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The Necessity Principle

How could we improve this task?

Fuller, Rabin, and Harel (2011) [pdf] define “intellectual need,” “problem-free activity,” and offer several ways to improve that task in one of the best pieces I read last summer:

When students participate in mathematical activities that stimulate intellectual need, we say that they are engaged in problem-laden activity. Unfortunately, many students are engaged in problem-free activity, in which they are driven by factors other than intellectual need and, as a result, do not have a clear mental image of the problem that is being solved, or indeed an understanding that any intellectual problem is being solved.

The piece features:

  • Dialog between teachers and their students that results in “problem-free behavior” and “social need.” There’s something in here for everybody. Everybody — myself included — will feel a twinge of recognition reading one or more of those exchanges.
  • Great suggestions for how to mend those scenarios, for queueing up intellectual need and problem-laden behavior.
  • Five categories of intellectual need. The need for certainty, causality, computation, communication, and connection. You can lean on any of those categories and watch several great lesson ideas fall out.

Featured Comment

mr bombastic:

The recursive part in the original question is especially annoying in that it sends the message that math is used to take something that is totally obvious (two more brick in the next row) and somehow make it seem complicated.

26 Responses to “The Necessity Principle”

  1. on 24 Oct 2012 at 4:09 pmlouise

    This makes me think of all those presentations where someone stands in front of the room and reads power points. You think we can’t count?
    How could we improve it?
    http://www.lefthandedtoons.com/77/
    How many cheese pieces do you need for a 6 inch sub? A 12 inch sub? a giant 6 foot sub? I bet students might be interested.

  2. on 24 Oct 2012 at 9:01 pmBowen Kerins

    Here’s my version of the task:

    http://www.pinrepair.com/tim/tim5.jpg

    I guess it’s not linear, and it’s not quite the same task, but whatever, it’s awesome.

  3. on 24 Oct 2012 at 10:28 pmblaw0013

    Can I “like” this? #oldguyintheroom

  4. on 25 Oct 2012 at 4:18 amMichael P

    What about some variation on this problem, featured in an earlier post?

    Split up the class into groups, give them a different pattern in a manila envelope, and have them text each other instructions on how to reconstruct the pattern in their own drawings.

    Another idea: just ask, how many bricks would be in the 10th row? The 100th? The 1000th?

    (Incidentally, the above strike me as very Dan-Meyerish questions. Especially the second.)

  5. on 25 Oct 2012 at 7:15 amJames Conron

    I’m just assuming print here.

    1) Take away the “The landscaper is..[blah blah blah]“. As context, it’s pretty weak, and probably pseudo-context.

    2) Take away the numbers next to the rows. Let students count it themselves.

    3) Simplify the question. “a) Extend the pattern to 10 rows. How many bricks would be in the 10th row?”, “b) How many bricks would be in the nth row? Provide two formulas. Both a recursive formula and an explicit formula.”

    Now normally I would say students should be able to solve a problem any way they can solve it, and if you wanted them to solve it a certain way, it should be the only obvious choice. However I’m assuming here that both explicit and recursive formulas are outcomes that need to be covered in this question.

  6. on 25 Oct 2012 at 7:28 amJames Conron

    BTW, in further to what I’m saying, if you’re not constrained by outcomes you could simplify this problem to a picture of the pattern and one simple question. Something like “Extend the pattern to a 1000 rows. How many bricks would there be in the 1000th row?”

  7. on 25 Oct 2012 at 8:34 amJason Dyer

    Doesn’t the pattern only have 8 rows, anyway? (I’ve never seen one of these where the “first row” is at the bottom.)

  8. on 25 Oct 2012 at 9:01 amDan Meyer

    James Conron:

    BTW, in further to what I’m saying, if you’re not constrained by outcomes you could simplify this problem to a picture of the pattern and one simple question. Something like “Extend the pattern to a 1000 rows. How many bricks would there be in the 1000th row?”

    No need to constrain outcomes. If you ask for the 1000th row, you’ll likely need the explicit form. Or, if the student uses a script or something, they’re using the recursive form.

    Harel’s point (I’m speculating) is that there’s no intellectual need for the two forms. The textbook treats them as ends, not as means to another more problem-laden end.

  9. on 25 Oct 2012 at 9:28 amBelinda Thompson

    Can I ask about the learning goal of the task? Is it to discover a distinction between recursive and explicit descriptions of sequences? Is it to practice writing formulas using patterns?Knowing what students should understand as a result of engaging in the task would help in the redesign/improvement.

    Also, I’m compelled to read more about “intellectual need”. For me, math is engaging and fun when it’s intellectually satisfying, not necessarily when it’s about some activity that I might be interested in or about something “real” like doubling a recipe (who cares how much sugar? Just use your scoop twice.). I suspect this is true of lots of students.

  10. on 25 Oct 2012 at 3:30 pmJavier

    Bowen’s bowling ball pyramid made me consider what this might look like with billiard balls if the opening arrangement was extended. Some have already suggested finding the number of balls in the nth row. You could ask what row certain balls fall in. How does the total number of billiard balls change if you are allowed to start in rows other than 1 (i.e. remove rows as well as add).

    Going to the idea of the recursive formula- students might benefit from using a spreadsheet to have the computer do the heavy lifting, they have to learn how to get the computer to do the calculation. Once there, they could explore how they might use the input value to get the output value rather than going through the process recursively.

  11. on 25 Oct 2012 at 8:38 pmBryan Meyer

    I really like Bowen’s visual image of a similar task. The original task (the one Dan posted) asks students to generalize…how many in the 100th row, 1000th row, nth row? I know I’m not suggesting anything ground-breaking here, but I like starting with Bowen’s picture and then (perhaps after having students figure out how many balls in the pyramid?) asking this question:

    “If you had would you be able to make this type of pyramid?”

    …which could lead to:

    “Which number of balls COULD you make this type of pyramid with?

    …and a host of other interesting follow-ups and extensions. The question just seems more intriguing and perplexing to me.

  12. on 25 Oct 2012 at 8:40 pmBryan Meyer

    Sorry, the first question should have read:

    “If you had ‘x’ number of balls would you be able to make this type of pyramid?”

    *where x is some ‘not-so-arbitrary’ number of balls.

  13. on 26 Oct 2012 at 7:33 amJames C.

    Regarding using the recursive formula, I can’t think of a single case where a recursive formula is easier to use than an explicit formula in solving these types of problems. At best I can think of problems that are more easily described using recursive formulas, but not solved. (Try solving, say, the 25th Fibonacci number recursively, and you’ll see what I mean.)

    Most students find explicit formulas much more intuitive. Even when programming, it’s really not until students already have a grasp of recursion and are learning more complex topics such as sort routines or functional programming that the recursive method becomes easier.

    For the high school level series and patterns, it’s not likely students would use recursion to solve the problem unless the teacher forced them to use it.

  14. on 26 Oct 2012 at 8:24 amDan Meyer

    James C.:

    Regarding using the recursive formula, I can’t think of a single case where a recursive formula is easier to use than an explicit formula in solving these types of problems.

    45, 57, 69, … ,

    What’s the next term in this arithmetic sequence?

    You’d reach for the explicit formula? I and most students I’ve worked with will treat it recursively. It’s only when you ask for the 2012th term that recursion starts to look really unattractive.

  15. on 26 Oct 2012 at 8:51 amJason Dyer

    There are also many (many, many, etc.) sequences defined recursively for which no explicit formula is known.

  16. on 26 Oct 2012 at 11:24 amDebbie

    Before reading through others’ replies my thought was to provide students with envelopes of black and white tiles, ask them to create a tiling design and provide an explanation for customers to know how many of each colour tile they should buy.

    I wonder how many would create “linear” patterns in rows and how many would go for a “radial” pattern? Also, how many would describe their instructions in words and how many would use symbols?

  17. on 26 Oct 2012 at 11:32 amJames C.

    I’m going to backtrack a bit here. When I was thinking of recursion, I was thinking of it in the computer science sense, where it’s usually referred to solving a problem top-down rather than bottom-up. Same “recursive formula” either way you solve it though. You’re both right that many sequences do not have explicit formulas and students would use a recursive formula to solve most sequences where an explicit formula is not obvious.

  18. on 26 Oct 2012 at 11:33 amDan Meyer

    @James, Word. Recursion in CS took me a lot of head-thumping.

  19. on 26 Oct 2012 at 1:11 pmmr bombastic

    I think it would be more interesting, and actually have something to do with brick patterns, if you provide trapezoidal areas with given dimensions, a couple or three different sized brick options, and ask the students to try and create designs like that shown. If it isn’t possible, they need to provide the specifications for a new brick of reasonable size that would work. Obviously you want to move towards very large trapezoidal areas requiring custom bricks.

    The recursive part in the original question is especially annoying in that it sends the message that math is used to take something that is totally obvious (two more brick in the next row) and somehow make it seem complicated.

    I disagree that an explicit formula is needed for 1000 bricks. I add 2 bricks per row. So add 999 x 2 to the first row. Many kids think and do work this way because it is much simpler and more natural than writing out a formula and plugging 1000 into the formula. Again, asking for an explicit formula in this question is sending the message that math complicates something that is actually fairly simple.

  20. on 26 Oct 2012 at 8:37 pmChristine Lenghaus

    Have you seen the Wii game of bowling where you practice starting with pins then after each bowl another row is added on? – then you can ask questions like “when will I be trying to knock down a total of 101 pins?” or “after 5 practice bowls, how many pins will be in the last (largest) row?” I knick-named it “Wii love maths”!

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