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Wondering about educational theory

December 4, 2007

A classroom assistant I know tells me that at her school, when kids are doing their homework (called “seatwork” if it is intended to be done in class), she isn’t allowed to help them much other than to keep them on task and give them a thumbs up-or-down when they’re done.  The little blighters are pretty much supposed to figure everything out on their own.  At the risk of sounding “old school” this seems wrong to me.

It isn’t much to go on, but a study described at Cognitive Daily on how kids conceptualize numbers at different ages got me wondering if that’s a good strategy. 

The study found that between second and fourth grade, kids seem to undergo a transformation in their understanding of number proportions from zero to one thousand.  Briefly, in the second grade, they couldn’t accurately position numbers like 150 or 366 on a number line.  By fourth grade, they could.  OK, so what?

Here’s what: with minor intervention – like less than an hour – a large minority of the second-graders were able to do the task as well as fourth-graders.  And the new ability stuck with them.  It reminds me of what Arthur C. Clarke said about classroom design: “The ideal classroom is a log, with a teacher on one end and a student on the other.”

There’s a fashion in education where kids are kinda supposed to re-invent the wheel on their own, but in my admittedly limited experience as a tutoring volunteer, for many of them it is only frustrating and confusing.  Just a few hints in the right place, a little nudge here or there, and they “get it”.  And keep on getting it.

Opfer and Siegler argue that a “rapid and broad” change like this can occur in a variety of different types of knowledge, ranging from fractions to biology. They claim that there are certain concepts which, when grasped, open up a whole different understanding or representation of how a system works.

Any thoughts on this, especially from teachers?  Am I missing something?


One final thought: I wonder if even adults would display a similar pattern to second graders, if the numbers were large enough. I’d be interested to see a study on adults using numbers, say, from zero to a billion. How many adults would properly place the value of one million just a third of a millimeter from the left end of a 30-centimeter line?

I find many adults have trouble with large quantities.  A million, billion, or trillion all sound sort of the same… a godsend to politicians who would rather the voters not think about it too much.

Categories: Education
  1. December 5, 2007 at 06:41 | #1

    I always ask my (mostly adult) algebra class how many millions are in a billion, and how many billions in a trillion. More than half seem to know, and most can figure it out given a few leading questions. Of course since they are taking a math course required for a college degree, they are not a representative sample.

  2. Ted
    December 5, 2007 at 08:07 | #2

    I find many adults have trouble with large quantities.  A million, billion, or trillion all sound sort of the sameā€¦ a godsend to politicians who would rather the voters not think about it too much.

    I may have mentioned this before, but I think the best way is to convert the numbers to smaller forms. For example when the politician says, “$1B”, I think, 1,000 millions. I can visualize a 1,000 although visualizing a million gets a bit more amorphous so I substitute “value units” instead of a number line approach.

    So if they say, “The GAO reports that we lost track of $12B in Iraq”, I’m thinking, “Gosh, what could 12,000 million dollar packets do to our democracy if they’re used in the wrong way by ideologues. How much does it take to bribe local technicians that service Diebold voting Machines? How much does it take to build the extra little room that uses splitter taps at the local telcos? Is 12,000 millions enough to build secret prisons? How many scholarships could 12,000 millions fund (at state schools I mean)?”

  3. Ted
    December 5, 2007 at 08:15 | #3

    Oops, wrong link on the unaccounted $12B. Here’s another.

  4. Mrs SEB
    December 5, 2007 at 12:50 | #4

    As far as “seat work” goes…

    As I understood (and used as an educator) “Seat Work” is supposed to be the activities at the end of learning a lesson/concept that reinforce the concept (help them commit it to memory).  The one wherein the students have had instruction,  examples, been allowed to ask questions, tried working some samples with guidance already and are
    now given some problems to do alone to re-enforce their learning.  Move that learning from short term to long term memory.

    Is it always used in this manner?  No, unfortunately, “seat work” has become synonymous with busy work. Busy work is a time eating, fluff activity intended to keep students occupied momentary and in their seats (out of the teacher’s and other students’ hair).

    When I taught the process was:
    1. assess the prior knowledge of students in regards to the concept.
    2. present the concept
    3. explain the concept
    4. practice the concept
    5. re-enforce the concept
    6. evaluate the students understanding of the concept

    If needed:

    Re-present, re-explain, more practice, more re-enforcement, re-evaluate.

  5. December 5, 2007 at 13:48 | #5

    As Ted pointed out, it’s exactly why Math education is sooooooooooooooooo important.

  6. December 5, 2007 at 19:46 | #6

    I can understand the concept of seatwork if practiced like Mrs. SEB suggests it was intended to be practiced.  Done like that, it makes intuitive sense to me that it would help me retain the lesson.

    Years ago, when I used to hire people, I would ask candidates a simple question: “What’s ten percent of a dollar?”  I’d guess at least half the people I interviewed didn’t know the correct answer.

  7. james old guy
    December 6, 2007 at 12:47 | #7

    I am beginning to believe that education has been fixed beyound repair.

  8. Ted
    December 6, 2007 at 17:19 | #8

    Another 1,000 millions gone. Gosh, wonder if that could have been used to develop alternative energy?

  9. james old guy
    December 6, 2007 at 18:38 | #9

    Having had to deal with those clowns before, I am willing to bet the equipment is there, its that some piece of paper is missing. Those audits are done by bean counters that wouldn’t know an M1 tank if it ran over them. I have no doubt some equipment is missing but not anything in that qty.

  10. aos
    December 7, 2007 at 18:48 | #10

    Despite scientific training and constant immersion in fairly large numbers, I find that to really appreciate them, I need to place them within a context or make a tangible comparison. 

    Though we live in a world swimming in very large numbers, we aren’t really built to work that way.  I think numeracy is important but even more is the responsibility of those advancing a concept or describing something to do so in an understandable and intuitive manner.

    For example, instead of saying the war in Iraq costs x number of dollars a day to run, say something like it costs x number of dollars which is the same number of dollars it would cost to provide free health care to all Americans for x amount of time.  Of course, comparisons open the door to political statements like this.  Pure numbers avoid that problem but then outside of everyday manageable numbers you are left with just “really big”, and “really small”.

  11. December 7, 2007 at 19:07 | #11

    That reminds me of a great quote from Richard Feynman -

    There are 10^11 stars in the galaxy. That used to be a huge number. But it’s only a hundred billion. It’s less than the national deficit! We used to call them astronomical numbers. Now we should call them economical numbers.

    Richard Feynman, physicist, Nobel laureate, (1918-1988)

    In ye olden days people lived in villages and built things one at a time.  Now we live in megacities and manufacture megaquantities in nanoscale.  Understanding quantities and proportions beyond our senses will be an important challenge for education, pretty much from now on.

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