Teaching

Here is another installment in my series reviewing the NY State Regents exams in mathematics.

The following question appeared on the June, 2014 Algebra 2 / Trig exam.

Graphs without scales are common on Regents exams (I’ve written about this before).  Personally, it’s not a huge deal to me–I’m a lazy grapher, myself.  However, a colleague of mine regularly complains about this, and she made an excellent point regarding the grading of this particular problem.

The solution to this problem involves translating the graph one-unit to the left and two units up.  But since no scale is given on the graph, it’s not clear what one unit to the left would be.  If we assume the box on the graph indicates one unit, then the red graph below would be appropriate.  But if we assume a box to represent half-a-unit, the purple graph would be correct.

If no scale is explicitly given, it seems like both graphs should be considered correct and receive full credit.  But the rubric doesn’t address this possibility, and it’s unlikely students were given the benefit of the doubt.

Here is another installment in my series reviewing the NY State Regents exams in mathematics.

Elementary statistics plays an increasing role in high school math curricula, but the ways these concepts are often tested raises some concerns.  After all, the manner in which ideas are tested can reflect how the ideas are being taught.

Here’s a question from the 2014 Integrated Algebra exam:  which of the following is not a causal relationship?

Causality is notoriously difficult to establish, but I’ll set aside my philosophical objections for the time being.  My primary concern here is with (2) being the correct answer.

First, correlation is a relationship between two quantities.  What quantity is population correlated with in answer choice (2)?  “The taking of the census” is an event, not a quantity.  This may seem like nitpicking, but what quantity are we supposed to assume in its place?  It seems natural to assume “the census taken” to mean “the number of people recorded on the census”, but then how could there be no causal relationship?  What causes a number to be written down for “population”, if not the actual population?

Here’s another question from the 2014 Integrated Algebra exam.

It’s important to talk about bias in surveys, but no substantial thought is required to answer this question:  three of the answer choices have absolutely nothing to do with campsites.  And for the record, the question should really be phrased like “which group is most likely to be biased against the increase?”.

And this is a problem typical of the Algebra 2 / Trig exam.

I know it’s pretty much standard usage, but no finite data set can be normally distributed.  The correct terminology here would be something like “the heights can be approximated by the normal distribution”.

I’m aware that some may see these complaints as minor, but as I’ve argued before, I think it is extremely important to model precision and rigor in mathematical language for students.  We expect this from our teachers and our textbooks; we should expect it, too, from our tests.

Here is another installment in my series reviewing the NY State Regents exams in mathematics.

The following question appeared on the June, 2014 Algebra 2 / Trig exam. To start, steeper is not a well-defined term, not in an Algebra 2 / Trig class, anyway.  I’m not against using the word in everyday mathematics conversations, but I’m not a fan of putting it on an official exam like this.  After all, I think these exams should model exemplary mathematical behavior.  But that’s not the real issue here.

Even if we accept what steeper means, it can not be said that either graph is steeper than the other. Take a look:  here, is graphed in red and is graphed in blue.

It seems pretty clear that the blue graph is steeper than the red on the right hand side, it also seems pretty clear that the red graph is steeper off to the left.

To be precise, the derivative of is greater than the derivative of for , thus making the red graph steeper for those values of x.

Thus, there really is no correct answer to this question.  The answer key originally had (3) as the correct answer, but it is no truer than (2).  Ultimately, a correction was issued for the problem, and both (2) and (3) were awarded full credit.

Mistakes are bound to happen when writing exams, and it’s good that a correction was ultimately issued.  But this is a pretty obvious error.  This question should not have made its way onto a high-stakes exam taken by tens of thousands of students.  A thoughtful student might have been frustrated, confused, or disheartened confronting this question with no correct answer.  Hopefully its impact didn’t extend beyond these two points.