Teaching

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

It’s probably unavoidable that large-scale standardized exams will contain errors.  But some errors are more serious than others.

The New York math Regents exams consistently contain errors that demonstrate a lack of mathematical understanding on the part of the test-makers.  These aren’t just “typos”, as administrators and politicians often suggest;  they are serious conceptual errors, and they call into question the validity of these assessments.

Consider number 24 from the Geometry exam demonstrates several different kinds of errors.

The test-makers indicated that (2) is the correct answer.  Presumably, they believe the acronym SAS stands for the Side-Angle-Side Similarity Theorem.  However, SAS typically stands for the Side-Angle-Side Congruence Theorem.  If a student interpreted SAS, SSS, and ASA in the usual way, i.e. as congruence theorems, then AA would be the only possible way to prove two (non-congruent) triangles similar, and thus (1) would be the correct answer.

Perhaps the test-makers might claim that the context of the problem, proving similarity, should have led students to assume the acronyms stood for similarity theorems.  But, alas, there is no ASA similarity theorem.  What were they thinking here?

More generally, “Which method could be used” is not a good mathematical question.  Lots of different methods could be used.  It’s not inconceivable that AA could be used to prove these triangle are similar, so how could that possibly be an incorrect answer to this question?  Ultimately this question was thrown out, but not before thousands of students across the state had already taken their final exam.  And even as they tossed the problem out, the state still refused to accept responsibility for publishing an erroneous question, hiding behind the old alternative methods defense:

Since there are alternative methods to prove that the two triangles given in Question 24 are similar, all students should be awarded credit for this question. (link)

Unfortunately, this is just the latest example of serious mathematical errors in NY State Regents exams.

[Update:  An earlier version of this post criticized #25 on the Algebra 2 / Trig Exam.  Thoughtful comments provoked me to re-examine my criticism, and also pointed to a different issue with this question, which can be found in a separate post.]

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

Here are two questions from the June 2013 exams.

These questions aren’t particularly interesting:  both give the graph of a circle and ask for its equation.  Since the questions are nearly identical, it would be strange to see them on the same exam.  It is even stranger, then, to see them on two different exams:  one was on the Geometry exam, the other the Algebra 2 / Trig exam.

Why does the same kind of question appear on the terminal exams of two different, sequential courses?  Are students supposed to learn this topic in Geometry, or in Algebra 2 / Trig?  The test-makers don’t seem to know, which calls into question the fairness of these exams.

And the question of fairness has implications for teachers as well as students.  Student exam scores now constitute a substantial component of a teacher’s yearly evaluation.  If students are supposed to learn how to find the equation of a circle in Geometry class, is it fair to use such a question to evaluate an Algebra 2 / Trig teacher?

The fact that test-makers don’t know which topics belong in which courses raises some serious questions about the validity of using these test results to evaluate teacher performance.  In addition, one wonders how facing the same question on sequential exams impacts student growth measurements so popular among educational policymakers nowadays.

Unfortunately, this is only the latest in a long line of inappropriate questions on math Regents exams.

A common debate in math education centers on the extent to which students should memorize things, like multiplication tables, the quadratic formula, or the division algorithm.

There are many sensible arguments against memorizing algorithms. While memorizing algorithms might produce good results in a typical math class, mathematics makes the most sense when it is understood as a coherent, interconnected system of thought. We want students to be resourceful and creative problem solvers, and this requires that students understand the context and the connections, not merely the steps, of the procedures they learn.

Occasionally this argument is taken to the extreme and teachers altogether discourage memorization of algorithms and procedures, claiming that it’s pointless for students to have the procedural knowledge without understanding the context.

While there is some merit to this argument, I was recently reminded how valuable it can be to blindly memorize algorithms.

A friend invited me to go rock-climbing, something I had never done before. We arrived at the gym, and my friend, an experienced climber, showed me how to identify the beginner trails on the wall and encouraged me to get climbing right way. Over the next hour I attempted a few trails, and met with more frustration than success.

After a particularly discouraging attempt, I sat down next to my friend to rest. He told me that he noticed I was getting stuck in the middle of the trail and was having difficulty finding the next hold. He suggested that I examine the trail before I start the climb and memorize the sequence of moves needed to make it to the top.

By memorizing the wall-climbing algorithm, I was freed from trying to do too many things at once. I could focus on developing the fundamental techniques — proper holds, balance, body position, different reaches — without worrying about what the next step should be.

Ultimately we want the ability to be on an unfamiliar wall, or in an unfamiliar problem, and have the confidence and skill to figure out what to do next. But this skill is really a combination of many skills, and it can be challenging, and frustrating, to try to develop them them all simultaneously.

Sometimes memorizing things — like a sequence of handholds, or the quadratic formula — can help us get to the top. And it can help prepare us for the day when memorizing won’t be enough.

This beautiful image is this year’s testament to trying new things and giving vague directions.

At the end of every school year I challenge myself to do something brand new in each of my classes. This could take the form of new mathematical content, a new kind of project, a novel technology, or something else entirely. What’s important is that it’s something I’ve never done before.

In this year’s Calculus courses, that new thing was Sage Mathematical Software. To get students familiar with plotting in Sage, I presented them with a mundane task. I gave them code that defined a square region in the plane and asked them to play around with some of the parameters to create a new graph, which they would then post in our class forum.

Naturally, I provided an example of what they might do.

Having never done this with students before, and being a novice with the software myself, I didn’t really know what to expect. I suppose I expected students to produce graphs similar to what I had done. They didn’t.

They played around with the functions, the colors, the domain and range. They inserted cotangents, exponential functions, and additional constraints. They found options for the region_plot function I didn’t know existed. I expected them to produce graphs of boring quadrilaterals, and instead, they produced beautiful, complex, and intricate contour maps like the one seen above.

This is what can happen when, instead of telling students to do A, B and C, you give them vague, open-ended directives like play around and share. More often than not, students rise to the occasion and create work that surprises and amazes. And this reminds us to keep trying new things in the classroom, if only to create opportunities for students to defy our expectations.

You can see more of the beautiful, diverse images the students created here.