Friday, January 24, 2014

Do the kids have rulers?

Arne Duncan has an Op-Ed in the WP this morning about our mediocre educational achievement, as measured by the NAEP. While reading this piece it occurred to me that most people my age don't have a sense of the NAEP because we didn't take it when we were in grade school.  So to satisfy my curiosity I went to the NAEP Web Site.  To have a look see, in the sidebar on the left with the pale blue background I found a link to Sample Questions, Analyze Data, and More.  Then on the next screen I clicked Questions Tool in the main section.  Then again Questions Tool and then Mathematics.  There are questions for the fourth grade.  Below is a screen shot of question 10 - draw a line segment of a given length.  (The screen shot shows only the top part of the box where the line segment is to appear but otherwise contains all the information that the question provides.) 
I've also provided a link to a larger view of the screen shot because the font is small below.

Full sized screen shot

The question is from last year's test and is rated hard, meaning most students didn't get it.  Here's my little analysis of it.

Q:  At what age do students first learn to use a straight edge to draw a line segment?  (People who were in fourth grade or thereabouts in 1963-64, do you have any recollection of doing this at school?)
A:  My memory of this is poor but it seems to me we would have done this in art using crayons.

Q:  Another and perhaps more subtle part of the question is that the length to draw is not an integer value.  So the student needs to know fractions and to read tick marks on the ruler to get the right length.  When did we learn fractions and to read the tick marks on the ruler?
A:  I'm pretty sure I did multiplication and division in 3rd grade with Mrs. Minsley.  So it makes sense to me that I learned fractions in 4th grade.  But I've got no clue about when my cohort learned to read a ruler's tick marks.  Again, that seems like something we'd do in art rather than in math.

The question wants the students to plop one corner of the ruler at point A, the corner coinciding with 0 inches.  Then the corner should be aligned so that increasing the number of inches keeps the line segment in the box - either moving down and to the right or straight down.  This seems quite straightforward now, but would it seem that way to a fourth grader?

Some reasons for getting the question wrong when the students conceptually understands what is at issue.

An answer is marked correct if the drawn line segment is within 1/4" of correct.  It's possible for the student to understand what's going on but make an error in construction that leads to a greater than 1/4" deviation.  Some of these errors I believe I might have made.
  • Parallax 1 - The ruler starts out near A but not at A
  • Parallax 2 - The segment ends either longer or shorter than required because of the angle the pencil is held.
  • The ruler slips during the construction
  • A mistake is made on the tick marks.  Here I want to note that fourth grade is when I first got glasses.  So it is possible that I could have made a visual error, rather than a conceptual one.
  • The student read the question wrong and draws an integer-valued length instead.
I have some report cards of my performance from these years.  They don't speak much at all to math, but on reading there are emphatic statements about needing to be more careful and not jump to conclusions.  One of my kids had similar issues when he was in grade school. 

Do we have a sense why most students are missing this question?

I made a point of emphasis above that for me the requisite skills were probably learned doing art rather than doing math.  But are kids doing as much art these days?  One gets the impression that they aren't.  Is anybody making the argument that art and math are tied in this way?

One more thing

Teachers who have enough job security that they can talk openly about these issues might want to opine whether they feel the NAEP offers an accurate assessment of their students' capabilities.  I would like to see some discussion about the standardized tests we took then (I believe in the NYC public schools that I attended those were called Iowa Exams) in comparison to what we have now.  And I'd like to see this done both on the fairness of the test issue and on how performance has changed longitudinally.  The sense one has is that things have gotten worse, but how things look depends on where you sit. 

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