I think we all know what happens when we assume. If you don't, then watch this short clip of Tony Randall in a famous episode of the Odd Couple:
If you're a mathematician, you already know how dangerous it can be to make unjustified assumptions. And yet, the NYS Regents exams are asking students to do just that.
For example, take this Model Response from the June 2019 Algebra I exam, question 36.
The student's response earns full credit (4 points), just as the Rating Guide says it should:
But wait a moment! Are we really sure that these are the inequalities represented by the graph? To me, it sure looks like the dashed line could represent
Can you state with any certainty that my inequality is incorrect? There's nothing in the graph that suggests that the slope and y-intercept are necessarily integers. Sure, we'd like them to be -- it looks like they could be -- but are we justified to assume that they are?
In fact, why should we even assume that these lines are straight, other than the fact that they do kinda look like straight lines, don't they, huh? If the folks who wrote this exam question called this a "system of linear inequalities," that's one less assumption we'd have to make.
We need to teach students to avoid making conclusions based on unjustifiable assumptions. Instead, the makers of the NYS Regents exams are encouraging it.
The following problem from the August 2018 Geometry Regents exam is even more egregious.
Well, at least the problem describes these figures as triangles, so we can safely assume that the ground, the support wire, and the poles are all straight lines.
A student can properly answer the question by showing similarity by AA; for example, the triangles share an angle at the stake and have a pair of congruent corresponding angles formed by parallel lines cut by a transversal (either the wire or the ground).
However, the Model Responses put out by NYS also gives full credit for the following response:
Whoa now! Where in the statement of the problem does it say that the telephone pole forms a right angle with the ground? Sure, it looks like it does in the diagram, but that angle could just as easily be 89 or 91 degrees. Even with a protractor, I could at best estimate the angle at 90 degrees, and who brings a protractor to the exam?
The test makers are allowing students to earn full credit by making unjustifiable assumptions. Certainly, as the problem is written, students who assume that this is a right angle should NOT be given full credit, despite the scoring of the Model Response.
When it comes to grading Regents, let's be more like Felix Unger and less like Oscar Madison. Let's keep sloppiness out of math education.
Donny Brusca
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