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.
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.