intended to be taught. It is quite a balancing act: to decide when there is a teachable moment that would justify changing the lesson plan a bit.

  3. I was joking, mostly, about CW and CCW being passe - though it's a struggle. When my 6th grade students ask to go to the bathroom they have to fill out a pass with the date and time. I have to stop and teach them how to tell time off the analog clock. (As if 'analog clock' was ever in *our* vocabulary). The funny thing is that when I began teaching I was surprised when they taught how to tell time in first grade, as there are so many 3rd grade skills involved. And now my 6th graders have never needed to learn.

    No, teachers don't use manipulatives near enough. It's also discouraging on sites like Linked In, where teachers go for tricks and shortcuts rather than conceptual understanding.

    The model our Connected Math Program uses for multiplying integers is running at a certain pace along a number line marked in intervals of 5, from -50 to 50. Time earlier and later is the other factor - very similar to your swimming pool - though in this one, I can have kids model problems in front of the classroom, or better yet, out in the hallway. (Given they sit, and slumber, all day - anything that gets them moving is good.

    But yes, you're at a real disadvantage as a sub.

    I'll find time to ask my students the license plate problem. The dynamic in my room is that I'll have 4-6 hands raised almost immediately, with the rest of the students sitting there waiting for the 'smart' kids to answer. Well, this is a topic for another discussion.

  4. Ha! I used the number line thing to do integer addition and subtraction. I took the kids outside, with sidewalk chalk and had them make number lines on the blacktop. Facing the positive numbers was addition, facing the negative end of the number line was subtraction. Adding a positive number was walking forward; adding a negative number was walking backwards.

    Yes, I know that the joke about CW and CCW was mostly that, but it might actually be time to rethink the nomenclature. Kids nowadays are much more familiar with driving - perhaps turning right and turning left would be more relevant.

    I think some curricula truly are better than others a driving home the underlying mathematical principals, but, of course, it still depends on the skills of the teacher - and the time factor. I actually like Everyday Math as a curriculum, but it takes an enormous amount of time to do it correctly. You really need to READ and ANALYZE the students' answers to the writing questions and make them COUNT toward the students' progress - and this is very seldom done consistently.

  5. Ah, I hadn't seen this reply; glad I checked back! So, I gave my students the license plate problem, and because my students are bright, and use the Connected Math Curriculum, and have me as a teacher, I figured I'd perhaps get a different response that you did when you subbed in 6th grade. Mostly I got ... the same response. Wow! Honestly though, I'm not surprised.

    But then this: I gave a couple hints to one of my three math classes, but then put off discussion, as well as the answer, for a few days. When I came back to it, I found that only a handful, out of 80 student, had been working on it. Why? "Well, you didn't *assign* it; we aren't being *graded* on it." Sigh.

    So that day, I assigned it, due Monday. On Monday about 20-30 of the students had worked on it. We discussed the strategy and the answer and everything, and I told them they were still responsible for getting it in, and most did, except ...

    ... one class - which looked totally baffled, even after I explained the problem. They were so convincing that I decided it was the better part of valor to just excuse them from the problem completely. I did tell them that in about a month, we'd be doing problems like this in our probability unit, but we'd start out much simpler. Anyway, I still think it's a great problem, fresh to students, and not beyond what confident problem solvers can tackle.

    btw, while I haven't taught Everyday Math myself - I transferred to middle school about the time we adopted it, my sense is your analysis of teachers' approach is likely valid.

  6. I actually like Everyday Math. That isn't the curriculum that I hate that I was referring to. The problem with EM is that there isn't quite enough practice of similar problems. Theoretically, this practice is supposed to come through the frequent use of games, which were developed and correlated to the curriculum. But most teachers find the games too time-consuming and to onerous to set up, so they leave them out and substitute practice worksheets instead. (Drop in the Bucket, Math Matters, etc.)

    The other problem with EM is that, if done correctly, there should be a lot of time spent on discussing the written parts of the workbook - the parts requiring answering in sentences and paragraphs. But this is very time-consuming for teachers to grade and, unless they are very sure of themselves mathematically, most teachers curtail these discussions.

    But the kids don't seem as confused by the different algorithms as teachers and parents are. They seem to especially like the lattice method of multiplication. And the column method of long division is MUCH easier to explain than the standard long division algorithm. Kids understand it much more quickly and can be encourage to shorten it as much as possible, once they understand what they are doing.

    I haven't had a chance to try the license plate problem on other 6th grades yet. I have been subbing at the high school level lately, where I have been flabbergasted that the students found graphing the sine function SO DIFFICULT. They were supposed to compute the values for various values of pi and graph them to make a sine wave. NOT ONE student in the two classes was able to produce a reasonable sine wave in a 90 minute class. Incredible.

  7. It looks like I wrote about EM twice. Oh, well, I am not going to edit the post.

    I have used Connected Math a little bit and I like parts of it, too. They have some good experimental lessons.