Thursday, October 9, 2014

Math problem of the week: a 6th grade Common Core-inspired "Performance Task"

Below is a problem that appeared in a recent article in Education Week, which explains that it comes

from a sample 6th grade performance task for the common-core math assessment being developed by the Smarter Balanced testing consortium. The full five-part problem asks students to find the volume of a cereal box, to label the dimensions on a diagram of a flattened box.., and to determine the surface area of the box. The task culminates in this open-ended question.

Notice the note at the bottom: "This problem has been edited for clarity."

The issue of clear communication comes up in a second way in the Edweek article. The article quotes Doug Sovde, director of content and instructional supports for PARCC, which, besides Smarter Balanced, is the other main institution that has been developing Common Core tests.
Being able to communicate a thought process in words is a critical skill, said Doug Sovde..."I think it's entirely reasonable for somebody to work through something on scratch paper and be able to share their reasoning after they worked through the task," he said. "It's not unlike a student making an outline before they write their research paper."
I'm confused: how is sharing your reasoning after you work through a task like outlining something beforehand?

Yes, clear communication is important, especially when you are inflicting something on millions of students--as opposed to solving a problem in private.


Hainish said...

"This problem has been edited for clarity."

Wow...I would hope all problems are edited for clarity before they're used in an assessment.

Auntie Ann said...

To do that problem well takes a lot of time. The easiest is to do a prime factorization and work with the factors, but if you don't know that when you start, you can waste time figuring it out.

It also helps if you know that the surface area will be smaller the more square the shape.

The score here could be grossly unfair. A kid who has done a couple of problems like this previously, would sail through it quickly; while a student who is coming to it cold, would take a lot of time doing it. The difference is not between their knowledge, but their experience.

This is why the Common Core *is* a curriculum. If you aren't teaching problems like this, your students are screwed.

Ze'ev Wurman said...

Auntie Ann seems to assume that (a) the new side lengths must be integers, and (b) that some other quantity (total sides length?) needs to be preserved. Neither seem true (even as integers are convenient). But the rest of her comment is spot on.

This is a typical half-assed Common Core (better, SBAC?) problem where students are expect to plug-and-chug (i.e., guess-and-check). Assign some new values, calculate surface area, calculate volume, plot (or fill a table entry) and repeat. After doing it for 4-6 times, try to guess the pattern and make an educated guess for the next trial and error or, if you happen to be lucky, one of the previous guesses happens to work out so declare victory and quit. Meanwhile one incessantly -- and foolishly -- repeats the calculations for getting volume and surface area (did I mention the "rote and mindless" calculations already?)

Now, for the current box, the surface area is twice (2x8 + 2x12 + 8x12) or 2x136=272 square units. Volume is 2x8x12=192 cubic units.

If one happened to learn that the more 2-D shapes are "like a circle" (or "like a square" for polygons) the more area they contain for a given perimeter, or in 3-D the more they are "like a sphere" (or "like a cube") the more volume they have for a given surface area, this is a no-brainer. Estimate the cube root of 192 -- 5 gives 125, 6 gives 216, so it should do. Interestingly, in a 6x6x6 cube the area and the volume are numerically the same, 216 square/cubic units. Neat. Since 216 is greater than 192 and less than 272, we are done. Except that we will be probably docked points because we didn't do enough "plug-and-chug."

Alternate approach: divide 272 by 6 (6 faces) ==> 45.33, so max area of a face is that. The closest square root is 6 (7 is too much), and the rest as above. If volume too small, increment side by 0.1 until the volume exceeds 192.

It is a bad problem to see if a kid understands math. If he was exposed to that particular fact, it's a no brainer -- not a "performance problem" by any reasonable measure. If not, it is a long, tedious, and error prone plug-and-chug problem. Awful.

Barry Garelick said...

Guess and check is justified by the math reformers as getting students to analyze and work with "patterns". Which ignores the fact that there are other more efficient patterns that one can master through fundamental procedures and facts as Ze'ev has demonstrated.

Auntie Ann said...

(I didn't know calculators were allowed until I was finished with the problem, so I was trying to come up with an integer result which wouldn't take a lot of time to work out the arithmetic.)